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Series on Advances in Mathematics for Applied Sciences - Vol. 41

MATHEMATICAL METHODS IN ELECTROMAGNETS Linear Theory and Applications

Michel Cessenat CEA/DAM Centre d'Etudes de Bruydre$-le-Chdtel France

World Scientific Singapore Â»New Jersey'London* Hong Kong

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Library of Congress Cataloguig-in-Piiblicatioii Data Cessenat, Michel. Mathematical methods in etectromagnetism: linear theory and applications / Michel Cessenat. 396 p. 22.5 cm. - (Series on advances in mathematics for applied sciences; vol. 41) Includes bibliographical references and index. ISBN 9810224672 1. Electromagnetism â&#x20AC;&#x201D; Mathematics. I. Title. II. Series. QC760.C43 1996 537\01'51-dc20 96-11628 CIP British Library Cataloguing-in-Publicatkm Data A catalogue record for this book is available from the British Library.

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MfiTHEMflTICfiL METHODS I I ELECTROMflGNETISM Linear Theory and Applications

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Acknowledgements I wish to thank Professor R. Dautray for drawing my attention to certain problems of electromagnetism. I also wish to thank Professor J.L. Lions for suggesting to me, one day at Orly airport, to write this book. Applying his ideas on asymptotic analysis to electromagnetism is an exciting challenge! I wish to express my gratitude to Professor A. Bossavit for constructive remarks on ways to improve this book. I am grateful to Professors M. Artola, P. Benilan, R. Petit and also to Dr. A. Gervat for useful comments. Dr. G. Zerah must be thanked for fruitful discussions on ferromagnetism. I am also gratefully indebted to Dr. J.-M. Clarisse, and O. Cessenat, and also to C. Averseng and C. Mares for their help.

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Preface Electromagnetism has numerous applications whether in energy transport or signal transmission in devices as diverse as antennas, waveguides, optical fibers, gratings, electrical circuits, light in lasers and plasmas. Thus, much research has been devoted to problems in electromagnetism essentially from a physical point of view. One aim of this book is to present a more global analysis encompassing both the physical and mathematical aspects of electromagnetism. In particular, powerful modern mathematical methods are emphasized, which lead to numerical applications. Phenomena under consideration are modelled as "distributed system" by the "electromagnetic field" which must satisfy Maxwell's equations. In general, media are described using the "continuum assumption" by introducing specific constants and thus, without entering their fine structures. This description leads to constitutive relations. Various conditions must also be treated such as: initial, boundary or transmission conditions, conditions at infinity, finite (or locallyfinite)energy conditions. In the absence of any coupling with other phenomena such as fluid motion, determining the electromagnetic field which satisfies all the necessary conditions is already a difficult problem. The student or the engineer may feel quite uneasy in face of these problems and think that usual mathematical books are helpless. Indeed, problems in electromagnetism can be very diverse, and differ from classical problems in partial differential equations in the vectoriai nature of the electromagnetic field. Thus, differential geometry is present at all stages whether when modelling, choosing a mathematical framework, or computing theoretical or numerical solutions. Another peculiarity is inherent to the constitutive relations and to the specific "constants" of the media. These constants may be complex or take the form of matrices for usual dissipative and anisotropic media, or be positive real numbers for free space (or ideal conservative media). Therefore, different methods must be applied depending on the media. It must also be the case that these specific constants strongly depend on frequency, thus leading to evolution problem with time delay, or depend on the electromagneticfielditself, givingriseto nonlinear effects. Last but not least, electromagnetism problems are wave propagation problems, therefore presenting classical features of such problems. Hence stationary problems in the frequency domain are often ill-posed, and become well-posed only by imposing radiation conditions at infinity or on the direction of propagation. This results in difficulties when treating scattering by infinite obstacles.

VII

VIII

PREFACE

One aim of this book is to provide the reader with the basic tools and the functional analysis concepts corresponding to the usual physical hypotheses of the modelling stage. We also present in the Appendix, certain properties of differential geometry, the corner stone of electromagnetism. As examples of application of these tools and concepts, we treat several fundamental problems of electromagnetism, e.g.: scattering of an incident electromagnetic wave by a bounded obstacle, scattering by a grating (periodically infinite obstacle), wave propagation in a waveguide or in an optical fiber. Several recent approaches suitable for solving electromagnetism problems and related to numerical methods are presented, in particular: Integral methods, resulting in solving an integral equation on a given surface (typically the surface of the obstacle in a stationary scattering problem) thus sparing one spacial dimension. However, the resulting matrices are full. Semigroup methods, allowing to treat evolution problems, and also stationary problems both with planar or cylindrical geometries, and a given direction of wave propagation. Variational methods (particularly suited to numerical applications), in the case of dissipative media. Spectral methods, when spectral decompositions (of normal operators) are possible. However, these approaches, well known in physics as modal decompositions, and which are the core of many textbooks on electromagnetism are often used beyond their scope. In many cases, hybrid approaches are preferable which combine the respective advantage of different methods. For instance, when solving a stationary scattering problem, a finite element method is applied to an inhomogeneous obstacle and coupled to an integral method for the outer domain. The determination of constants for a sample in a waveguide is another typical example: a finite element method is used in the sample and coupled to a spectral method for the rest of the homogeneous waveguide. One of the main ideas developed in this book is the use of Calderon projectors and operators (also called impedance or admittance surface operators). These operators are surface operators containing all the information relevant to the electromagnetic properties of the domain bounded by the surfaces where they are defined. Such operators are especially useful when tackling asymptotic problems in electromagnetism: e.g. the scattering by an obstacle of high conductivity or with fine periodic inclusions. We hope that this book is useful for students or engineers having to solve problems in electromagnetism. We also hope that mathematical notions will not conceal physical concepts from the reader, but rather allow a better understanding of modelization in electromagnetism, and emphasize the essential features related to the geometry and nature of materials. Some prerequisites in functional analysis may be useful as for example Dautray-Lions [1].

Contents Chapter 1 - Mathematical Modelling of the Electromagnetic Field in Continuous Media: MaxweU Equations and Constitutive Relations I - Evolution Maxwell Equations 2-Stationary Maxwell Equations... 3 - Constitutive Relations 3.1 - Linear isotropic dielectric media 3.2- Linear anisotropic dielectric media • 3.3- Linear chiral media 3.4-Nonlinear constitutive relations

1 1 4 7 7 20 22 23

Chapter 2 - Mathematical Framework for Electromagnetism 26 1-Spaces for curl and div: Trace Theorems 28 2- Jump Formulas Across a Bounded Hypersurfaee T in Rn 32 3-Differential Operators on a Regular Surface T 33 4 - The spaces H ~ 1/2(div,D, H " lT2(curl,r). Trace for H(curl,Q) 35 5-TraceforWl(div,Q) 40 6-Some Complementary Results. Polar Sets 41 7-Traces on a Sheet 42 7.1 -Trace problems on sheets (the scalar case) 43 7.2 - Trace problems on sheets (the vectorial case) 46 S-Some Regularity Results 49 9 - The Hodge Decomposition 52 10 - Interpolation Results 57 II -Some Variational Frameworks 58 11.1- Variational frameworks based on Hf curl, Q) spaces 58 11.2- Variational frameworks based on H (Q) and the Laplacian .... 60 12 - First Problems with Inhomogeneous Boundary Conditions 62 12.1 -Problems with H~1/2(div,D boundary conditions 62 12.2 - Problems with H1/2(r) boundary conditions 66 13 - Boundary Problems of Cauchy Type. Uniqueness Theorems 67 14-Whitney Elements; Numerical Treatment 68 Chapter 3 - Stationary Scattering Problems with Bounded Obstacles 1 - Stationary Waves due to Sources in Bounded Domains 1.1- The main properties for Helmholtz equation in R3 1.1.1 -Local regularity properties 1.1.2- Outgoing and incoming Sommerfeld conditions 1.1.3-Elementary outgoing (incoming) solution 1.1.4-Fundamental properties 1.2-The main properties for Maxwell equations in R3 1.2.1-Relation to the Helmholtz problem •• — •..-. 1.2.2- The Silver-Mulier conditions * IX

71 71 71 72 72 73 73 75 78

CONTENTS

1.3- Transmission problems and surface integral operators 1.3.1- Helmholtz problems with charges on "regular surfaces" 1.3.2- Maxwell problems with currents on "regular surfaces" 1.3.3-Some regularity results 1.3.4-Integral method for a sheet 1.3.5-Incoming waves 2 - Scattering Problems with a Complex Wavenumber. Limiting Absorption Principle 2.1 - Helmholtz equation in R3 for a complex wavenumber 2.2-Maxwell equations with complex coefficients 2.3 - The Calderon projectors for a complex wavenumber 3 - Vector Helmholtz Equation, Knauff-Kress Conditions 3.1 - Knauff-Kress conditions at infinity 3.2 - Vector Helmholtz problems with jumps conditions 4 - Boundary Problems with a Real Wavenumber 4.1 - Limiting absorption principle 4.2-Exterior boundary problems 4.3 - Some consequences; the exterior Calderon operator 4.4-Interior boundary problems 4.5 - Some consequences; the interior Calderon operator 4.6-Integral equations and boundary problems 4.7-Some consequences for the integral operators 4.8 - On the numerical solution of some scattering problems 5-Scattering Problems by a Dielectric Obstacle 5.1 -A general variational formulation 5.2-Solution using an integral method 5.3 - Behavior at infinity. Radar cross section. Optical theorem 6 - Scattering and Influence Coefficients for Several Obstacles 6.1 - Decomposition of the trace spaces 6.2-Screen effect. Extinction theorem 6.3 - Coefficients of mutual influence of antennas 7 - Mutual Influence of Sheets 7.1-Introduction 7.2-Electromagnetic fields due to currents on a sheet 7.3-Matrix elements; influence coefficients 8 - Fields due to Currents on a Line 9 - Scattering Problems by a Chiral Obstacle 10 - Conclusion on the Calderon Operators for Scattering Problems 11 - Multipole Expansions. Rayleigh Series 11.1 - Multipole expansions for Helmholtz in R2 11.2-Multipole expansions for Helmholtz in R3 11.3-The source of a wave un 11.4 - Multipole expansions and analytical functionals 11.5- Multipole expansions for the electromagnetic field

80 80 85 90 92 95 98 98 101 101 102 102 103 104 104 106 108 110 112 114 116 118 121 121 122 123 127 128 129 130 130 130 131 133 133 136 138 142 142 143 147 152 156

CONTENTS

12-Scattering by a Dielectric Ball 12.1 - Scattering with Helmholtz equation 12.2-Scattering with Maxwell equations 13 - Addendum. Compactness Properties in Scattering Problems

159 159 161 166

Chapter 4 - Waveguide Problems 1 - Waveguides with Helmholtz equations 2 - Waveguides in electromagnetism

167 167 172

Chapter 5 - Stationary Scattering Problems on Unbounded Obstacles 1 - Plane Geometry 1.1 - Plane geometry with Helmholtz equation 1.1.1- Helmholtz problem in a half-space with Dirichlet condition 1.1.2- Transmission Helmholtz problems in R3 1.1.3- Transmission problem with two different media 1.1.4 - Some examples of applications of the Calderon operator 1.2-Plane geometrywith Maxwell equations 1.2.1 - A typical problem in a half-space 1.2.2- Scattering problems with two different media 1.3-Theslab 1.3.1- The scalar case with Helmholtz equation 1.3.2-The slab with Maxwell equations 2 - Periodic Geometry; 2D Gratings 2.1 - Periodic geometrywith Helmholtz equation 2.2-An integral method for gratings 2.3-Periodic geometrywith Maxwell equations 2.3.1-Mathematical framework 2.3.2-TheCalderon operator 2.3.3-Some scattering problems 3 - Conical Geometry 3.1- Properties of some unbounded operators associated to A 3.2-Solution of Helmholtz problems Chapter 6 - Evolution Problems 1 - Cauchy Problems 1.1 -Scalar wave Cauchy problems 1.2-Cauchy problems in electromagnetism 1.2.1 - Cauchy problems in free space R3 1.2.2 -Cauchy problems in a domain Q. 1.3- Some hyperbolic properties of wave evolution 1.4-Radon transforms 1.5- First applications of the Radon transform method 2 - Scattering Problems - Incoming and Outgoing Waves 2.1 - Another application of the Radon transformation 2.2 - Incoming and outgoing waves in electromagnetism

187 187 187 .. 187 193 195 198 200 200 206 210 210 213 214 215 223 227 227 228 230 235 236 242 251 251 252 254 254 255 256 258 262 264 264 267

XII

CONTENTS

2.3 - Incoming and outgoing waves with a bounded obstacle

269

3 - Causal Problems 276 3.1 - Some examples with the wave equation 278 3.1.1 -A typical example of causal* problem 278 3.1.2-An example with boundary conditions 279 3.1.3 -An example with transmission conditions, integral methods ..280 3.1.4-An example with a waveguide 284 3.2-Some examples with Maxwell equations 286 3.2.1 - Causal Maxwell problems in the whole space 286 3.2.2-Causal problems with currents on a surface 287 3.2.3 - Causal Maxwell problems with boundary conditions 289 3.2.4-A causal waveguide problem 291 3.2.5 - Uniqueness at most 294 Appendix - Differential Geometry for Electromagnetism 296 1 -Introduction. Mathematical Framework 297 1.1-Manifold with boundary 297 1.2- Riemannian manifold M with or without boundary 299 1.3-Definition of the codifferential 304 1.4 - The gradient, divergence and Laplace-Beltrami operator 305 1.5- Decomposition of the space of tangent p-vectors 306 1.6- Currents; generalized r-forms (or distribution r-forms) 310 1.7- Application: Jump formula of exterior derivative 311 2-Sobolev Spaces of r-forms; Maxima Spaces 312 3 - The Hodge Decomposition for a Compact Manifold with Boundary 315 3.1- Variational frameworks for the Laplace-Beltrami operator 317 3.2 - Variational frameworks for the operators d5 and 6d 327 4 - The Hodge Decomposition for a Compact Manifold without Boundary.. 333 5 - The Hodge Decomposition in L2 for Unbounded Manifolds 337 5.1-The Hodge decomposition for L2f(Rn) 337 5.2-The Hodge decomposition for L2r(M) 340 5.3 - Comparison between the cohomology of M, its complement and its boundary 342 6 - Application to 3-dimensional and 2-dimensional Cases 344 6.1 - The 3-dimensional case 344 6.2-The 2-dimensional case 347 7-Maxwell Equations with Differential Forms 351 7.1 - Maxwell equations in R3 351 7.2 -Maxwell equations in R4 351 7.3 - Transformation laws. Lorentz and evolution transformations .... 353 References Index Notations

365 373 375

MfiTHEMflTICfiL METHODS I I ELECTROMflGNETISM Linear Theory and Applications

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CHAPTER 1

MATHEMATICAL MODELLING OF THE ELECTROMAGNETIC FIELD IN CONTINUOUS MEDIA MAXWELL EQUATIONS AND CONSTITUTIVE RELATIONS 1. EVOLUTION MAXWELL EQUATIONS

Electromagnetic phenomena in a domain Q £ R 3 occupied by a medium and for all time t of R (or only for positive t or for a finite interval of time) are described with the help of four functions D, E, B, H of (x,t) € Q x R with values in R 3 . These are fairly generally called (but there is no universal agreement on these names): D: electric induction, E: electric field B: magnetic induction, H: magnetic field. These functions are related to two functions (or distributions) defined on £1 xR, for all t (or only some interval), with values in R and R3 (resp.), called charge density p, and current density J (resp.), by the Maxwell equations: a n

i) - -g-r- + curl H = J, (I)

+ cur

* ^ = 0, Maxwell Faraday law

iii) div D = p, iv) div B = 0,

Maxwell Ampere law

Gauss electrical law Gauss magnetic law

with the following usual notations, in a Cartesian system of coordinates, for E = (Ei,E2,E3), x = (x,,x2,x3): 1

1 MATHEMATICAL MODELLING „ BEj

(2)

divE=2^,

9E,

9E, 3E,

9E,

c u r l E = ( ^ . ^ ^ 1 . ^ 2 , -i.^J).

The densities of current and of charge satisfy the continuity relation or charge conservation: (3)

| ^ + divJ=0.

The system of units employed here is the "systeme international" formerly called "MKSA". The units for the quantities p, J, D, E, B, H, and their relations with the fundamental units M (mass), L (length), T (time), Q (charge) are: [p] ■ Coulomb/cubic meter « L" 3 Q, [J]=Ampere/square meter=L~ 2 T~ *Q [D] = Coulomb/square meter « L " 2 Q, [E] = Volt/meter = MLT" 2 Q ~ l , [B] ■ Weber/square meter = MT" l Q ~ ! , [H] = Ampere/meter = L"" lT~ 1 Q. From the identity, for all vector fields u: div curl u = 0, we see that equations (l)iii) and iv) are partially redundant: applying the divergence to (l)i) and ii), then taking account of (3), we obtain (4)

| ( d i v D - P ) = 0,

g(divB) = 0,

which implies the relations (l)iii) and iv) for all time t if these relations are satisfied at some initial instant. Besides there are additional relations, which are seen below. To some extent, it is possible to act on p and J to produce a required electromagnetic field as in transmitting antenna. A typical example is a thin wire modelled by a line, with p and J concentrated on this line. Besides, they may also be partially known and subject to a random process. Free space case. For free space in a domain Q (with only given charges and currents in it), the quantities D, E, B, H are linked by the following "constitutive relations" (5)

D = c0E,

B = M0H,

with £Q and jiQ called respectively permittivity and permeability of the free space (6)

€ =

o 5gJ 10~9 Farad/meter, ^ o = 4«. 10~7 Henry/meter, c0 JIQ c 2 = 1,

with [eQ] » M" 1 L" 3 T 2 Q 2 , [jiQ] = MLQ"'2, c being the velocity of light in free space. The electromagnetic field, reduced to (E,B), satisfies Maxwell equations

1.1 EVOiWTION MAXWELL •\ (IY

1 9E al

EQUATIONS

i <n + c u r i B =

ii) | | + curlE = 0 iii) divE=^-p iv) div B = 0,

which are also the equations of electromagnetism for the microscopic scale, naturally posed in all the space R . By homogenization or by any averaging process, these are believed to give Maxwell equations (1) in continuous media for the macroscopic scale. When there is no charge and no current, using the relation curl curl u = - Au + grad div u (for all vector fields u), we deduce from (1)' that E and B satisfy the wave equation (with velocity c) which shows the hyperbolic nature of Maxwell equations: 3

- c at 2

Energy Balance in free space (without charges or currents). We define the electromagnetic energy W(t) in a domain CI of free space by (7) W(t) = | / ( D . E + B.H)dx, with the density of energy: f t

(8)

w = | ( D . E + B.H) = j ( E E + p H ). 2

The Poynting vector is then defined with the vector product A by (9)

S = E A H ,

i.e., Sj E2H3 =

—

E3H2, S2

— EjHj — EjH^>

E|H2 — E2HJ.

Using the formula: div ( E A H ) = curl E . H - E.curl H , (10) and Maxwell equations, we have (for J = 0): | ^ + d i v S = 0. (ID Integrating on CI gives: (12)

^ ( t ) = -/ n.SdT, r

where n is the exterior normal to the boundary r of CI. Thus the time derivative of the energy in CI (i.e. the power P(X)) is equal to the opposite of the flux of the Poynting vector through the boundary T of CI.

1 MATHEMATICAL MODELLING

These relations are a priori fonnal and must be justified but we already know that E and H must be square integrable on Q, in order to have a finite electromagnetic energy which gives the natural mathematical framework for the study of the electromagnetic field. When there are particles (at rest) in the free space, which are modelled by point charges, the electromagnetic energy is not finite. This is generally viewed as a defect of the modelling. In the SI unit system, the unit for energy is Joule and the unit for power is Watt, with [W] - M L 2 ^ 2 , [P\ = MI^T" 3 . ® Remark 1. When the medium that fills O is not the free space, formula (7) is not representative of the electromagnetic energy but it is still possible to balance the power with the expression: (13)

P(t)=/a(^.H+^.E)dx

in the framework of square integrable fields. The problem of defining the energy of the electromagneticfieldby some formula with respect to D, E, B, H, comes from the difficulty of separating it from the energy of matter. In any case, writing that D, E, B, H are square integrable fields on every compact set is believed to impose that the electromagnetic energy be locally finite. This property often ensures uniqueness of the solution, notably for problems with wedges. 2 . STATIONARY MAXWELL EQUATIONS

If we assume that the electromagneticfieldis defined and satisfies (1) for all t in R (which is not so obvious in comparison with diffusion phenomena), it is possible to apply a Laplace transformation (see Dautray-Lions [1] Chap. 16). To be specific, let S*(R) denotes the set of tempered distributions, we define, for each distribution f on R, the following set (14)

If={«€R, e-*f€S*(R)}.

It is an interval of R. The bilateral Laplace transform of f is defined by (14)'

Lf(p)=F(e- |t O(n),

p=*+in, *€l f , n€R,

where Fg denotes the usual Fourier transform of the tempered distribution g.

1.2 STATIONARY MAXWELL EQUATIONS We write formally: *gft)=/* °° e-intg(t) dt, LRp) =/*~ e"pt f(t) dt. •■00

aOO

In electromagnetism it is more common to use the Fourier-Laplace transform: (15)

?(<o) = Lfi[-io))=j*~eitoiTt)dt, » = ip = - n + «.

Let H and z" be the real and imaginary parts of every complex number z = z' + iz". Thus for CD a a)' + ia)", (15) is defined forco" =t € If. Restricted to real numbers co, \ is the usual Fourier transform of f (up to a change of sign): f (co) s Ff( - co). Then using the following Fourier-Laplace property: (16)

( ^ ) \ o ) = - icof(co), to € I f (the interior of If),

the Fourier-Laplace transform of (1) is: i) icoD(co) + curl H(co) = J(co), ii) - io)B(co) + curl E(o>) = 0, (17)

iii) div D(co) = p(co), |iv) divB((o) = 0, with: (18)

-i(op(a)) + divJ(a)) = 0.

These equations are called the stationary (or steady-state) Maxwell equations. Note that, using (15), the real fields D, E, B, H are transformed into complex fields satisfying (19) D(co) = D( - S), E(o>) = E( - w), co € C with to" € I D , I E ,... since (15) implies

(19)'

f^)=rooe-iStf(t)dt = h-S). A

Thus the knowledge of D(co),... for positive numbers co also implies that of D( - co), ... when Im co = 0 is allowed, i.e., when D,E... are tempered distributions in time. The variable co > 0 is the (angular) frequency (or pulsation). We keep this terminology for a complex co. The variable v = co/2n is the frequency. Its unit is Herz (Hz). Frequencies are often given in MHz (106 Hz) or GHz (109 Hz). In free space, the wavelength X is define by X = c/v; then 1 GHz corresponds to 30 cm. The visible domain corresponds to 1015 Hz (X = 0.4 to 0.8 pm, 1pm = 10 " 6 m). Frequencies ofy-rays, X-rays are higher than 1015 Hz; hyperfrequencies or microwaves frequencies are about 10n or 1010 Hz, \ is about millimeters or centimeters. Frequencies for radio, TV, radar are less than 1010 Hz.

1 MATHEMATICAL

MODELLING

For electromagnetic fields defined for positive times only, we can use a unilateral Laplace transformation defined by: L u f(p)=/°° e"pt f(t) dt. Then the initial values of the field D(0), B(0) would appear in (17). But this would be inconsistent with the usual constitutive relations (see (22) below). Remark 2. We could also consider the Maxwell equations (1) for complex fields and notably for vector functions of the form: EKt) = D 0 e- i w t ,

(20)

E(t) = E 0 e- iwt , with D 0 , E 0 € C 3 ,

then equations (1) and (3) give equations (17), (18) for DQ, EQ directly. It must not be forgotten that the modelling of electromagnetic phenomena imposes real fields. In fact, stationary fields correspond to the real part of (20). ® Remark 3. The modelling of physical situations prevents us from having any distribution for p and J: they are nothing worse than first-order distributions. Remark 4. Some properties of Fourier-Laplace transforms We recall the analyticity property of some Fourier-Laplace transforms: i) If f is a distribution with compact support, then f (o>) is a holomorphic (or analy tic) function on all the complex plane C. ii) If f is a distribution on R (with If * 0 ) with its support in [a, oo [, then its FourierLaplace transform is holomorphic in a half-plane w" > £0,and satisfies: I f(o>)| < *-»** Pol( I <oI), a) = a)' + io>", (a" > £o , where Pol is a polynomial with positive coefficients (see Schwartz [1] p. 310). Converse statements are Paley-Wiener theorems. o o If S and fare distributions with support bounded to the left, with: I s and I f * 0 , then I s < f 2 I s n i f and: S*f(co) = S (o)).f(o>), for<o" € lsD I f . Some useful examples of Fourier-Laplace transforms Let Y be the Heaviside function: Y(t) = 1 for t > 0, 0 for t < 0; we will also use the sgn function: sgn t = 1 for t > 0, - 1 for t < 0, and the constant function 1, l(t) = 1 for all t. Then their intervals (14) are respectively given by: I Y = [0,+oo),

I 1 . Y = (-cx,,0], I sgn = {0}, I,={0}.

Denote pv the Cauchy principal value, 6 the Dirac distribution, their FourierLaplace transforms are i)

for f =s Y, £(<!>) = i/<i), a)" > 0; for the boundary value o>" = 0, we have: A

f(a)) = ipv--? + 7I6((D,),

1.3 CONSTITUTIVE RELATIONS

ii) f = 1 - Y, f(a)) = - i/o, <■>"< 0, and for a>" « 0: f(o) = - i pv ^ + n^w'), iii) f = sgn, f(a>) = 2i p v ~ , for a>M= 0, iv) f= 1, ?(<»)-2ff 3»'). fora>" =0. ® 3 . CONSTITUTIVE RELATIONS

In a given medium occupying the domain Cl, we define the vector fields P and M the (electric) polarization and the magnetization respectively by: (21)

P + e0E = D, M + noH = B.

For certain media, P and M depend only on E and H, this gives relations by (21), called constitutive relations expressing D and B by means of E and H. They may be of very different types: linear or not, local or not. So we give only some of them. We will assume here that the medium is at rest. Moving media will be considered in the Appendix. These relations may come either from a microscopic theory and (or) from homogenization, or from results of experiments (by solving some inverse problems or by mathematical modelling as in hysteresis phenomena). They enable us to partially avoid the description of the electromagnetic field interaction with matter. This is certainly a weak point of the modelling. The mere distributional framework which was sufficient for Maxwell equations (1), does not allow us to give a meaning to the constitutive relations even in very simple situations. The framework of (locally) square integrable functions (with respect to x) will be useful to give a meaning to such situations. 3.1. Linear isotropic dielectric media 3.1.1. Conditions obtained from invariance by time translation and causality In a linear dielectric medium, the constitutive relations have to be: i) linear (and local, free of x-derivatives) between D and E on the one hand, and between B and H on the other hand, ii) invariant by time translation, therefore expressed by means of a convolution in time (here we assume that the electromagnetic field exists for all time): (22)

D a E * E, B = ]x * H, for certain distributions in t, z and ji,

iii) causal, i.e., E(x,t') and H(x,t') cannot influence P(x,t) and M(x,t) for t' > t. These conditions imply (for isotropic media) that e and p are distributions with respect to t with support in R + . The relations (22) are formally written as: (22)' D(t) ^ / ^ ( t - t')E(t')dt\ B(t)=J4[ji(t - t')H(t')dt\

1 MATHEMATICAL MODELLING

We furthermore assume that the relations are local in x, and that e(t) and p(t) are (bounded) functions of x. If the medium is homogeneous and isotropic, the real valued distributions e and y are independent of x in Cl; t and p correspond to the permittivity and the permeability of the medium. The Fourier-Laplace transforms of relations (22) are: (23)

D(a>) * i(a>) E(a>), B(a>) = £(a>) H(a>).

The quantities e(a>) and £(<*>) are called permittivity and permeability of the medium at the frequency a). Their units are the Farad/meter and the Henry/meter. Then we define, 5 being the time Dirac distribution, (24) K = c-e 0 6, x ^ P - ^ S , and thus: (24)' K = e-c 0 , x ^ p - j V Comparing (24) with (21), we see that P and M are related to E, H by: |i)P-«.E,

i.e., P(<D) = S(<0)E(O>),

ii)M-x«H, i.e., M(a>) = x(a))H((D); K(O>) and x((o) are called electric and magnetic susceptibilities at o>. We often replace them by €0Kr(a>) and n0Xr(w), hence: t = c0er = e0(l + Kr), \x = ^ =* |io(l + x^: (25V

P(»)-«o^*)E<«), M(<o) = H0Xr(co)H(a>);

V^)* Xr(») are called relative electric and magnetic susceptibilities at <o; er, ^r are the relative permittivity and permeability of the medium. In order to give a sense to the convolution product in time (22) we have to make various hypotheses on (E,H), and on (e,p). Hypotheses on (E, H). We will study two possibilities: HI) E, H are distributions whose support in time is bounded below: supp E and supp H c [t0,+oo) for some t0; we write E and H € D|(R) . In this way the convolution product (22) has a meaning since the supports of £ and ji are also bounded from the left. Before the time tQ, E(t) = H(t) = 0, and we assume also J(t) = 0, p(t) = 0, t < tQ; the history of the medium begins at time tQ. We often take tQ as the initial moment. H2) There is p, 1 < p < oo, such that E, H € LP(R,L2(Q)3). We will make additional hypotheses on e and p (see for example (27) below) which will give a meaning to (22): in this case, all the past of the electromagnetic field intervenes in the domain Q.

1.3 CONSTITUTIVE

RELATIONS

The hypotheses HI) and H2) exclude electromagnetic fields of the form: E(t) = Re(E 0 e la,ot ),

H(t) = Re (H0em°X),

Vt€R,a>^>0,

(EQ, HQ independent oft), although their Fourier-Laplace transforms exist! Hypotheses on (E, JJ), or on (K, X). We will study four possibilities. In all of these, E and ]x are, at least, tempered distributions oft, thus I£ and Ip 2 R+. H3) E(O>) and j5(a>) are continuous functions of w on R (o>" = 0), with: (26)

lim

E(G>) = E0,

|u)|—»oo

lim £(<*) = u . |a>[—»oo

A hypothesis, a priori a little stronger, giving the continuity of the mappings: E - * E ♦ E, and H-> ji ♦ H in the spaces Lp(R+,Lq(Q)3), V p , q * 1, is: H4) K and x € L 1 ^ ) , with: supp K and supp x c R* = [0, oo). Let CQ(R) be the space of continuous functions on R, tending to 0 at infinity. The Fourier-Laplace transforms of E and ji with H4) are: (27) E = E 0 + K, iU*i o + x, defined for a>">0, with: S(a>), x(o>) € C^R^), V*>">0. Many other hypotheses, even stronger than (26) can be made, for example H5) K, x, § | , g | € L 1 ^ ) with support in [0, oo). This hypothesis implies: K(CD), X(*>), O>K(G>), O>X(O>) € Co(Rw0, for all CD" > 0, then (28) thus:

ici)K((o) and io>x(o>) — 0 for M - oo, w € R or C,

This hypothesis H5) also implies that if E, H € LP(R, L2(C1)\ then: D

> W ' B, f

€LP(R,L2(Q)3).

H6) E and p have a compact support, that is the medium keeps the memory of its past only for times t such as: - T < t < 0. In this case no hypothesis on (E,H) need to be made since in this case the convolution product is always defined. From Remark 4, we have: e (a>), £(a>) for a) € R are restrictions to the real axis of analytical functions with exponential type, which is important for the determination of E, JI by sampling: the question is to determine a function f from the family of numbers (£(j5)), j € Z, the results of experiments for angular frequencies with the same spacing 5 > 0.

1 MATHEMATICAL MODELLING

The answer is given by the "first sampling theorem" (or Shannon theorem), in fact a dual form of it; see for example, Meyer [1] p. 10: Let T m = inf {T, supp f £ [ - T,+T|}; if 0 */T m , then the family (f(j5)), j € Z, does not determine f; the tighter the spacing of the sampling, the greater the memory T of the material is. It is moreover often implicitly assumed (see Landau-Lifschitz [1]) that: H7) K and x are positive functions, which implies: H7)' K(CD) and x(u) are positive decreasing functions on the imaginary axis (note that K and x are real valued on this axis). 3.1.2. Consequences of the hypothesis H4). Kramers-Kronig relations The hypothesis on the support of e and ji, therefore of K and x, being in R+, along with H4) implies: (29) (l-sgnt)K = 0, ( l - s g n t ) x = 0. The Fourier-Laplace transform of (29) is (the product is changed into a convolution, and the Fourier-Laplace transform of sgn is given in Remark 4): (30)

(l + iff)K = 0,

(l + iff)x = 0,

with H the Hilbert transformation on R, defined by the Cauchy principal value:

(31)

fflM-ipvr'^IdStfilim/.-

. 1^-dS.

Thus the functions K and x (and thus i and A) are holomorphic functions of co in the complex half-plane: (32)

C* = {co€C,co" = Imco>0},

with K and x € L°°(C+), and are given thanks to their restrictions to the real axis by (33)

ft^.^/^^IdS

(similarlyforS), o>€C+.

By decomposition into real and imaginary parts (with usual notations): (34)

£(o>) * f'(») + if "(co),

A(o)) » M » ) + i£"(co),

co € C + ,

1.3 CONSTITUTIVE

RELATIONS

the relations (30) give, for all real <■>: i) *»<•) - e 0 =|pvjr~ i ^ d S

= ff8-<«),

(35) ii)f(«o) = - i p v /

"^^dS (0-0)

= -ff(6'-e 0 X»),

.v«\ .. »•***£ its and similarly for. .ji'fa) - P 0 , and P"((D). These relations are called Kramers-Kronig relations: the imaginary parts of z and A are fixed by giving the real parts, and the opposite is true. Furthermore as e and M are real, e and A satisfy (19), i.e.,

(36)

8 ( - 5 ) * I ( 5 j , A(-S) = AM,

co€C, c o ' - e l ^ , . . .

thus: (36)'

« ( - « ) - f ^ ) , P(-a)) = 0 5 ) , Vo>€R,

therefore: (36)"

l i ) f ' ( - o ) ) = £'(a>),

A'(-<«>) = A*(<«>),

jii)?»(-a>) = -g"(a>), A"(-<«>) = -AH(<«>), Vo)€R.

Substituting (36)" into (35), we obtain:

i)^(«)-« 0 4pvJ?* ,, (3) :r J5_dS f u

(35)'

- or

ii) f(«,) = . £ pv/~ (?'(S) - e0) - ^ - j d S, 0)

and similarly for A\ A" which gives the imaginary (resp. real) parts of 2, A from the real (resp. imaginary) parts of 2 and A for <o > 0 only. A medium with 2 "(o>) = 0, for all real o>, i.e. 2 real will be such that 2'(<t>) is constant, independent of a>. This is similar for y. Such a medium is called perfect. In fact only the free space is perfect. Other consequences ofH4) Notice the following property due to the continuity of the Hilbert transformation in FLl(R)cCQ(R). If lie* - e\II ! < x\, which implies | £'(«>) - ? \M | < n, for all real a>, then: |2"(a>)-2\'(a>)|<

fc"-«j|

,5n,

Va) €R,

or also ifK € Ll(R+), then: m Sup R |S(<o) M S Bc

*/ 0 ~|K<t)|dt = ,hc»Ll(R+).

1 MATHEMATICAL

MODELLING

3.1.3. Constitutive relation between J and E: Ohm's law In general, electromagnetic field produces currents and charges in media: these currents are called eddy currents. In certain materials (called ohmic conductors) we can add to (22) a relation constitutive between J and E, (of the same type, for the same reason): (37)

J = o«E (orJ = o*E + J e , Je a given exterior current)

with o a real function (or distribution) of t (for an isotropic medium) with support in R + also for causality reasons; o corresponds to the conductivity of the medium. The Fourier-Laplace transform of (37) is: (38)

J=aE

Ohm's law;

a(m) is called the conductivity of the medium (at m); its unit is ohm" Vmeter. It seems fairly natural to have hypotheses on o similar to that on e,y, notably to H4): H8) o = oj 8 + o 2 ,0! is a positive constant, with: supp a2 £ R+> °i€ L (R), giving a sense to (37), with the hypothesis HI) or H2) on (E,H), and the properties: supp J Q supp E with HI) when supp Ec[t i f +oo[), orJ€L p (R, L2(Q)3)withH2). Modelling an energy loss by Joule effect (see below) leads one to assume: H9) Re0 = Re(0 l +d 2 )>O. 3.1.4. Stationary Maxwell equations with Contitutive relations From (17) with (23) and (38), we obtain the stationary Maxwell equations in the form: | i) (ia>€() - 0(cu))E(o) + curl H(co)» 0 , ii) - ia>j*(<o)H(a>) + curl E(co) = 0, (39) jiii) div(£(a>)E(c«))) = p((i)), jiv) div(0((!>)ft(a>)) = O, with: (40)

- ia>jS(o>) + div (3(a>)t(a))) = 0.

Then we can change D and t into D c and ec defined (formally) by: (41)

Dc(t) = D ( t ) + / J(s)ds = € * E + / 0«Eds, "°° t -« t D C = D + Y * J = E * E + Y « ( 0 » E ) = (£ + Y , O H E .

L3 CONSTITUTIVE

Let: (42) and (43)

RELATIONS

I(t)=/ o o o(s)ds=/ o o(s)ds=:(Y*aXt),

supp2£R+,

£c = e + X = c + Y . o .

The Fourier-Laplace transforms of (43) and (41) are: (44)

£c(o>) = ?(o>) + i ^ r ,

DC(CD) = tc

(co)E(o)), co" > 0.

The stationary Maxwell equations (39) without exterior currents and charges may be written, according to (44): |i)i(oc c E + curlH = 0, (45)

u)-i»f»A + cuiift«O f 01) div(£cE) = 0, |iv)div(AH) = 0.

Then in a homogeneous medium (£c and A x-independent) the components of fe and ft satisfy the Helmholtz equation: Au + k 2 u = 0, withk2 = <o2£c£. 3.1.5. General constitutive relations with Ohm's law With the hypothesis H3) on e and H8) on o, we see that: (46)

E C ^ E ^ + OJY + K + Y *

looks like e but does not satisfy (27): for instance Y * o 2 is a bounded measurable function, but in the general case it is not integrable: (47)

KC dJf K + oxY + Y * o 2 € L*(R) + L°°(R), but KC £ L*(R).

However the Fourier-Laplace transform of KC is such that: (48)

KC(O>) = K(O>) + i (o((i>)/a>)-»

0 for | a> | - > oo, a>» > 0.

The hypotheses on E and o are not at the same level, since dt^dx = de/dt + o. Thus we have to make one of the following hypotheses: H10) Oj = 0, o = o 2 is the time derivative of a function 0, 9 € L (R), supp 0 Q R+. Then I = Y ♦ o = Y ♦ 9' = Y' * 0 = 6 * 0 = 9, the prime denoting the derivative. Thus: cc = e + e = £o5 + Kc,withKc = K + 0 €L l (R), thus ec = e0 + (cc, KC = K + 9 €C 0 (R^). With this hypothesis we see that EC has exactly the same properties as E, there is no singularity of E C at 0, and therefore it is not valuable for all media.

Hll)

1 MATHEMATICAL MODELLING

oj * 0, and o2 is the time derivative of a function 92 with 02 € L*(R), supp 82 Q R+, with the hypothesis H8),

then we have, as in (46) and (47): (49)

cc * 605 + a{Y + vc, vc = K + 82 6 Ll(R), supp vc Q R+,

and thus e0 + vc also satisfies the Kramers-Kronig relations (32). Note that the hypotheses H5) on (K,X), and H8) on o also imply that dx^dt, hence (vc,x) also satisfies the hypothesis H5). The notions of conductor or of insulator are generally defined with respect to their microscopic properties (the possibility of having "free electrons" or not). At the macroscopic level it seems that a conductor has o{ different from 0, whereas c^ = 0 for an insulator. Besides, a good conductor (at the frequency <o) has a high value for £ c(<°)> anc* a perfect conductor would correspond to the limit e£(<o) —* <x> (or oy—» oo). These terminologies may be somewhat confusing; sometimes the word "dielectric" concerns only insulator. 3.1.6. Conditions on t and \x for a dissipative medium in a bounded domain Usually the "energy (power) balance" in a bounded domain Cl is obtained in the following way. We consider the expression: (50)

P(t) = / Q ( ^ .E + | £ .HXx,t)dx.

Using the Maxwell equations (1) and relation (10) (similar to (11) (12) for free space), we obtain: p

(51)

® » / Q < curl H E - H c u r l E ) <** " fa J E * = -/adiv(EAH)dx-/nJ.Edx.

With the Poynting vector S (see (9)), we define: (52)

P s (t)=/ a div(EAH)dx=/ r n.EAHdT=/ r n.SdT,

/>j(t)=/ Q J.Edx.

Thus (51) with (52) gives the "power balance": (53)

./Xt) = Ps(t) + />j(t).

The usual interpretation of (53) is as follows (Jones [1] p.51): the left-hand side of (53) is the rate at which electromagnetic energy decreases in Cl; Ps(t) is the rate at which electromagnetic energy crosses T; Pj(t) is the rate at which electromagnetic energy is converted into heat in Cl.

1.3 CONSTITUTIVE RELATIONS

We also define: (54) thus (54)'

3Dan /» c (t)=/ n (- gF S.E+||.H)dx=P(t)+ ?j(t), P c (t)=/ f t (c 0 f .E + ^ . H ) d x + / f l ( f . E + ^ . H

J.E)dx.

Then with W 0 (t)=| / f l (E0 E2+t«0H2) dx, we define: (55) (56)

Po(t)=/fl(£of.E^of.H)dx=^(t), Pm(t)=/fl(f.E+^.H

J.E)dx,

and thus (57) Pc(t)=F0(t) + Pm(t). At this point, we note that: i) PQ(t) is the derivative of an energy, which looks like the energy of the electromagnetic field in free space. This is natural when comparing to the microscopic situation, ii) Pm(t) corresponds to the interaction of the electromagnetic field with matter. As Pj(t) is a part of Pm(t), it seems fairly natural that Pm(t) correspond to a transfer of electromagnetic energy into matter, by Joule effect, absorption,..., that is, by electric or magnetic loss. Generalizing the case where J = a{E with oj > 0, thus giving Ps(t) > 0, we can think that the modelling of a closed system (without exterior currents) must satisfy H12) P m (t)>0, V t € R , with Pm(t) defined by (56), and thus with (27) and (37): (58)

Pm(t)=/a[((|

* 0) • E).E * ( g * H).H] dx.

Recalling that by (54), (53), (57): (59)

- P c (t). - P0(t) - Fm(t)=Ps(t),

then H12) gives: (60) -P 0 (t)2:P s (t). Remark 5* If electromagnetic energyflowsout of Q by crossing T, the flux of the Poynting vector through T is positive and we have:

(61)

1 MATHEMATICAL MODELLING Ps(t) > 0,

thus by (60), PQ(t) is negative and the "electromagnetic energy" WQ(t) is decreasing in Q, using (55). Related to H12), but not equivalent to H12), the following hypothesis may be made: H13) tlMmhnijM^O, £"((*) = lmji(a>)>0, V*>> 0, that is also (with (35)*): H13)' Re(ia)£c(co))^0, Re (ia>|i(a>)) < 0,

V<o€R.

The link between H12) and H13) may be viewed in one of the following ways: i) Assume that: (62)

E, H€L2(R,L2(Q)3), and ^ , ^

2 €L

(R,L 2 (Q) 3 ),

then Pm(t) € L*(R) and using the fact that the Fourier transformation in L2(R) is an isometry (up to a constant factor) we have: (63) = ^Re/R[(ia>£(a>)+ 8(a>))|E(co)|2 + i<a£(a))|H(a))|2] dec. We see that hypothesis H13) gives, when (62) is satisfied: (64)

/RPm(t)dt>0.

ii) Assume that the electromagnetic field is stationary, that is, D, E,... are the real parts of (20), for real <■>. Then we have: (65)

imt

1C0U

t j2jZiiat 2 .-2iuti 2 2iu>t j? 2 - 2 i ]<ix+-^falH'0e A

Thus the mean value of P0(t) over a period T=2n/a> is zero. Then P and M defined by (25), are: I P(t)=K . E (t)=/°° «(s)E(t - s)ds = Re (E_ f ic(s) e io)(t " s)ds) (66) = Re (E0 S(- <o) e1"1) = Re (E0*(<o) e"1"1), (67) M(t)=X « H (t) = Re (H0xX - <*) ei,ot) = Re (Hjfto)) e-i(ot). Similarly: (68)

J(t)=o . E (t) = Re (E08( - o>) e1"") = Re (E03(<o) e~lmt),

1.3 CONSTITUTIVE RELATIONS

and thus Pm(t) is given (according to (56)) by: |i>m(t)*iRe[e-^-^^ I

+Re[((oft>)+S'((D))/ fl |E 0 | 2 dx+(o>X>))/ Q |H 0 | 2 dx].

For real o>, we have from (44), (27): (70)

Re(- io> ec(a))) = a>S"(a)) + o,(c«>),

Re(- io>£(a>))= x"(a>).

Thus hypothesis HI3) implies that the mean value of Pm(t) over a period is always positive. It is not true that PJit) is positive for any t, but it is true with H13). It seems that the discrepancy oetween H12) and H13) may come either from time delay due to convolution or from matter giving (during a short interval of time) energy to the electromagnetic field. We define: (71)

C+ = {z€C,0<argz<n,z*0};

C* = {z€C, zM = Imz>0}.

We generally adopt inequalities a little more precise than HI3): H14) <D£c(co)€C+, o>ii(o))€C+, Vo>€C+. Thus for real positive o>: H14)' Vc(<*)>0 withe'c(<*)>0fore c(«>) = 0, similarly for £(<■>), V o » 0 . Remember that we have, thanks to (36) and (36)': (72)

e^(- o>) = - f » < 0,

£"(- o) = - p(o>) < 0 for a) > 0.

If we suppose that H5) and H7) are satisfied, the functions (73)

<K^>) = Im(coftc(a>)), (K(o) = Im(a)x(a>))

are imaginary parts of holomorphic functions of a> in C+, which tend to zero when | a> | -—» oo. Thus 0 and (> | are harmonic functions on C+ (i.e., for o>" > 0). Since their boundary values are non-negative, then from the (strong) maximum principle (see Dautray-Lions [1] chap. 2), we have (74)

«<«>)> 0, («<■>)> 0, Vo>€C+ (o>">0),

and thus we deduce H14) from H14)\ Remark 6. With above notations (see H8) it is often separately assumed that:

1 MATHEMATICAL

H15) giving: (75)

MODELLING

c"(a)) = ic"(a))^0,fi M (a))>0, o'(a>) > 0, o' 2 £ 0, V « > 0, tmc(m) = ?M(a>) + (SWco) > 0, for a) £ 0.

With hypotheses H4) and H8), we can apply the strong maximum principle in: i) the first quadrant of the complex plane: C^fcossco' + io)", C0'>0, 0 ) H > 0 } ,

to K" and A". Recall thatft"(a))= 0 for o> = ia>" from (36), thus: (76)

ftM(a>)

= ?M(a>)>0, aH(a>)>0

for<o€C++9

and then: (76)' ? M < 0 and A"(o>) < 0 for o> in the quarter of plane o>' < 0, o>" > 0, ii) the half-plane C + (<o" > 0), to flj, thus (77) d^(a))>0, Vo)€C + . Note that although £c(o>) is singular at 0, we can apply the maximum principle to ££(a>) in C**, because as a> —* 0, id(a>)/a> » i8(0y<a = id'(0)/<*, because dM(0) = 0, (78)

with Im(id,(0)/a>)Sr(«),(d,(0)/|a)|2)>0, andd'(O) * 0.

On the imaginary axis with o>" > 0, d(a>) is real, e c(<o) = £ "(a>) = 0. Thus: (79)

£»>0

in C++.

Note that H14) implies the inequality: (80)

o>'?^(<o) + a>Me c(a>) ^ 0 for a>" > 0,

similarly for A and thus: (SOY

£»

> - ^ fc(a>) (or - aig a) < aig ? ^ 7T - arg a>) fora>" > 0.

With the hypothesis (see H7)) that K (or KC) and x are positive functions oft, KC and X are decreasing functions of co along the imaginary axis, and thus e c and A have the same properties; since tc and A tend to e0 and p (resp.) as | o> | —» oo, we have (81)

2c(o>) > 60, A(o>) > }iQ, V a) = io", o>» > 0.

This implies, thanks to H14)' that e c and A never vanish in C + , as asserted by H14).

1.3 CONSTITUTIVE RELATIONS

Often a stronger hypothesis than H14)' is made (see Landau-Lifschitz [1])) H15) €;(o>)>0,A"(o))>0, V<o>0. With H7)\ this implies that ec and y never vanish in C + . Thus they have inverses which tend to z~l and \Tl at infinity (with H4) and H11)). Hence we can inverse the causality relations (22)' and (41)! & 3.1.7. Some other definitions From the permittivity and the permeability of a medium at the frequency a>, we usually define (with hypotheses H14): i) the wavenumber k = k(») by: (82) l^ = o>2€cix, 03argk<*, thus, with (71), k € C+ for © € C+, and also: k=Jo)2ecfi, withargk=2(aig<oec + aig<■#); ii) the wavelength X = 2n/\ k' |, k* = Re k, but generally defined only for real k; iii) the complex index of the medium n = n(«) by n « ck/a>« ± j£c(o))(i(a>)/c0|io, (with the choice n = c Jf^, with 0 S aig n < n, we would have ck = ± <on). For real positive w, we can take n = c ^I~p and ck = no>; iv) the impedance of the medium Z « Z(a>) by: A

(83)

Z = ^ = 4 - = $ / ^ with -*/2<argZ<ii/2, G>£C

sinceargZ = arg(co(i)- aigk=2( ar 8( a) &)~ ^"^(^c))' thusZ' = ReZ>0. Note that Z exists for all « in C+ (with Z *0), since Sc(a>) and ji(<o) never vanish according to H14). We have ec(<o), £(a>) from k, n, Z by: (84)

o>6c = k/Z, a>A = kZ, £c = n/cZ, £ = nZ/c,

but we cannot choose arbitrarily k and Z, with 0 £ arg k < n, - n/2 < arg Z < n/2, to have ec andftin C+. Note that the impedance unit is [Z] = Ohm = ML2!-"1©"2. The admittance of the medium is defined by Y = 1/Z. Remark 7. Using (36), the quantities k, n, Z as functions of o satisfy: (fl(- S)) 2 * (f(a>))2, forf^k, n, Z, thus: f(- 5)= ± fiCco). With the previous definitions of k, n, Z, we have:

(85)

1 MATHEMATICAL MODELLING k(-S) = -k(<o), Z(-S)=Z(S),

n(-5)=ip:

Thus Z and n satisfy (36) and are the Fourier-Laplace transforms of some real functions or distributions. Besides: i)when |«>| -»oo, k(a>) « ci>/^gr*©c, Jim^ Z(<*)=Z0> and J m ^ n(&>)= 1, with Z 0 at I|io/c0 the impedance of the vacuum, Z0 «120 *; ii) when w —»0,and 0| * 0 (see H8)), we have: Z(<o) —»0, k(a>) —• 0, n(co) is singular atO. 3.1.8. Summary of the main properties for a linear isotropic dielectric medium The electromagnetic behavior of a linear isotropic dielectric medium is given by the "constants" of the constitutive laws, the permittivity c or tc (linked by (42)) and the permeability p, with the following "natural" properties (see H4), HI 1)) (86)

£C = E06 + OJY + VC,

}is|io5 + x.

Y the Heaviside function, oj constant > 0, KC, and x € L (R), real, with support in R+. Real and imaginary parts of the Fourier-Laplace transforms of vc and x have to satisfy the Kramers-Kronig relations (with H the Hilbert transformation, see (31)): (87)

O^ffO;,

0 ^ - f f O ; . , simUarlyforx.

Finally the energy loss in the medium is expressed by relations H14) (or H14)*). Apart from vacuum and perfect media, we have to assume that: (88)

c»>0,

£"(<*)>0,

V»>0.

There is no natural hypothesis of positivity for real parts of ec or KC and A or x. H14) requires positivity only when the imaginary part is null. Note that these real parts are Hilbert transforms of positive functions, in the "regular" case. 3.2. Linear anisotropic dielectric media Certain media called anisotropic media have their physical properties which depend on the direction taken at each point. In the linear anisotropic dielectric media, the constitutive relations are given by (22), with e and ]x real matrices, functions (or distributions) of time with support contained in R* (from causality). Their Fourier-Laplace transforms will be complex valued matrices, with properties similar to those of isotropic media, under similar hypotheses, see H3): we only have to replace eQ, fiQ by e l , JI I, I being the unit 3x3 matrix, in (26), (27),... We have as in (36), (26):

1.3 CONSTITUTIVE RELATIONS

(89)

£ ( - £ ) = S(o>), £( - i ) = y(o>), 0) € C, o>" € Ie, 1^,...

(thus t (a>) and p(o>) are real matrices on the imaginary axis) and (90)

lim

£(<o) = £0I,

|o)|-*oo

lim £(*>) = u I. (u>|—»oo

Decomposing £(o>) and £(<o) into real and imaginary parts: £(0>) = 6'(o>) + ic »(o>) ,

fa)

= y'(o>) + i£"(o>), 0) € C + ,

we again obtain the Kramers-Kronig relations as in (35) or (35)' between $' - e0I and ? ", orft'- )x I and p", and also for the components of these matrices: (91) 8j(«)-« 0 ^-i5f8j|(«) f tj(«) — H ( ^ - e 0 « ^ X « ) f V i J . Similarly for "ohmic anisotropic conductors", an electric field creates interior currents as in (37), but with a a real matrix. The Fourier-Laplace transform of (37) is Ohm's law with 5(<t>) a complex matrix also satisfying (89). We can thus define quantities D c and ec as in (41) and (43), giving the stationary "homogeneous" Maxwell equations (45) for anisotropic media, with matrices ? c , £. The only interesting new point is the positivity relations according to dissipation and energy losses in the medium, corresponding to H12) or H13) for all o> > 0:

H16) Re( - i»2*c«Mj> ~ °'

Re(

- kSflyMj) " °'

VE^I-^cC3,

or stronger (see HI5)): there is a = a(a>) > 0, for all o> > 0, such that: H17) R e ( - i c o ? c ( a > ) U ) > a | t | 2 ,

Re(-ia>£(a>)U) > a |£| 2 ,

VUC3,

that is: for all <o > 0, the following matrices are definite positive: £

>)=i( ? c-?;xc«>), AI(CO)=JJ(A. a*x»>-

But note that for o> —► +oo, fc(o)) —♦ c 0 l, P(o>> —»ji I, thus necessarily a(o>) —»0. For Q) — 0, A((o) -♦ A'(0), a real matrix, - mlc = - ia>£ + 8 -> 6'(0), thus a(o>) — 0. From the strong maximum principle, we can assert that HI7) implies that these relations are also satisfied, with hypotheses at infinity, for all a) € C4"*". Example. When the medium is a ferrite (with an applied magnetic field parallel to the z-axis), the permeability matrix is, with complex p., j = 1, 2, 3 J»!

(92)

*2 fcol - f c 0 M3

The matrix r - JJ (fl - (0)) is positive definite if and only if: (93)

u» > 0 , i = 1,2,3 and fr")2 <ni'n" 2 3

1 MATHEMATICAL MODELLING

Remark 8, If the matrices ?c(a>) and |i((o) satisfy H17), they are invertible, and their inverses also satisfy: there is a 0 depending on a> so that for all o> > 0: (94) Re(-(i(0? c (a>)r 1 U)>a o |5| 2 , Re( -(icafcca))" * U ) :> aQ |$| 2 , V t € C 3 . The importance of such linear anisotropic media comes notably from the fact that they are stable by homogenization: mixing periodically two linear homogeneous dielectric media (isotropic or not) with H15) or H17), we obtain a linear composite anisotropic medium with H17). 3.3. Linear chiral media A medium whose constitutive relations are given by: (95)

D=£*E+?.H,

B - j a , E + |i,H,

with real functions (or distributions) of time c, T , p. , p with support in R+, is called a chiral medium. The Fourier-Laplace transform of (95) is: (96)

D(<D) -1(«) E(o>) + t ((«)) H(O0, §(<*) = y (a>) E(o) + A(o) ft(a>),

with e, t , £ , A also satisfying (36), and with the usual analytic properties (see 3.1). Note that if Ohm's law is also satisfied, we can replace D and c by D c and cc resp., as in (41),(44)),in relations (96). Furthermore we generally assume that 7 = - **. Modelling dissipation and loss of energy in a chiral medium leads to the following hypothesis, similar to H12),H13),H15) or H17): for all co > 0, H18)

| R e ( : i ( o 2 9 a p £ a Ip)>0,

^ V * - « j , y , £a = « a j ) € C 3 , a = l , 2 ,

I with 9 U = c (or ec), 9 12 = ? , 921 ~ ? , 9 22 « 5, or a stronger one: there is a=a(w) > 0, for all w > 0 so that H19) R e ( - i o > 2 9 a ^ j I p k ) > a | $ | 2 = a Z U a j i 2 ,

V£cC3xC3,

giving therefore: (97) c "((«>) > 0, £"(CD) > 0, and: | X(o>) | 2 > 0. More generally E, 7 , JI, M may be real matrices corresponding to a chiral aniso tropic medium; then each (9ag) is a 3x3 matrix, satisfying (36), with some usual holomorphic properties, and with a hypothesis generalizing H19): there is a > 0, depending on o>, so that, for all o> > 0: H20)

Re(- io, 2 e aj(Jk * aJ V) * a 2 | y 2 ,

VUC3.

1.3 CONSTITUTIVE RELATIONS

Note also that this family of media is stable by homogenization: by mixing periodically a linear chiral medium with another linear dielectric medium, chiral or not, isotropic or not, we obtain a linear anisotropic chiral medium (with H17) for dissipative media). 3.4. Nonlinear constitutive relations The coefficients of the constitutive relations in general depend on the state of the medium, notably on the temperature, on the strain (in an elastic medium), and also on the electromagnetic field itself, thus giving nonlinear constitutive laws. If the medium is near equilibrium and if the electromagnetic field is not too strong, we can often linearize these relations. We give some examples where nonlinearity appears. In thefirstexamples, nonlinearity implies anisotropy: i)" birefringence" in an electricfield;the permittivity is given by: €

ik =£ °ik + a ^i^k» a constant; ii)"magneto optic effect"; the permittivity is a (non-symmetric) tensor function of H but satisfies: eik(H) = €ik(-H);

iii)"Hail effect": the conductivity is a (non-symmetric) tensor of H: «ik( *!) = %(-H). iv) ferroelectric media and ferromagnetic media have fairly similar properties, with respect to the electric field or to the magnetic field; here we develop only the question of ferromagnetic media. A ferromagnetic material may be defined as one that possesses a spontaneous magnetization, and its magnitude M depends only on the temperature of the material and not on the presence or absence of an applied magnetic field. Induced magnetization may be neglected. The magnetization vector M(x) is: (98)

M(x) = MsU(x), with |U(x)| = 1, xeCl.

In a static case (that is without evolution) the magnetization field is deduced from H thanks to a minimization principle, as follows. We consider a ferromagnetic medium occupying a domain Q in R ; we assume that the exterior domain O' of D is occupied by free space. An "exterior" magnetic field HQ (a constant vector) is applied onto this medium. Then the magnetic field and the magnetic induction must satisfy: (99)

divB = 0,

curlH = 0 inR3,

with the following conditions:

1 MATHEMATICAL MODELLING

i)inQ' = R 3 \Q: (100)

H = H0 + H r ,B = noH = Ho(H0 + Hr), withH r (x)-0for | x | - o o .

ii) in Cl, we set H = H0 + H* and we have: (101)

B = PoH + M,withM = MsU, U€S 2 .

From (99) and (100), H,. is given from a potential ♦ with: (102)

Hr a grad <> t in Q\ with Ao = 0, * -> 0 at infinity.

From equations (99), (since we search localfiniteenergy solutions) we have the following transmission conditions across the boundary f of Cl: (103)

[nAH]r = 0,

[n.B]r = 0,

where [u]f is the jump of u across T. We assume that the domain Cl is simply connected. Then from (99) there is a potential v (in the Sobolev space H*(Q)): (104) H' = gradv inQ, so that, from (99) and (101): (105)

divGiQgradv + MsU) = 0 in a,

with the boundary conditions on I\ from (102), (103) and (104):

<«*>

V l r

'oK^Mlr-'.Slr

We define the exterior capacity operator Ce (see Dautray-Lions [1] chap. 2) by: (107)

^l r ) = ^ l r » 0 such that A o = 0 in Q', *(x)->0 for |x|—► 00.

Then v must satisfy the following boundary condition:

<108>

"oil r - , , o c e ( v lr ) = - n M lr = - M s n U lr

The magnetization vector M is determined from a minimization principle in the following way: U has to minimize the Gibbs function G (or free energy) (109)

G = Ee + Ea + Ed + E z + E tt , with Ee »/ f t w e dx, Ea =/Qwadx,

Ee the exchange energy, Ea the anisotropy energy, Ed the demagnetizing energy, E z the Zeeman energy, Eas the surface anisotropy energy. They are defined for a cubic crystal (for example) by:

1.3 CONSTITUTIVE RELATIONS

(110)

w^ClgradUl^CZIgradUil2,

(111)

wa = K^ufyf + U^uf + Uf U\) + K2(U?U|U|),

(112)

| E d = - 3 M s / n u - H ' d * w i t n < 1 0 4 > . E z = -M s / f l U.H 0 dx,

C>0, U ^ U ^ U ^ ) ,

I EÂťs - 5 Ks SQQ < n - u ) 2 C Ks constant. In the case of a hexagonal crystal, we and wa are given by { C , ( 2 &2)

(110)'

we=A 2

(111)'

wa = K 1 (l-U^) + K 2 (l-u5) 2 , Kjconstant,j = 1,2.

+C 2

\ ^ \ \ vnthCuC2>0

We have a problem with two unknowns U and v, with (105), (108). Plotting the component of the volume average magnetization in the direction of HQ against HQ gives the magnetization curve which is typical of a hysteresis phenomenon. For other situations in ferromagnetic media, see Brown [1]. These problems are not very different (at least from a mathematical point of view) from problems relative to liquid crystals where the main unknown is a vector field n of unit modulus, the "director'1 which gives the orientation of the "anisotropic axis". In a liquid crystal, a magneticfieldH and an electric field ÂŁ induce respectively a magnetization M and a polarization P M = Xl H + x2(n.H)n,

P - (^ - 1)E + c2(n.E)n,

with constants xv x?> *v *2 c o r r e s P o n c l i n g to the magnetic susceptibilities and the dielectric permittivity with the field parallel and perpendicular to the director. For the modelling of such situations, see Ericksen [1] and Leslie [1]. In the evolution case, we have to write thermomechanical balance laws; observed phenomena are typical of hysteresis phenomena which correspond to a certain nonlinearity with a certain memory of the past. We refer to Kranosel'skii-Pokrovskii [1], and also to Mayergoyz [1], for Preisach's hysteresis modelling. Many physical phenomena are based on a coupling of continuum mechanics (elastic solids or fluids) with electromagnetism; see Eringen-Maugin [1]. CONCLUSION

The use of constitutive laws generally avoids taking into account the interaction of electromagnetic field with matter: this macroscopic modelling cannot represent all thefinesituations in physics, notably when matter is not in (local) thermodynamic equilibrium. Then we must use finer modelling, coupling Maxwell equations with other equations (Vlasov equation, or Schrodinger equation for quantum effects). Such situations lead to nonlinear equations and thus are outside the scope of this study.

CHAPTER 2

MATHEMATICAL FRAMEWORK FOR ELECTROMAGNETISM

INTRODUCTION

An electromagnetic field given by the electric field E and the magnetic field H, in a bounded free space domain Q, has finite energy if E and H are square integrable, which immediately implies, at least in the stationary case, that their curls are also square integrable. Thus the L2-functional framework is a natural one for electromagnetism. Owing to the repeated use in electromagnetism of operators grad, curl and div, we are naturally led to define functional spaces based on these operators. For the grad operator the natural functional spaces are the Sobolev spaces. At first we briefly recall the main properties of these spaces, referring to DautrayLions [1J, Brezis [1], Adams [1], Necas [1] for developments. Then we give functional spaces based on the operators div and curl. Sobolev Spaces Let Q be an open set in Rn. We briefly recall the definition and the main properties of the Sobolev spaces on Cl; at first order (1) Hl(Q) = {u€L2(Q), |^€L 2 (Q),i=lton} = {u€L2(Q), gradu€L2(Q)n}, du

with grad u = (g--). _

and more generally using multi-index notations 26

INTRODUCTION

« = («1...-,«„), | a | = a , + . . . + a w i t h a e N , D°u=-Sj n

-5—,

H*(Q) = {ii€L (Q), D"ueL (Q), V a , | a | s k } , k e N . They are complex or real Hilbert spaces for the norm U

(

D a U | 2 d X ) , / 2

" ^ , f ,c «l We denote by D(C1) the space of C°° functions in CI with compact support in CI; let H £ ( Q ) be the closure of D(C1) in H ( Q ) . Then with a

j £

(2)

D(Q) = {v, B u e Z X R ^ v - u y ,

and supposing that CI is "regular" (locally on one side of its boundary I\ with suitable regularity), we have: Lemma 1. The space D(Q) is dense in H\Q), V k € N . For the proof, see Lions-Magenes [1], Necas [1], Grisvard [1]. Recall also the result under the hypothesis T Lipschitzian: the space ofLipschitzfunctions on Q, C is dense in H*(Q). 0,

The Sobolev spaces on CI for real k (and not only positive integer) are also defined by duality and interpolation, thus allowing one to define Sobolev spaces on the boundary T. We now give thefirsttrace theorem in Sobolev spaces. n

It 1 1

Theorem 1. Let CI be a bounded open set in R ofclass C " ' (see App.l). Letnte a unit normal to T. The trace mapping u —• (u( , | ~ | »---»~ol ) defined on r

1/2

D(Q), extends continuously from H (Q) into H " (0 x... x H (0. k

The kernel of this mapping is the space H (Q). We give here two interesting trace results (see Lions [2]); we define the spaces l

(3) H (A,Q) ={ u € H\CI), Au € L (Q)}, H(A,Q) #{u € L (Q), Au € L (Q)}. l

1/2

Propositionl. LetueH (A,Cl)

1/2

3/2

Let ueH(A,Q), then u | € * T ( D , ^l €H" (r). r

1/2

thenu| €H (r),gJ| €H- (r).

2 MATHEMATICAL FRAMEWORK

1. SPACES FOR CURL AND DIV; TRACE THEOREMS

Let Q be an open set in Rn. For v in L2(Q) or L2(Q)n, let (4) IMI = (/Q|v(x)|2dx)1/2 be the norm of v in these spaces, and we denote by (u,v) the corresponding scalar product of u and v. We define the following Hilbert spaces with their "natural" norms: H(div,Q) = {v€L2(Q)n, divv€L2(Q)}, (5) M

(6)

H(div.n) =(IM|2+lldivv,|2)1/2:;

H0(div,Q) * the closure of D(Clf in H(div,Q).

In a similar manner, we define for n = 3: (7)

H(curl,Q) = {v€L2(Q)3, curlv €L2(Q)3}, WI

(8)

H(corW s(M2 + ICUriV,2)1/2;

H0(curl,Q) = the closure of D(C1)3 in H(curl,Q).

In definitions (5) and (7), div v and curl v are taken in the sense of distributions. We also have to consider open sets in R2, and curl operators in this case. We define two "curl" operators as linear differential operators, one (scalar) from D\C1)2 into D\C1), and the second (vectorial) from D'(C1) into D'(C1)2 by: (9) (10)

curiv

=SST-a5r> Vv-(v lf v 2 )€D»(Q) 2 ,

cSri#.^.^)

V*€D'(Q).

Note that for all o € D(C1) and v € D9(Q)2f we have: (11)

<c7rU,v> = <^,v 1 > + < - ^ , v 2 > = <0,.^+^>=:<0,curlv>.

Then we can define the spaces H(curl,Q), HQ(curl,Q) for Q in R2 as in (7). We make the following regularity hypothesis: HI) The open set Cl in Rn is locally on one side of its boundary I\ with T bounded and Lipschitzian (i.e., defined by Lipschitzian local charts). We define: (12)

CQ (O) the space of (restrictions to Cl of) Lipschitzian functions with compact support in Rn.

2.1 SPACES FOR CURL AND DIV

Then we have a first property of spaces H(div,Q) and H(curl,Q): Lemma 2. Spaces C$l(Q)n and D(Q)n are dense in H(div,&) and H(curl,Q), with n = 3or2. PROOF. Using Lipschitzian charts we first prove the Lemma for the half-space, then we operate by truncation and reguiarization by convolution, as usually done in Sobolev spaces (see Duvaut-Lions [1], for T of class C1). We can also prove that the orthogonal complement of the space of the regular functions with compact support (in Rn) in H(div,£l) and H(curl,ft) reduces to {0} (see GiraultRaviart [1], Dautray-Lions [1], chap. 9). We also prove extension properties using similar arguments: Definition 1. An extension operatorfrom a functional space X(Q) into X(Rn) is a continuous operator Vfrom X(Q) into X(Rn) such that: Pu| ft = u , Vu€X(Q). Lemma 3. With hypothesis HI), there are continuous extension operators from H(div,Q) into H(div,Rn) (for any n) and from H(curl,Q) into H(curl,Rn) n = 2, 3. The following trace theorems are basic in functional electromagnetism and fluid mechanics (see Duvaut-Lions [1]).

analysis for

Theorem 2. Trace in H(div,Q). With hypothesis HI) the trace mapping: Yn: v —* n.v| , defined on D(Q)n, has a continuous extension, also denoted by y from H(div,Q) onto H~1/2(r). The kernel of this mapping, keryn, is H0(div,Q). Let us denote by n A v the vector product of the unit normal (outgoing or incoming) with vector v; let nTv be the projection on the tangent plane to T of the restriction v| of any element v € D(Q) to T (in the Appendix we will note t r for nT). We have: (13)

7i r v=-nA(nAv| r ), Vv€D(Q).

Theorem 3. Trace in H(curl,Q).

With hypothesis HI), the trace mapping:

YT: v —♦ n A v| (resp. nr) defined on D(Q)n has a continuous extension also denoted by YT (resp. 7ir), from H(curl,Q) into H" m (T) m with m = 3 for n = 3, m = 1 for n = 2. The kernel of this mapping, ker yf (resp. ker nr) is the space H0(curl,Q).

2 MATHEMATICAL FRAMEWORK

Remark 1. On the equivalence of the trace theorem for y% and nr i) With only the hypothesis HI), it is not obvious that we have n A u € H"1/2(D3 for all u in H"1/2(r>3, because n is in L°°(r)3 only. ii) If T has C ,n regularity (n > 1/2), then the normal n is C ,X] (i.e. there exists C > 0, so that |n(x) - n(y)| < C |x - y|n), the mapping ♦ —» nfl is continuous in HS(T) for |s| < 1/2 (see Grisvard [1] p. 21), so that the trace theoremsfory andnT are then equivalent. iii) Note that the decomposition of any vector function u into u = nru + (n.u)n, gives the orthogonal decomposition of L2(On into L2(On = L 2 (OeL 2 (r), but does not a priori (with only hypothesis HI)) decomposes the space H (On into H1/2(r)n = Ht1/2(r)© Hi /2 (0. But obviously this is true for the C1,n(n> 1/2) regularity of the boundary, and then by duality the H" (0 space is also decomposed. PROOF of Theorem 2.

(14)

i) We start from the Green formula

(v, grad<►) + (divv, ♦)=/ r n.v* dT,

forallvinC^W 1 , ♦incS,1(^)(orin/>(S)nandinZ)(Q)), and by density of d*0'\Q) in H^Q), for all ♦ € H^Q). Thus we have: (15)

U r n.vodT | < W H ( M ) .IWIHl(a), Vv€D(5), *€H l (n).

The left hand side (l.h.s.) depends on <► only through its trace ji = $| f on T. Since (16)

H_ 1 / 2 H

= inf W , , <n ♦€H1(a>,0|r=»i H(Q >

we get (17)

|/ r n.vndr | SlMIH(div>a). M H „ 2 ( i r

Thus the mapping y^. v —»n.v| defined on D(Q) equipped with the H(div,Q) nonn, 1/2

is continuous into the space H" (0 and (from Lemma 2) has a natural continuous extension also denoted by y from H(div, Q) into H" (0. ii) We have to prove that y is onto. Let n be in H" 1/2 (0. Then there exists u in H (Q) solution of the Neumann problem: (18)

-Au + u = 0 i n Q ,

| | l r = M on T.

Then v=grad u is in H(div,Q) and satisfies ynv=p, which proves that yn is onto. Remark that the Green formula (14) is also true for v in H(div,&) and 0 in Hl(Cl).

2.1 SPACES FOR CURL AND DIV iii) Proof about the kernel of y. Let w be in ker yfl orthogonal to Z>(Q)n in H(div,Q). Then w satisfies: (19)

((v,w)) = (v,w) + (divv, divw)«0,

Vv€Z)(Q)n.

Let w0 s div w. Then w0 is in L2(Q) and satisfies: (v, w) + (div v, w0) = 0, V v in D(Clf. Therefore w=grad w0, thus w0 € Hl(Cl). Applying the Green formula with ♦=w 0 , and v * w, we obtain (v, grad w0) + (div v, w0) = (w, w) + (w0, w0) = 0, thus w=0. PROOF of Theorem 3. i) We start from the Green formula: (20)

(curl v, *) - (v, curl ♦)=/ r n A v. ♦ dT,

for all v and * in JD(Q)3 or also v and ♦ in Hl(Q)3. Thus (21)

l/rnAv^dri^lMI^^.IW^^,

Vv€D(Q) 3 , M H W .

The left hand side of (21) depends on <> t only through its trace M = 4> | _ on T; thus (21)' hence

1/rnAv.MdTI ^M H(curin) .IW Hl/2(r)3 , Vv€Z)(5) 3 , P eH 1 / 2 (D 3 ,

(22)

llnAv« H . 1/2(n3 ^Bv8 H(cnria) , WinZ)(Q)3.

Thus the mapping yx: v—> n A v| defined on D(Q) equipped with the H(curl,Q) norm, is continuous into the space H"l/2(T) and (from Lemma 2) has a natural continuous extension, also denoted by y , from H(curl,£2) into W (0Note that the Green formula (20) is also true for all v in H(curl,Q) and * in H*(Q), with the duality bracket of H 1/2 (0 3 and H"1/2(03 in the r.h.s. instead of the integral. ii) Proof about the kernel of y . Let u be in ker y, orthogonal to D(Q) in H(curl,Q); then u satisfies (23)

(curl u, curl ♦) + (u, ♦) = 0, nAu| r =:0.

V 0 € D(Cl)3,

2 MATHEMATICAL FRAMEWORK

Let v be in L2(Q)3, with v a curl u; v satisfies: (v, curl 0) + (u,4>) = 0, V 0 € D(Q)3, thus curlv+u = 0, therefore v € H(curl,Q). Then the Green formula gives: (u, u) + (v, v) = - (u, curl v) + (curl u, v) = 0; hence we have u » v=0. <8>

Remark 2. We emphasize that the mapping y is not onto H " 1/2(T)3. A more precise trace theorem is seen later on. 0 2 . JUMP FORMULAS ACROSS A BOUNDED HYPERSURFACE T IN R n

Let Q be a bounded open set in Rn, with a regular (Lipschitzian) boundary T; let $ and u be "regular" functions on Rn, (resp. with values in C and Cn), that is of C1-regularity on each side of T, having limits up to the boundary. Then the jump of a function 0 (or u) across T is denoted by:

(24)

M r t f *l r _-*l r + .

with#| , 0L the boundary value of 0 on each side of r, T. being the interior face. The normal n to T is oriented from r_ to T+. We denote by dp the derivative in the sense of distributions in Rn of 0 with respect to X| and by (3^) the derivative in the classical sense of 0. Then denoting by 6r the Dirac distribution on T, we have: Proposition 2.

(25)

With the above regularity hypotheses on Cl, 0, u, we have:

a i o=(a i 0)-M r n i 6 r , grad 0 - (grad 0) - [0]f n 8 r , div u = (div u) - fn.u]r 6r curlu = (curlu)-[nAu] r ^fornsS.

These formulas are also true for 0 and u satisfying:

♦|Q € H!(0), *in, € 4^(5'), Of = R"\5, (26)

u | 0 € H(div,n), u|Q, € Hloc(div,Q') u | a € H(curl,Q), u| a , € Hloc(curl,Q,)forn = 3 ,

with the notation for X = H1, H(div,.), H(curl,.) (27)

Xloc(Q') V {* € D'(Q') (or € D'{ttf), ?<> | € X(Q*), V C € D(R%

2.3 DIFFERENTIAL OPERATORS

Formulas (25) are straightforward from the Green formulas. As an example, we have with 0 a d ^ 0 | Q , ♦Q»^%I 0 , • PROOF.

=48^Cdx^/a^Cdx./^QCdT+/^a,Cdr =<(9 i 0) f C>+/ r ii i (# n ,-^)Cdr. The generalizations to spaces (26) follow naturally from Theorems 2 and 3. Remark that in the r.h.s. of (25), (8^) denotes the function in LJ^R*1) whose restriction to Q or Q' is equal to the derivative in the sense of distributions in QorQ* of*nor*Q,. & 3 . DIFFERENTIAL OPERATORS ON A "REGULAR" SURFACE T

We define here some differential operators on a surface T in Rn, with fairly weak regularity hypotheses, that is T Lipschitzian. But we often have to suppose more regularity, for example that r is Cl,ot, for 0 < a < 1 at least on pieces Ij whose union is T (see App. 1 for these notations). The ideas on differential operators are in fact more natural from the point of view of differential geometry, so we refer to the appendix at the end of the book. But we generally have to assume some more regularity hypotheses (C1,1). In order to be self-contained we define here all what we need for "usual" electromagnetism. Let dr be the surface measure on I\ up = up „ be the orthogonal projection on the tangent plane at the point x of T, which exists almost everywhere for dr. It is possible to define differential operators on T either by restriction or by extension: for every "regular" function 0 (or vector function u) on the closure of Q, with trace 0O on T (with projection on the tangent plane uQ), we have: Definition 2.

The operators gradp, divp and curlp are defined by

I i) gradr 0O = nT grad 0 on T, andfor n = 3: curlr u0 = n. (curl u) | , (28)

ii) then by duality: <divr v, t> = - <v, gradr £>, V C regularJunction, andfor n » 3, <curir 0, v> = - <$, curlr v>, V v regular vector field.

We can also operate by extension: for every "regular" function $0 on T, we assume that there is a "regular" extension 0 of 0O in Q such that 0 = 0O on T. Then we have to verify that the l.h.s. of (28) does not depend on the chosen particular extension; but this is a direct consequence of the Green formula.

2 MATHEMATICAL FRAMEWORK

These definitions are somewhat formal; we can give a precise sense in Sobolev spaces in the case of Lipschitz T; let (29)

Ht"1/2(0 be the closure in H'V2(Tf of 1^(0 ¥{u € L2(T)n, n.u = 0}.

The mapping *0 -+ gradr 0O has a continuous extension from H 1/2 (0 into H^ 1/2(T). PROOF. Using the continuous mapping 4> —»grad 0, from H (Q) into H(curl,Q), and the continuous mapping nT from H(curl,Q) into H^ (O of theorem 3, we obtain the property. 1/2 Then by duality we see that divr is a continuous mapping from Ht (T) (the dual space of H^1/2(r» into H~m(T). The operator curlr is continuous from H^n(T) into H" l/2 (r), and then curlr is continuous from H 1/2 (0 into H^ l,2(T). We can give immediately some properties of these operators. At first we have: Lemma 4. For regular T (for example C1,1), and n = 3, we have for all u in H(curl,n) and * in Hn(Q) (or with (26)): (30)

curlr 7tru = - divr (n A u | ) ,

curlr (n A U | r ) * divr (nTu),

curl r 0| r = nA(grad0)|r = nAgradr(0|r)

inH" 1/2 (0, and thus: (30)'

cwl r 0o = n A gradr * 0 ,

V 0o € Hm(T).

PROOF, i) Wefirstapply the div operator to the curl formula of (25). Remember that v a (curl u) is in H(div,Q) and in Hloc(div,£2'), since div curl u = 0 in Cl and Cl\ hence [n.v]r € H"1/2(I>, then (31)

- [n.curl u] 5r - div([n A U J ^ S O .

For any 0 in D(R ), we have: <div ([n A u] r 8r), o> = - <[n A U ] ^ , grad 0> = - <[n A u ] r gradr *0>r = <divr ([n A u]r), oo>r, by duality, thus: div ([n A u] f 5r) = divr ([n A u]f) 6p Then (31) gives: (32)

[n.curl u] r = -div r ([n A u]r),

inH'm(T).

2 A TRACE FOR H(CURL,Cl)

With definition (29)i), we obtain the first formula in (30). ii) We apply the curl operator to the grad formula of (25) (recall that: v=r(grad♦)isinH(cu^l,Q)andHl0C(curl,a,), thennAgrado | r €H~ 1 / 2 (0 3 , from Theorem 3). Thus: (33)

[n A grad ♦) bj- + curl (n Wffy)* 0.

Now for any C in D(R3), we have: <curl (nMjj&r), £> - <nMr6r, curlfc>= <[<*]r n.curl O r = <M r curlr n^> = - <curlr M r C>, therefore: (34) curl (nMr&r) = - (c7rlrWr) *r Then using formula (33), we obtain the last formula in (30). Note that we can easily verify that curlr gradr ♦ = 0, divr curlr ♦ * 0. <8>

Then we define the Laplace-Beltrami operator Ar by: (35)

Aj* = divr gradr 0 = - divr (n A curlr ♦) = curlr curlr ♦.

4. THE SPACES H " 1/2(div,r), H " 1/2(curl,D. TRACES FOR H(curl,Q) Let O be a regular open set, with boundary I (for example C1' *), then we define (36)

H-1/2(div,r) = {v€H- 1/2 (0 3 , n.v=0, div r v€lT 1/2 (r)}, |H-I/2(curl,r) = {v€H- 1/2 (r) 3 , n.v«0, curl r v€H" l/2 (r)}.

Theorem 4. The trace mappings yx: v —* n A V | , and wr: v —> nTv are continuous from the space H(curl,Q) onto H~1/2(div,0 and H"1/2(curl,r) respectively. Remark 3. It follows easily from Theorem 4 that for the exterior domain Cl9, the above trace mappings defined on: (37) Hloc(curl,Q') = {v € L?oc(0\ C 3 ), with vC € H(curl,Q'), V C € Z)(R3)} are also continuous and onto. Theorem 4 implies continuous extension properties, like Lemma 3. & Remark 4. Regularity C1 *1 of r does not seem to be necessary (see Remark 1 for some regularity conditions), but optimal regularity conditions are not known. ®

2 MATHEMATICAL FRAMEWORK

PROOF of Theorem 4.

From Lemma 4, we know that for u in H(curl,Q), we have

1/ div r (nAu|J and curlr(7iru)€H"1/2, in

(38)

Thus we only have to prove that the trace mappings are onto, hence to make a lifting of the trace. We use a proof due to Tartar, for the mapping n- only. We note that we can localize and that the property is preserved by Lipscnitz charts with Lipschitz inverse. Thus we are reduced to the case of the half-space Cl R3+. Let u = (ulfu2) on R2 (forx3 = 0) with: (39)

u , , u2, ^ . ^

6 H

, / 2

(n.

Let w be the Fourier transform of w, in R or R ; w(£)=/ n e" 1 x * w(x)dx, with I=(£', 5j) = (Sj.tj.tj) the dual variables to x=(Xi,x 2 ,x 3 ). W e define v by: 1 *3 u (40)

W'>V=T7M2(*ll+h2

with ♦ € D(R), a n d / R o(X) dX = 1. Then w e have, with K = / R 0 2 ( X ) dX:

XR3l-j|2<i«=K:XR2^r-rT^T75l«j|2<«'<-«>. l-al^^-^^TT^Cl-il-l^l)Thus the inverse Fourier transform v of v satisfies: v=(v lf v2, v3) € L (R ) . We also have t v , - «,v. = «.fi2 - U , )

^-^2 ♦ (

KTJJ)

€ L2(R3),

dv. dv, , , thus g f - ^ f €L (R ); furthermore

tf,

^(UU,3|2)1/2(|v1| + 1 ^ - « l V 2 | )

l « i V W =^-^2 K-v, -fr+tfrty A

^ 3

^ 1

Thus (£jV 3 - £ 3 Vj) is in L ( R ), whence (§£- - ^ ) is in L (R ); similarly w e have

{

w~w)e

L2(R3)

and thus curi v €

L2 R3 3

< >-

We only have to verify that the trace of Vj o n the plane x 3 = 0 is Uj for j = 1 , 2 , that is / R Vj($\ S 3 ) d£ 3 = flj(£'), which is a consequence o f / R $(X)dX = 1. <8>

2.4 TRACEFORH(CURL9Ci)

Remark 5. Regularity result. For a regular T (for example C1'1) we easily see that the trace mappings yT and nr are also continuous from the space (41)

H W l Q ) ^ {U € H W , curl u € Kl(G)\

onto the spaces H1/2(div,r) and H1/2(curl,0 respectively, with for a = 1/2: (42)

H ^ d i v . I ^ v c H ^ r ) 3 , n.v=0, divrv€H"(r)}, ^ ( c u r l ^ ^ l v e r f 1 ^ 3 , n.v = o, curlrv €H"(r)}. <8>

Corollary 1. The trace mappings yT o/wf nr are a/so continuous from the space: (43)

H(curl, div0,Q)d*{u € H(curl,Q), divu = 0},

onto the spaces H~1/2(div,r) and H"1/2(curl,r) respectively. PROOF. We know from Theorem 3 that the kernel of the trace mappings is the space HQ(curl,Q). Therefore the trace mappings are isomorphisms from the orthogonal space H0x(curl,Q) to Hp(curl,Q) in H(curl,Q) onto their ranges; H0x(curl,Q) is the set of u in H(curl,£fc) satisfying, with K a real constant:

(44) that is: (44)'

^(curlu.curlt+i^u.Odx^O, VC€JD(Q)\ curl curl u+K2 u = 0 in D'(Cl).

Hence this is equivalent to: (44)" divu = 0, thanks to: (45)

-AU + K ^ O ,

curl curl u = - Au + grad div u.

Therefore H x(curl,Q) is also the set of u in H(curl,&) which satisfies (44)", and & thus is contained in the space H(curl, div 0,Q). Remark 6. From the corollary, the following problem:findu in H(curl,Q) such that I curl curl u+i^u^O inD'(Cl) (46) n A u| = j, j given in H" 1/2(div,0, has a unique solution. It is also possible to prove this result using a lifting of the boundary condition (which exists from Theorem 4), then using a variational <g> method, with the Lax-Milgram lemma.

2 MATHEMATICAL FRAMEWORK

Corollary 2. H"

The mapping v —»n AW is an isomorphism from H" 1/2(div,r) onto

(curl,r)» **fc inverse w - » - n A W, anrf we Aave

(47)

curlr v = - divr (n A V),

divr w=curl r (n A W),

forvinH- 1/2 (curl,r), andwinH" l/2 (div,r). The isomorphism property is straightforward on regular functions, then by closure. Formula (47) follows directly from (30) and Theorem 4. PROOF.

0 Proposition 3. The space H " 1/2 (div,0 is naturally identified with the dual space o/H~ 1/2(curl,r), when we use L 2 t (0 as "pivot" space. PROOF.

(48)

We apply Green formula (which results from (20)): fQ(cut\ u .v - u .curl v) dx=/_. n A U .v dT,

whose l.h.s. is defined for all u and v in H(curl,Q). Thus in the r.h.s. the integral is in fact a duality <,>, and we have: (49)

|<nAU,v>| ^llullH(curl,a)llvllH(curl,a)-

Then since the l.h.s. of (48) depends only on traces YTU = n A U | and nTv of u and (49)'

|<nAu,v>| ^llnAuB H . 1 / 2 ( d . v 0 llv« H . I / 2 ( c u r l D .

This implies that the duality is continuous. Note also that from Corollary 2 and Proposition 3, we have the duality property (50)

<nAU,v> = -<u,nAV>,

V u , v€ H"I/2(curl,lT),

as a consequence of the Green formula (48). 0 The duality of Prop. 3 may seem fairly surprising, if we refer to the usual dualities for Sobolev spaces. We can verify this duality from two points of view. i) At first we use local charts to work on space R2, then we use a Fourier transform method as in Theorem 4. We have u a (u {, u2) in H" 1/2(curl, R2) if and only if its Fourier transform satisfies: (51)

/R2^7ppnfiii2+ifi2i2+i«2fli-M2i2]d«<+«>.

2.4 TRACE FOR H(CURL,a)

Let M be the matrix defined by:

we see that u is in HT1/2(curi,R2) if and only if:

-ratf)

)<+oo .

The dual space is thus composed of elements v s (vj, v2) whose Fourier transform satisfies:

((;;},«-'(;;)><♦», with

Thus v=ty, v2) has to satisfy: (54)

At^^r^rt1^11^*1** ■^»la* l^îêâl^lctt^+oo.

thatisv€H"1/2(div,R2). ii) We use a "Hodge decomposition" of the space H- 1 / 2z/(div,H as follows: Lemma 5. With the C1'1 regularity of the boundary I\ every element u in H ~ l/2(div,r) (resp. v w H " 1/2(curl,D) has a decomposition of the form:

(55)

u = gradr0 + curlr(|> + a, with + € H 3/2 (0, <> l € H 1/2 (0, divr a = 0, curlr a = 0 (resp. v=gradr J + curlr$ + a, with J € H 1/2 (0, $ € H3/2 (f), divr a = 0, curlr a = 0).

PROOF. The two decompositions are equivalent from Corollary 2. Thus we will only prove decomposition (55) of H~'/2(div,r). Applying successively divr, curlp to (55), we obtain:

(56)

-1/2,

Ar o m divr u € H" 1 , z (0,

Ar <> | = curlr u €-3/2,i H" *U(T).

2 MATHEMATICAL FRAMEWORK

This gives (up to a constant on each component of T) * in H 3/2 (0 and <> | in H1/2(0» (thanks to a variational method for example), and then development (SS). From the two decompositions (55), the scalar product of u and v is given by: (57)

<u, v> =s <gradr ♦, gradr ♦> + <curlr <|>, curlr $> + <a, a>,

and thus, using the duality between Hm(T) and H"1/2(0: (57)'

<u, v> =5 - <Ar ♦, J> - «|>, Ar $> + <a, a>.

5 . TRACES FOR W*(div,Q) (FOR THE POYNTING VECTOR)

Let u and v be regularfieldsin (the closure of) Cl. Then we can apply the Green formula in the following form: (58) / (curlu.v-u.curlv)dx=/ r nAU.vdT=-/ u.nAvdT=/ n.uAvdT. When u and v are in space H(curl,ft), the last formula must make sense. Note that we have: (59)

div (u A v) SB curl u.v - u.curl v.

Therefore if we define the space: (60)

W*(div, Cl)6£[weLl(Clf,

divweL 1 ^)},

we have w=u A V € W!(div, Q), and then we have to justify a trace formula for w. Proposition 4. Let Clbea bounded open set in Rn, with Lipschitzian boundary T. Then the trace mapping w —• n.w| is continuous from W (div, Cl) into the dual space ofLipschitzian Junctions on t. PROOF.

(61)

It is based on the Green formula: / 0 (u.grad* + divu.*)dx=/rn.u<frdT,

for all u in W (div,Q) and all bounded o with bounded gradient, hence <J> € W ' (Cl). Wehave^l €C°,!(r)» andn.u| has a sense from (61) by duality with Lipschitzian functions on T. Application to electromagnetism. Let (E,H) be an electromagnetic field in Cl with finite energy, then E and H are in H(curl,Q) and the Poynting vector S = E A H (in the real case) is in W^div,!^), and therefore has a trace n.S on I\ so that

2.6 POLAR SETS (62)

/QdivSdx=/rn.Sdr=/rn.EAHdr=/rnAE.Hdrss-/rE.nAHdT.

6 . SOME COMPLEMENTARY RESULTS. POLAR SETS

In order to have a better understanding of finite energy for an electromagnetic field, we give the definition (see Lions [2], Dautray-Lions [1] chap. 2 for some applications): Definition 3. Let A be a dosed set in Rn. Let H^R 11 ) be the space ofdistributions in H" l(Kn) with support in A. Then A is a polar set if: (63) HX l (R n )-{0}. Lines and points are polar sets in R3, but surfaces are not polar sets. One of the main properties of polar sets is: Ho(R"\A) = Hl(Rn) ifand only ifA is a polar set. We can prove a similar result in the framework of spaces on curl and div: Proposition 5. Let Cl be a bounded open set in R3, A be a bounded polar set in R3, contained in Cl. Then: (64)

H(curl,n\A) = H(curl,Q), H0(curl,Q\A) = H0(curl,Q) H(div,Q\A) = H(div,Q), H0(div,Q\A) = H0(div,Q).

PROOF, i) Let u be in H(curl,£i\A). Since A is a polar set, it is a set of Lebesgue measure zero, and thus u is in L2(C1). Let G = curlA u, with curlA u in the sense of distributions in C1\A, that is <u,curU> = <G,<fr>, V0€Z)(Q\A)3. Then G € L2(Q) and curl u (in the sense of distributions in O) has the following form curl u = G + nA, with HA a distribution with support in A. We have div G = vA = - div nA in D9(Q)y and vA € H~ (Q), with support in A. Thus vA = 0, and then (see Theorem 8 in sec. 9) G is a curl (up to a regular vector function if Cl is not simply connected): G = curl u with u € L (Q) , hence: PA = curl v, v=(u - u) € L (Cl) . Therefore jiA is in H" (Cl) , with support in A, thus yA = 0, and we have proved thefirstpart of Proposition 5. ii) Let u € H(div,Q\A); thus using similar notations, we have u € L2(Q)3, and divA u € L2(C1). Thus we have in D'(C1):

2 MATHEMATICAL FRAMEWORK divu = divAu + vA, with suppvA£A;

but div u € H" *(Q), thus vA e H" l(Q) with supp vA Q A, therefore vA = 0, and we have proved the last part of Proposition 5. Application to electromagnetsm. Let j A be a current concentrated on a line A (or a point), at an angular frequency a>, then the electromagnetic field (E,H) created by this current in a domain Q being a solution of the Maxwell equations: curl H + io)£ E = j (65)

inQ,

- curl E + io>p H = 0

cannot be offiniteenergy in Q. In electrostatics, if we have some constant electric charge density p concentrated on a bounded line A, the electric field E is the opposite of the gradient of the electric potential u which is the solution going to 0 at infinity, of (66)

Au = - p 8 A inR3;

u is the "Newtonianpotential" (in electrostatics it is called the "Coulombpotential"). Since 5A € H"l " V(R3), for all v > 0, then u € H,1^V(R3) and therefore E € H,"VC(R3). A very simple example (but for an unbounded line A, the z axis) is the case of a constant charge density (taken equal to 1) on A; then we have: S ^ r X r ^ + y V ^ u ^ 7 . TRACES ON A SHEET

We often have to consider domains whose boundary is an irregular surface, such as a cube, a tetrahedron (and it is therefore convenient to decompose these surfaces into regular parts) as well as very thin obstacles that we have to model as sheets (that is surfaces with a boundary). So we study some trace theorems for these surfaces. Let I\ and T2 be two regular sheets (C 1,1 up to their boundary) in Rn, with the same boundary A. We assume that: H) There is a neighborhood Kg of A in I\ and T2 and a symmetry operator RQ exchanging the neighborhoods in r } ana TY and preserving distances on the surfaces

RoCnnr^-Kanrj^cFinr^-^nra.

2.7 TRACES ON A SHEET

Thus, we assume that we have local charts so that: R0(x,,Xn.1) = ( x ' , - ^ . j ) , x'€R n-2, X ^ ,

Xn.j€R.

Let 02 be a function on VhnT2; we denote by R ^ the function on V6nTl defined by R^^oR;1. Thus the use of the operator RQ reduces the problems of matching two functions on the two different sheets T{ and T2 to problems on the same sheet I*o, but with two faces To+ and rQ _ . At first we study matching problems in scalar cases. 7.1. Trace problems on sheets (the scalar case) Since functions in the usual Sobolev space H*(Q) for a bounded domain Cl with boundary T have traces in the space H 1/2 (0, we study matching problems in such a framework. At first, we give some definitions. We denote by p (a function equivalent to) the distance of the point x to the boundary A of the sheet I\ (a regular open set in the boundary T of Q), in a neighborhood of A. We define, as usual (see for example Lions-Magenes [1]): (67)

Hi/2(r,) dJf {u € H1/2(r,), P- 1/2u € L2(Ti)} = {u € H 1 ' 2 ^), u € H1/2(T)},

where u is the extension of u by 0 on the outside of rj. We denote by: (68) IT 1 7 2 ^) the dual space of H 172 ^), (H^ 2 ^))' the dual space of H*' 2 ^), that is also: (68)'

( H ^ ) ) ' = {f = f0 + fx, f0 € H" 1 ' 2 ^),

p1/2fj € L2(TX)}.

Let T2 = I\f j be the (regular) complementary sheet in T. For every function v on T, we denote by Vj the restriction of v to Tj for i = 1, 2. Lemma 6.

We have the following equivalences:

I i) w € H1/2(r)~Wi € H1/2(I\), i = 1, 2, and w{ - R > 2 € H ^ f l U (69)

ii) f € H - 1 / 2 ( D ^ fi€(H*/2(ri))\ i= 1, 2, a n d V i +f 2 €H" 1/2 (r 2 ), (orf^R^cH"172^)),

^AR^^R^R;1. PROOF. We first reduce to a neighborhood Vh of A, thanks to a regular function 05 such that: *d(x) = 1 if d(x,A) > 8, oa(x) = 0 if d(x,A) < 5/2. Then for every f inH"1/2(r), vinH 1/2 (0, we have:

(70)

2 MA THEM A TICAL

FRAMEWORK

<f, v> = <f, vo 5 > + < v ( l -0 8 )>,

and we have only to considerw=v(l - * s ) € H 1 / 2 (0 with supp w g Vh. Then i) is a consequence of properties of space H * 7 2 ^ ) , see Lions-Magenes [1]. For part ii), we write (first at least formally for all f € H" 1 / 2 ( 0 , w € H 1 / 2 (0): (71)

<f, w> = <f, w> r = <fj, Wj>r + <f2, w 2 > r *" <*!■ w l - KW2>TX + <% f l + f2> w 2>iy

with the transposed operator l R 0 of R 0 . From formula (71), we see that: f € (Hl/2(T)Y = H" 1 / 2 (0 if and only if % f, + f2 € I T 1/2 (r 2 ), oralsoifandonlyif f 1 +( t R^r 1 f 2 €H' 1 / 2 (r 1 ),thatis(69)ii).

<8>

Remark 7. The difference in sign of matching formulas (69) follows from the change in orientation of the sheets T{ and I"2 by R Q (in the terminology of differential geometry, see the appendix, we can consider the elements of H (O as even O-forms, and those of H ~ 1 / 2 ( 0 as odd (n - l)-currents). ® Let T0 be a regular bounded orientable sheet, T0 its canonical oriented covering, that is identified to the union of the two faces Tx = r o + and T2 = r o - of the sheet r o (see Schwartz [1] pp.315, 316 and Dieudonne [1] T.3 p. 145, for these notions). Let o be the symmetry which exchange the two faces. Then we define the following Sobolev spaces: (72)

i) H 1 / 2 (f0 ) tf {w, Wi € H 1 ' 2 ^ ) , i = 1, 2 and Wl - w2 € H ^ ) } ii)H-1/2(f0)d=ef{f, ^ ( H ^ ) ) ' , i=l, 2andf! +f 2 €H- 1/2 (r 0 )}. 1/2 ***

The two spaces H (r o ) and H"

Ml ***

(r o ) are dual spaces by: def

(73)

<f,w> = <f!,Wj>ri + <f 2 ,w 2 > r2 = <flfWj - w 2 > Fi + <f2 + fi,w 2 > r .

Let Q be a bounded open set in R n , which contains r . Then we have the trace property (adapted to fissured materials): Corollary 3 . H

The trace mapping yQ: u —* u | - is continuous from Hl(Cl\T0) onto

(r o ). The trace mapping y . : u —♦ §jj | is continuous from on r 0 l d ef (74) H (A,a\r0) = {u € H W o ) . AU € L 2 (Q\r o )} 1/2 o/iwH" (? 0 ).

2.7 TRACES ON A SHEET

We only have to consider a regular open set Clx contained in

PROOF.

the closure of Q, with T0 contained in its boundary. Let Q2 = ®&u ^l ^ e s u c ' 1 that dQx = F0 U r x . Then u € Hl(Q\T0) implies u | Q € H 1 ^ ) , i = 1, 2. Therefore the traces: u|

« 1 l ro au| r 0 . (reSp - u| « 2 l ro =u| rJ andu| n I ir 1 =u lfl 2 lr 1

must satisfy the matching conditions; thefirstpart of the corollary follows. The same argument applies to the matching of the traces of YjU on T0. & Remark 8. In the situation of Lemma 6, let Rx be a lifting of H 1 ' 2 ^) in H l/2 (D, and let R{ be a lifting of ( H " 2 ^ ) ) ' in H" 1 / 2 (0. Then we can write:

u ^ u ^ M u - * ^ ) ) , Vu€H l/2 l(H,

(75)

-1/2/

f—Ji,(f| r )+(f+*!(f| r )), Vf€H- 1,z (r),

thus we obtain the decomposition of spaces Hiy (0 and H" ' ( 0 into the direct sum rl/2, l H \n, ^(0-(Hi/j(r1)yeH-1•1/2/1 ^(r2)t

rl/2,

rl/2/ 1,z Hfl/2/i (r)-H 1/z (r l )eH^(r 2 ),

(76)

and we can identify the orthogonal (or polar) space of H1 (Tx) to (H* Q ^ ) ) ' , and the -1/2,1 rl/2r rl/2, orthogonal space of H^(r 2 ) to the space H " ' ^ ) in the duality Hr*(T), H"1/2(T), thanks to the formula, forallwinH1/2(r), finH'U2(T): (77) <w,f> = <R{(w\r ),f> + <w-tf1(w|r ),f> = <w|. ,t/{1f> + <w-JR1(w| r ),flr >. r r M

Remark 9. In the case of the orientable sheet of Corollary 3, we have the natural mappings: S:

|w€H 1 / 2 (f 0 )-*i(w| r

+w| r )€H 1/2 (r 0 ),

f€H-1/2(?0)-i(f|r

)€(Hi /2 (r 0 ))\

+ f| r

and: D:

weHuri/2,; \T0)-+±(w\1, f€H1/2(f0)-i(fi L

-w| 0+

-fi o+

rl/2/i )€H^(r 0 ), 0-

)€H" l/2 (r 0 ).

The mapping w—• (Sw, Dw) is then an isomorphism from space Hrl/2;(ro) onto H1/2(ro)x Hi/2(ro) and from H- l/2 (f 0 ) onto (Hj/2(ro))' x H"1/2(ro). This allows us to identify these spaces. &

2 MATHEMATICAL FRAMEWORK

7.2. Trace problems on sheets (the rectorial case) Here we assume, in order to avoid technical details, that the sheets are very regular (that is, they are infinitely differentiable manifolds with boundary). Then we have the chain of spaces (for i = 1, 2): (78) D^)-*

H ^ ) - H1'2^)- L 2 ^ ) - H - 1 ^ ) - (H^r^^^'ffi)-

Thus we can naturally use surface operators in the sense of distributions on T. (or in the sense of generalized differential forms or currents, see the Appendix). Let Q be a regular bounded open set in Rn, with boundary I\ We define: (79)

H"s(div,Q) V{u € H " W \ div u € H"S(Q)}, s € R.

Then we have: Proposition 6. For every s with 0 < s < 1/2, and every u in H~s(div,Q), n.u has a trace on the boundary TofQ, such that: (80)

n.u€H- s - 1 / 2 (0.

PROOF.

We use the Green formula (with n the exterior normal to T):

(81)

(divu, 0) + (u, grad 0) = / r n.u 0 dl\

Let 0 be in Hs+1(fl), then by the usual trace theorems, we have 0 | f € H 5 * 1 ' 2 ^. Then for u in H"s(div, Q), the l.h.s. of (81) has a sense and defines a continuous mapping on H^^Q). Thus from (81), n.u | is a continuous form on H**"1 (T). Replacing the domain Cl by the sheet Tj into definition (79), we have: (82)

H ^ d i v , ^ ) dJf{u € H-'Fxf, n.u = 0, divr u € H'S(T{)}9

and then we obtain a proposition similar to the preceding one, with v the exterior normal to the boundary A of r j : Proposition 6 \ For every s with 0 < s < 1/2, and every u in H"s(div, rj), then v. u has a trace on the boundary A of 1^ so that: v.u € H"s" (A). PROOF. We only have to replace the Green formula (81) by: (83)

(divr u, 0) + (u, gradr 0)=/ v.u 0 dA.

2.7 TRACES ON A SHEET

Remark 10. F o r n = 3, we also have the following trace result: the space (84)

H-\curi,r,)¥{o€H"W\

n.u = 0,

c u r l u eH' (Ti)}, s

is such that for every s € [0,1/2], v e H ^ c u r l j r j ) , then (with T a unit tangent vector to the boundary A of rj) T . V has a trace on A such that T . V € H ~ "

(A).

Proposition 6' and Remark 10 allow us to define the following spaces: (85)

H^^divJjJ^lueH^^div,^), H ; (curi,ri) = {u € 1/2

v . u | = 0} A

H^VUi),

t . u | = 0}. A

F o r T the boundary of a regular open set in R , let j be in H " (div,r). Then by restriction to T. we obtain that j . for i = 1, 2 is in the following space: n

(86)

X(divJ )^ {f€(H^(r ) y, f

n.f*0,

1/2

div f€(H^(r ))'} r

with d i v f in the sense of distributions in r$. F o r n = 3, we also define: r

(87)

X ^ u r l J ^ ^ l f e ^ f f i ) ) ' , n.f=0, 3

curl f€ ( H ^ ) ) ' } . r

Contrary to the elements of H~ (div,I\) and H~ (curl,r.), the elements of X(div,r.), or of X(curl,r.) have no trace on the boundary of I\. This may be seen from Green formula (83). F o r tangent vector fields (or 1-forms) we have matching properties similar to the scalar case i n the framework of H spaces, as i n Lemma 6; this may be verified on each component of the field. F r o m now on, we have to specify the behavior of "fields" under orientation change, so using the language of differential geometry (see the Appendix) we consider the elements of space H ~ ( d i v , r ) as odd 1-currents (or also as generalized odd 1-forms), and the elements of H " " ( c u r l , r ) as (generalized) even 1-forms. We only recall that the pullback R * w of the 1-form w on T^resp. the pushforward R f of the 1-field 0 by the symmetry operator R is defined by: 1/2

l/2

1 / 2

1/2

(88)

RoW ( > = w ^ C I T ^ v ) , x

<R f, w> = <f, R*w>, Q

for all tangent vector v to I\; T R corresponds to the Jacobian of the transformation R (see A p p . (326)). In the situation of Lemma 6, we have Q

Theorem 5. (Matchingproperty in H ~ ( d i v , D and H " We have the following equivalences: 1/2

i/2

(curl,D).

2 MATHEMATICAL FRAMEWORK

(89) f € H"l/2(div,r)<-* fi € X(dfr,rft), i = 1, 2 and R0fj + f2 € H;1/2(div,r2), andfor n = 3, (90) w€H~1/2(curl,I>* WjCX^urUj), i = 1, 2andR^w2-w1 cH^^curl,^). We only have to show for (89) that (fpf2) as in the r.h.s. of (89) defines an element fin H~ 1/2(div,D. i) Atfirstthe last condition of (89) implies that: R0fj + f2 € Hf l/2 (r 2 ), and thus f defined by f| = f{ satisfies f € Ht"1/2(0 from Lemma 6.

PROOF.

-1/2,

ii) The element g defined on T by g | = g. = divr (f| ), i = I, 2 is also in W *i

(T)

from Lemma 6, and from commutation: divr R0fj = R0divrfx. iii) We have in the sense of distributions (or currents) on T: (91) divrf=(divr0 + M A 5 A , with (divr f) the surface divergence in the sense of D*^) i = 1,2 and [v-flA the jump of v.f across A, which is zero from hypothesis. Therefore we have proved that divr f » g, and (89). Equivalence (90) is similar, and can also be deduced from (89). In the situation of Corollary 3, with a sheet r , we have: Corollary 4.

The mappings u —> n A U | - and u —• «- u are continuous from

H(curl,Q\ro) onto H~ 1/2(div,f0) and onto H"1/2(curl,f0) respectively, with: (92)

H - ^ d h r J ^ ^ l f ^ - f l j . €X(div,r i ),i=l,2and f 1+ f 2 €H; 1/x (div,r o )}, H-1/2(curi,f0)dif{f, f. = f| eX^urlJj), i= 1, 2andf, - f2€H;1/2(curl,ro)}.

Remark 11. Using a lifting Rx of X(div,Ij) into H" 1/2(div,f), we can write: (93)

f= J R 1 (fl) + (f-JR1(f! )).

Thus we have the following decomposition into direct sum (like in Remark 8): (94)

H-l/2(div,r)=x(div,rj)©H;l/2(div,r2),

and similarly, for the curl: (95)

H"l/2(curl,r) = X(curl,rI)©H^,/2(curl,r2).

2.8 REGULARITY RESULTS

We verify that in the duality H"l/2(div,r). H_1/2(curl,r), the dual space of XCdiv,^) is H^curLrj), and the dual space of H;1/2(div,r2) is X(curl,r2). Therefore the orthogonal (polar) space of XCdr/.Tj) (resp. H~1/2(curl,ri)) is the space X(curl,r2) («sp. H;1/2(div,r2)). ® Remark 12. We note that gradr and curlr are continuous operators from HQ QU^) into (ul^T^y and by duality divr and curlr are continuous operators from H^CTj) into (H*'2^))', where as usual the index t refers to tangent fields Application to electromagnetism Let jj. be some current concentrated on a (regular, bounded) sheet I = To in a free space domain Q, with angular frequency «. Then the electromagnetic field (E,H) created by this current, which is solution of the Maxwell equations: curl H + io)£0 E = L (96)

inQ,

- curl E +toy H = 0

is of finite energy in Q\Z, i.e., E and H are in H(curl,D\X), if and only if j ^ satisfies (97) J2€H;1/2(div,Z). Therefore in the scattering of an incoming electromagnetic field (EpHj) by a thin obstacle that we model by a sheet I, the jump of the scattered field across this sheet necessarily satisfies (97). 8. SOME REGULARITY RESULTS

It is often interesting to give up the H(curl,Q) or the H(div,Q) space for the H^O) space, (k > 0), at first to use regularity results in Sobolev spaces and better to use the compacity of the natural injection of Sobolev spaces in L2(D) for (regular) bounded sets O. We will use the following lemma extensively; for its proof, we only refer to Peetre [1], Tartar [1], Lions-Magenes [1]. Lemma 7 (Peetre). Let EQ, E r E2 be Banach spaces, A{9 A^ be two continuous linear mappings from EQ into E{ and E2 respectively, with: i) A~ is a compact mapping; ii) mere is a constant c> 0 such that: (98) Then:

IMIE ScfllAjvllg +IIA2vlIE }, W € E 0 . l 2 °

2 MATHEMATICAL FRAMEWORK

i) ker A{ hasfinitedimension, and Im Aj is closed ii) there is a constant c > 0 such that: (99)

inf

llv+wllp < c n IIA,vllp , V v € En.

Theorem 6. Let Cl be a regular (Clfl) bounded open set in R3. Let u be in L2(Q)3 wi*/icurlu€L2(Q)3, divu€L2(Q), a/irfn.u|r€H1/2(r)(resp. nAu| r €H t 1/2 (D). Then\xeH\Q)3. PROOF,

i) We first reduce to "homogeneous " boundary conditions with a lifting:

I r = Re/ r {-curlv.nAV-divv(n.v)+^.v}dT.

We can obtain the last term in many ways, for instance using the covariant derivative or covariant differentiation along n (for this notion, see Gilkey [1] p. 104, Dieudonne [1]) and then the formulas (310) of the Appendix on div and curl. With the notations of the App. 6, Remark 20, with R the mean curvature, v r and v11 the projections of v on T and n (and using (47) or (30)), we have: Ir = Re/ r [-(div r v r )v n + (c7rrlrvn).nAv]dr+/r(fv,v)dT (100)"

= Re/ r [grad r v n .v-(curl r (nAv)v n ]dT+/ r (fv,v)dr, with(rv,v) = . / r ( i 2 « i | v i | 2 + 2R m |v n | 2 )dr, ai = g - 1 ^ , a n d v = 2 v i e i

vnn

in an orthonormal coordinate system (elf e2, n), e{ = g." 3j. Thus for all v in H*(Q)3 with n A v| = 0, or with n. v| = 0, there exists C > 0 so that (101)

/^|gradv| 2 dx^/ ft [|curlv| 2 +|divv| 2 ]dx + C/ r |v(x)| 2 dr.

2.8 REGULARITY RESULTS

iii) We only have to apply Peetre Lemma, with EQ equal to the space: (102) H ^ Q ^ u e H W , n A u l j . - O K r e s p . H ^ ^ ^ l u e H W , n.u|r=0}), and E, = L2(Q)3 x L2(Q)3 x L2(Q)3, E2 = L2(D3, and the mappings: A , : v e E 0 —(curlv,divv,v)eEj, A 2 :veE 0 — v| r eL 2 (D 3 (the compactness of A2 is due to the compactness of the mapping H (O —»L (O). 1 3

From Peetre Lemma again, we have proved: the norm of H {&) on Hto(Q) and on H^0(Q) is equivalent to the norm: IMId^lfft[|curlv|2+ |divv| 2 + Iv^dx}"2. From Theorem 6, we obviously have: Corollary 5. With hypotheses of Theorem 6, we have the following equivalences'. (103) u€H ! (Q) 3 ** u€L2(Q)3, curlu€L2(Q)3, divu€L2(Q), n.u| r €H 1/2 (r) (resp.nAu| r €H t 1/2 (r», with equivalence ofthe norms: lu) % 3 , and: H (n) [llcurl uH + lldiv ull + Hul + ta-ttLi/juJ and [lcurl uB + l,div uB + ,u * + lln A u,l H 1/2 (n^' Then we define very useful spaces (see section 10 below and the Appendix) Definition 4. We call cohomology spaces the following spaces: (104)

|^i( Q ) djf {« € L 2 (Q) 3 ,curlu = 0,divu = 0,n.u| r = 0}, ^ 2 (Q) d i f {u€L 2 (Q) 3 ,curlu = 0,divu = 0,nAu| r = 0}.

From Peetre Lemma, we have: Corollary 6.

The cohomology spaces arefinitedimensional.

We have also further regularity results for n = 3 (or also for n = 2). For the proof, see Dautray-Lions [1] chap. 9: Proposition 7. Let Cl be a regular (Ck+1 •l) bounded open set in R3. Then: u€L2(G)3, c u r l u c H W , divu€H\Q), n.u| r €H k+I/2 (OornAu| r €H t k+I/2 (r) satisfies ueHM(G)3.

2 MA THEM A TICAL FRAMEWORK

9 . THE HODGE DECOMPOSITION

It is often very useful in electromagnetism to know (that is to characterize) the range and the kernel of the operators grad, curl and div in the space of square integrable fields in a domain Q. This is part of differential geometry, known as the Hodge decomposition. Here we will give only main results, referring to the Appendix for developments, and to Dautray-Lions [1] chap. 9. Proofs are often an easy application of Petree Lemma. Theorem 7. Let Clbean open bounded set in R3, with boundary T. The following sequences are so that the range of one operator is contained in the kernel of the following in the sequence, withfinitecodimension: (105)

I _

«rad

l ^

*rad

curl

div

Ul(Cl)—> H(curl,Q)—> H(div,Q)—* L2(Q) ~

cud

div

7 ^

H ' ( Q ) — > H0(curl,Q)—> H0(div,Q)—> L2(Q), and (therefore) the ranges of the operators grad, curl, div (defined on their natural domains H l (Q), H(curl,Q), H(div,Q), or also with vanishing boundary conditions) are closed in the corresponding \?(Q) spaces. Remark 13.

Let r be an open set in F; we define

j Hlr (O) d={ u € H^G), u = 0 on ro}, (106)

Hr (curl,0) V { u € H(curl,Q), n A u = 0 on ro}, Hr (div.ii) ^|u£H(div,Q), n.u = 0 on TJ.

Then we can prove that the following sequence has the same properties as (105): i

grad

curl

div

Ul (O)—> H r (curl,Q)—> Hr (div,Q)—> L2(Q). <8> A simply connected domain Cl is such that grad Hl(Q) = ker curl. More generally ■

in the domain Cl let liy i = 1 to N, be cuttings of Cl so that the domain Q = Q\UZi be simply connected. Let Fj, j = 0 to m, be the components of r. Then we have Proposition 8. Characterization of the cohomology spaces. The cohomology spaces (104) are also given by Hx(Cl) = {ueL2(Clf, u = (grad q)inQ, q €H 1 ^), with Aq = 0 i n Q , | £ l r = 0, [q]^ = qconstant, [|^]2. = 0} H2(C1) = {u € L2(Q)n, u = grad o, * e Hl(Q)y A ♦ = 0, ♦ | f = Cj constant) with

Idim HX(C1) » N (the "cutting" number ofCl), dim H2(C1) = mi[m+listhe number ofcomponents of T.

2.9 THE HODGE DECOMPOSITION

Remark 14. Then we define: (107)

ker0curid^"{v€L2(a)3, curlv=0, nAv| r = 0} ker0divdJf{v€L2(Q)3, divv = 0, n.v| r = 0}.

We have: I #!(&) = ker curlO ker0 div CcurlH 1 ^) 3 , but is not in Im grad, H2(Q) s ker0 curl Oker div £grad H*(Q), but is not in Im curl. Characterization of the intersection of the kernels. We have the decomposition: (108) ker curl O ker div=HX(Q) Sim gradO Im curl ®H2(Cl), with: (109)ImgradOImcurl = {u€L2(Q)3, 3 * € H 1 ^ ) , v€H I (Q) 3 , u = grad+ «curlv}. Thus 0 (resp.v) is a harmonic (resp.vector) function on Q. We can choose v with: divv = 0, n.v| r = 0, / 2 n.vdl = 0, i = l t o N , so that * (up to a constant) and v are unique; they are conjugated harmonic functions in Q. Theorem 8. Characterization of kernels for curl and div (with Cl in R 3 ). With the hypotheses of Theorem 7, let u be in L2(Q)3 with i) curl u = 0; then there exist o € H*(Q), 1^ € HX(Q) such that: u = grad * + hx . Moreover, 0 is unique up to constant on each connected component ofCl; ii) curl u = 0, n A u | = 0 ; then there exist unique * 0 € H^(Q), h2 € H2(C1) such that u = grado0 + h 2 ; iii) div u » 0; then there exist v € H^Q)3, h2 € H2(Q) such that u = curl v + h 2 ; v is unique under the supplementary conditions: divv=0, n.v| r =0, / i j n.vdl = 0, i = l t o N . iv) div u = 0, n.u I = 0; then there exist v0 € Ht!0(Q), h{ € HX(C1) such that: u = curlv0 + hj; v0 is unique with the supplementary conditions: div v0 = 0, / r n.v0 dT « 0, j = 0 to m. j

Theorem 8*. The Hodge decompositions of the kernels of curl and div. The spaces ker curl, ker div, ker curl, ker div have the orthogonal decompositions in L2(&)3: I ker curl = Im grad 0#x(Q)f ker div = Im curl 0# 2 (Q), (110) J ker0 curl = Im grad0 ®H2(£l), ker0 div=Im curl0 QH^Cl),

2 MATHEMATICAL FRAMEWORK

with: Im grad = grad H*(Q), 1

Im grad0 * grad H*(Q),

Im curl = curl H ^ ) = curl H(curl,Q), Im curl0 = curl H^Q) 3 = curl H0(curl,Q). Definition 5. Let J(Q), J(Q), V=J^Q), V = Jtl(Q) be the following spaces: |j(Q) = {u€L 2 (Q) 3 ,divu = 0,/ r n.udT = 0,j = 0tom}, J(Q)«{u€L 2 (n) 3 ,divu = 0,/_ n.udX = 0 , i = l t o N } (111)

V = JT1(Q) = {u€H(curl,Q), divu = 0, nAu| = 0 , / n.udT = 0,j = 0tom} j

V=J^(Q) = {u€H(curl,Q),divu = 0 , n . u | r = 0 , / . n . u d Z = 0 , i = l t o N } . r I n Theorem 9. The ranges of the curl operator are given by: (112)

curl H^Q) 3 = curl V=J(Q),

curl H* (Q)3 = curl V * J(Q);

the curl operator is an isomorphism from space V onto J(C1) andfrom V onto J(Q). Then from theorems 7 and 8 we obtain the following decomposition theorem: Theorem 10. (113)

Every u in L2(C1)3 has the decompositions:

u = gradp +h 2 + curlv,

with p €H*(Q), h2eH2(Cl)9 v€V,

u s grad p -i- hx + curl v0, with p € H^Q), hx € Hx(Cl), v0 € V. Moreover with these conditions, pQ, h2, v, p (up to constants on the components of Q), h., v are unique. Theorem 10'. The Hodge decompositions of L2(0)n. the orthogonal decomposition: (114)

The space L2(Q)n has

L2(ft)n = grad Hl0(Cl) 0 ker div=grad Hl(Cl) ® ker0 div, L (Q) = Im curL 0 ker curl = Im curl 0 kerQ curl, for n = 3,

and thus (114)'

L2(C1)3 = grad H* (Q) 0 H2 0 curl Hl(Cl)3 L2(Q)3 = grad Hl(G) © ^ ® curl Ul0(Q)3.

For n a 2, we have similarly (but with dim HX(C1) = dim H2(C1)): (114)" L2(Q)2 = gradHl0(Q) 0 # 2 0curl 1^(0) = gradUl(Q) ®HX 0curlH£(Q).

2.9 THE HODGE DECOMPOSITION

We can detennine p 0 , h2, v, p ,hj, vQ in (113) by a variational method: i)/wp o andp: I (grad p o , grad ♦,,) = (u, grad ♦„), V ♦„ € H£(Q), I (grad p, grad ♦) = (u, grad ♦),

V* € H 1 ^);

u)for v0: (116)

(curt v0, curl o0) = (u, curl ♦„), Vo 0 eV.

The sesquilinear form a(v0, *0) = (curl v0, curl *0) is coercive on V from Peetre Lemma (with E 0 = V, E t = E2 = L2(Q)3, A{ = curl, A2 =* I); in) for v: (117) (curl v, curl ♦)«(u, curl ♦), V ♦ € V. The sesquilinear form a(v, ♦) = (curl v, curl ♦) is coercive on V from Peetre Lemma (with E 0 = V, E{ = E 2 = L2(Q)3, A{ = curl, A2 = I); iy) for h2, we first use a basis of the space i/2(Q), from the elements: (118)

Vj = gradXj, withXjSH^Q), AXj = 0ina,

Xjlr

=*y, j = 0tom.

Note that 2 V: = 0, and the rank of the basis is only m. Then we define the capacitance matrix: (119)

dx. C^^gradXi.gradXjdx^/j.^dT.

This matrix is symmetric positive definite, of rank m (see for example Dautray-Lions [1] chap. 2). Then h2 is given by: in

(120)

withyJ = pZ l| , and h2 = gradp = Sygradx,, J J j

^ j=0

/ n u.grad X i dx=^ o C i j y j . Since we can determine p 2 only up to a constant, we can choose y = 0, and then the system (121) has one solution. v)for hj we use the following basis of H^Q): Vj =grad q., with q. € Hl(Q) satisfying: (121) Aq. = 0 i n 4

= 0, [q.]2i = 5ij>

],.=<>.

2 MATHEMATICAL FRAMEWORK

Then we define the inductance matrix: (122)

L i j S / f t gradq j .gradq i dx=/ 2 i ^ d l ,

which is a symmetric positive definite matrix. Then h{ is given from: N

(123)

h, = (grad q), in Q, q = 2 y.q.. with y. given by: N

/ Q u.gradq.dx= j 2Li j y j . <8>

Remark 15. For "static" electromagnetism, that is for frequency equal to zero, the above matrices very naturally occur: i) in electrostatics: in a cavity, that is, in a free space bounded domain with a perfectly conducting medium on its boundary T, the electric field E satisfies curlE = 0, divE = 0, n A E | r = 0, that is E €H2{Q)\ E is obtained by (120) from the values of the potential p~ on the surfaces I\, ii) in magnetostatics we have to find the magnetic field H (or the magnetic induction B) solution of: curlH = 0,divH=:0,n.H| r =:0, thatisHcff^Q), and then H is obtained by formula (123), with the N values y. to be determined. Remark 16. The Hodge projectors. Let us define the following operators G°, G, R°,R, P1, P 2 by: I G°u = p given by (115)i), Gu = p given by (115)ii), (124) Ru = vby(117),R°u = v0by(116), |P l u«h l f P2u = h 2 . Note that we have G° = (- AD)~ *div, but we cannot give such a formula for the other operators, since the space (D(C1)) is not dense in H (O), and the same for V and V. Then the operators: (125) P£ tf grad G°, P r d=if curl R, P 2 (resp. Pg d #grad G, P° d #curl R°, P1) are orthogonal and complementary projectors in L2(Q)3; they are called the Hodge projectors associated with the Hodge decomposition (113)i) (resp. (113)ii); they are also (the Hodge) projectors in H*(Q)3; thus: (125)'

P^ + Pr + P 2 ^ ! (respPg + P^ + P ^ I ) . <8>

2.10 INTERPOLATION RESULTS

Remark 17. The Hodge decomposition for a "closed" surface. We recall that the Hodge decomposition for a surface T (the boundary of a regular bounded domain Q) have been given by Lemma 5 for the trace space H~ l/2(div,r); see also the Appendix. Here we recall only the decompositions: I L2(T) = gradr Hl(T) © H(T) © curlr H!(T), (126)

H71/2(r) = gradrH1/2(r) ©H(T) ©curlr H 1 / 2 (0, ^ H~I/2(div,0 = gradr H 3/2 (0 © H(T) © curlr H 1 / 2 (0, I H" 1/2(curl,r) = gradr H 1/2 (0 © H(T) © curlr Hm(T).

These decompositions are associated with the Hodge projectors (orthogonal in L ): (127) P^gradj-A^divr, Pr = curlrAf:lcurlr, P c =P# ( r), with: (127)' Pg + Pr + Pc = I. ® Remark 18. For an unbounded domain, with bounded complement, we have similar results, but the Sobolev spaces must be replaced by the Beppo Levi spaces; we refer to the Appendix and to Dautray-Lions [1] chap. 9. ® 1 0 . INTERPOLATION RESULTS

We often have interesting results by the interpolation theory, see for example Lions-Magenes [1], Triebel [1]. We give here some results useful in electromagnet ism, for the trace and regularity properties. With the usual notation [X,Y]e the interpolated space of 0-order between X and Y, we have: Proposition 9. Let Clbe a bounded regular open set in R3, and 0 a real number with 0 < 9 < 1. Then with usual notations (see (79), (42), (84)): (128)

Hs(curl,n) = {u € HS(Q)3, curl u € HS(Q)3}, s € R,

we have, with s = 1 - 0: (129) I [H^curl^XHfcurl,^ = Hs(curl,n), [Hi(curl,G),H0(curl,a) ]Q = H*(curl,Q), I [H1(div,Q),H(div,n)]0 = Hs(div,Q) , [H£(div,Q),Ho(div,Q)]0 = H*(div,Q), andfor the boundary T: (130)

[HWurxHfcurl,!^ = Hs(curl,0, [H1(div,r),H(div,r)]e = Hs(div,D.

2 MATHEMATICAL FRAMEWORK

PROOF, i) Results on the domain Q can be obtained by extension to the whole space R3, thanks to the trace theorem 4: every u in H(curl,Q) has an extension Ru in H(curl,R3). Then we have to prove the results for Q = R3; this is easily obtained by Fourier transformation. Taking the restriction to Q gives the first result in (129). The other results easily follow. ii) We can also use the Hodge projectors (see (125)): since they are the same in L2(Q)3 and in H1(Q)3t they operate also in the interpolating spaces, so we have (129), as a straightforward application to retractions and coretractions properties with respect to interpolation (see for example Triebel [1] p. 22, and 118). This gives also results (130), using the Hodge projectors for the surface I\<8> Prop. 9 implies that the interpolation results (129) are also true for every real positive 8, and in the case of (130) for every real 9. Then we obtain a generalization of the trace result of Prop. 6 Proposition 10. Interpolation trace results. Let Q be a bounded regular open set in R3, and s a real positive number. Then the following trace mappings are continuous and surjective mappings u € Hs(curl,Q) —nAu| r €H s " 1/2(div,H, (131) u € Hs(curl,Q) — nru € Hs" 1/2(curl,r), u € Hs(div,Q) — n.u | € Hs ~ m(T). 1 1 . SOME VARIATIONAL FRAMEWORKS

Here we give variational frameworks useful in electromagnetism, which are particular cases of more general frameworks in differential geometry. These are given in Appendix, sections 3.1 and 3.2 (see Morrey [1]). There are essentially two types of variational frameworks in electromagnetism: either based on the H(curl,Q) space (or some subspace) with the curl curl operator, or based on the Sobolev space H^Q)3 (or some subspaces) with the Laplacian. We refer to Dautray-Lions [1] chap. 7 (for instance) for the definition of a variational framework, and for many examples with usual functions spaces. 11.1. Variational frameworks based on H(curl,Q) spaces 11.1.1. A variational framework based on HQ(curl,Q) Let Q, be Lipschitz open set in R3. Then let V » HQ(curl,Q), and a(u,v) be the sesquilinear form on V: (132) a(u,v) ssf (curl u. curl v + u. v) dx. Then (V, H = L2(Q)3, a(u,v) ) is a variational framework which defines the unbounded (selfadjoint and positive) operator A in H by: D(A) = {u € V, such that v—» a(u,v) is continuous on V (133) equipped with the topology of H}, (Au,v) = a(u,v), V u € D(A), v € V.

2.11 VARIATIONAL FRAMEWORKS

A sesquilinear form a(u,v) is coercive on V if there exists C > 0 such that Rea(u,u)^CIIu»y, Vu€V. Then the Lax-Milgram lemma implies that the "standard" problem: given fin H or in V, find u in V so that (134)

a(u,v) = (f,v),

VvcV,

has a unique solution in V. From (133), and the Green formula, we characterize the operator A for (132) by: (135)

Au = curl curl u + u, u€D(A) D(A) = {u € V = H0(curl,Q), curl curl u € L2(Q)3}.

Since V is a space of distributions, (134) with (132) is equivalent to the problem: I curl curl u + u = f in Q, (136)

nAu| r = 0.

Since the sesquilinear form a(u,v) given by (132) is coercive on V (by definition), there is a unique solution u in V of (135) thanks to Lax-Milgram lemma. 11.1.2. A variational framework based on H( curl, ft) We can also consider the same framework (V,H,a(u,v)) but with V = H(curl,Q). This defines a new operator A with i) Au = curl curl u + u, (137)

|ii)D(A) = {u€V, curl curl u€L2(Q)3, nAcurlu| r = 0}.

This is easily verified from (133) using the Green formula. Then we can solve other problems similar to (136), but with the boundary condition n A curl u | = 0. More generally, V may be any closed space with H0(curl, Q) Q V Q H(curl, O). For example if r o is a subset of I\ V = {u € H(curl,Q), n A U | =0}. r o Example in electromagnetism. We consider a less "standard" problem in electromagnet ism: let Q be a domain bounded by a perfect conductor and Cl is occupied by a "dissipative" dielectric with permittivity e and permeability n, with given electric and magnetic current densities J, M, with J and M in L2(Q)3. We have tofindthe electromagneticfield(E,H) withfiniteenergy in £1, so that: (138)

i) - curl E + io>jiH = M, iii) nAE| =0.

ii) curl H + icocE = J,

2 MATHEMATICAL FRAMEWORK

Then we can define a variational framework on the electric field E, with the following sesquilinear form in V = HQ(curl,Q): (139)

a(E,E)=/ a (- 5 ~curlE.curiE-ia>eE.E)dx, V E , E c V .

We see that problem (138) is equivalent to finding E in V solution of: (140)

a(E,E)ss/a(i^M.curlE-J.E)dx,

VEeV.

We define H by (138)i). Then (140) is equivalent to: (141)

/ Q (-H.curlE-ia>£E.E)dx=/ f t (-J.E)dx,

VEeV,

which is equivalent to (138)ii) thanks to the Green fonnula. Then from the usual hypotheses on a dissipative medium (see H14 in chap. 1), we have z" = Im z > 0 for z s E and y with o> > 0. Thus there is a positive number a so that: (142)

j a x>^ a i lnE' ^ ^ ^ ^ , R c a ( E , E ) = / f t ( £ ^ 5jricunni |curlE| 2 ++<oe"iiii <oe"|E|2)dx |a>|i|

that is, a(E,E) is a coercive (but not symmetric) sesquilinear form. Thanks to the Lax-Milgram lemma, this implies that problem (138) has a unique solution. 11.2. Variationalframeworksbased on H*(Q)3 and the Laplacian Let Cl be a (regular, i.e., C1'1) domain in R3, either bounded or the complement to a bounded set. We define the spaces: (143)

| v | = Ht1(a) = {u€H 1 (a) 3 , n.u| r = 0} V^H^QMucH 1 ^}) 3 , nAu| r = 0 (or*ru = 0)},

(with nT the projection on the tangent plane) and the sesquilinear form: (144)

a(u,v) s (curl u, curl v) + (div u, div v) + (u, v).

Theorem 11. The sesquilinearform a(u,v) defined by (144) is coercive on V| and VJ". PROOF. When Q is a bounded domain, this is equivalent to Corollary 5 (see (103)). When Cl is the complement of a bounded domain, this is also easily deduced from Peetre lemma (lemma 7), and from the inequality (101). Theorem 11 is generalized by Theorems 4 and 15 of the Appendix, in differential geometry. 0 2 3 2 3 The variational framewoiks (Vj, L (Q) , a(u,v)) and (VJ\ L (Q) , a(u,v)) define selfadjoint operators A* and A" by (145)

EKA±) = {u€V ± , (A ± u,v) = a(u,v),

VveVfH

2.11 VARIATIONAL FRAMEWORKS

with: | i ) A ± u = - Au + uinQ, (146)

I ii) EKA+) - {u € H 1 ^) 3 , curl u € H*(Q)3, div u € H!(Q), I n.u| r = 0, nAcurlu| r = 0}, l 3 | ii)' D(A") = {u € H (G) , curl u € H W , divu € Hl(Cl), nAu| r = 0, (divu)|r = Oj.

PROOF of (146) for A~

only. Taking v in D(Q)3 in (145), we obtain (146)i).

Taking v in V]" with div v = 0 we have this: for u in D(A~), v -♦ (curl curl u + u, v) is continuous on the subspace {v € L2(Q)3, div v = 0}. But it is also continuous on grad HQ(Q) (since (curl curl u, grad ♦) = 0 for ♦ in D(Q). Therefore we obtain that curl curl u is in L2(C1)3. Thus v=curl u is in L2(Q)3 with curl v in L2(Q)3, div v=0, and n.v| = div^n A u | ) = 0. Thus v is in H 1 ^) 3 . Since Au is in L2(Q)3, we obtain from (45) that grad div u is in L2(Q)3. Then we have div u = 0 on T thanks to: (147)

(curl u, curl v) + (div u, div v) + (u, v) = (- Au, v) + / r div u (n.v) dl\

Proposition 7 implies that for regular Q, D(A+) and D(A~) are in H2(Q)3. A more general result is given in the Appendix (see (156)"). ® In order to solve the problem (138) in H!(Q)3, with suitable regularity of M and J, we use the following sesquilinear form: (148)

a 0 (E,E)=/ Q [-|j~(curlE.curlE + div€E.div?E)-ia>€E.E]dx.

For a dissipative medium, aQ(E, E) is coercive on Vj = {E € H (Cl) , n A E | = 0}. Then, instead of (140), we consider the problem:findE in V! such that (148)'

a0(E,E) = LMJ(E),

VE e Vlf

withLM j(E) ^fQ ( j ~ M . curl E +-j~ div J. div ? E - J . E) dx. If M € H(curi,Q), J € H(div,Q), with oo = div J € Hl0(Cl)$ then LMJ is continuous on L2(Q)3. Thus the probler problem (148)' has a unique solution E in D(A") (see (146)) and it is equivalent to (138). Remark 19. An obstacle to compactness. One of the main differences between the spaces H*(Q)3 and H(curl,Q) for regular bounded Q is: the imbeddings into L2(Q)3 of H(curl,G), H(div,Q), H(curl,div,Q), and their intersections with ker curl or ker div are not compact since all these spaces contain the space

(149)

2 MA THEM A TICAL FRAMEWORK ff°(Q)

= kercurinkerdiv={u€L2(Q)3, curlu = 0, divu = Oj,

which is a closed subspace of L2(Q)3 with infinite dimension. Thus it prevents all greater spaces to have a compact imbedding in L2(£l)3. Using the (Hodge) decomposition of L2(Q)3 in the form (with (143)): L2(Q)3 = grad H„(Q) ® H°(Cl) © curl HJ[(Q)3 we easily obtain the decomposition of the space H(curi, div, Q) = grad D(AD) © H°(G) ® curl EKA"), where D(AD) is the domain of the Dirichlet Laplacian, D(AD) = H2(Q)n HQ(Q), for a regular domain Q, and with D(A") given by (146) which is also contained in H2(Q)3. Thus the spaces grad D(AD) and curl D(A~) are contained in H 1 ^) 3 , with compact imbedding in L2(Q)3. Now we easily obtain the result when Cl is a regular bounded open set in R3 (in fact it is true with a Lipschitz-continuous boundary, see Costabel [2]). Proposition 11. The imbedding of each of the following spaces into L2(Q)3 is compact (u€H(curl,div,Q), nAu| r €L 2 (r) 3 }, {u € H(curl,div,Q), n.u| r €L 2 (0}, and more generally (see Weber [1], Hazard-Lenoir [1]) if C € L°°(C1) with Re t > a > 0 {u € H(curl,Q), div Cu € L2(Q), n A U | € L2(03 (resp. n.Cu | € L2(0}. We refer to the Appendix, A.2 Remark 5 and to Theorem 9 A.3 for other compactness results, for example for the compensated compactness theorem. & 1 2 . FIRST PROBLEMS WITH INHOMOGENEOUS BOUNDARY CONDITIONS

12.1. Problems with H " 1/2(div,0 boundary conditions i) Previously (in section 4), we have solved some boundary problems in the H(curl,Q) framework (with "regular" Q, bounded or the complementary of a bounded set). Proving that the trace mapping yT is onto (see Corollary 1), we solved (see (46)):findu in H(curl,Q) satisfying, with m given in H ~ 1/2(div,D (150)

curl curl u + i^usO inQ, nAu| =m.

This is equivalent to the system (of Maxwell type): (151)

-curlu + v = 0 , curlv+1^ = 0 inQ, nAu| =m,

or also to: (151)' -Au + ^ussO, divu = 0 in£l, nAu| =m. We denote by Cm and C+m the boundary values (with K = 1): (152)

Cm as n A curl u | = n A V| ,

C*m = TifCurl u,

2.12 INHOMOGENEOUS BOUNDARY CONDITIONS

with u the solution of (151), and v in (151)'. Then we define: Definition 6. The Calderon or Capacity operators. The mapping C: m —»Cm and C+: m —»C*"m are called the Calderon or the capacity operators. We verify that C is a continuous isomorphism in H ~ (div,H with C 2 = - I. ii) More generally, thanks to Lax-Milgram lemma, we know that there exists a unique solution to the following problem in variational form: find u in H(curl,Q) such that (153)

(curl u, curl v) + (u, v) * <f, v> for all v in H(curl,Q),

with f given in the dual space (H(curl,Q))' of H(curl,Q), which is not a space of distributions. So the only point is to recognize a boundary problem in (153). This problem may be understood in the following way: using the orthogonal decomposition (154)

H(curl,Q) = H0(curl,n)® H^ (curl,Q),

we have by duality (154)' (H(curl,Q))' = (H0(curl,Q))' ©(Hj1 (curl,Q))\ and since H^ (curl,Q) is isomoiphic to the trace space H" (div,r), see Corollary 1, we can identify dual spaces, that is, (H^ (curl,Q))' with H"1/2(curl,r)Thus we can identify f with the couple (f0,g), with f0 in the space of 1/2

distributions (H0(curl,£2))\ and g in H" (cur!,0. i.e., we can write: (155) <f, v> = <fQ, v0> + <g, n A v>r, V v € H(curl,Q), vQ being the projection of v onto HQ(curl,Q) and <,> the duality bracket in Cl and T. First taking v in H (curl,Q) in (153) we have: (156)

curl curl u + u =o*f

Then taking v in H^ (curl,Q), (153) gives (156)*

(curl u, curl v) + (u,v) = ^g,nA v>r .

Using the Green formula (on the "unusual" side !), we have: (157)

<nAu, curlv>r = <g, nAv> r

Using (152) we have: (157)'

<n A u, C+(n A v)>r = <g, n A v>r.

2 MATHEMATICAL FRAMEWORK

Thus by duality (see (173)' below), we have: (158)

C+(nAu) = g,

thus nAcurlu| = nAg.

Then we see that the variational problem (153) is equivalent to (156) and (158). Note that the dual space (Ho(curl,Q))' of HQ(curl,Q) is characterized by: (H0(curl,Q))' = {w=curlv+u, veL2(Q)3, uâ&#x201A;ŹH0(curl,Q), oru = grado, oeH^Q)}, since the operator curl curl + I is an isomorphism from H (curl,Q) onto (Ho(curl,Q))\ Remark 20. Using the same method we can interpret the inhomogeneous Neumann problem for the Laplacian, thanks to the corresponding capacity operator. More precisely for every uQ in H 1/2 (0 let CuQ be defined by: (159)

Cu0=||r,

with u the solution in H (Cl) of the Dirichlet problem: - Au + u = 0 in Q, u | = uQ. We call also C the capacity operator for the Laplacian, see Dautray-Lions [1] chap. 2; it is an isomorphism from H 1/2 (0 onto space H"~ (OThen we consider the inhomogeneous Neumann problem: (160)

/^(gradu. grad v +u v)dx = <f, v>,

Vvin Hl(Cl).

with f given in (H 1 ^))' which is not a space of distributions. We use the orthogonal decomposition: (161)

H^Q) = Hl0(Q) 0 Hl(Cl), with Hl(Cl) = {u â&#x201A;Ź Hl(Cl), - Au + u = 0},

which is isomofphic to space H1/2(D. Then its dual space is identified to space H~ 1 / 2 (0. Thus by duality we have: (162)

(u\ci)y = H-\CI) 0 (Hl(Q))\

and then f is identified to the pair (f ,g) with fQ in H" l(Q), and g in H~ 1/2(T). Note that if f is in L2(Q), we have to decompose f into (fQ,g) also, i.e., we have to consider l?(Q) as asubspace of Hl(Cl)\ not of H" l(Cl). Thus we have: (163)

<f,v> = <f0,v0> + <g,v>r,

VvcH^Q).

First taking v in H^(Q) in (160), we see that u satisfies: (164)

-Au + u = f0 inQ.

2.12 INHOMOGENEOUS BOUNDARY CONDITIONS

Then taking v in Hl(Q), and applying Green formula to (160), we obtain: (165) that is: (165)'

<u,95>r = <&v>r, <u, Cvr> = <g, v r>r ,

V vr â&#x201A;Ź H1/2(D,

using the capacity operator C given by (159). Thus by duality, since *C (or C*) is identical to C: (166)

C(u| r )=g.

Therefore the solution u of the inhomogeneous Neumann problem (160) is also the solution of (164),(166). We note that in the two cases (problems (153) and (160)) there is no trace for curl u, with u in H(curl,Q) or for the normal derivative of u on T for u in H*(Q). Remark 21. A very interesting property of these capacity operators is given by their relations to the quotient norms. More precisely: i) Norms in H 1 / 2 (0. Let * be H 1/2 (0. Then its quotient norm is given by (167)

IWI2 H

inf J [|gradu|2+ lutfdx. ueH^.ul^o"

The infimum is reached at u solution of the following problem in H*(Q): (168)

-Au + u = 0,

u| r = 0.

Thus thanks to the usual Green formula, we obtain (formally): (169)

M^ 1/2

=/r|.dr,

or using operator C, see (159), and the duality bracket <,> for H 1/2 (D, H (170)

IW^i/2(n=<C0,0>.

ii) Norm in H" 1/2(div,D. Let m be in H " 1/2(div,D. Its quotient norm is: (171)

llml2 v,

inf

.Ltlcurlu^+lutfdx.

The infimum is reached at u solution of the problem in H(curl,Q): (172)

curlcurlu-f u = 0inQ,

nAu| =m.

Thus using Green formula in (171), we obtain:

1/2

2 MATHEMATICAL FRAMEWORK

(173) and also: (173)'

ltaV/2,(div,

=/ r curiu . nAU dT = <C+m, m>r = <nACm, m> r ,

( K , ^ > H - i / 2 ( d i v 0== <mu» c+m v> = <C*moi m^.

r-1/2/(div,D Thus we see that it is possible to evaluate norms in H11/2/(T), or in H~ thanks to the capacity operators, using (169), or (173), in correspondence with positivity properties of these operators. ®

12.2. Problems with H 1/2 (0 boundary conditions Corresponding to the H*(Q)3 variational framework, we have the boundary problem: find u in H*(Q)3 satisfying, with given m and g i)-Au + u = 0 (174)

inQ,

r ii)nAu| r = m, nicH^U2(T)

iii)(divu)| r = g, g € H

i/2, U

\T)

The uniqueness of the solution is a consequence of theorem (11). Then in order to solve (174), we proceed in two stages: i) Lifting of the boundary condition (174)ii). We note that V]" = Hn(Q) is a closed subspace of H(curl, div, Cl). Let V orthogonal space. It is characterized by: (175)

be its

= {u € H(curl,div,Q), - Au + u = 0, (divu)| = 0}.

Recall that the trace mapping y: u —♦ n A u | is continuous from H(curl,div,Q) onto X

1/2

H" (div,0 with kernel VJ"; therefore y is an isomorphism from V" onto space H" (div,r). Moreover, thanks to Theorem 6, we have u € H (Cl) when n A u | is in H t l/2 (0. Thus y is also an isomorphism from V" X n H 1 ^ ) 3 onto H t 1/2 (0. Therefore for all m given in Ht (T), there exists a unique U in H (Cl) such that: (176)

|i)-AU + U = 0 inQ, |ii)nAU| r = m, iii)(divU)| r = 0.

ii) Thus by difference we have tofindw = u - U, w in Hl(Cl)* satisfying: i ) - Aw+w=0 inCly (177)

I,z ii)nAw| r = 0, iii)(divw)|r = g, g € H "1/2/ (n

This is solved by a variational method: using the sesquilinear form a(u,v) given by (144), w must satisfy

2.13 BOUNDARY PROBLEMS OF CAUCHY TYPE (178)

a(w,v)=/ r gn.vdT,

VvcVJ".

Thanks to Lax-Milgram lemma we know that (178) has one solution. Remark 22. The difference in tackling the two boundary conditions is very meaningful: the first one (174)ii) which is included in the Sobolev space HJ(Q) is said to be stable, the second (174)iii) which is obtained by duality, by the & weak formulation of the problem is said to be unstable (see Necas[l]). Conclusion on the difference between the two types of boundary problems 12.1 and 12.2: in the first one, more general boundary conditions are allowed; the second gives a regularity result with respect to the first, and is based on the usual Sobolev space H*(Q)3 and thus compactness properties (for regular bounded Q); also it seems easier to apply in numerical methods. We will study other more difficult boundary problems in chapter 3. 1 3 . BOUNDARY PROBLEMS OF CAUCHY TYPE. UNIQUENESS THEOREMS

Here we give without proof (according to the case, proof may be quite obvious or quite difficult) some results which are fundamental to a deep understanding of boundary problems in the Helmholtz case and also in the Maxwell case. They concern only the question of uniqueness of the solution and not (a priori) the question of existence; furthermore there will often be no solution at all for such problems. Therefore they do not have (a priori) a physical flavor ! In all this section, Q is a "regular" domain in Rn, (n = 3 for Maxwell equation) bounded or not, with boundary r. These uniqueness theorems are of Holmgren type (see for instance Dautray-Lions [1] sec. 2.1.5); see also Saut-Scheurer [1] and Muller [1] for other uniqueness results. Theorem 12. The Helmholtz case. The following problem:findu with i)Au + k2u = 0 inQ, ii)u| = 0 , gjj l r = 0, or only on some regular part TQ ofT,

(179)

iii) u € LJ^fi), i.e., tu € L2(C1) for all ? in Z>(Rn), (or more generally with u in a functional space so that the trace condition ii) makes sense) has 0 as unique solution. When r is a part of r we prove that u is null in a neighbourhood of To, then we use analyticity of the solution. Corollary 7. The following problem:findu in L*oc(Cl) satisfying lyAu + k^u^f inQ, (ISO)

I ii)u

lr = uo'§Hlr = u i '

2 MA THEM A TICAL FRAMEWORK

with given f in L (Q), u0 in H (O, ux in H"1 (T), can have at most one solution. Theorem 13. The Maxwell case. Thefollowing problem: find (E,H) with i)curlH + i<DeE = 0,

(181)

ii)- curlE + icopH = 0 inQ,

iii)nAE| =0, nAH| = 0 (or only on some part T0 of T),

and with (cp) constant on Cl (and (E,H) in a functional space so that the trace condition iii) makes sense), has (E, H) = (0,0) as unique solution. Corollary 8. The Maxwell problem:find(E, H) satisfying I i) curl H + icocE = J, ii) - curl E + iwiiH = M , on £i, iii)nAE| =M°, nAH| =J° (or only on somepartT0 of 0>

(182)

iv) (E,H) withfiniteenergy, i.e., E and H in Hloc(curl,Q), with given J, M in L (Cl) , J°, M° in H"

(div,r), can have at most one solution.

1 4 . WHITNEY ELEMENTS; NUMERICAL TREATMENT

Let Q be a bounded open set in R3 where we have to solve an electromagnetic problem; we have to approximate vector fields, or n-forms, and to take into account the tangential (or the normal) continuity of the fields. Here we briefly describe finite elements, known as Whitney elements, adapted to these requirements. First we specify some notations; we essentially follow Bossavit [1]. Let (T{) be a tesselation of Q by a finite number t of tetrahedra, that is: U ^ Q , TDT^0 J

for i * j ,

with the usual properties of a finite element mesh. Note that tetrahedra at the boundary may be curved. Then we call N, E, F, T respectively the set of nodes (vertices), edges, facets and tetrahedra which constitute the mesh. Let a., i = 1 to 4, be the nodes of a reference tetrahedron TQ, and let (w.(x)), i = 1 to 4, be the barycentric coordinates of the point x; then: (183) x= 2 Wj(x) 2L, that is also X:= 2 w ^ a :J: , 1= 2 w iW, i=lto4

i=lto4

or in matrix form:

C (183)'

C wi<x> ^

C a " ai2 ai3 ai4 ^

2.14 WHITNEY ELEMENTS

then we can inverse relations (183)':

(184)

=A r

[Si] " u?J'

Wi(x) {A

^ ^

therefore we have for x in T : o

(185)

3w ^r(x) = A^ i j ,

i = l t o 4 , j = lto3, Vxin7*0.

Since w.(x) is a linear function of x in T , its gradient is constant in T . Then in the mesn, for n in N, wn is a "node-element". For every edge e = 0,j) in the mesh, we define "edge-elements": (186)

we = Wj grad w- - Wj grad Wj;

then for evey facet f=(i, j, k), we define "face-elements": (187) wf Âť 2 (Wj grad Wj A grad wk +WJ grad wk A grad wt + wk grad w4 A grad WJ) and for every tetrahedron r=v=(i,j,k,l), we define "volume-elements": (188)

wv=6gradwi.(gradWjAgradwk).

Note that these elements w depend a priori on the chosen orientation for the edge, facet, tetrahedron, and that "edge-elements" and "face-elements" are vectorfields.We can prove fairly easily the following properties: (189)

/^.VdUSe^w^

with "v* a unit vector along e*, or normal to f. Definition 7. We call Whitney space of order i, i = 0, 1, 2 , 3 respectively and we denote it by W!(Q), the vector space generated by (wn), (we), (wf), (w^. We also denote by WlQ(Q) the subspace ofelements o/W!(Q), vanishing at the boundary. The elements of Whitney space are linear combinations: (190)

u= 2 usws,withS = tf,ÂŁ,F, f, seS

where complex numbers u denote the degrees of freedom. From property (130) it follows that the dimension of space W\C2) is n, e, f, t, that is the number of nodes, edges, facets, tetrahedra for i = 0, 1, 2, 3 respectively.

2 MATHEMATICAL FRAMEWORK

The elements of W!(Q), for i = 1 to 3, are certainly not continuous on Q, but they have the following important properties: curlwc = 2gradwiAgradw.,

fore = (i,j),

divwf=6gradwi.(gradwJAgradwk),

forf=(i,j,k)

in the sense of distributions in Q, that is they are bounded on Q: thus edgeelements have no jump of their tangential part across facets, and face-elements have no jump of their normal part across facets. Thus we have the inclusions: (192) W°(Q)cH1(Q), W^QJcHteur^Q), W2(Q)cH(div,Q), W3(Q)cL2(Q). Moreover, since we have from the definition of the barycentric coordinates, for example in TQ (with a suitable choice of the edge orientations): 1m 2

Ulto4

Wj(x), then 2 wV,J; (i j} = - grad Wj. j*i

Therefore we also have the inclusions: (193)

gradW^QJcW^Q), curlW^cW^Q),

divW2(Q)cW3(Q),

and we have the sequence, with the properties of (105): A

(194)

grad

curl

div

W°(Q)—> W'(Q)—> WZ(Q)—> W3(Q).

Whitney elements are a generalization of mixed elements of Raviart-Thomas and ofNedelec elements; see Girault-Raviart [2] and Roberts-Thomas [1]. For some developments on these Whitney elements and their use in computation for electromagnetism, we refer to Bossavit [4].

CHAPTER 3

STATIONARY SCATTERING PROBLEMS WITH BOUNDED OBSTACLES

1. STATIONARY WAVES DUE TO SOURCES IN BOUNDED DOMAINS

First, we study in the whole space R3 stationary waves, of wavenumber k, produced by a source f localized in a bounded domain. We have to specify conditions at infinity corresponding to outgoing or incoming waves; these conditions are known as the Sommerfeld radiation conditions. Then we study electromagnetic fields produced by charges p and currents J localized in a bounded domain, at a given angular frequency co or at a given wavenumber k, in free space. Conditions at infinity corresponding to outgoing waves or incoming waves are called the Silver-Miiller conditions. Then it will allow us to solve (by a surface integral method): i) boundary problems, in a bounded domain or its complement, ii) scattering problems for a bounded obstacle due an (exterior) incident wave. 1.1. The main properties for the Helmholtz equation in R3 Here a stationary wave is a solution u of the Helmholtz equation in the whole space (1) Au + k ^ - f inR3, with k > 0, and fa distribution with compact support (i.e., fin E '(R3)). 71

3 STATIONARY SCATTERING PROBLEMS

1.1.1. Local regularity properties First it follows from the hypoelliptic property of the operator (A + k2) that u is C°° in any open domain Cl where this is true for f. It is even an analytic function in Cl where this is true off, since (A + k2) is analytic-hypoelliptic (see Treves [1] p. 23); thus u is analytic outside the compact support off. Using Sobolev spaces we have from the elliptic nature of (A + !r), see Lions-Magenes [1]: u€H^(R 3 )forf€Hf oc (R 3 ), V s € R . 1.1.2. Outgoing and incoming Sommerfeld conditions The energy balance for the wave equation: (2)

-^l^-Aw-ginR^xR3,

for a given "regular" function g, in a bounded domain Cl with boundary I\ at timet, is: (3) with:

the r.h.s. of (3) is the energy provided in Cl by the source g (see for example Dautray-Lions [1] chap. 2.8.7); n being the exterior normal to Cl. For a source g of the form: g = e" la)t f (for f in L2(R3), with compact support, and with angular frequency o>) and for a stationary wave w = e " u, u satisfies the Helmhotz equation with the wavenumber k = co/c : (4)

l^u + Au^-f,

and the energy balance for the (bounded) domain Cl is (5)

/ I <u) < l5 f .Re(ia)/ r gudr) = Re(ia)4fudx).

The condition: "f having a compact support K" implies that u is regular outside K and allows us to define: Definition 1. We call outgoing {resp. incoming) Sommerfeld condition the condition at infinity: (6)

g | - i k u = o ( j ) withu = 0 ( £ ) , r = | x | - o o ,

3.1 WAVES DUE TO BOUNDED DOMAINS

((6)'resp.)

| | +fcu= o(A) 3n

with

u = 0(j) , r=|x|-oo,

with o and O uniform with respect to a = x/r, and we call outgoing (resp. incoming) wave a solution u of the Helmholtz equation that satisfies the outgoing (resp. incoming) Sommerfeld conditions. The flux of energy of an outgoing (resp. incoming) wave through every boundary Sf of a ball Bf containing the support of f is positive (resp. negative): (7)

/Sr(u) = Re(-io>/ s |judS r ) = Re(ck 2 )/ s |u| 2 dS r + o(l) > 0 (resp.<0).

Note that Definition 1 is also valid for fa distribution with compact support. 1.1.3. Elementary outgoing (incoming) solution The function O = <&out defined on R3 by: (8)

p ikr

GOO-aW-j^,

r=|x|,

which satisfies: (A + k*) = - 8 , (9)

f-M--£.o<|>.

corresponds to an outgoing wave due to the (point) source 5. It is called the elementary outgoing solution of the Helmholtz equation (or Green function by physicists). Its conjugate is the elementary incoming solution <f>m. Note that: (10)

<D € L?oc(R3) for 1 < a < 3,

grad O € Lfoc(R3)3 for 1 < 0 < 3/2.

1.1.4. Fundamental properties Let f be a distribution in R3 with compact support. Then the convolution product: (11) u = 4>*f satisfies the Helmholtz equation: (12)

(A + k2)u = (A + k 2 )0*f=_5*f=-f.

We prove (see for example Dautray-Lions [1] chap. 2.8) that u given by (11) is an outgoing wave. Then we have: Theorem 1. The problem: find u, a distribution in R3, satisfying

(13)

3 STATIONARY SCATTERING PROBLEMS OAu + l ^ u ^ - f , f given in £*(R3), k>0, ii)(§S-ikuXx) = o()),

withuCxJ^CX^whenr^lxl-oo,

has a unique solution, given by (11). We only need to prove uniqueness, which is a consequence of the following lemma: Lemma 1 (Rellich). Let Cl' be a connected domain which is the complement of a bounded domain, and let u be a wave satisfying: i)Au + k2u = 0 (14)

inQ',

«)/ s 2 |u(ra)|2da = o ( r 2 ) , i . e . / s |u(x)|2dTx = o(l)whenr- oo.

Thenu = OinCl\ The "standard" proof of the Lemma uses a "multipole expansion", or "Rayieigh expansion" (see section 11). For such a proof, see Muller [1] p. 87. We also prove it in chap. 5.3. PROOF ofthe uniqueness in theorem I. We only have to prove that f = 0 implies u = 0. We use the energy balance (5) in the domain Q = Bp, with c = 1. Thus:

R e ( - i k / s §j|udS r ) = 0,andthusk 2 / s |u(x)|2dSr=o(l)whenr—oo, with the Sommeifeld condition (6). Therefore u = 0 from Rellich Lemma. ® Proposition 1. Behavior at infinity. The outgoing solution of the Helmholtz equation (13) has the following behavior at infinity (uniformly in a): (15)

u(ra) = 0(r)f(ka) + o(^), r=|x| , a = x/r,

f being the Fourier transform of f, given by f(£)=/f(x) e"^x dx. PROOF. There are many ways to prove this proposition. We give here a proof using the Green formula (see Muller [1] p. 114). Let *x(z) ■ *(x-z) be the elementary outgoing solution centered at x. When r goes to infinity, ♦ has the behavior: x

(16)

94> x

^ ( z ) = - ik4>(r) (n.a) e-** 2 + o(l), <Dx(z) = *(r) e"**2 + o(l),

3.1 WA VES DUE TO BOUNDED DOMAINS

with r = |x|, a = n=x/r. Let Cl be a "spherical" domain Q = BR \B R with Rt > R2, with supp f c BR . Then: (17)

9* ^ ( ( A + ^ . u - ^ A + l ^ D d z - Z ^v ^ u - ^ g ) ^

therefore (since the boundary 3D of Q is 3Q = SR uS R ): -

3*»

(18) -u(x) = -I(S R 2 ) + I(SRi), with KS)=/s(-gif u-* x g[)dr z . First let Rt —»<»; then: (19)

u(x) = I(SR) forR = R2.

Then let r= |x| —» oo; using (16), we obtain (20)

u(x) = u(ra)=*(r)/s e- t o y (ik(n.a)u+||)dr + o(l). R

But we also have, using the hypothesis that f has a compact support, and the Green formula: f(ka) = <f,e- ,ta -S B = / ((A + k2)u . e-aa,Jt - u . (A + k V * * * ) dx R "R (21)

hence giving (15).

® A

In scattering problems, it is very important to characterize the source f from f(ka). The properties of f(ka) will be specified at section 11. Here we only note that if f(ka) s 0 for all a in S , then u(ra) = o(£) when r —»oo. From the Rellich Lemma, u(x) = 0 for r > R, outside the support off, i.e., u € E '(R3) which is possible only if f(£)/(£ - k ) is an entire function, see Hbrmander [1] p.78. Such sources are perfectly stealth, since they create waves which vanish outside a bounded domain. 1.2. The main properties for Maxwell equations in R3 Here we study the following Maxwell problem in the whole space R3: find (E, H) in Z>'(R3)3 x £>'(R3)3 satisfying (22) * '

P curlH + ia>£E = J, ii)-curlE + io>yH = M inR3

3 STATIONARY SCATTERING PROBLEMS

with E and p the permittivity and the permeability of free space (e > 0, y > 0); J and M are electric and magnetic currents, and are distributions with compact support in R3. Very generally, magnetic currents do not exist, but it is very useful to introduce them, so that Maxwell equations become more symmetric with respect to E and H. 1.2.1. Relation to the Helmholtz problem We can eliminate in system (22) the electric field E or the magnetic field H, in order to obtain equations on H only (resp. E). Applying the curl operator to (22)i), or to (22)ii), we obtain: (23)

i) curl curl H - k H = curl J + ia>e M ii) curl curl E - k2 E = - curl M + ia>p J in R3,

with k2 = (o2t]i. Using the relation: (24)

curl curl u = - Au + grad div u,

and applying the div operator to (23), we obtain: (25)

I - k2 div H = icoe div M -l^divEsicojidivJ,

and thus (23) gives: (26)

- AH - k^H = curl J + i<DÂŁ (M + k" 2 grad div M), - AE - k2E = - curl M + icop (J + k"2grad div J).

Let m and j be defined by: (27)

m d=f curl J + icoe (M + k"2 grad div M), j dJf - curl M + i|i (J + k"2 grad div J).

We have obtained the (vector) Helmholtz equations: (28)

AH + k2H = - m ,

AE + k ^ - j ,

3 3

where j and mâ&#x201A;Ź2s'(R ) , i.e., are given distributions with compact support. As in the case of the Helmholtz equation, it is necessary to introduce conditions at infinity in order to obtain a well-posed problem. We make the hypothesis that E and H satisfy the outgoing Sommerfeld condition for u = E. or H.. Then from Theorem 1, E and H are obtained by the convolution product:

3.1 WAVES DUE TO BOUNDED DOMAINS

(29)

E=$.j,

H=*.m.

The behaviors of E and H at infinity are given by: (30)

E(ra) = <Kr)j(ka) + o ( j ) ,

H(ra) = *(r) m(ka) + o( J ) ,

with (from (27)): I A

j (ka) s - ika A M(ka) + ia>p IlaJ(ka), (3D m(ka) = ika A J(ka) + ia>£ riaM(ka), with n a the projection of the vector v on the tangent plane to the sphere Sk at ka: (32) n a v = riv = v - a ( a . v ) . We immediately verify the following properties: i) a. j (ka) = 0 ,

a.m(ka) « 0 ,

V a € S2,

(33) ii) - a A m(ka) = ^ j (ka),

a A j (ka) = Z m(ka),

with Z the free space impedance: (34)

Z = oJe=^r,

Z =j.

We can also obtain these formulas in the following way. First applying the div operator to (7), we have: (35)

v d = f divE = 4>*divj,

with the behavior at infinity: (36)

v(ra) = <D(r) div j (ka) + o (j) = O(r) ika. j (ka) + o (j).

Since div J has a compact support, we have from (25) (37)

v = divE = 0 forr>R 0 ,

hence a. j (ka) = 0. Operating similarly for div H, we obtain (33)i). Then applying the curl operator to (29), we have: (38)

w d = f curlE = *curlj,

whose behavior at infinity is given by: (39)

w(ra) = O(r) curl j (ka) + o ()) = <D(r) ika A J (ka) + o (|).

3 STATIONARY SCATTERING PROBLEMS

Using the Maxwell equation (22)ii) outside the support of M, that is: (40)

-curlE + iamH = 0,

we get through (30) and (39) (41) - ikot A j(ka) + ia>M m(ka) = 0, that is (33)ii).

Conversely, condition (33)i) implies through (36) div E ■ o( j) when r —» oo, thus from the Rellich Lemma div E = 0 on the outside of a ball Ba. Then if E satisfies the Helmholtz problem (42)

i) AE + l^E = - j with j € E '(R3)3, and a. j (ka) = 0, ii) ^ - i k E * o ( | ) w h e n r — o o , def 1

we have div E » 0; if we define H on the outside of the support ofj by H » ^ curl E then (E,H) satisfies Maxwell equations, since curiH

=iiir c u r i c u r l E s -iijr A E =^ E = - i < D £ E -

1.2.2. The Silver-Miiller conditions Conditions (33)ii) imply the following behavior at infinity: <0EaAE(ro)-kH(nx) = o(y) with E(ra) = 0 ( 7 ) (43)

uniformlyinaeS . <opaAH(ra) + kE(ra) = o(f) with H(itt)-0(j) 9

These conditions are called the Silver-Miiller conditions for the Maxwell equations (22). They are equivalent to the outgoing Sommerfeld conditions for the Helmholtz equation (if we only change i into - i in (22), then (43) would correspond to the incoming condition). This is a consequence of the theorem: Theorem 2. The Maxwell problem: find the electromagneticfield(E,H) in the space ofdistributions in the whole space R3, satisfying i) Maxwell equations (22) with given distributions with compact support (J, M), ii) the Silver-Muller conditions (43), has a unique solution, given from (29) and (27) by (44)

E = j * 0 = [ia>p (J + k"2 grad div J) - curl M] * O H = m ♦ * = [ia>e (M + k"2 grad div M) + curl J] * <&.

Its behavior at infinity is given by (30) with (31), (33)

3.1 WAVES DUE TO BOUNDED DOMAINS

Note that we can write (44) in many ways, since: (45) (curlM)•<!> = -NT,grad«>, with(M A »grad<l>) l3 :M 2 ^- M 3 , | J - , . . . . The electric and magnetic charges pc and pm are obtained through J and M by (46)

icop = divJ, iwp = divM.

Then (44) may also be written: (47)

E = io>p J*$ + M*grad$- |-pe*grad, H = icocM*0- J^gradO- ppm*gradO.

These formulas are often called Stratton-Chu (or also Kirchhoff) formulas. When M = 0, if we define the vector potential A and the scalar potential 0 by 1 1 c2 A = jiJ*0, ♦ = £p*0 = j~divJ**=!^divA, withp = pe, then (E, H) is given by: E = io> A - grad <>, H = p curl A. PROOF. We only have to prove uniqueness of the solution of the Maxwell problem with Silver-Muller conditions at infinity, for J and M = 0. This is obtained through the energy balance equation: let T be a surface which is the boundary of a regular bounded domain Q; then thefluxof the Poynting vector S = Re (E A H) through T is zero

(48) since (48)'

P = Re(/ r n.EAHdT) = Re(/ a div(EAH)dx) = 0, div(EAH) = curlE.H-E.curlH = io>>iH.H-ia>£E.E.

Then taking T = Sr, with high r and using (43), we have: (49) P = R e ( / s n.EAHdS r ) = Re(/ s nAE.HdS r )=^/ s |H| 2 dS r + o(l); therefore / | H | dSr = o( 1). From the Rellich Lemma, we have H = 0, and then E = 0 in the whole space. Remark 1. The Silver-Muller conditions imply: (50)

a.E(ra) = o(i), a.H(ra) = o ( | ) , i.e., x.E(x) = o(l), x.H(x) = o(l),

(and also: a.curl E = o( \), a.curl H = o( \)). Thus (x, E, H) is an orthogonal frame at infinity. If E2, HL are the projections of E and H onto the tangent plane to the sphere rl, then with the usual notations we have:

(51)

3 STATIONARY SCATTERING PROBLEMS

curl2 E 2 = o( j ) ,

curlz H 2 = o( j ) .

If we compare the Silver-MuUer conditions with the outgoing Sommerfeld conditions, we also obtain: (52)

clTrl 2 E rS =o(|),

cwl 2 H r = o(±),

which cannot be proved straightforwardly. 1.3. Transmission problems and surface integral operators Here we study waves due to distributions of charges or currents located on a surface T which is the boundary of some bounded domain Q, first in the Helmholtz case, then in the Maxwell case. 1.3.1. Helmholtz problems with charges on "regular" surfaces First we recall (see Prop. 2, chap. 2) some jump formulas for a regular function u on each side of a (regular) surface I\ Here by "regular" we mean that u is C1 (up to the boundary). This is necessary so as to speak of the derivatives of u in a classical sense. Let 5 r be the Dirac distribution on I\ We have (53)

grad u = (grad u) - [u]r n Sj-,

and if u is a (regular) vector function of x in R , with values in C : (54)

curl u s (curl u) - [n A U] 6 r ,

div u = (div u) - [n.uL 8p

Expressions in brackets (grad u),... in (53),(54) correspond to derivatives in the classical sense, whereas the l.h.s. of these equalities are derivatives in the distribution sense; [v]r is the jump of the function v across the surface I\ i.e., the normal n to T being oriented from T. to Te (55)

[v] =v|

-v|

We can easily prove (53), (54) from Green formula. As an example for all ♦ € D(R3)3, we have: <curl u, ♦> 1?<u, curl♦> 5= /

u.curl ♦ d x + / n u.curl ♦ dx

=/ nf curlu.0dx+/ Q curlu.dx+/ r nAu| r d T - / r n A u | r d T , giving (54). These formulas (53), (54) are also valid in the Sobolev framework, with the following regularity conditions; let X = X f be the space defined by: (56)

X d J f {u€l4 c (R 3 ), u| n €H ! (Q), A(u| n )€L 2 (Q),

«la.€H/0C(S'), A(u|fl,)€l4c(Q%

3.1 WAVES DUE TO BOUNDED DOMAINS

In these formulas, we recall that the Laplacian is taken in the sense of distributions in Q and Q\ and that (57)

H^Q') = {u, *u | a € H\C1'), V » € I>(R3)}, 4 c (fi') = {u, *u | n> € L2(G'), V ♦ e 2XR3)}.

When u is in X, then from the usual trace theorems (see Prop. 1, Chap.2), u and its normal derivative have traces on each side of T, with (57)'

€H 1/2 (D, H i

u|. i,e

€H"1/2(r). *i,e

Then the Green formulas are valid for u in X and therefore also the jump formulas (53),(54). Let y., ye be the trace mappings on X (58)

y. (u) = y.u = (u | r , gj | r ), and ye similar to y.,

and let G{ =f y.X, Ge =f yfiX. Then we have Gl * Ge, with G{ and Ge * Y, with (59)

YdJfH1/2(r)xH"l/2(r).

Thus y. and ye, which are continuous from X into Y, cannot be onto. Let 7^ (resp. n2) be the projection on thefirst(resp. second) component. Then we can prove (from Lions [1]) that ^y. and *jye (resp. 7i2y. and 7i2ye) are surjective mappings onto H1/2(D (resp. H ' 1/2(D). The set G^HG^ corresponds to the traces of functions in X (which we can take with compact support) which are "continuous" with their normal derivatives across T; thus these functions are in H2(Rn): we can verify it by a Fourier transform method. Therefore we have (for regular T) (59)'

G{nGz = H 3/2 (0 x H 1/2 (0.

Also there is no extension mapping from H^AjQ) (nor from H(A,Q), see (3) chap. 2) into the space H^R 11 )) = H(A,Rn) = H2(Rn). Proposition 2. The mapping y = y_ = y.-y e is continuous from X onto Y. PROOF. In order to have a lifting of y, we can use a variational method, with a lifting of the jump condition [u]f = p, and with the bilinear form:

3 STATIONARY SCATTERING PROBLEMS a(u, v)=/ R n (grad u. grad v + uv) dx

which is coercive on H^R11). Then we have a lifting u of (0,Uj), as the solution of: (60)

a(u,v)=/ru1vdT,

VveH^R 11 ).

We obviously have the same propertyfor y « y. + y , and then: G{ + Gc » Y. Let u be a (stationary) wave with locally finite energy in Q and its complementary Q' up to the boundary, which is discontinuous (with its normal derivative) across the surface I\ Therefore, using formulas (53), (54), we see that u satisfies: (60)'

Au + k2u = - ([§j|] 8r + div ([u]^)).

Conversely, we have: Proposition 3. The following problem:findu satisfying

(61)

lOAu + l A i s - f inR3, with f^p'Sp* div (pnfip), with given p, p' resp. in H 1/2 (0 and H" 1/2 (0; ii) the outgoing Sommerfeld condition,

has a unique solution u in X, which is given by the Kirchhoffformula: (62)

u = O * f=4> • (p' 6r + div (pn8r)),

and the jumps ofu and its normal derivative across T are given by:

(63)

[u]r-p,

lg]r-P'.

PROOF. Thanks to Theorem 1, we only have to prove that u given by (62) is in X. Let Uj be a lifting of the jump conditions (59) (this comes from Proposition 2); then Uj satisfies: (64)

AUl + k2u1 = - g - <p' 6r + div (pn6r)), with g € L?oc(*3)-

We can choose u} with a compact support, so that g also has a compact support. Then v = u - Uj satisfies: (64)'

Av+k 2 v=-g,

as well as the outgoing Sommerfeld condition; thus v is given by v = 4> * c with g in L2. Thus from the regularity properties (see 1.1.1) we have v in Hj oc (R 3 ), therefore v in X, and thus u = v + Uj is in X. ®

3.1 WAVES DUE TO BOUNDED DOMAINS

Proposition 4. The mappings P. and P e defined by: (65)

Pi(p,p>) = y.u, resp. Pe(p,p') = - Yeu,

with u the solution of (61), are continuous complementary projectors in the space Y defined by (59). The fact that these operators are continuous is a simple consequence of Propositions 2 and 3. They are obviously projectors. Furthermore the uniqueness of the solution in Proposition 3 implies that the intersection of their kernels (or of their ranges) is reduced to 0. From (63) we obtain: PROOF.

(66)

P. + P e = I.

<8>

Definition 2. The operators P. and P e are called interior and exterior Calderon projectors (relative to the Helmnoltz problem). Then if we denote by Y. and Y e their images in Y, we have the decomposition of Y into the direct sum (but it is not an orthogonal decomposition, since the Calderon projectors are not Hermitian): (66)'

Y = Yi©Yc.

Single and double layer potentials. potentials respectively by: (67)

£p' = O * (p'6r),

We deflne the single and double layer

Pp = • div (pnfy),

or also, forx outside T, by: (67)'

Le\x)=/r

p'(yWx - y) dr^

Fp(x) = - J r p(y) f^ (x - y) dr y .

Therefore the solution u of (61) is given by: (68)

u = us + u d ,

withu s = Lp', u d = F p .

Proposition 3 implies that u s and u d have the following properties: 8u

(69)

i)[u s ] r = 0,

[ ^ S] l r = P\

|ii)[u d ] r = P,

[-§if] r = 0,

^d,

since Aus + k 2u s = - p'6 r , and Aud ++ k 2 u d = - div (pn5r). Note that u s € H ^ R 3 ) thanks to (69).

3 STATIONARY SCATTERING PROBLEMS

Then we define the following surface integral operators L, K, J, R:

U>>=I*\rLp>\r,

V - K ^ + j ^

|Kp = Pp| r+ Pp| r ,

Fp-iMr-4lV|r •

(70) I

We also define the matrix operator S by:

<"> - t i t ) Then.we obtain the relations between the Calderon projectors and the operator S: (72) or:

P ^ G + S^P.-^O-SXS.Pj-P,,

(73)

P.^Pj.PZ-Pe,

P ± - £ < I * S ) , S = P_-P + ;

therefore S is also defined (using Proposition 3) by: (74)

S(p>.]=yiu + y e u,

and since P+ and P_ are projectors, S satisfies (74)'

S2 = I.

Thus S is an isomorphism in Y. This implies the following relations and properties between the operators L, J, K and R: i) the continuity properties are given by: (75)

H- 1/2 (r)^H ,/2 (T), H , / 2 (r)£ H"1/2(r) H -l/2 ( r ) J, H -l/2 ( r ) > H l/2 ( r ) K H i/2 (r)>

ii) the algebraic relations: (76)

KL + LJ = 0, RK+JR=0, K2 + 4LR=I, J2 + 4RL = I.

Other properties, useful for the limit problems, will be given below. We only note that these operators are (at least formally) given by: (77)

Lp'00 = / r P'(y) <Kx - y) dr y , Rp(x) = - / r p(y) ^ Jp'(x) = 2/ r p'(y)^(x-y)dT y ,

(x - y) dTy

Kp(x) = - 2 / r p ( y ^ ( x - y ) d r y .

3.1 WAVES DUE TO BOUNDED DOMAINS

1.3.2. Maxwell problems with currents on "regular" surfaces First we define the functional space: (78)

X 0 = {u€L?oc(R3)3, curl(u| ft )€L 2 (Q) 3 , curl(u| a ,)€L? oc (Q') 3 }

with notations (57). Then, from the trace theorem (see chap. 2), if u is in XQ, it has traces on each side of T: (79)

€H"1/2(div,r), or* r u€HT1/2(curl,r).

nAul r

*'c

i,e

We also have trace properties, if we define the trace mappings y., y f y., ye by: (80)

y ie u = n A u | r

Proposition 5.

andy. e = 7i r . c u.

The mappings y., ye, y. + ye, Yj-ye (resp. the same with tilde)

are continuous from XQ onto H ~ 1/2(div,D (resp. H " I/2 (curl,r)). Then we will use the product spaces X = XQ x XQ and (81) Y div =H- l/2 (div,OxH- 1/2 (div,r), Ycurl = H- 1/2 (curl,nxH- 1/2 (curi,r). Let u = (E,H) be an electromagnetic wave with locally finite energy in Q and its complement CT up to the boundary, u being discontinuous across the surface T. Therefore, thanks to formula (54), u = (E,H) is in X and satisfies: i) curl H + io)£ E = - [n A H] 6p, (82)

ii) - curl E + iwji H = [n A EL 8r.

Remark 2. Redundant matching conditions. Note that we have from (54): (83)

divE = - [n.E] r 5 r ,

divH = - [n.H] r 5 r .

Applying the div operator to (82), we obtain that the jumps of n.E, and n.H are related to the other jumps through: (84) since: (85)

icoe [n.E] r = divr ([n A H]r),

- top [n.H]r = divr ([n A E]r),

<div (v 6 r ), <» s - <v 6p, grad d» = - / v. gradr <»l dr =/ r div r v. 0dT = <divrv8r, 0>,

V*€Z)(R3),

3 STATIONARY SCATTERING PROBLEMS

for all (regular) tangent vectorfieldsv. Conversely, we have:

<8>

Theorem 3. The following so-called "transmission" electromagnetic problem: find (E, H) with locallyfiniteenergy (i. e.,EandH in L2loc(R3)3) satisfying I i) curl H + i(0€ E = 0

ii)-curlE + i<o)iH = 0, in ft and Q\

(86)

iii)[nAE] r =M r , - [ n A l f l ^ J p with Jr, Mr€H"1/2(div,r), iv) the Silver-Muller conditions at infinity,

is equivalent tofinding(E,H) with locallyfiniteenergy satisfying i) curlH + icoeEsJpSr ii) - curl E + icop H = MrSr in R , iii) the Silver-Muller conditions at infinity,

(86)'

and has a unique solution (E,H) with locallyfiniteenergy up to T, that is (E,H) is in X, and it is given by the convolution product (47), which is written {formally): E(x)=/r[i<o|i<l>(x - y)J,<y) - \ grad *(x - y)pc(y) - grad *(x -y) A M^yfldTy (87) H(x)=/r[ia>€^x - y)Mj<y) - £ grad <t>(x - y)pm(y) + grad <D(x -y) A Jj<y)]dry with:' (88)

ia)pe dir divr J p

iu>pm dJf divr Mr.

The formulas (87) are also called Stratton-Chu formulas. PROOF.

From Proposition 5, there is (EQ, HQ) in X, with a compact support, so

that (89)

[n A E 0 ] r - M r ,

[n A HJ r = - J r ,

thus there are J and M in L (R )3 with compact support, so that: (90) Let: (91)

curl H0 + ia>c E 0 = Jr 5r - J, - curl E 0 + io>ji H0 = Mr hT - M. EjsE-Eo,

Then(Ej, Hj) satisfies:

Hj = H - H 0 .

3.1 WAVES DUE TO BOUNDED DOMAINS

(92)

curlH,+ io)£E,=J - curl El + ki>|i Hj = M.

Thus from theorem 2, (E., Hj) is given by the convolution product (44), and we have (E,, H,) in X, thus from (91), (E,H) is in X. Then Theorem 3 follows easily. ® We define the mappings P{ and Pe by:

(93)

\ d = l - n A H , o , p i Jr J = "l- nA <J'

with (E, H) the solution of (86). We have: Proposition 6. The mappings P. and P are continuous complementary projectors inY^iseeW)). The proof is very easy, as in the case of the Helmholtz equation, see Proposition 4. The jump condition (86)iii) directly gives: (94)

VPe5*1-

Definition 3. The operators P{ and Pe are called interior and exterior Calderon projectors (for the Maxwell problem). Then if we denote by G{ = Yd. . and Gc = Y» e their images in Y«v, we have the decomposition of Ydiv (see (8l)) into the direct sum (but not orthogonal, since the Calderon projectors are not hermitian): (94)'

Y^-GiOGe-Y^eY^.

Electric layer and magnetic layer. We define two new operators in a similar way to the Helmholtz problem (see (67)), associated with the electric layer and the magnetic layer respectively by: I Lm ur * ikfaj^p + k"2grad div (ur Sp)) * 0 = ik (L ur + kf2grad L divr ur) \Pm ur = - curl (ur 6r * O) = - curl (Lur), 1/2

for all u r in H"

(div, T), or also for x outside T, (formally) by:

I Lmur (x) = / r [*(x - y) Uf<y) + k"2 grad <t> (x - y) divr u^y)] dTy, Pm ur (x) = - / grad 4> (x - y) A u^y) dTy.

3 STATIONARY SCATTERING PROBLEMS

Therefore the solution (E,H) of the Maxwell problem (86) is given by: (96)

|E-PmMr + ZItaJr .

(with Z given by (34)), or in a matrix form: (96)'

CHK *)(:)■

Then from Theorem 3, we see that for u r in H" 1/2(div,0, we have L m u r and PmuT in X 0 , and also: Proposition 7. The operators Lm and Pm have the following properties across T: [nAL m u r ] r = 0, [n.L m u r ] r = -j5div r u r , (97)

[nA/>mur]r = u r , [n.P m u r ] r = 0,

that is: the tangential component of LmuT is continuous, whereas its normal component is discontinuous, and the tangential component of Pmur is discontinuous, whereas its normal component is continuous. The tangential jump formulas are implied by the definition of the operators L and Pm, since for u r * J- with M r = 0, the jumps are obtained from the Maxwellequations (86)' and (86)ni). The normal jumps are due to formulas (84), with (34). ® PROOF.

Remark 3. The tangential continuity of L m u r implies that curl LmuT is in L2l0C(R3)3, and thus L m u r is in Hloc(curl,R3)3. Note that the following relations between Lm and P are satisfied: (98)

|curi/>mur=ikLmur-ur8r |curlL m u r = - i k P m u r ,

and that: (99) div Pm u r = 0,

div Lm u r « ^ (divr u r ) %.

We can prove these relations by simply applying the curl operator to (96) or (95) and then the div operator to (96) or (95).

3.1 WAVES DUE TO BOUNDED DOMAINS

Then we define using (80): (100)

Tur^-ty+y^/Vi,.,

Ru r = 2nAL m u r | r .

Thus (at least formally) I Tuj<x) = 2 / r n, A (Uj<y) A gra^ 4>(x - y)) dTy, (100)'

Rur = 2ik n A [Lur+-^ gradr L divr ur]. The traces of the electromagneticfield(E,H) solution of (86) satisfy (from (96)): (101)

l(y+Y e )E = - T M r + ZRJ r , . |-(Y i + Ye)H = - ^ R M r + TTr.

If we define the matrix operator S by: r -T

(102)

S=^.^R

ZR^

_TJ,

then we obtain:

we deduce the relations between the Calderon projectors and S: (104)

P ^ d + S), Pe = i ( I - S ) ,

S-Pi-P^

Since P. and Pe are projectors, S satisfies: (105)

S2 = I,

and thus S is an isomorphism in Ydiy. This implies that the operators T and R are continuous in H" 1/2(div,0, and also: (106)

T 2 -R 2 = I,

TR + RT=0.

Remark 4. If we consider transmission problems across a line in the complex plane for the Cauchy-Riemann operator, we also obtain an operator S which is the Hilbert operator, and which is very similar to the operator S defined by (102) for the Maxwell equations. ÂŽ

3 STATIONARY SCATTERING PROBLEMS

1.3.3. Some regularity results a) The Helmholtz case We assume that Q is a bounded domain with boundary T with convenient regularity. We define for real positive s the following spaces: X$ = X8rd#{ueD'(Rn), u| f t €H s («), ul^eHf^Q')}, (107)

X s = Xj<A)d^{u €D'(Rn), u| fl € HS(Q), Au| fl e HS(Q), u|Q.€Hf0C(Q'), Au|a>eH?0C<5')}. ~5

If u is in X and satisfies the Helmholtz equation in Cl and Q', then u is in X , and more generally, we have Amu € Xs and X s , for all m € N. Proposition 8. Let u be in Xs satisfying the Helmholtz equation in Cl and CT; then: u| r (108)

i,e

-l/2/] €Hi 5S—0T),

e H s-l/2 ( r ) >

Amu| i,e

! gdu, | r €H,s-3/2, (0,. " M,e

3Amu

ur'ni,e

6Hs-3/2(r>,

for interior traces andfor exterior traces (and then for the jumps). SKETCH OF PROOF. For s = 0 and 1, the proof is a consequence of the trace theorem in spaces H°(A,Q) ■ H(A,fi) and H*(A,Q), see Prop. 1. chap. 2. For s with 0 < s < 1, we have the result by interpolation, but we have to prove that:

(109)

[H1(AVQ),H°(A,Q)L = HS(A,Q) = {u € HS(Q), Au € L2(Q))>

s - 1 - 0.

We identify H°(A,Q) with the graph G0 of the Laplacian in L2, and H^A^) with the graph G{ of the Laplacian in H1 x L2. These graphs are closed subspaces in L2 x L2 and H1 x L2. The orthogonal space of G0 in L2(Q) x L2(Q) is the space: G 0 ± -Kf.g>.f--Aft fandg€L2(Q), g| r = 0, § | l r = 0} = H2(Q). Then we can prove that the orthogonal projector P0 in L (Cl) x L (O) on G 0 satisfies: P0(H!(Q) x L2(Q)) C Gj. Now (109) is a consequence of theorem 1.2.4 of Triebel [1]. Conversely we have: Proposition 9. Letpe HS(D, P' e Hs_1(r), then Pp e X s + 1 / 2 , Lp' € X s + , / 2 , withs ;> - 1/2. Thenfor(p,p')eHs(T)xHa-\r), wehave u^Pp+Ip'eX 5 * 1 ' 2 .

3 A WAVES DUE TO BOUNDED DOMAINS

PROOF.

We define, for all real positive s:

(110)

X s d J f {u€D'(R n ),

Then for all * in D(R\ H

3/2

u|Q€Hs(Q), u | n , € H s ( Q %

the mapping p —♦ *Pp is continuous from H

(T) (resp.

(0) into X! (resp. X 2 ). Thus by interpolation it is continuous from H s (r) into

pC 2 ,X 1 ] 0 = X a w i t h s = | ( l - e ) + | 8 , o = 2 - 0 . T h u s P p € X s + I / 2 f o r l / 2 < s < 3 / 2 . In a similar way we obtain the result for - 1/2 < s < 1/2, then Lp' € X s * 172 . Proposition 9 for half integer s, s > 3/2, is due to regularity results. ®

Remark 5. The single layer and double layer potentials also have the jump properties for every integer j : I^PtaJr-C-k2)!^,

[|jA j Pu o ] r = 0 ,

(HI) [Aj Lu!]r = 0 ,

[^^LuJj.-C-k^Up <8>

We define the space, for real s: (112)

Y s = Y s (r) d # H s + l / 2 (r)x

HS-1/2(0.

Then Proposition 9 implies: Proposition 10.

Let (p,p') be in Y5, s a positive real. Then the solution u of the

Helmholtz transmission problem (61) has its traces y.u, yeu in Y*; thus the Calderon projectors P. and P e (see (65)) act in Y8, a/irf the space Y8 /IOS f/ie decomposition: (113)

Y s = Y?0Y^,

w i t h ^ = PiYs,

Y^P;**.

Remark 6. By duality P. and P e (see (65)) also act in Y8 for negative real s, and then decomposition (11 J ) is true for all real s. <8>

Therefore S defined by (71) acts in V s for all real s and then we have the fbllowing properties for the operators L, R, J, K which is a generalization of (75) (114) H S (H - i HS(H, H s (0 -£ H s (f). <8>

3 STATIONARY SCATTERING PROBLEMS

b) The Maxwell case Regularity results for Maxwell equations will be obtained as a consequence of the above properties. We define the following spaces for s real, with (128) chap. 2: (115)

I X^ ri tf{u e (X s )3, with (107), curl u| Q € H8(G)3, curl u| f f e H^C(Q >)3},

I^^H^di^OxH^div.D. Then we have: Proposition 11. Letj € Hs(div,0, s > - 1/2; then L J and P m j€(X s * 1/2 ) 3 . PROOF.

From hypothesis we have j € HS(D3, divr j € HS(T).

Applying Proposition 9, we obtain Ij € (Xs*372)3 and L (divr j) € X8*372, thus: grad L divr j € (Xs*"2)3, and from (95) we have L J and P J € (Xs*172)3. (8) The case s » 1/2 gives H1 regularity. From Proposition 11, we have: Proposition 12. Let (Jr, Mr) € Y^iv; then the solution u = (E, H) ofthe problem (86) (or(86)*)*arfr/ks: (116)

u = (E, H) € (Xs + 1 / 2 ) 3 x (Xs*172)3.

Proposition 13. The Calderon projectors P. and Pc defined by (93) (relative to the Maxwell problem) operate in Y 8 ^ for all real positive s, and by duality for all reals. This gives the decomposition: (117)

^ d i v - Ydiv, i^ Y div, e» W** Y div, i = ^Ydiv> Ydiv, e * P e Y div •

As a consequence we have: Corollary 1. The operator S defined by (102) operates in Ysdivybr all real s, and the operators T and R (see (100)) operate in Hs(&\vJ)for all real s. 1.3.4. Integral method for a sheet a) The Helmholtz case Let To be a "regular" compact sheet, that is a regular part of the boundary T of an open set Cl in R3.

3.1 WAVES DUE TO BOUNDED DOMAINS

We recall (see chap. 2) that the elements of H^r,,) and H" 1/2(ro) extended by 0 -1/2/, on I\r 0 are elements of HA12,( 0 and H" "*(T) respectively. We define: (118)

xj. ^{ueOXR^CueHVRVo^CAueHVRVo), VCcZXR")).

If the tilde denotes the extension on T by 0 of an element defined on ro, then Proposition 9 implies Proposition 14.Letpe Hl0'l(T0), p' eH"1/2(roX thenPp^PpandLp'

= f Lp'

are in xj. with Lp' e H^R11 ). Corollary 2. Letpe H„/2(ro), p' € H~1/2(ro), then the problem:findu with locallyfiniteenergy in K\T0, i.e., u e HJ^R'Nr,,), satisfying li)Au + k2u = 0 i n R V o , (119)

ii)[u]r =P.[§Hl r =P', iii) the outgoing Sommerfeld condition at infinity,

has a unique solution u in X r , is given by u = Pp + Lp\ and the traces satisfy: o

(120)

u = (u,|S)|

€H,/2(ro)x(Hi/2(ro)>'-

andYr u = ( u , g ) |

Remark 7. We note that the mapping Sr : (p,p') —> yr u + yr u is continuous and injective from H^(r o ) x H~1/2(ro) into Hl/2(ro) x (H£/2(ro))\ and related to the operator S for T by: Sr (P,PJ) = S(p,p')L . The operators Pr *o

defined by Pr (p,p') = Yr u, Pr (p,p') = Yr u, satisfy Pr l

O+

'o+

'o-

, Pr O+

o±

O-

= A (I ± S r ), £

but they are not projectors! With the notations Kr p = (Kp) | _ ,..., these mappings ° 'o are continuous in the spaces: (121)

H^(r 0 ) - ° H1/2(r0), Hl0n0(T0) - ° (H'/2(r0)>)

H - 1/2 (r 0 ) - ? H 1/2 (r 0 ), H - ,/2 (r 0 ) - ? (ul0'20(T0)y and then we have (122)

r Kr» 2Lro ^| l ro Jr0J

STr o== 2 R

<8>

3 STATIONARY SCATTERING PROBLEMS

b) The Maxwell case Since the elements of H~,/2(div,ro) can be extended by 6 out of r o into elements of 1/2

H" (div,0 (see (85) chap. 2) we have from Proposition 7: Proposition 15. Ieru r be in H~m(divfT0), then Lmur andPmuT are in the space H/0C(curl,R\ro). Proposition 16.Let(JT , Mr )€H; 1/2 (div,r o )xH; 1/2 (div,r o ). Then the problem: find (E, H) with locallyfiniteenergy on R\T0, satisfying i) curl H + io)E E = 0, -curlE + tojiHsO i n R \ r o , (123)

ii)[nAHL « - J r , [nAE] =M r iii) the Silver-Muller condition at infinity,

has a unique solution given by (96) in H/oc(curl,R3\ro) x H^curljR 3 ^). We denote by y r , yr the trace mappings: yr

(E,H) = (n A E, n A H)|

each side of r o . We have: Proposition 17.

77ie mapping Sr : (Jr , Mr ) -> yr (E,H) + yr (E,H)from *0

1/2

*o+

1/2

H; (div,r o )x H; (div,r0) into X(div,ro) x X(div,r0) with X(div,r0) defined by (86) chap. 2, is related to the mapping S by: (124) S r (Jr ,M r ) = S(Jr ,M r ) | . With the operators Tr , Rr from H; l/2(div,ro) into X(div,ro) such that: (125)

Tr u r =T(ur )|_ , Rr u r =R(ur )| ,

we also have: f

-Tr

and the operators Pr (127)

ZRr "\

defined by Pr

(Jr , Mr ) = yr

= i ( I ± S r ) , with ImPr

(E, H) are also given by:

Q X(div,ro)xX(div,ro).

3.1 WAVES DUE TO BOUNDED DOMAINS

1.3.5. Incoming waves In scattering problems we often disregard incoming waves. However they intervene at different stages, involving holomorphic properties with respect to the wavenumber k: for the spectral resolution of the Laplacian, like in the Limiting Absorption Principle (see sections 3.2, 3.4, and Wilcox [1]), or for resonance states (see chap. 6.2), or for inverse problems (to find the source of a wave from its behavior at infinity). Incoming waves also appear as conjugated to outgoing waves. For the same evolution in time (here given by e~1(ut), the incoming solution of the Helmholtz equation (4) (resp. Maxwell equations (23)) is given by: (128) u t e » * ^ , f (resp. E ^ O ^ j , Hin = Oin*f, with(27)). The relations with the conjugated functions are given for real k, e, y by: (129) uout(f) = u^f), Eout(M, J) = - E ^ - M, J), Hout(M, J) = l A - M, J), and the behavior at infinity of incoming waves is given by (130) u^ra) = O^r) f(- ka) (thus E^ra) = O^r) j (- ka), H^ra) = O^r) m(- ka)). Since 4>m = O, the integral operators for the Helmholtz equation K, J, L, R and S = S out (see (70), (71)) are changed into their conjugated, thus S m = S o u \ In the Maxwell case, the integral operators T=T° ut , R = Rout = ikRo, and S = S om (see(100)\ (102)) are changed into T ^ f , Rin = - R = ikR^, and:

iit^K'-U^h'--^

(131) Sfa =

Note that in the decomposition (66)' of Y (or (94)' of Y ^ , the interior space Gi does not depend on the outgoing or incoming condition, contrary to Ge and the Calderon projectors. Thus we have the decompositions (132)

Y(rep.Ydiv) = G i ®G° u t = G i ©G^, with G° ut = Ge,

associated with I = Pjn + Pf = P?ut + P° u t , with P?uel = Pie. Thus (133)

Gj = ker (I - pi") = ker (I - P?ut).

This implies the following relations: (134) p; n .p° u t =o, P ^ . P J ^ O . Then: (135) P'n.(P?ut - PJn) = P°ut.(P?ut - P?1) = (P°ut - pfypf" = ( P ° u t . pf).pj> = 0, thus

3 STATIONARY SCATTERING PROBLEMS

<135)> i f .(Sout - S h = P°ut.(Sout - Sta) = 0, (Sout - Sh-Pf*- (Sout - Sta).Pf1 = 0. and thus: (136) (Smt - S*1)2=0, (21 - (Sout + S111)) (Sout - S111) = 0. Therefore: (137) S^S01" -i- S^.S111 = 21, S^.S01" - Sout .Sta - 2(Sout - S"). Of course, this gives relations between K, J, L, R (resp. T, R) and their conjugated operators. Relations (135)' imply: (138) Im(Sollt - S*1) c G j C ker(Som - S^). Remark 8. We see that the operator S om - S™ have a leading role in all these questions. This is due to the following properties. Let $ e n be the entire solution of Helmholtz equation defined by:

(139)

* e n 4(* 0 U t -* t a )=^(*- * ) = ^ , r=|x|.

Then for all sources f (or (M, J)) with compact support, the function U cn = $ e n , f is an entire solution of the Helmholtz equation (resp. E cn « ♦ , j , H « # , m for Maxwell equations). When f is given as in (61), U en tends to 0 at infinity as 0( 1/r), and is decomposed from the surface T into the sum of an incoming wave and an outgoing wave. Furthermore its traces on T are given by

[^]4<pr-^)(pp04^-pr)(pp04(sr-sr)(Ppo with a similar result in the Maxwell case. Now we can also obtain this function U from the behavior at infinity of the solution u of (61) given by (IS). Let: (140) U«x) =*/s2 f(ka)e**'*da- JVj<f, +***><P**da .<f, f$2e**ix~*da>. Nowwithx = r0, (141)

/s2e^^d«=/^eitopd«=2«/1eik™du=4n5i^=^*en(r).

Thus Uf = (4it)2 k~1 Uen, with a similar result in the Maxwell case.

LetG•*en be the space of traces: (142) G ^ - R U J , * ^;Pl r )» U cn = $ cn ,f, f=p% + div(pn«r)> (p, p')€ Y(see(59)}. Then G**1 is regular (C00 if T is C00, analytic if T is analytic) and contained in G|. Proposition 18. The operator (S out - S"1) is a regularizing (thus compact) operator with the essentialfollowing properties (in the Helmholtz and the Maxwell cases), with (Smt - S™)2 = 0 ker(S 0Ut -S in )«G i , I m ( S o u t - S ^ G * 1 -if^O^-Pf^Gj 1 )isdenseinG { . Moreover (Sout - Sm) is a one-to-one mappingfrom G? and G^ut onto G**.

3.1 WA VES DUE TO BOUNDED DOMAINS

We first prove that Gf flG°ut = {0}. Let ( v ^ ) e G^nG°Mt. Let u be the solution of the exterior Helmholtz problem with (u | _, g—| ) = (v0, Vj). PROOF (for Helmholtz).

Then the behavior of u at infinity is (up to o(l/r)) given by *(r)f(ka) and <D(r) f(- ka); thus u(ra) s o(l/r); hence u = 0 in Cl' thanks to the Rellich Lemma, thus (v0, v{) = 0. i) Let U = (UQ, Uj) € ker (Sout - S*). Thus (P* - P°ut) U = 0; hence F? U = P°utU is in G*nG° ut = {0}. Thus U € G^ ii) Density of Gcn in G.. This result will be obtained by duality. a) The Helmholtz case. In Y=H 1/2 (DxH"" I/2 (0 we introduce the duality pairing (so that we can identify Y with its dual space Y') by (143)

<( J°i ) , ( $ )>y, Y = <u0, vl> - <u1, v°>,

where <,> denotes the duality H1/2(r), H"1/2(0. b) The Maxwell case. Now in V = H" 1/2(div,D we introduce the duality pairing (so that we can identify V with its dual space V = H ~ 1/2(curl,0) by: (144) <u,v> = <u,v> vv = <u,nAv>y v ,=/ r u.nAvdT, Vu, vinV. Then we define the duality pairing in Ydiv (see (81)) (so that we can identify Ydiv with its dual space Y9 = Ycurf) by

(145)<[ Y ] , f *Jo ]>Y,Y = <M, J°> + <J, M°> = <M,nAJ% v + <J,nAM°>vv.. Now if (E^H1), (E2,H2) are two solutions of Maxwell equations (both incoming or outgoing in Cly), we have on each side of T, using Green formula: (145)' / ^ ( n A E ^ t f + nAH^E^dTsrO. Taking the difference, we obtain with the notations rmE = (y. + YC)E (see (103)),... / r {ymE^[H2] + ymH^[E2]. + [n A E ^ m H 2 + [n A H1] ymE2) dT = 0; thus using (103), the pairing (145), (144) and taking Mj = [n A Ej], Jj * - [n A Hj]

Thus in the Maxwell and the Helmholtz cases we have: X (146) S = - S and thus: % = P{, % = Pe, with S = S* or Sout andtL = L,tR = R,tK = -J, t J = -K, and also: l T=-T, lR = -R. Now (146) implies that the polar set of Gen = Im (Sout - S"1) is ker (Sout - S"1), i.e. G., and the polar set of G. is again G.. This proves the density of Gen in G..

3 STATIONARY SCATTERING PROBLEMS

Remark 9. We can also use an antiduality pairing by (144)* <u,v>=<u,nA ?>yy, = / u.nA ? dT, giving: (146)' S = (Sout) = - Sm, and thus T = (T°ut) - - 1", R - (R om ) = - R m . These formulas are again obtained using Green formulas in Q and Cl\ but with $ two waves, one incoming and the other outgoing. 2 . SCATTERING PROBLEMS WITH A COMPLEX WAVENUMBER k

Except in the free space case, physical media are dissipative (see chap. 1 section 3.1.6, with the H14) hypothesis). Of course dissipative media are bounded but in a first step we assume that such a medium occupies the whole space. This allows us to define the Calderon projectors and then to tackle physical problems in free space by the "limiting absorption principle". 2.1. Helmholtz equation in R3 with a complex wavenumber We again consider the Helmholtz equation (1) with a complex wavenumber k. Proposition 19. When Im k > 0, the Helmholtz equation (I), where f is a distribution with compact support, or a tempered distribution in R3. has a unique solution u in the space of tempered distributions, with u in H2(R ) when f is in L2(R3). PROOF.

(147)

Using a Fourier transform method we have: (-5 2 + k2)u = - f inR 3 .

Since(-5 2 + k 2 )*0foraUUR 3 ,with(-e 2 + k 2 )" l -^0when|£| — oo, wecan divide by (- S2 + k2), thus obtaining: (148)

= - (- £2 + k 2 )'* \ € S *(R3)

with fi € L2(R3) when ? € L (R ). The inverse Fourier transform is the solution u of (l)inS'(R 3 ); forf inL2(R3), we write: (149)

u = RAf,

withRA = (-A-X)" 1 , X^k2;

Rx is the resolvent of the positive selfadjoint operator ( - A) since \ is not in the spectrum of ( - A). When f is in S '(R3), we also have by convolution (150)

u = <l>k,f,

w i t h ^ ^ - Z - V - ^ + k2)"1, O k ( x ) = ^ , r = | x | ;

k is the tempered elementary solution of (1). With Im k > 0, it is exponentially decreasing with (high) r; then for f in E '(R3), u satisfies, with n = Im k:

3.2 COMPLEX WA VENUMBER

(151)

u(x) =

0(|SF),

(g-iku)(x) = o ( ^ F ) ,

whenr-oo.

There are no conditions at infinity to impose on u: Sommerfeld conditions appear when X tends to a real number with Im X positive or negative. This is easily seen on the elementary solution: Lemma 2. The elementary solution of the Helmholtz equation satisfies \^<b^'mD'(R3)oTS\R3)when\ (152)

k2^]^withlm\>09

O k ^ O ^ = *_^in^R 3 )or5XR 3 )wtenX==k 2 ^k^wir/fImX<0.

Then using the continuity properties of the convolution product (see Schwartz [1] p. 157 and 247) we obtain: Proposition 20. Limiting Absorption Principle in R3. When the complex number k tends to a real value k , with Im k2 positive (resp. negative), the solution u. of the Helmholtz equation (1) (with f a distribution with compact support) tends to the solution of the same equation for kQ with the outgoing (resp. incoming) Sommerfeld condition in S '(R3); thus (153)

ukd=fk*f—u.

= f O ko *fin5 , (R 3 )whenk-»k o withImk>0.

When fis in L2(R3) with compact support, (uk) converges in H2loc(R3). Obviously we cannot have Rxf-> RA f in L2(R3) for all f € L2(R3) when X0 = k2 is in o(- A), the spectrum of operator (-A). PROOF of Lemma 2. As an exercise, we prove (152) by a Fourier transform method, to specify the Fourier transform of the elementary solutions. For all * in £(R3), we have: (154) < ^ , < > > = / R 3 ? l - ? <K5)d*=/R+xs2 - i - j ^ ^ d p d a ,

with p = 11|, Im k * 0. Let 6(p) be defined on R by: (155) Then:

9(p) = p 2 M«(|p|), withM«(p)=/s24>(pa)da, p>0.

(156)

<<& k ,*>4/ R i^dp.

The limit for k —* k0 with Im k > 0 or Im k < 0 is given by the Plemelj formulas with respect to the real axis in C (see Dautray-Lions [1] chap. 11A.2; we could also use the variable t = p2, and the half-line R* instead of R). When k0 * 0, we have

100

(157)

3 STATIONARY SCATTERING PROBLEMS

k . v fco, t >o

^'*

A >=<

>= 1 pv

9(0)

+ in91 k

\-* i 4^*> i < o).

and 9(P) (158) k - n , - l ! % , . > o < ^ » # > - < * t ' * > - ^ ^ « p r^*-2tw<W Thus (159)

<\>*>

= <PV-T*TZ ,♦> + !*< 8G 2 -!£),♦>, 5 -k;

and )' VO-pv^r+it^-k^). (159)' r - ksolution ; The incoming elementary <&]£ is given by: K in

(160)

* j£«) = P v p - U " * *« " >£>•

PROOF of Proposition 20. Lemma 2 implies (153), and thanks to (158)', the Fourier transform of Uj. is:

(161)

\(i)=\(Qte>=Pvp-Trfto+fc**1- $ ««>• r-»5 r2 3.

n Note that this implies when f € L /(R ) with bounded support: Re ( u v f ) = (2n)"3 Re (fl^, f) = (2n)"3 P V / R 3 - J -1^ - J i^«|2 J f«)| dS, (162) I m ( u v 0 = (2n)-3Im(flko, f) = ( 2 « r 3 ^ / s 2 |t l (k o a)| 2 da>0.

Now we prove the convergence of (uk) in H20C(R3). Let Ba be a ball which contains the support off. We have: (163)

(u k -u ko Xx)=/ B

tlx')dx',

Fork = ko + in,withn>0,wehave: \-—^—| ikr

r=|x-x'|, s|

r"'|

^n.thus:

^o^

(164) |(u k - Uko Kx)| 2 ^C/ Ba Ie 4~f

| 2 |Rx')| 2 dx^Cn 2 / B a |«x')| 2 dx\

hence uk —»Uv in L (R ), and in Lj0C(R ). Thus: A (uk - u^) converges to 0 in L^R 3 ), hence in LJ^R3), since A (uk - u^) = - k 2 ^ - u^) + (k2, - k2) u^. Furthermore: (165)

grad(u k - U k o )=/ B a^ffcOdx',

3.2 COMPLEX WA VENUMBER

101

ikr

**o**

with: a = £ ^ £ , and <Kr)=V <*-?>-

(166)

V - ^ - T ) -

But we have

Kr)|<[ 1 ~ e " V f J T ! r e T *'-g^i^—]« n 0(l),

thus grad (uk - u^ ) ~> 0 in L°°(R3), hence in Lfoc(R3). 2.2. Maxwell equations with complex coefficients Proposition 21. Let <D (or c, or ji) te complex, with Im k 2 * 0 . TAe/i the Maxwell equations (22) wil/i J o/wf M given I/I E '(R3)3 or i/i S '(R3)3 have a unique solution (E,H) with E, H in S '(R3)3. For M, J in L2(R3)3 (re$/>. in H(div,R3)) we have E, H I/I H(curl,R 3 ) (resp. in H*(R3)3). When a> tends to a real mQ with Im a> > 0 (resp. < 0), and M, J are in L 2 (R 3 ) 3 with compact support, the solution (E,H) = (E ,H w ) of (22) converges in 5*(R3)3 x 5'(R 3 ) 3 to the outgoing (resp. incoming) solution (EQ,H0) of (22) for mQ and EQ, HQ are in Hloc(rot,R3). PROOF. There are two ways to prove this proposition either by a Fourier transform method, or by convolution like in section 1.2.1 reducing to the Helmholtz equation (see (28)') with j and m in E '(R3)3 (or in H " 2 (R 3 ) 3 ). 8

2.3. The Calderon projectors for a complex wavenumber k We consider the "standard" problems (61) in the Helmholtz case, and (86)' in the Maxwell case with charges concentrated on a regular surface T as in 1.3.1. Proposition 22. Let w (or e, or y) be complex with Im k 2 * 0 . Then Proposition 3 in the Helmholtz case and Theorem 3 in the Maxwell case are also valid without condition at infinity if we replace the space X by the space: i) Helmholtz case: X = {u€D9(R\ (167)

€H*(A,Q), ul

CH^O')}

ii) Maxwell case: X = X 0 x X 0 , with: X 0 = {u€D'(R 3 ) 3 , u| Q €H(curl,Q), u| a ,€H(curl,n'), divu| n €L 2 (Q), divu| Q I €L 2 (Q%

Then this allows us to define the Calderon projectors P. and P like in (65) and in (93) and thus the operators S (see (21) and (102)) with thee same properties. Furthermore we have for instance in the Helmholtz case as a consequence of the limiting absorption principle:

102

3 STA TIONARYSCA TTERING PROBLEMS

Corollary 3. When the complex number k tends to a real value kQ, with Im k2 positive {resp negative), then (168)

Sk( J . j - S j J £,] (wp.s£( £,])inH1/2(OxH-,/2(T),

and thus the corresponding Calderon projectors and the integral operators Kk, 1^, RR and J k strongly converge in their naturalfunctional spaces. PROOF. Let u = u k be the solution of (61) with k a complex number with Im k > 0. We use a lifting U of the jump conditions (63), with U in H^AjQ) and in H ^ A , ^ ) with compact support. Then vk = u - U satisfies: (169) Av k +k 2 v k =F,

withF€L 2 (R 3 )H£'(R 3 ).

Then v k =R^F converges to a limit denoted by R%F thanks to Proposition 20.

3 . VECTOR HELMHOLTZ EQUATION. KNAUFF-KRESS CONDITIONS

Using the (vector) Laplace operator A (as in chap. 2 section 11.2) instead of the curl curl operator in electromagnetism leads to some differences in the scattering theory by integral methods. We again assume that k is a real positive wavenumber. 3.1. Knauff-Kress conditions at infinity Let f be a given vector distribution with compact support in R3; then let u be the unique (vector) solution of the Helmholtz problem (13) with Sommerfeld conditions. Its behavior at infinity is given by (IS); then we have: (170)

curl u = 0 * curl f,

divu = **divf,

and thus their behaviors at infinity are given by: (171)

(curluXra) * *( r ). ika A ?(ka),

(divuXra) » 0(r) ika.f(ka);

then we obtain: (172) (adivu-aAcurlu-ikuXro) » <D(r)ik[- f + a ( a . f ) - aA(aA?)]=:0. Thus u s E (or H) must satisfy the following, uniformly in a: (173)

(adivu - aAcurlu - ikuXra) = o(y) whenr-*<».

These conditions are called Knauff-Kress conditions at infinity.

103

3.3 VECTOR HELMHOLTZEQUATION

3.2. Vector Helmholtz problems with jumps conditions Let u be a solution of the (homogeneous) vector Helmholtz equations in Q. (a bounded open set in R3) and its complement Q\ We suppose that u satisfies (174)

u | a € H\C1)\ u| ft , € H ^ Q ' ) \

with the Knauff-Kress conditions at infinity. Recalling the jumps formulas (53), (54), and then applying the curl and div operators to these formulas we obtain, with the same notations: (175)

curl curl u = (curl curl u) - [n A curl u] &J- - curl ([n A U] 6r), grad div u = (grad div u) - [div u] n5 r - grad ([n.u] 6j.).

With the usual relation: curl curl u = - Au + grad div u, we obtain that u satisfies inDXR3)3: (176)

Au + k2u = g, g = [n A curl u] f 6r - [div u] r n6r + curl ([n A u] r 6r) - grad ([n.u]r 8j-).

Then u is of the following form: (177)

u = -*g.

This formula is analogous to the KirchhofT formula (62) or the Stratton-Chu formula (88). We give some mathematical frameworks for these jumps: i) In the "regular" case (based on the H2 regularity): (178) [nAu] r €H t 3/2 (0, [n.u] r €H 3/2 (H, [nAcurlu] r €H t 1/2 (r), [divu] r €H 1/2 (r). ii) In the "variational" case (based on the H1 regularity): (179)[nAu] r €H t 1/2 (0, [n.u] r €H 1/2 (H, [nACurlu] r €H- 1/2 (0, [divu] r €H" 1/2 (r). Note that the space Ht1/2(0 and its dual space HJ" U2(T) are so that: rl/2,

r-1/2,

1/z f-1/2/. Ht1,z(r) = H- 1,z (div,r)riH- 1/z (curl,r), Ht"1/2(n = H(div,r) + H- i/z (curl,D.

j-1/2,

l/2/i

Of course we can develop the integral method in these frameworks and define Calderon projectors. We can also consider complex wavenumbers k. Then we have the analogue of Proposition 19 in a domain Q.

104

3 STATIONARY SCATTERING PROBLEMS

4 . BOUNDARY PROBLEMS WITH REAL WAVENUMBER k

In chap. 2, section 12, we first studied some boundary problems which are "coercive" or "elliptic", that is, they may be solved thanks to a variational method and to Lax-Milgram lemma. This is possible for pure imaginary wavenumbers k and more generally for complex wavenumbers, but the case of real wavenumbers is more difficult. We can study it from various methods; here we develop the method based on the limiting absorption principle. 4.1. Limiting absorption principle We first consider "exterior" problems, i.e., in the complement Q* of a regular bounded set ft in R3. We assume that Q* is a connected set. Either in the (scalar) Helmholtz case with the Neumann (or also Dirichlet) condition or in the (vector) Helmholtz case, these problems can be written as: (180)

Au-Xu = f,

where A is a positive selfadjoint operator with a continuous spectrum only and X is in the spectrum of A. We specify in the scalar case, then in the vector case, with the definition (3) chap. 2: i)A = - A , withEK^ÎucHVA^XulÔor^lÔ}, ii) A = A" defined in (147) chap. 2. When f is a square integrable function with bounded support we have the following result, for which we refer to Wilcox [1], Bendali [2], Hormander [2]. (180)'

Lemma 3.

Let RQ , R be real positive numbers RQ < R with Q, contained in the

ball B Ro . Let 0 < a < b, a > 0, J a [a,b] x (0,o]. Then there exists M » M(J ,RQ, R), M > 0, so that: (181) IIR(z)fll , <Mllfll 2 , V f € L t o , s u p p f s Q ' R , V Z € J , l L H (*&R) <°') ° with R(z) = (A - zl)~l the resolvent of A, and with: Q'R = Q' n BR. Moreover the mapping: z € J —»R(z)f € H/^A,^) is uniformly continuous. Theorem 4. Limiting absorption principle (scalar case). The following problem: flndu in Hloc(A,Q'), satisfying, with k real and given f with compact support: i)Au + k2u = - f inQf = R \ Q , (182)

feL^ni^R3)

ii)u| r = 0 (resp.|j| r = 0 ) , iii) the outgoing Sommerfeld condition,

105

3.4 BOUNDARY PROBLEMS

has a unique solution u+, given thanks to the resolvent R(z) ofA= — A by: (183)

u* = R£f =

lim

R(z) f in H/0C(A,Q').

Obviously the theorem is true if we substitute incoming for outgoing and - 0 for 0 in (183). In the vector case we have: Theorem 4 \ Limiting absorption principle, vector case. Thefollowing problem: find u in Hj0C(A,Q')3, solution ofthe Helmholtz problem with k real and given f

(182)'

liJAu + k ^ - f uiQ', f€L2(Cl9?nE'(R3)3, u)nAu| r = 0, (divu)|r = 0, iii) u satisfies the KnauffiKress conditions (173),

has a unique solution u + which is given thanks to the resolvent R(z) ofA -A" (183)'

u+=

lim

R(z)f inH{0C(A,Qf.

z = lr + iaf o>0, a—»0

PROOF, i) Wefirstprove uniqueness in the scalar case. Using the Green formula on the domain Q'R = Q? n BR, we have (with f=0) (184)

(|gradu| 2 -k 2 |u|2)dx)

Re(-ikf

= R e ( - i k / r g u d T ) - R e ( - i k / s g u d S R ) = 0. Then using the Sommeifeld condition we have (185)

Re(-ik/ s HudS^^kVg |u| 2 dS R + o(l) = 0.

We conclude using the Rellich lemma. ii) In the vector Helmholtz case we can develop a similar proof, using the Green formula with: (186) Re(-ikf t (|curlu| 2 +|divu| 2 -k 2 |u|2)dx) = 0, Q

and then using the Knauff-Kress conditions and the Rellich Lemma. iii) The existence of the solution follows easily from lemma 3: uz « R(z)f tends to a limit u+ when z —♦ k2; u2 satisfies all the properties of the theorem but the conditions at infinity. These conditions are deduced from the KirchhofTfonnula (obtained from (62) and (177)) which gives the solution uz thanks to its traces on a sphere SR (successively in the scalar then in the vector case):

106

3 STATIONARY SCATTERING PROBLEMS

i) u(x)=/SR(u(y) §^(x - y) - |M><x - y)) dTy, (187)

ii) u(x) = - \fs (n A curl u + n div u) <&(x - y) dSR(y) + curl/ s nAiiy*(x-y)dS R (y)+grad/ s n.Uy*(x-y)dSR(y)}, R

with u = u and * « * 2 (in order to simplify). Passage to the limit in these formulas wnen z —» k2, gives the corresponding (Kirchhof!) formulas for k2 that is u + satisfies the required condition at infinity. 4.2. Exterior boundary problems Then we can solve some "usual" exterior boundary problems with inhomogeneous boundary conditions: Theorem 5. ExteriorscalarHelmholtzproblem. The following problem: find u in H/0C(A,Q') satisfying the Helmholtz problem with k real and given u0, or u{ i)Au + k2u = 0inQ', (188)

i0u|r-uo€HI/^(iwp.g|r-u1€irl/2(n)f iii) the outgoing Sommerfeld condition,

has a unique solution. PROOF. Uniqueness is proved in Theorem 4. In order to prove existence of the solution, we use a lifting to the boundary conditions obtained by the (variational) solution U in H^Q') of the problem (188) when k = i (for instance). Let C be in D(K3) with C(x) = 1 in a neighborhood of r. Let f be defined by: f = [2 grad C .grad U + U (At + (1 + k2) C)], then f is in L2(Q*) with compact support,

Thus v=u - CU is in HJ^.CA.Q'), and satisfy: |i)Av+k 2 v=-f, (189)

|ii)v|r-0 (resp.g|r = 0), iii) the outgoing Sommerfeld condition.

Then the limiting absorption principle gives the result. 0

Theorem 5 \ Exterior vector Helmholtz problem. The following problem:findu in r W satisfying the Helmholtz equation with k real and with given (m,g) Hl0C(A,Q')

107

3.4 BOUNDARY PROBLEMS lOAu + k ^ O inQ\ ii)nAu| r = m,m€H t 1/2 (0, (divu)|r = g,g€H- 1/2 (r),

(190)

iii) the Knauff-Kress conditions (173), has a unique solution. This is similar to the scalar case: uniqueness is proved in Theorem 4'. Existence is proved thanks to a lifting of the boundary conditions, which is the (variational) solution of (174) chap. 2 and then applying Theorem 4'. PROOF.

Then we can solve exterior Maxwell problem in the free space: Theorem 6. Exterior Maxwell problem. Let c, \nbe positive real numbers. The problem:find(E, H) satisfying, with a given m i) curl H + icoc E = 0, ii) - curl E + io>ji H = 0 in Q\ (191)

iii)nAE| r = m, m€H"1/2(div,0> iv) the Silver-Mutter conditions (43) at infinity I v) (E,H) € Hloc(curl,Q') x Hloc(curl,Q'),

has a unique solution. PROOF, i) We first prove uniqueness directly for zero boundary conditions. Using the Green formula on Q'R, like in (48), with the Maxwell equations, we have: Re(/ Q'

(192)

div(EAH)dx) = Re(/Sc n. E A H d S R - /l r n . E AHdT) R

[icop | H | 2 - i<oe lEl^dbO^O.

= Re(/ a

Since we have /_ n. E A H dT = 0, thenS / n. E A H dSR = 0. Thus using the Silverr R Mu Her conditions (43) we obtain: (193)

|* r H| 2 dS R = / s

f$ K

|n r E| 2 dS R = 0. K

Since we also have (50), then: / s Rellich lemma and we have E

| H | 2 dSR = / s

| E | 2 dSR = 0. We can apply

A-o.

ii) The existence of the solution is proved thanks to a lifting of the boundary condition. Using the solution (u,v) of problem (151)' chap. 2 with K = 1, and a regular function £ with compact support equal to 1 in a neighborhood of T, we define:

108

3 STATIONARY SCATTERING PROBLEMS

(194) E = E-Cu, H = H-tv. Let(J,M)be: J = (-io)£ +l)Cu-(gradC)Av, Then (E, H) must satisfy:

(195)

M = ( - i<o|i +1) Cv+(grad C) A U.

| i) - curl E + ioji H = M, ii) curl H + iwt E = J infl', Ui)nAE| r =0, iv) the SUver-Miiller conditions (43) at infinity, I v) E and H e Hl0C(curl,Q')-

We note that M and J are in H(curl,Q') with bounded supports. Furthermore, if T is smooth (C00): (196) divJanddivMeHs(Q')( VseR; E must satisfy the vector Helmholtz problem, with f given in L(£}') flF(R) :

(197)

|9AE + k 2 E - - f inQ\ ii)nAE| r = 0, divE| r = 0, iii) the Knauff-Kiess conditions (173) iv)E€Hloc(curl,Q').

1 Since divE=r^divJ €L2(Q^ with the boundary condition ii), then E € Hjoc(Q')3. Thus we can apply the limiting absorption principle (theorem 4'), which proves the existence of the solution of the Maxwell problem. & Remark 10. An application to scattering. Solving the boundary value problem (191) amounts to solve the following scattering problem: given an incident electromagnetic wave (EJ,HJ) on a (regular) bounded obstacle Q which we assume to be a perfect conductor,findthe reflected wave (E .H ). Since the total electromagneticfield(Et,Ht) (the sum of (EpHj) and (E^H^)) must satisfy, on the boundary T of Cl: n A Et | = 0, then the reflected wave (Er, HJ must be a solution of problem (191), with: n A E r | = m = - n A E j | given. We give below (see 4.8) another method (a surface integral method) to compute this electromagnetic field. # 4.3. Some consequences; the exterior Calderon operator We note that in the Maxwell problem (191), the magneticfieldH also satisfies (198)

nAH| r €H- |/4t1/2/. (div,r)

109

3.4 BOUNDARY PROBLEMS

Definition 4. The exterior (outgoing) Calderon operator. The mapping: (199)

C e (orC^): m = n A E | r ^ n A H | r

with (E,H) the solution of (191), is called the exterior (outgoing) Calderon operator. We also denote by C£ the mapping m = nAE|—»|g(nA curl E | ), thus C£=Z Cc does not depend on Z the impedance of the (exterior) medium, see (34), but only on the wavenumber k. There are many properties of these Calderon operators. First we have: Proposition 23.

The exterior Calderon operator is an isomorphism in the space

1/2

H" (div,r), with: (200)

(Cc)2 = - ^ I, (Ctf = - 1 ,

and (C^T 1 = - C£.

Furthermore the exterior Calderon operator is an isomorphism in Hs(div,0/br all real s i/T has a C00 regularity. PROOF. First we note that if we exchange c and ]i in the problem (191), then the solution (E,H) of (191) is changed into (H, - E); this easily gives (200). Then we prove that Cc is an isomorphism in H1/2(div,D, and the regularity result:

If m € H1/2(div,0, then the solution (E,H) of (191) is in H/OC(Q')3 X HJOC(Q')3. This is a consequence of Theorem 6 chap. 2,firstapplied to £E with C € D(R3), C = 1 in a neighborhood of r (since n A E | is in Ht1/2(0), then to H using (28), (30) chap. 2 io>P n.H | = n.curl E | f = curlr nTE = - divr (n A E | f ) € HV2(T). Then we have the regularity result for all half-integer s thanks to Proposition 7 chap. 2 and then for all real s by interpolation and duality. ®

Proposition 24. Positivity properties. The exterior Calderon operator satisfies: (201)

Re/ r C e m.(nAm)dT>0,

PROOF.

We apply the Green formula like in (192). Then we have:

(202)

Re/ r n. EAHdT = Re/ s n.EAHdS R >0, K.

Vm€H~l/2(div,r), m*0.

3 STATIONARY SCATTERING PROBLEMS

110

from the Silver-Muller conditions (and the Rellich lemma). This gives (201) (thanks to Propositions 3 and 4 chap. 2). We can give a more precise result: thanks to the formulas (30), (33), we have (202)'

Re/rn.EAHdT*-^/s2|j(ka)|2da,

with j(ka) = ik[aAJ(ka)+^n a M(ka)], J=mAH| r = C c (nAE| r ), M = - n A E | r . We can define, instead of (199), other Calderon operators such as: m —* icrH, which is an isomorphism from H~"1/2(div,D into H~ 1/2 (curl,D, and also an exterior incoming Calderon operator C c,in using incoming instead of outgoing conditions in Definition 4. Then using for instance the antiduality pairing (144)' we have the "reciprocity formula": (203) <nAEl9Ce^(nAE2)>^^<C\nAEl\nAE2>9

nAE U €H" 1 / 2 (div,D,

thus C c = C c ' out = - (Ce'm)\ with C'tin = - C'_k = - C£. With the duality pairing (144), we obtain: ' C ^ - C * . We can also define an exterior Calderon operator C e k for complex k, with similar properties, and the limiting absorption principle gives: (204)

lim

c£m = C £ m , Vm€lT 1 / 2 (div,r).

The exterior Calderon operator above defined in (199) has numerous properties and is a basic tool in many applications to the scattering in electromagnetism (see below). 4.4. Interior boundary problems We first consider the inhomogeneous vector Helmholtz problem in a (regular) bounded domain Q. Since the selfadjoint operator A" defined in (147) chap. 2 has a compact resolvent (because the natural injection of the Sobolev space Hl(Q) into L2(Q) is compact), we know that the following interior Helmholtz or Maxwell problems depend on the Fredholm alternative: Theorem 7. Interior vector Helmholtz problem. The following problem: given a function f in L2(Q)3 and a complex number Y,find u in Hl(Q)3 satisfying (205)

OAu + l A i ^ - f inQ, ii)nAu| r = 0, (divu)| r = 0,

3.4 BOUNDARY PROBLEMS

111

depends on the Fredholm alternative: either k2 is not an eigenvalue of the selfadjoint operator A", then there exists a unique solution; or \? is an eigenvalue of the selfadjoint operator A", then there exists a solution u if and only iffis orthogonal to the corresponding eigenvector space, and then u is determined up to this eigenvector space. Theorem 7 \ Interior inhomogeneous vector Helmholtz problem. The following problem with given boundary conditions (m,g):find u in H*(Q)3 satisfying: (206)

i)Au + k2u = 0inQ, ii)n Au| r = m, (divu)|r = g, withm€Ht1/2(0, g€H- 1 / 2 (0,

depends on the Fredholm alternative: either k2 is not an eigenvalue of the selfadjoint operator A ' , then there exists a unique solution; or k2 is an eigenvalue of the selfadjoint operator A", then there exists a solution u if and only i/(m, g) satisfies'. (207)

J r (m.curl v + g n. v) dT = 0,

for all eigenvectors v relative to k2, and then u is determined up to this eigenvector space. The proof is classical using the Green formula: (208)

- / (Au + k2u) v - u (Av + k2v) dx=/r[(n A curl u). v - n A u. curl v - divu n. v + n.u div v] dT.

Theorem 8. Interior Maxwell problem. The problem: find the electromagnetic field (E,H) withfiniteenergy, i.e., E, H in H(curl,Q), satisfying, with given m i) curl H + io)£ E = 0, (209)

ii) - curl E + io>ji H = 0 in Q, 1/2

iii)nAE| r = m, m€H" (div,0,

depends on the Fredholm alternative: either k2 = co2€^ is not an eigenvalue of the selfadjoint operator A" or of A (see (134) chap. 2), then there exists a unique solution; or k2 is an eigenvalue of the selfadjoint operator A" (or A), then there exists a solution (E, H) if and only ifm satisfies: (210)

/ r (m.curl v)dT = 0,

for all eigenvectors v relative to k2, and then E is determined up to this eigenvector space.

112

3 STATIONARY SCATTERING PROBLEMS

PROOF. Thanks to a lifting of the boundary condition, using the solution (u,v) of (151)* chap. 2, we can reduce this problem to an inhomogeneous Maxwell problem which is directly relevant of the Fredholm alternative, like for the exterior Maxwell problem (see (191)). We can also reduce (209) to the vector Helmholtz problem (203). $ If c or fi is not a real number (i.e. if the medium is dissipative), the problem (209) has a unique solution for all m as in (209)iii). We can also solve the interior Maxwell problem with some heterogeneous media with € and p matrix valued and dependent on x, with the hypotheses H17) chap.l only. We can also solve problem (209) using a variational method on the unknown vectorfieldH. 4.5. Some consequences; the interior Calderon operator First we note that in the Maxwell problem (209), the magnetic field H also satisfies (198), then we can define in the "regular" case, (i.e., when k2 = ©2ey is not an eigenvalue of the selfadjoint operator A" or of A): Definition 5* The interior Calderon operator. The mapping: (211)

C^orCJp: m = n A E | r ^ n A H | r

with (E,H) the solution of(209), is called the interior Calderon operator. We also denote byC^ the mapping j m n A E| —»|g(n A curl E| ), thus Cj^ Z C1 does not depend on Z the impedance of the (interior) medium, see (34), but only on the wavenumber k, and satisfies (CJ)2 * - 1 . Note that it is possible to define interior Calderon operators when k2 is an eigenvalue of the selfadjoint operator A, we only have to eliminate the corresponding eigenspace but with these technical difficulties, it is less useful. There are also many properties of these Calderon operators. First, Proposition 23 is true for the interior Calderon operators in the regular case. Then, we also have positivity properties: Proposition 25. Positivity properties. The interior Calderon operator satisfies (212)

-Re/rCim.(nAm)dT>0,

Vm€H" 1/2 (div,0.

PROOF. This follows from the Green formula: (212)'

Re/ r n.EAHdT=/ ft [Re(-io)?)E.E + Re(ia>fi)H.H]dx<0.

If we have a conservative medium (the free space), then (212) is always zero. & From (201), we have the positivity property of the operator R (see (100)) for k > 0 -Re/ r RM.n A MdT=Tkf^^^ [M(x).M(y) - \ divrM(x) divrM (y)]dTxdry > 0.

3.4 BOUNDARY PROBLEMS

113

For a dissipative medium (e" = Im c> 0, p" > 0), then with (209) we have (213) -Re/ r C i m.(nAm)dTiC 0 IIEIlJ, (curlft) ,

VmeH~ ,/2 (div,n,

with C 0 a constant, and thus with the quotient norm: (214) -Kef

&m.(nAm)&

±C0knt_iniJt

_, Vm€H" 1/2 (div,r).

Therefore the sesquilinear form (215)

a(m,m') = - J r C1 m. (n A nV) dT 1/2

is coercive on VT (div,T)for a dissipative medium. Remark 11. Calderon operators for the Helmholtz equation. i) First in the scalar case, we can define the exterior Calderon operator Cc by:

(216)

c e (u 0 )=g;i r

u being the solution of the Helmholtz problem (188); n is the exterior normal to the bounded domain Q; Cc is an isomorphism from H1/2(r) onto H" 1/2 (D, and from H s (0 onto Hs ~ l(T) for all s if r is C°°; Cc has the positivity property (from Green formula, since Cc corresponds to outgoing waves), for all k > 0, (217)

Im(Ceu0, u0) = I m / r | ^ u dT>0,

Vu 0 €H 1/2 (r),

u o *0.

Then we can define the interior Calderon operator C1 like in (216), u being the solution of the interior Helmholtz problem corresponding to (188), generally with complex k. When k2 is not an eigenvalue of the interior Helmholtz problem, neither for the Dirichlet condition, nor for the Neumann condition, C1 is an isomorphism from H 1/2 (0 onto H" l / 2 (D, with the same regularity property. Furthermore it has the positivity property, from the Green formula (218)

Im (C 1 ^, u0) = - 2k,k" fQ \ u | 2 dx,

V u0 € H 1/2 (0, with k = k' + i k".

This implies the positivity properties for the operators L and R of (71) for k > 0 Im(Lp\p')>0, Vp'€H- 1/2 (r),P'*0,

Im(Rp,p)<0, Vp€H 1/2 (H, p*0.

We can verify easily using (71), (72), that the Calderon operators Cl and Cc are related to the integral operators L, J, K, R by: (219) Ce = - i L - 1 ( I + K) = - 2 ( I + Jr1R, Ci = - | L - I ( - I + K ) = - 2 ( - I + Jr 1 R.

114

3 STATIONARY SCATTERING PROBLEMS

ii) Then in the vector case, we can define Calderon operators. The exterior Calderon operator C c is given, with u the solution of the Helmholtz problem (190)) by: (220)

Cc(m,g) = (nAcuriu| r , n.ulp;

^isanisomorphfcmfromHj^ from H*/2(T) x Hl/2(T) onto H*/2(r) x Hm(T).

and also

4.6. Integral equations and boundary problems We can reduce the exterior or interior Maxwell problem (191) or (209) to an integral equation on the boundary in many ways. Here we give some of them. The general method is to choose an extension to the whole space, of the unknown (E,H) solution of the Maxwell equation. Then we only have to determine the jumps of E and H across the boundary T, or also the surface electric and magnetic currents J r and M r such that: (221)

Jr»-[nAH]r,

M r =[nAE] r

i) Solution with an electric layer only. The chosen extension (E9H) is such that there is no jump of the tangential electricfieldE across the boundary T: thus we have tofindthe surface electric current J p so that (using (101)) (222)

ZRJ r =m,

withM r =[nAE] r =0.

Then (E, H) is obtained from J r by (the restriction to Q' or to Q of) (223)

E^ZI^Jr,

H= -FmJr.

We note that this is valid in the exterior case as well as in the interior case. ii) Solution with a magnetic layer only. The chosen extension (E,H) is such that there is no jump of the tangential magnetic field H across the boundary T: thus we have to find the surface magnetic current M p so that a) for the exterior boundary problem (using (89), (93)): (224)

n^^1"]—m,

withJ r =-[nAH] r =0,

rij being the projection on thefirstcomponent, so that with (104) and (102): (224)'

|(I+T)Mr«-m.

b) for the interior boundary problem: (225)

|(I-T)Mr«m.

115

3.4 BOUNDARY PROBLEMS

Then in the two cases, (E, H) is given by (the restriction to Q or Cl' of) (226)

E = ZPmMr,

H^LmMr.

iii) Solution by extension by 0 a) for the exterior boundary problem, we have to find (227)

Jr=*nAH| ,

withMr = - m .

Then (Mr, Jr) must satisfy:

<» <?;>('4* ")(*)■•• b) for the interior boundary problem, we have to find (227)'

Jr = - n A H | ,

withMr = m.

TTien (Mr, Jr) must satisfy

(229)

fMr-\

f I+T

-l'rJ-U*

- Z R y M

I*TJL

and we obtain the electromagneticfieldin Q or Q' by formulas (96)'. Remark 12. There are other possibilities; for instance (see remark 10), when the exterior boundary problem is associated with a scattering problem of an incident wave in free space (which is naturally defined on the domain of the obstacle, like a plane wave) by a perfect conductor, then we can extend the reflected electromagnetic field by (-Ej,-H x ) in the obstacle. Then we are reduced tofindthe jump: Jr = -[nAH] r = nAH t | ,

whereas: Mr = [nAE]r = - nAE t | =0,

and then Jr must satisfy;

thus (230)'

ZRJr=-2nAE,|r,

(I-1)J r »2nAH I | . <8>

116

3 STATIONARY SCATTERING PROBLEMS

Remark 13. In fact for a given m, it is not obvious that the solution of the exterior problem is obtained only by an electric layer or a magnetic layer, and then it is possible that equations (222) and (224) have no solution, whereas method iii) (extension by 0) is always allowed. Therefore the system (228) is always solvable (but it is a system!). The difference is due to the eigenvalues of the selfadjoint operator A (see Theorem 8); these values are called irregular frequencies (for the exterior scattering problem). Thus to apply these methods, we have to avoid these irregular frequencies; for instance in a scattering problem, we use a linear combination of (230)' (see below), with a complex constant. 0 4.7. Some consequences for the integral operators i) From Theorem 6 (existence and uniqueness of the solution of the exterior boundary problem) we deduce (231)

Im(I-*D = ImR,

kerR H ker(I-T)*{0}.

ii) The "regular1* case. From Theorem 8 (existence and uniqueness of the solution of the interior boundary problem in the "regular" case) we deduce: (232) PROOF.

Im(I+T) = ImR,

kerR H ker(I+T) = {0}.

Let Ge be the set of the exterior boundary values for the

electromagnetic field: Gc ■ Im Pc » ker P.. Then for all u in X = H~ 1/2(div,D, there exists v in X so that (u,v) is in Gc, thus: (233)

Pi(;) + Pi[})-0.

Therefore the mappings: u € X - > P | ( J ) and v c X - ^ P j f J ] are injective and have the same range. Thanks to (102), (104) this gives (231). The proof of (232) is similar. Proposition 26. In the regular case (k2 is not an eigenvalue of A), R, (I + T) and (I -T) are isomorphisms in the space X = HP l/2(div,r)t and more generally (i/r has the C°° regularity) in Hs(div,D/or all real s. Thanks to relations (106) we have to prove the properties for the R operator only. That R is onto is a consequence of equation (222) and theorems 6 and 8 (or also of (231), (232), since X • Im (I + T) + Im (I - T)).

PROOF.

117

3.4 BOUNDARY PROBLEMS

Letj be such that Rj = 0; thenwehave: m = n A E | = 0 , thusE = 0 i n l a n d Q' (in the regular case), thus H = 0 and j = 0. ®

iii) The "singular" case (k2 is an eigenvalue of A) We define the following spaces: S k d # { n A H | r , H = icurlE, E C H W ^ E + I ^ E ^ , divE = 0, n A E | r = 0}, (234) 1 3 2 = {nAH| r ,H€H (Q) ,AH + k H = 0,divH = 0,nAcurlH| r = 0}, and (234)'

etf 5 ddef E k = (nAc,c€S k ),

E k H?{ u € X,/ r u.v dT = 0,

VveSJ.

Then instead of (232), we have (235)

kerRnker(I + T) = Sk, R(Sk) = (I+T)S k , ImR=E k |ImR + I m ( I - T ) 2 S k .

As a consequence we have: (236)

kerR = ker(I+T) = S k ,

ImR = Im(I-T) = Ek

(andalso: ker(I-T) = C e E k , Im(I + T) = C e S k ). Remark 14. From the properties of the S operator, we also have: (237)

kerR n kerT={0},

ImR + ImT=X.

<8> Remark 15, Let G and G. be the sets G„ = Im P = ker P., G. = Im P. = ker P . corresponding to the exterior and interior boundary values for the electromagnetic field, then their projections on their components are: (238)

Tlx Ge = fl2 Ge = X, with:

i) in the "regular" case: ITj Gx = n 2 Gj = X, ii) in the "singular" case: Il 1 G{ = n 2 Gj = S k .

These results are easily proved from the "reciprocity" formula: (239)

/jjiAE. H ' d r = / r n A H . E'dT

for all (E,H), (E\H') satisfying the Maxwell equations in Q; this is proved using the Green formula, with:

118

(240)

3 STATIONARY SCATTERING PROBLEMS

div (E A H*) * div (H A E').

Then taking E' an eigenvector of A, gives (238)ii). In the "regular" case, we can give the Calderon operators as function of the operators R, T, thanks to the formulas (228): (241)

cj-R"la-T)--a-,n"lR-(i-i-,i)R"1—Ra-i-T)"1, CIk=.R-1(I+T)=(I+T)"IR=-(I-T)R"1 = R(I-T)"1. ®

4.8. On the numerical solution of some scattering problems 4.8.1. The problem of irregular frequencies In order to solve the scattering problem of the remarks 10 and 12 thanks to an integral method, we have to solve equations (230)'. If k2 is (near to) an eigenvalue of the operator A, then we cannot solve only one of the two (vector) equations (230) since R and (I-T) are not invertible. This is the problem of "irregular" frequencies. We develop a method implemented by BonnemasonStupfel [1]. Let a be a (complex) number with Re a * 0. Then we replace (230) by the linear combination (242)

- ( I - T ) J r + aZnARJ r = 2f I H f 2[-nAH I | r + an r E I ].

First we know, thanks to Theorem 6 and Remark 10, that (242) has a solution. Then we prove that this problem always has at most one solution. i) First we consider the interior problem:find(E,H) with E in H!(Q)3 so that (243)

i)curlH + iwcE = 0, ii)-curlE + m\iH»0 inQ, iii) n A H | - a IIrE « f j, 1/2

with fj given in Ht (0 (for instance), with e and i* real positive numbers. Let a(E,E') be the sesquilinear form on Hl(Q)3 (244)

a(E,E')=/ n - [j^curlE .curll' + ioeE. E'Jdx- a / ^ E . E'dT.

Then we can write problem (243) in the variational form:findE in Hl(Q)3 with (245)

a(E,E')=/ r f f . E'dT,

VE'€HW.

Then taking the real part of (245) with E'» E, we obtain: (246)

Rea(E,E) = R e ( - a ) / r | n r E | 2 d r = Re(/ r f,.Edr).

This implies uniqueness: when fj« 0, (246) implies nrE = 0. Thus from (243)iii) n A H | = 0, and then, from section 13 chap. 2, E = H = 0 in Q.

119

3.4 BOUNDARY PROBLEMS

ii) Let Q 0 be defined by:

(247)

Qo£Yrr) —Jr+fltnAMr.

and thus

<*«>

C nAElr^ Qo[-nAH|jJ=nAHlr-aI1rEf nAE|.^

Fromtoi), we know that Q I _ n A j j i

I = 0 implies E = H = 0.

iii) Then applying Q to equation (230) gives:

< 249 >

QoPi[j0r]=-Qo(-nAH;![)=f..

and thus we obtain equation (242). Therefore uniqueness of the solution J r of (242) is deduced from that of (243) in the following way: (250)

Q o P i [ j ° r } - 0 implies Pi( £ ] = < > , thus ( ^ J € G e ,

and then J r = 0 from Theorem 6. We can show that problem (243) (and thus (242)) depends on the Fredholm alternative. This gives another proof of Theorem 6. 4.8.2. A saddle point method Here we briefly give a variational (and integral) method for solving a scattering problem, which is due to Bendali [1]. We consider the scattering problem of Remark 10, which is equivalent to (230). We only take the first equation of (230) (we assume that k2 is not near an eigenvalue). Let

(251)

c^-^IIrE,,

p^Jp

Then we have tofindp with given c in H" (252)

(curl, T) so that

- n A R p = 2ikc.

Thanks to the definition (100) with (95) (see also (144)), we have to solve (253)

<Lp, q> - \ <Ldivrp, divrq> = <c, q>,

Then we define: (254) and

\ = -^divrp, k2

V q € H" 1/2 (div,0.

120

3 STATIONARY SCATTERING PROBLEMS

(255) HT 1 / 2 (0 = {♦ e H" ,/2(T), / ♦ oT * 0 on all connected component Tj of rj. Thus X € KV2(T). Then we define, with* 0 (x)=^p, r= |x|: (256)

L0Mx)=/r*0(x-y)X(y)dry;

L0 is a continuous operator from H" 1/2 (0 into Hl/2(T). Then with these notations, problem (253) is:findp € H-,/2(div,r) and X e Km(T) so that: i) <Lp, q>+<LX,divrq> » <c, q>, (257) ii) <L0divr p, ♦>+IT <L0X, ♦> * 0, for all q € H~,/2(div,r), and all ♦ € H7I/2(f)- Using the sesquUinear forms: a(p, q) = <Lp, q>, (258)

a0(p, q) * KL^, q>,

b(X, q) = <LX, divr q>,

b0(X, q ) . <L0X, divr q>,

c(X,*)=k2<L0X,<»>, withp, q 6 H~1/2(div,r), K ♦« Hr1/2(T), we can write (257) in the form: find(p,X) so that for aU q € H_1/2(div,r), and* € (259)

Km(T),

i)a(p, q) + b(X, q)*<c, q> H)5^p5+c(X, ♦) = <).

This system is solved by perturbation of the system:find(p,X) so that, for all q in H- 1/2 (div,0, and ♦ in H;1/2(r), (260)

i) a ^ , q) + b0(X, q) = <c, q> ii)bo(*,p)»0(or=<x,<»), 1/2

1/2

with given c in H" (curi,0 and x in the (anti) dual space of H.

(T) that is:

(HZ 1/2(T))' = Hi/2(D 4 e H1/2(D, / r L; V dT, V r4 component off).

3.5 DIELECTRIC OBSTACLE

121

Then we prove (see Bendali [1]) that the system (260) satisfies the "inf-sup condition" or "Brezzi condition11 (see for instance Girault-Raviart [1] p. 39) and thus (260) has a unique solution for all given (c,x) in H " 1/2(curl,H x H*1/2(H. Then we obtain the solution of the system (259) by perturbation of the system by a (regularizing) compact operator, and thus we can apply the Fredholm alternative. 5 . SCATTERING PROBLEMS BY A DIELECTRIC OBSTACLE

Let (Ej,H.) be an incident electromagnetic wave with an angular frequency o>, on a (regular) bounded obstacle Cl. Let the medium inside Q be a dielectric with permittivity and permeability t{ and iÂť1; then let kx be the wavenumber inside Cl; outside Cl is the free space (or some "conservative" medium) with permittivity and permeability t = eQ and p = u and wavenumber k. The problem is to find the electromagnetic wave (Ef,H ) reflected by this obstacle. Let (E,H) be the electromagnetic field inside Q; (Er,Hf) must satisfy the Maxwell equations in free space, with the Silver-Muller conditions at infinity and with locally finite energy; the problem is then reduced to find (E,H) satisfying (261)

I i) curl H + io)â&#x201A;Źi E = 0, iO-curlE + iwjij H = 0inO,

and the continuity relation on the boundary T of O (262)

nAEr + nAEpnAE,

nAHr + nAHj = nAHonr.

The fact that (Er,H ) is a reflected wave, gives us a relation between their boundary values on r as seen in section 4.2. We can obtain it in two ways. 5.1. A general variational formulation Here we could assume that inside Cl the medium is inhomogeneous and even anisotropic with usual dissipative conditions, see H14) chap. 1: (263)

Re (- io^) > 0,

Re (- mv{)> 0.

Using the exterior Calderon operator Cc, we have: (264)

nAH r | r -C e (nAE r | r ),

and then with (262), we obtain the boundary condition on T: (265)

nAH-C^nAE^f^nAHj-C^nAE!).

122

3 STATIONARY SCATTERING PROBLEMS

The problem: find (E,H) in Q satisfying (261) and (265), is solved by a variational method. Let a(E,E') be the sesquilinear form (266)

a(E,E*)=/Q - [ |jL curl E. curl E' + im E. I f ]dx - / r Cc(n A E). I* dT,

for E, E* in H(curl,Q). The problem (261), (265) is equivalent to:findE € H(curl,Q) so that (267)

a(E,E')=/r f,. E'dT, V E' € H(curl,Q).

The proof is "classical" with the Green formula. Thanks to (263) (if the medium is inhomogeneous we have to assume a slightly stronger hypothesis), and to the positivity property of the exterior Calderon operator Ce (see (201)), a(E,E') is coercive: there is a constant aQ so that (268) Rea(E,E) * fa(Re(-±)

|curlE| 2 + Re(-ia*)|E| 2 )dx fc a0IIEI^(curl$n).

Since a(E,E') and the form: E*-+/r fj.E'dT are continuous on H(curl,Q), we can apply the Lax-Milgram lemma: thus the scattering problem has a unique solution. Remark 16. We can solve other scattering problems, for instance when there is a perfect conductor with domain Qj in the obstacle. Then the electric field must satisfy the boundary condition on Ti9 the boundary of Qj: n A EL =0. We apply the variational method with V={E € KKcur^QVQj), n A E | = 0}. 5.2. Solution using an integral method Let (M, J) be defined by: (269)

M = nAE| r ,

J= -nAH|r

Let Pf be the interior Calderon projector associated to the exterior wavenumber k. Then we have (by definition):

(270

f n A Er| ^

^[-nAH^J*0

Then using (262) and (269) we have:

thus (with the integral operators T, R associated to k, and the impedance Z of free space):

3.5 DIELECTRIC OBSTACLE

123

|i)(I-T)M + ZRJ = 2nAE,, (271V i i ) - 2 R M + ( I - T ) J = -2nAH I onT. Now we assume that the medium is homogeneous with constant ej and v{. Let P* be the exterior Calderon projector associated to the interior wavenumber k{. Then we have (by definition): (272)

Pjf^^O,

that is also with Tl9 Rlf andZj relative to the interior medium: iHI + ' t y M - Z ^ J - O , (272)'

ii)j-R|M + (I + T|)J-0.

Thus the scattering problem amounts to find (M,J) solution of the integral equations (271) and (272), that is also (271)' and (272)'. We can take (270) into account in a weak form, (thanks to (100) with (95), see (253) for R): taking the scalar product of (271)'i) with n A J\ and (271)41) with n A M' and adding, give us for all NT, J' in H~ l/2(div,D iZk y (273)

[J(x).J'(y) - k-2 divrJ(x).divrJ'(y)] 4Kx -y) dTxdTy

- f/pfr [M(x).M'(y) - k~2 divrM(x).divrM'(y)] <D(x -y) dTxdTy - /pfr [J(x) A M'(y) + M(x) A J'(y)]. grad0 (x - y) dTxdTy -i/ r [M(x)AP(x) +J(x)AM,(x)].ndTx = J r [- HjW.M'to + EjW.J'WJdTx.

We can also write equations (272)' in a weak form. The case of a dielectric layer against a perfect conductor would be solved in the same way. These problems depend on the Fredholm alternative, and uniqueness follows from the Rellich lemma. For some numerical developments on these methods, we refer to Bendali [1] and Heliot [1]. 5.3. Behavior at infinity. Radar cross section. Optical theorem The behavior at infinity of the reflected electromagnetic field (Ef,Hr) is given by the formulas (30), (31) with: (274)

M = -nAE r | r & r , J = nAH r | r 6 r .

Then we define the differential cross section for the direction a by ,2

(275)

o(a)V^ljm 4irp

|2'

IE,!'

withp = |x|.

124

3 STATIONARY SCATTERING PROBLEMS

Thus using (30) and (33), we have:

(2?6)

^^li^^li^'

since the incident wave is a plane wave, that is (EpHj) is given by: (277)

E,(x) = E,e ° , H,(x) = H,e ° ,

where a0 € S is the direction of the propagation of the plane wave, and Ev H{ arc constant vectors in C , Z being the free space impedance: (278)

a o ! , = 0, ao.H, = 0, H^J^oAE,,

1,^-Z^AH,.

The radar cross section, or back scattering of is defined by of » o( - aQ) where aQ is the incident direction of the plane wave. It is a surface and its unit is m2 (in the S.I.) but it is generally expressed in dbm2 (db decibels) by 10 log10 of. Then the total cross section or scattering coefficient o s is defined by: (279)

* ,a . o S = / s 2 ^ « ) d « =1 i _JL 7 ^ / s 2f l l I*?«/ I)~|U2 d

IE,I :

We also define the following powers: i) the scattered power Ps, as the real part of the flux of the complex Poynting vector of the reflected wave on the surface T: (280)

Fs = Re(f1 n . E r A H r d I ) = - ^ / 2 |5 j ( k a ) I 2 d a , (4it) Z

(see (202)')t and thus 05=4nZ | El1" Ps is proportional to the scattered power; ii) the absorbed power P t as the real part of the flux of the complex Poynting vector of the interior wave, on the surface T: (281)Pa = - R e / r n . E A H d T = / Q ^ (with H13, chap. 1); if Hj is real, Pa corresponds to the dissipation by Joule effect. iii) the toted power Pm = Ps + P . Using relations (262), we have: (282) Ptot = Re/ r [nAH r Ej - n AE r HJ dT=Re/ r [nAH. E, - nAE. H,J dr, since the flux of the Poynting vector of the incident wave plane on the obstacle is zero: Re (fr n.E, A H, dT) = 0. Then with (30), (31) and (269), we have:

3.5 DIELECTRIC OBSTACLE

125 -iko„.x

(283)

Ptot = - R e / r [ J . E , + M.HI]e" ° dT = - RfilJOw^^ + MOw^.H,]

=4rIm0(k«o).Ei)=4lm(A(kao).HI). This relation which gives the total power as a function of the forward scattering is generally called the optical theorem. Since the total power P is greater than P 9 we have the inequality s

(284)

l / s 2 lJ(M 2 da S £lm( j(ka0).Ii)-

(4*)'

Note that in the scalar Helmholtz case, we also define, with usual notations, a scattered power, an absorbed power, and a total power by /,

s = Hn/ r Sfn r dT,

Pa*-Im/r^udr=(Imk?)4|u|2dx,

|/»t = P s + i'a = - I m / r ( | u I . u ^ ) d r = I m / n ( k 2 - k g ) u u I d x , which we can also write in a form similar to (283) for an incident wave plane. Remark 17. Is it possible to have a differential cross section vanishing for all directions? First note that the reflected electromagnetic field would be OU/r2) at infinity, and thus would be zero from Rellich lemma. Then the electromagnetic field (E,H) in Q satisfies:

(285)

i)curlH + i<o£j E = 0, ify-curlE + iuiij H = 0, iii)nAE| r = nAE,| r , nAH| r = nAH,| r ,

which is a Cauchy problem (see chap. 2). Let: Ex = E - Ej, Hj =* H - Hj in Q. Then (E^Hj) satisfy: I i) curl Hj + icoej Ej = iu>(E - zx) El, (286)

") - curl Ex + io^ Hx = iai(p - pj Hj, in Cl, |iii)nAE 1 | r = 0, n A H , | r - 0

Thus we can extend (E^Hj) by 0 to the whole space, and then apply formulas (30) and (31) with (M, J) given (thanks to the characteristic function of Q, lQ) by (287)

M d#i<o(£ - «,) E,l n , J dJf icofc* - nr) H,l^.

Then j (ka) = 0, m(ka) = 0, V a € S% imply with (277):

126

(288)

3 STATIONARY SCATTERING PROBLEMS ^[^(e-cÊj-kfti-âAHji^kâO,

V«€S 2 ,

when €., pj are constant; thus C| = e, p. = p, i.e., Q is occupied by free space! If the domain Q is occupied by a dissipative medium (with (213)), we can verify using the power balance (that is the variational method), that the problem (28S) cannot have a solution; (E,H) must satisfy (289)

/ r [Re(i^)H.H^Re(-ici)?)E.E]dx*Re(/p.E I AH I )dr*0,

that implies E ■ H = 0 (at least where the medium is dissipative), and then E = H = 0 anywhere, in contradiction to (285)iii)! Remark 18. It is possible to have a differential cross section vanishing for certain directions (that is for isolated directions) but it cannot be zero in the neighborhood of a direction (that is, in an open cone). This means that there is no perfect shadow domain. In other words: (290)

j(ka)*0 Va€S 2 sothata.a,>ii foraaj €S 2 , andn>0,

also implies Er = Hr = 0 ! We can prove this result, as an easy consequence of Muller's lemma 115, 117, pp. 341 and 342 (see Muller [1]). As a matter of fact, (290) implies: (291)

j^k^sOandn.dOc^AKka^srO, V«€S 2 suchthata.aj>n

(that is, the wave is circularly and linearly polarized respectively) and this implies thanks to analyticity properties (see Muller [1]) that (291) is satisfied for all a of the unit sphere. This immediately implies that (290) is also satisfied for all a of the unit sphere which proves the property. ® Remark 19. In radar problems, we are interested in having a zero radar cross section. This is possible for particular geometries (symmetry invariance with respect to an axis), as asserted by the well-known (among radarists at least) Weston Theorem; here we give a (small) generalization. Proposition 27. "Weston Theorem". Given an incident electromagnetic wave (EpHj) with an angular frequency ©, on a (regular) bounded obstacle Q which is invariant under a 90° rotation S about some axis. We assume that the direction of incident propagation is parallel to this axis. Furthermore we assume an "impedance" condition on the boundary T ofQ: (292)

UAH*!

-COtAE^

sothat 5" ! CS—2T 2 C" ! t

3.6 SEVERAL OBSTACLES

ill

with (Et,Ht) the total electromagneticfield,Z the impedance of free space\ C an operator on the boundary and S the rotation. Then the radar cross section is zero. We note that this contains the case where the medium inside Q is a dielectric with permittivity and permeability t{ and px with tx/tQ = P/PQ, since the operator C is then the interior Calderon operator; thus the impedance of the inside medium must be equal to that of free space. This also contains the case of the following usual impedance condition (which requires n = 1): (293)

7irE = nZnAH onT, n€C,

PROOF. First we note that the differential cross section is given by the following formula for all directions a, where (E,H) denotes the reflected field (294) o(a)=:—^-j r«mo ^ a A E f r a ^ H f r o ) ^ ^ ^ r !^4*^E(m).aAH(itt). l E ll r ~ > °° l E ll r~*°° This a simple consequence of (30), (31). Then we note that the problem relative to the reflected field is invariant under the 90° rotation about the incident axis so that (5E,5H) defined by (5E)(x) = SE(Sx), (SH)(x) = SH(Sx), and (ZH, - (1/Z)E) are solutions of the same problem. From uniqueness of the solution we deduce that (SE,SH) = (ZH, - (1/Z)E). Then this relation implies: o(- a 0 ) ^ - - ^ f Urn. 4m2 E(- ra0).SH(- ra0) (295) ^

rlim^

(- ^

E(- ra0).E(- ra0)) < 0 f

iE,r 6. SCATTERING AND INFLUENCE COEFFICIENTS FOR SEVERAL OBSTACLES

Let T be a "regular" bounded nonconnected surface with p components (rA), \ in I. We assume that T is the boundary of a bounded domain Q, with N components, Q j , . . , ^ . Let & be the (open) complement of Q with N' + 1 components Q'0, Q ^ , . . ^ ' ^ (with Q'0 unbounded). Note that Cl\ is not the complement of Q., nor is T. the boundary of Q.. We denote by 8Q. (resp. 3Q\) the boundary of Cl. (resp.Q*.). From Alexander's relation (see for instance Dautray-Lions [1] chap. 2.4.5) we have p = N + N \ We assume that there are electric and magnetic surface currents on T, that is, on each surface I\, (M., J.); we assume that the whole space outside T is occupied by free space. We study the electromagneticfielddue to these currents in R3 and its trace on the surfaces T., and the influence of currents on I\ onto another surface T .

3 STATIONARY SCATTERING PROBLEMS

128

6.1. Decomposition of the trace spaces The trace space X = H ~ 1/2(div,0 has the following decomposition: (296)

X=H _1/2 (div,I> $ XA» © H"l/2(dhr,r^.

Let (M, J) a (M v Jx), X € I, be magnetic and electric currents on T in this space. The electromagneticfield(E,H) induced by (M, J) is given by formulas (96): E=2EX, H = | H X , (297) |with E^PJA^ZI^^,

HK={LmMK-PmJk.

Then taking the traces on the boundary, we have (see (101) to (103)):

(29,)

1-<V^H >?H

t\***»\ .4^ - v J

We see that S is decomposed into (S^); and S^ is a continuous operator from X x xX x intoX y xX y , and it is regularizing for X*p, t h a t i s I m S ^ i s i n H ^ r ^ x H ^ ) for all s if the surface Tp is C00. This implies that T^, R^ are also regularizing. Relation (105) implies: (299) 2SykS^6wi and on each surface Tk: (300)

VY,H€I,

CSJU^-I.

Remark 20. On the orientation of the normals to the surfaces I\. Let us denote by H. and Cl\ the domain of R3 inside and outside of Tk. We note that the natural onentation of the normal to the surface TK (from inside to outside) does not necessarily agree with that of the domain £1. Let S w be the matrix operator S given by (103) with (100) for a surface T with orientation t of the normal n. We have: S ( ~ € ) »-S ( i ) , and thus 5 = eS w is orientation independent. Choosing: e' = - e for i, e' = e for e, relations (93) and (104) are written: (301)

[-^Hir;J=-^<'-'«[J.

and thus the Calderon projectors are given by (302)

P€^(l-t>S(t))=%(l-s'*S),

w i t h e r - lfori, s'*=+lfore.

We denote by SA the operator S » S relative to the surface r x (for the natural orientation), and by P \ . the corresponding Calderon projectors. 0

3.6 SEVERAL OBSTACLES

129

Writting SA ^? S^, we have SA = t Sx according to the orientation of the normal to I\ with Cl. Thus the operators (303)

Pf=±(I + SA) = ±(I + eSA), P^=i(I-S A ) = i(I-eS A )

are the (exterior and interior for QA) Calderon projectors relatively to the surface Ty Then the (total) Calderon projectors relatively to the domain Cl are decomposed into operators (Pe .)A but only their diagonal terms are projectors. We denote by G c . (resp. G x c .) the graphs of the operators P c . (resp. PAc .), by riA the restriction to the surface TA. Then we have:

(304) GfeG^XxxX^Gfcn^G^cn^, or Gfcn^G^cnxGi, according to whether the natural orientation of the normal to the surface TA is the same as that for Cl, or not. Note that generally we don't have equality in (304)! We easily see that the graphs Ge . have the decompositions: (305) Gi =. ©Q ^ ( Q j ) , Ge = Gc(00)©.s ^

G ^ V ' with Cl0 = R\fi', o,

(with obvious notations), but GACL) is different from the sum

AGf.

rA£aa.

6.2. Screen effect. Extinction theorem In the general case, if (M,J) is in Ge (resp. G.) then the electromagnetic field (E,H) due to (M, J) is zero in the domain Cl (resp.Q'). This is generally known as the extinction theorem. Thus for each X, if (M.,J.) is in GA (resp. G\), the electromagnetic field (E.,H.) ._.

A A

satisfies: (306) EA(x) = 0, HA(x) = 0 forx€Q A (resp.a\). Thus currents in G\ (resp. Gxe) do not influence the exterior (resp.interior) of Clx. This implies: (307)

S^Pf = 0 if rM£Q'A,

S ^ O if rMcQA.

More generally, if (MA, JA) is in G^Cl^) for all X so that r x £9&j, then the field: (E,H) = 2 (EA,HA) is zero in R\£1J. Let Clyj0 be its unbounded component. Thus currents on its boundary prevent interior currents (on the boundary of Cl.) to act on points ofCl9. .

130

3 STATIONARY SCATTERING PROBLEMS

6.3. Coefficients of mutual influence of antennas If k2 is not an eigenvalue of the operator A in any bounded domain (Q, or Q'.), then we can define Calderon operators Ce>1 as above. We can write, using standard notation, for the exterior and interior domains respectively: (308)

C e - J ( (n AE| rx ) xeI ) = (nAH| r t e i ) A e I );

thus C e and C1 are matrices: C*'1 = (C*'') (309) d a .

C^Qj), C'-C^Q,,)©

- w - We have the decompositions: ©

CXCTj), but cVQj) *©€*(%).

Notice that the exterior Calderon operator for Q (the complementary set of the unbounded component of Q') always exists. This operator takes into account the mutual influence of the different bounded obstacles, i.e., the components of 7. MUTUAL INFLUENCE OF SHEETS

7.1. Introduction Let r be a (Lipschitz) connected surface which is the boundary of a bounded open set Cl in R3. Let I*A, X € I, be a partition of T. This partition can be made in order to apply a numerical method, or to decompose T into more regular pieces (r being a polyhedral surface), or into surfaces enlightened by a wave plane or in the shadow, in order to consider high frequencies problems. Then to a current J in H~1/2(div,D, corresponds the family (JA) X€l, with J. eX(div,r.) (see (87) chap. 2) with matching conditions in the case of two pieces see (89), in the more general case these matching conditions are more tedious and technical to write (obviously H""1/2(div,T) is not the sum of the spaces X(div,rx)). If we assume that each current on Tk is regular, that is in Hs(div,rA) for some positive number s, then these conditions are easy to write: Proposition 28. Let (TJ, X € I, be a partition ofY, with Tx regular. Let (JA) X € I, be regular currents on Tx (JA in Hs(div,rA) with positive s). Then (JA) X € I defines a current J in H " 1/2 (div,0 if the following matching conditions are satisfied (310)

vA. JA.vM.Jy = 0onAAM = rAnf>J, VX,n€l,

with vA the exterior unit normal in I\ to its boundary. PROOF. This results from the usual jump formula of the divergence on surfaces (the analogue to (54))

(311)

divrJ= 2 (divr JA)+ 2 exvA.JA,

131

3.7 MUTUAL INFLENCE OF SHEETS

with (divr JA) in the classical sense, EA = +1 or -1 according to the orientation of 8I\. Then we have J in L 2 t (0 and divf J in L 2 (0, thus J is in H°(div,D, and therefore inH" 1/2 (div,D. « Obviously we have an analogous proposition by changing divf into curl f , with the matching conditions: (312) VAAJA-VMAJ^0 onA^^^nf^, VX,n€l. 7.2. Electromagneticfielddue to currents on a sheet Here we study the electromagneticfieldproduced by a regular current J on a sheet r o which does not satisfy v.JaOon the boundary A of r o , and thus is not in the space H^1/2(div,ro ) (see (85) and (92) chap. 2), so that we know that (E,H) is not of locally finite energy in the neighborhood of the boundary A of r o . The electromagnetic field (E,H) produced by J is given by (96) (with M = 0). Its expression will be given thanks to the jump formula (which is a result of differential geometry, see (94), (273) and (278) in the Appendix and also for instance Schwartz [1] (IX, 3; 24)) (313)

curl J a (curl J) - v A J A 5 A , V the exterior unit normal to A in T0,

where ( ) cl is the "classical" part (on To), whereas the last term is concentrated on A. We can also prove (313) using the formula (see (310) in the Appendix) (314)

<curlJ,O d = f <J,curlO=/ r J. [|j(nAC)-curl r C , l + 2R m nAC]dr,

with C € D(R ) . Thus we can separate curl J into its transverse and tangential part. Applying the Stokes formula, we obtain the boundary term, and the expression of() cl isgivenby: (315)

<(curl J) c l , S>=/ r (curl^ J .Cn + J.| i (nAC) + 2RmJ. nAC)dT0 ,

for all C € D(R3)3. For the gradient we also have: (316) with

grad p = (grad p) d - v pA 5 A ,

(317) <(grad p) d , O = / r (grad^ p. ? - p ^

- 2Rmptn) dT0, V C € D(R3)3.

The expressions of (grad p)cl and (curl J)cl contain the transverse distributions G{p and G2J: (318) <G l P ,0 = - / r o P ^ d r o ,

<G2J,0=:/roJ^dro,

VC€Z)(R3)3,

132

3 STATIONARY SCATTERING PROBLEMS

with the properties, for all regular p and J: + (* • Gjp) € H'CRVo), ♦ (* • G2J) € H ^ R V O ) 3 , V ♦ € D(R3). Proposition 29. The electromagneticfielddue to a "regular" electric current J on a sheet TQ which does not satisfy v. J = 0 on the boundary A ofTQ is decomposed into its regular part (E »H ) and its singular part (E sing »H si ) according to: I (E, H) = (Ereg, H^g) + (Esjng, H^g), (319)

Ereg = ikZ(ZJ+gL(gradp)d), H r e g ^ ^ E

sing s - * f

PWVPA*A

+ «wd (v. JA8A)], H^g = - L (v A J A S A ).

with (320)

♦ E^and^H^cHkRVo) 3 , V*€D(R 3 ), E^andH^€C~(R\A)3, withE^W-O*}), H ^ O f l o g r ) , r*d(x,A).

and thus ♦ EandtHeH^RVj3, (321) PROOF.

V*€D(R\A).

The electricfieldE is given by (96): |E*ikZ(U+4gradLdiv r J)

(322)

= ikZ(IJ+4gradL(div r J) cl )-i|grad£(vJ A 8 A ). We also have: (323) gradL(divrJ)d = i<DgradjLp = ico**gradp = ico**[(gradp)d-vpASA]f which gives (319) for the electricfieldE. For H, this results from: (324)

H = curlU = curi**J=<&*curlJ,

and then we apply (313). Properties (320) on (E ,H ) follow from the usual regularity properties of the single layer potentials (see section 1.3.1). For ^sing'^sinP' tkis is a consequence of the properties of the Newtonian potentials relative to a charge density concentrated on a closed line (see below). We have a more precise result (325) E ^ x ) ^ . ^ ^ ^ ) ! + 0 ( l o g r ) , H stag (x) = :-^vAJ A logr + 0(l) > with a=j (x - nAx), r ■ d(x, A), nAx=the projection of x on A.

133

3.8 CURRENTS ON A LINE

7.3. Matrix elements; influence coefficients th sheet TQ9 we can decompose RJ and TJ into their regular and singular On the parts RJ = v(RJ)'reg +(RJ). , x 'sing'

TJ = (TJ)'reg +(TJ). 'sing',

(326) (RJ)rcg = 2iknA[U+gL(grad r p) c l ], (RJ)

2i siiuf " f

n A L [ ia,Vp

A*A +

(TJ)^ = - 2nAUcurl J) d>

g r 3 d (V J

" A8AM' Casing =

2 f l A UV

A J

A8A)-

When we have to calculate the matrix elements of the Calderon projectors, or of the S operator, we have to calculate matrix elements of R and T (see (102), (104) and (273)) between the sheets I\ and r (giving the mutual influence between Tk M andT) (327)

<RJ,n A J>Âť 2 <RJx,n A J >, <TJ,n A J> = 2 <TJA,n A J >, Ap

V J, J piecewise "regular" currents on T. We have four types of elements according to: i) I\nf M = 0 , ii) r A nr p has one point, iii) fA and fj, has a common line, iv)TA = T^ In cases ii) to iv) there are singular integrals to calculate for the matrix elements of T. We can use Stokes formula: <TJA,n A MM>=/r ^ gradr yQ(x - y) (JA(y) A Mp(x)) dTydTx 'A^p

= / r x r *(x-y)divry(JA(y)AMll(x))dTydrx

(328)

+/8rxr ^(x-yJv^J^AM^dLydr,, A

and then with AA = drA, the last term can be calculated by: (329)

dLy(nyAJA(y))./r <D(x-y)M(x)dTx. ^A

We refer for instance to Bendali [1], Heliot [1] for the explicit computation of such matrix elements with P{ functions. 8. FIELDS DUE TO CURRENTS ON A LINE

Let J a J $A be an electric current concentrated on a (closed, i.e. without boundary) bounded regular curve A, with JA regular, tangent to this curve. Then the electromagnetic field (E,H) due to this current in free space, which satisfies the Silver-Muller conditions at infinity is given by (44) with M = 0, that is: (330) E = io>p(A + k- 2 graddivA), H = curlA, with A = , J = ZJ = I(jeA5A),

134

3 STATIONARY SCATTERING PROBLEMS

with JA = j e A , eA unit vector tangent to A; A is called the Hertz potential. We have: (331)

divJ=§8A,

with s the usual curvilinear coordinate on A; we can write (330) in the form E = io>ji ( U + IT 2 grad L ( ^ 8A)),

(332)

H = curl (U).

We know that (E,H) cannot be of finite energy in the neighborhood of A, so we have to study the behavior of (E,H) in the neighborhood of A. This follows from the Lemma: Lemma 4. Let p dA be a distribution concentrated on the closed line A with regular density. The behavior of the Newtonian potential u 0/-p Q dA, in the neighborhood of A is given by (with x = (poc,s) in load coordinates): (333)

u ( x ) = J ? / A ] 7 - ^ p o ( y ( s ) ) d s = -p o ( J t A x)iLogp + 0(l),

nAx being the projection ofx onto A, p = d(x, A) = | x - wAx|, a.grad u(x) = - PQ(*Ax) TT~- + 0(log p), a unit vector orthogonal to A, (334)

a.grad u(x) = - PQ(*AX) ^ log f>, a = eA(s) along the tangent to A. PROOF.

(335)

First with the Lipschitz hypothesis on pQ we have: *u(x)-po(*Ax)/A]Iz^<b=/Aj3^^

Therefore the behavior of u(x) in the neighborhood of A is given by that of v(x): (336)

V(X) = P O (* A X)^W(X)

with

wfr^/^^jds;

with a < 0 < b and y(0) = 0, y*(0) » (0,0,1) = eA(0) = e 3 (choosing s = 0 for *Ax), we have:w(x) * J ^ + y J ) " 1 ' ^ * - 2 log p. The gradient of u satisfies: (337)

^(x)=^/A Ix-y^r^MsWds, y.-Xj

with p. =-^7—, r= |x - y|, and thus (with y=y(s)): (338) 4 * | | ( x ) - p o ( * A x ) / J x - y ^

135

3.8 CURRENTS ON A LINE Ifp i s C ' 1 on A, we have P„(y(s)) - Po(y(0)) = po(y) - Po(nAx) = p' (nAx) s + 0(s2), thus:

(339) 4«^(x)-p o (it A x)/ A P. |x-y|- 2 ds-p;(i. A x)/ A P. | x - y | - 2 s d s = 0(l). The gradient of w(x), given by ^ = /

| x - y | ~ 0. ds, i» 1 to 3, satisfies:

| ^ « - 2 ^ f o r i = i , 2, Xj orthogonal to A, 04 = (x-it A x)./p, (340)

| ~ « 0 along the tangent to A (03 is to first order an odd function of s). Then we have for i = 3: (341)

/ ^ | x - y | - 2 sds .

^ ^ d y

= ^ ^ d t . -21ogp,

and for i= 1, 2:

(34iy / A Mx-y|- 2 sds * ^i^f^s^-^i^^^ra^- T? from which we obtain (334). &

Proposition 30. The electromagneticfield(E,H) due to a line electric current J on a closed bounded curve, by (332) has the following behavior in the neighborhood of the line, with notations ofLemma 4 and with local coordinates x = (p<x,s): Etr(x) « - i ^ k - 2 i | ^ ,

Htr(x) ~ ^ ( « A J e s )

(342)

Es(x) = i < ^ l o g p [ - j - k - ^ ^ ] ,

Hs(x) = (Xlogp),

with J = jes 5A, es = eA(s) a unit vector along the line A, and (Es, Hs) the components of (E, H) along es, (Etr, Htr) the transverse components. PROOF. Formula (332) for H gives: B ikr

(343)

H(x)=J A §f(|JAJe r )ds,

with f j . £

, r=|x-y|.

Then we obtain (342) for H, and also for E due to Lemma 4. ®

136

3 STATIONARY SCATTERING PROBLEMS

The electromagnetic field (E,H) due to a line current J has the following behavior at infinity: (344)

E(ro) * icoM [J(ko) - a (a. J(ka)] Q(r), a e S2, H(ra)sikaA?(ka)«(r)

with J(ka) * <&-*** J> =/ A c -

t e

forr= |x| -♦<», ^ j (s) es ds.

These formulas are an easy consequence of (30), and of: (345)

|(divJXka) = <e- a t o x ,divJ> = ika<e- itex ,J>*ikaJ(ka), (curl J)(ka) = <e" ikttX , curl J> = ika A$(ka).

Remark 21 • So far we have described a radiating antenna. The modelling by a line of a thin receiving antenna with a very conducting medium and the modelling of the induced current by a Dirac distribution concentrated on a line (according to the Faraday's effect) is outside the scope of this book.

9 . SCATTERING PROBLEMS BY A CHIRAL OBSTACLE

We consider the scattering problem of section 5, but with a linear chiral obstacle (see chap. 1.3.3) instead of a dielectric medium. Then we have to find an electromagnetic field described by (D,B), (E,H) in the obstacle Cl satisfying: (346)

i)curlH + io>D = 0, ii) - curlE + icoB = 0 inQ,

with the constitutive relations (see (96) chap. 1, where e, p, e, £ may be matrices): (347)

i)D = cE + eH, ii)B = jiE + nH,

with hypothesis HI8) chap. 1 of a dissipative medium, and with the boundary conditions (262), giving (265) on E. We emphasize that the boundary conditions are unchanged with the constitutive relations (at least if the medium is at rest), because they are due to the curl operator! We can easily prove uniqueness (at most) of the solution of this problem as follows.

3.9 CHIRAL OBSTACLE

137

Multiply (346)i) by E and ii) by H, integrate and add, we obtain: (348)

I R e ~ to KD'E> + <B'H)1 = Re/ 0 (curl H. E - curl E. H)dx I =-Re/adiv(EAH)dx=-Re/rn.EAHdT.

To prove uniqueness of the solution, we can take Ej = Hj = 0 and thus Re/rn.EAHdr = Re/rnAE.HdT=Re/rnAErHrdr>0 (see (202)). But from the dissipativity hypothesis H18) chap. 1 the l.h.s. of (348) is strictly positive except for E = H = 0, which proves uniqueness! Then proving that this problem depends on the Fredholm alternative (like in section (4.8.2)) implies existence and uniqueness of the solution. Here we develop a variational method with (slightly) different hypotheses. We write Maxwell equations (346) with (347) with E and H only: (349)

i) curl H + io)€ H + i<oc E = 0 ii) - curl E + i<oji E + i<o|i H = 0 in Q.

We can eliminate H applying -(curl + mt) (icon)"1 to (349)ii) adding to i) (350)

- (curl + iweXiw)*)" (- curl + ia>fi)E + icoe E = 0.

It is better to use a variational framework. We define the sesquilinear form (351)

a(E, F) = - f [(top)" ^curl E - ic*p E). (curl F + ia> xz F) + io>e E . F ] dx -/rCc(nAE).FdT,

with l€ the transposed matrix of?, and E, F in H(curl, Cl). The scattering problem is

(352) a(E,F)=/ r f,. FdT, VF€H(curi,n). We assume that (353) ? « ? • , and Re(- i<oic£ . {) > a |£| 2 , V U C 3 , forK = €, |if with a > 0 (see H17 chap. 1). Thanks to (201) (and to Remark 8 chap. 1) we have (354)

Re a(E, E) 2> a [llcurl E - ia>? EH2 + HEII2],

V E € H(curl,Q).

But we can obtain easily: there are positive numbers ot0, f>, so that (355)

llcurl E - icojl EH2 < a0 [llcurl Ell2 + IIEII2] < P [llcurl E - icojl EH2 + IIEII2]

for all E € H(curl,Q), and thus the norms:

138

3 STATIONARY SCATTERING PROBLEMS

E l , 1?[Ilcurl E - icoP EH2 ♦ 1EII2]1'2 and IIE«H(curla) d=*f[iicurl Ell2 + IE!2]1'2 are equivalent. Therefore the sesquilinear form a(E,F) is coercive on H(curl,Q). The Lax-Milgram lemma implies existence and uniqueness of the solution of the problem (352) like in section S. 1. Remark 22. The constitutive relations are often written in different forms, for instance: (356)

D = c c E+Y cm B,

H^B-rmcE,

with y _ - - y _ , which is equivalent to (347) with: (357)

P - ^ S - ^ y ^ «^c^cYme' * = W c = - 5 '

1 0 . CONCLUSION ON THE CALDERON OPERATORS FOR SCATTERING PROBLEMS

Let a be a bounded domain occupied by a homogeneous linear isotropic dielectric with permittivity e and permeability p. Then all the electromagnetic properties of this obstacle (with respect to an incident wave at frequency m) are contained in the interior Calderon operator C* (see (211)), if © is a "regular" frequency, that is k2 = o>2€|i is not an eigenvalue of the operator A defined by: Au»-Au,D(A)={u€L 2 (Q) 3 > curlu€L 2 (Q) 3 ,divu = 0,nAu| r =0}. Using a more physical minded terminology we can call it a surface admittance operator. The essential property of C1 is that its graph corresponds (up to the vector product with the normal) to the set of tangential boundary values of the electromagnetic field in Q. We recall also the positivity property (212). Less usual for physicists is the similar notion for the exterior domain, which is generally occupied by free space. Then all the electromagnetic properties of the exterior domain (at frequency CD) are contained in the exterior Calderon operator C e : this is a surface admittance operator for the free space domain; when a> = 0, this corresponds to the "usual" notion of a capacity operator. Like for C1 the essential property of C c is that its graph corresponds (up to the vector product with the normal) to the set of tangential boundary values of the electromagnetic field in the exterior domain Q'(taking into account the Silver-MQller condition and the local finite energy up to the boundary T of Qv). We recall also the positivity property (201). The main differences between C\ C c and the Calderon projectors P., P c are: i) P. and P keep informations on both sides of T, and exist in all frequencies, ii) P. and P c are integral operators (whereas C1 and Cc are pseudodifferential operators ofzeroth order, for this notion, see for instance Hormander [1], Gilkey [1], Taylor [1]),

3.10 CONCLUSIONONCALDERONOPERATORS

139

Hi) the range spaces G c = Im P e and G l * Im P l are (up to a minus sign) respectively the graph of C e and C1 for regular frequencies. Let r be a regular surface which is the boundary of a bounded domain Cl. The space Y of magnetic and electric currents (M,J) on T which produce an electromagnetic field (E,H) (at angular frequency <o) in free space, with locally finite energy on both sides of T and satisfying the outgoing wave condition at infinity, is decomposed into spaces corresponding to tangential interior and exterior boundary values of the electromagnetic field (E,H) (up to the vector product with the normal) G l and G c 0 according to: (358) Y = H"1/2(div,r)x H"1/2(div,r) = GoeGj,. Now let the interior domain Cl be occupied by a medium with characteristics (e^jij) at a), and let Gl{ be the space corresponding to boundary values of the electromagnetic field in Cl. Let (E., Hj) be an incident wave at frequency o>. The scattering problem is to find the reflected wave (Ef, Hf) and the transmitted wave (Et, Ht) inside Cl so that: (359)

n A Ej = n A Et - n A Er, n A HJ = n A Ht - n A H r ,

that is to decompose space Y into the direct sum: (360) Y^G^eG^. Using the interior Calderon projector P? for the exterior wavenumber k0, we see from (271) section 3.5 that this problem is reduced to: the restriction off? to G\ is an isomorphism from G\ onto Gj,, i.e., there exists a > 0 such that IIP?(M, J)ll > a ||(M, J)H, V (M, J) € G/ . We can use the Calderon operators C l,e to solve numerically scattering problems by using special basis. We develop this in the Helmholtz case, for C c = Cj. Let ^ ( x ) as 4>(x - XQ) be the outgoing elementary solution of the Helmholtz equation centered at XQ. Let F be a subset of R3. We obtain the following, setting: (361)

MF-K«Sblr^l^«0€F|.

Proposition 31. We assume that F is a dense subset of the boundary TQ of an open subset Cl of Cl, strictly contained in Q, and such that - k2 is not an eigenvalue of the Dirichlet Laplacian in C1Q. Then Af- is a total family in G e = G(Ce), i.e. the vector space [Mp] generated by finite linear combinations of elements ofMF is dense in G e . Proposition 31 implies that given (u^u 1 ) in G , (thus u1 = Ceu°) and n > 0, there exist (x.) in To and complex numbers (a.) such that (362)

* llu -2aj*\»:Sn, 0

*i ta^ZcjWLysn.

140

3 STATIONARYSCATTERING PROBLEMS

of Proposition 31 (adapted from Petit-Cadilhac [2]). We show that the polar set [Mp]° of [MFJ for the pairing (143) is equal to the polar set of Gc, which is Ge. Let (vV 1 ) be in [MF]°; (v°\vl) satisfies: PROOF

(363) thus

<*%v^-<^lriS-0,

Vx 0 €F,

(363)'

/ r (v 1 (y)«Ky-x o ).v o (y)^(y.x o ))dr y S 0 > V x 0 € F .

Settingf=(vl Sr+div^nfij-)), andu=$«f, (363)*is:u0^)-0, Vx^,€F. Thus u satisfies the Helmholtz equation in Q, hence in Q0> with u | . Since - k2 is not an eigenvalue of the Dirichlet Laplacian in Q0 by hypothesis, this implies that u = 0 in Q0> thus in Q. Therefore (v°,v1) € Gc, i.e. the Proposition. ®

For the interior Calderon operator we have the analogue of Proposition 31: Let F be a dense set in the boundary Tx of an open set Qx with Q strictly contained in Qj. Then M^isa totalfamily in G. ■ 0(0*). Instead of elementary solutions we can use plane waves with wavenumber k for G.; we prove like in (363)' (but with Rellich lemma) (364) thesetM^{(eikaJi\T9 ika.ne****^), acS 2 } is a totalfamily in Gv Numerical methods based on expansions with elementary solutions such as (361) are called Fictitious Sources Methods (see Petit-Cadilhac [2]). 0 Calderon operators are useful for modelling many physical situations (see section 3.9 for example). We give another example. Excitation of a proper mode of a cavity by an incident wave. We assume that the cavity Cl is occupied by the free space (or a conservative medium) and is bounded by a perfectly conducting medium (occupying a bounded domain Qx) with a hole TQ so that an exterior incident wave (E{y Hj) at frequency <D can go into the cavity through the hole. Then the electromagnetic field (E,H) in Q is a solution withfiniteenergy of the Maxwell equations (365)

i)curlH + icncE = 0,

ii)-curlE + MiH«0 inQ.

Let T{ be the boundary of the conducting medium for Q. Let Cc be the Calderon operator exterior to the domain Q occupied by the cavity and the conducting medium, with boundary f. Then let C* be defined by c;<nAE|r)-CWE|j)|r.

3.10 CONCLUSION ON CALDERON OPERA TORS

141

Recall that n A E is zero on I\r o . With the notations (SS), (86) of chap. 2 we have 1/2 H A E Ll €H; (div,r0), andC^(nAEL)€X(div,r 0 ). l o

Then (E,H) must satisfy the boundary conditions ji)nAE| r =0, '1

(366)

|ii)nAH|. - ^ ( B A E L ) - ^ ! . , with f ^ n A H , - C c (nAE,U).

We have proved (see section 5) that the scattering problem for a conducting obstacle has a unique solution and thus problem (365), (366) also has a unique solution. Now we assume that k2 = o>2ey is a simple eigenvalue for the operator A (see (147) chap.2). Let (EQ,H0) be a corresponding eigenmode with unit energy / a ( e | E 0 | 2 + n|H 0 | 2 )dx=l. Then (E,H) has an orthogonal decomposition: (E,H) = c 0 (E0,H0) + (E,H), with (E,H) orthogonal to (E0,H0). Then in many physical situations we are interested either to generate at best the eigenmode (E0,HQ), that is to maximize the modulus of c over all possible holes and incident waves with given energy or to minimize it in order to protect the interior from all incident beams. We note that the problem:find(E,H) satisfying Maxwell equations i) curl H + i(i>€ E = J,

ii) - curl E + io>y H = 0 in Q,

with the boundary condition on T = 8Q: nAE| =0 or nAE| =nAE 0 | and with given J and EQ, depends on the Fredholm alternative and does not generate eigenmodes! Remark 23. Interior Calderon operator and its Fourier transform. Let (E, H) be an electromagneticfieldin a domain Cl occupied by a medium with (complex) characteristics (c, p). Let M = n A E | , J = - n A H | . Generalizing (31) to complex k, we have (367)

- ika A M(ka) + ai>M riaJ(ka) = 0

that is also, with the (complex) impedance Z = cop/k, (367)'

naJ(ka) = Z"1 a A M(ka), with J = - C1 M.

142

3 STATIONARY SCATTERING PROBLEMS

This can be obtained from the behavior at infinity of the extension of (E, H) by 0 out of Q. In other words, the mapping: (M,-J) —• j(ka) (and &(ka)), given by (31) vanish on the space G1 (for the wavenumber k only). Of course we have similar properties in the scalar case for Heimholtz equation. 0 Remark 24. The Calderon operator for Heimholtz or for the Laplacian is also called the Dirichlet-Neumann operator or the Poincare-Stekloff operator. The Calderon projector for a domain Q is defined for instance in Chazarain-Piriou [1]. • Finally note that the Calderon operators and projectors have all the symmetries of the problem. The Calderon projectors well adapted to computation when we have a few number of homogeneous and linear media. 1 1 . MULTIPOLE EXPANSIONS. RAYLEIGH SERIES

Here we develop essentially a spectral method, based on expansions which are very familiar to physicists. But there are fine questions of convergence, well studied by Muller [1], which we follow. We begin by the simplest case, that is the scalar case with Heimholtz equation in two dimensions. 11.1. Multipole expansions for Heimholtz in R2 Let u be a solution of the (homogeneous) Heimholtz equation for the wavenumber k > 0 in a domain Q in R2. We assume that 0 is in Q, and thus there is a disc Ba of radius a contained in Q. Since u is regular on circles Cf of radius r ^ a, its restriction to Cf may be developed in Fourier series, and we have, in polar coordinates: (368)

u(r,9) = 2 cn(f)e™, n€Z

where cn(r) satisfies the differential equation: (369)

1^+1 ^

(k2-^)cn =0

for r<a,

and thus c n is proportional to J n the Bessel function of order n: (370)

cn(r) = cnJn(kr), withc n €C.

Furthermore we have: (371) 2 Ic n (r)| 2 *2 |c n J n (kr)! 2 <+oo for r * a . When u is a solution of Heimholtz equation in the domain Q* (the complement of a bounded domain Q), and satisfies a Sommerfe i condition at infinity, cn(r) is proportional to the Hankel function of order n:

3.11 MULTIPOLE EXPANSIONS

(370)'

143

cn(r) = c n H f W with cn € C,

K = 1 for the outgoing condition, K * 2 for the incoming condition. We also have (371) (replacing the Bessel functions by the Hankel functions) for r > a. Let un be defined by: (372) un(r,0) = Jn(kr) e™ (resp. H J W ) e™). Thus un satisfies Helmholtz equation in R2 (resp. R2\{0}, and (A + k2)un = p is concentrated at the origin); u = I cnun is called the multipole expansion of u (or also a Rayleigh series). If u does not satisfy the homogeneous Helmholtz equation in the whole space R (or R2\{0}), then we see that the expansion u = I cnun cannot be convergent everywhere. Thus we have the question offindingthe domain of convergence of this series. The answer is similar to that for entire series I rV110: there is a radius of convergence p so that the series is convergent inside (resp. outside) the disc B for the expansion with Bessel functions (resp. Hankel functions), and is divergent outside (resp. inside) this disc. We recall that the Rayleigh hypothesis is that these series are convergent everywhere inside (resp. outside) Q, for any domain Q; we know that this Rayleigh hypothesis is wrong. We study this question of radius of convergence for three dimensions only. 11.2. Multipole expansions for Helmholtz in R First we recall some essential properties of spherical harmonics. For the study of such properties, we refer to Dautray-Lions [1] chap. 2.7.3 and 9.B 1.1, to Muller [1], and to Stein-Weiss [1]. These properties originate from the representation of the rotation group in R3. 2

We use a spherical coordinate system; let a = (aiya29a3) € S , with <Xj = sin0cos$, a2 = sin0sin4>, a3 = cos0,

O<0<n, O<0<2TI.

The set of functions ( ^ ■), n€N, - n ^ j < n , defined with the Legendre functions (373) Knj(a)==(2nrl/2p|l(cos0)e1J0(also often denoted by Y/m(0,0), / = n, m=j) which are the traces on the unit sphere S2 of the homogeneous harmonic polynomials in R3, is an onhonormal basis o/L2(S2). Furthermore they are the eigenfunctions of the Laplace-Beitrami operator on S2 according to A

s2Kn,j(«)

= -n(n + l)K n<j (a),

and thus the eigenspace Vn corresponding to the eigenvalue n(n + 1) has 2n + 1 dimensions. The elements of Vn are called spherical harmonics of order n and often denoted by Kfl. Using spherical harmonics allows us to find the solutions of the Helmholtz equation with separate variables (r,a) thanks to the usual expression of the Laplacian in (polar) spherical coordinates:

144

(374)

3 STATIONARY SCATTERING PROBLEMS

Au(nx)=|^+?f+^Aau,

w i t h A , , ^ , r= |x|, « - ? ,

we see that cn(r)K„(o) is a solution of the Heimholtz equation if y=cn(r) satisfies: d?

(

-j—)y=0.

The solutions of this equation are well known: they are the "Bessel" function jn(kr) (for the regular solution up to 0) and the Hankel functions hn(kr), related to the previous J„(kr) and HQ(kr) by: (376)

j„(D=(£) 1/2 J ,(r),

hJt)(i)-(£),/2H<"),<i),

wtthic=l,2.

Then let ^ be the function: (377)

uE(w)-hf)fla)Kll(«) (wsp.^kr^a)).

Expansions of the form Iu ft are called Rayleigh series (or also multipole expansions). We recall the main property of convergence of these series (see Muller [1], Theorems 11, 12) which will be a consequence of the following Lemmas. Theorem 9. Let (Kn) be a sequence ofspherical Harmonics. Let (378)

c i . / ^ l ^ a ) ! 2 * , c^^O.

We assume that there exists rQ so that the series: (379)

2 |h£W 0 )j 2 c* < + *

Then the series: (380) u(ia) - 2 hfW) ^(a) n "

(«*p. 2 U n 0» o )| 2 i^<+ » ) . (resp. u(ra) = 2 jJM ^(a)) n

converges absolutely and uniformly on every compact set K in R \Bt (resp. in Br ). This is equivalent to: if the series (380) is convergent in L2( Sr ), then it is also convergent on every compact set K as above. Furthermore it is also convergent in all "reasonable" topologies, for all Sobolev spaces for instance. This implies that the series u given by (380) satisfies the Heimholtz equation with wavenumberk. In the "exterior" case, that is for the expansion with Hankel functions, we will verify that such a expansion satisfies the Sommerfeld condition (outgoing or incoming for K s 1 or 2).

3.11 MULTIPOLE EXPANSIONS

145

Lemma 5. On any finite interval [a,b] with 0 < a < b < oo (resp. [0,DJ) we have (denoting by T the Eulerfunction): (381) h?(r) * < - l ) « ^ r ( n 4 uniformlyforn—oo, a ^ r

(?f

(resp. y

^ H S T ^ (§)\

(resp. 0 : £ r < D ) and then:

<*»• M^ -

$-c«'——

Corollary 4. IWift hypotheses (379) die coefficients c 0/ f/ie expansions (380) satisfy respectively fl

Lemma 6. ITie spherical harmonics satisfy the following inequality: (383)

iVa)! ^^!/^!^)! ^,

VneN.

PROOF of theorem 9. From the above lemmas 5, 6, there are an index N and a constant C such that for n > N we have: Q

I h^wK^coi

s c 42inrr ( ^ )

(res . i ^ W K ^ C O I < c 42KTT [ £ ] ), n

which implies the convergence of the series in the conditions of the theorem. <g> 2

Remark 25. It is possible to replace the topology of the L norm on the sphere of radius r by some other topology with the same convergence results; for instance we can assume that the expansion (380) converges in the sense of distributions on the sphere. In R this is equivalent to: there exists a number p such that Q

c |J (lcr )| ( l + n r - 0 w h e n | n | - o o . n

Then we define the radius ofconvergence of the series (with (378)) by: Ree^taffr, 2

(384)

IhSWc^+oo),

R c i ^ s u p f r , 2 |j (kr)| 2< oo}. n

146

3 STATIONARY SCATTERING PROBLEMS

Then if Q is a bounded obstacle which contains the origin and if u satisfies the Helmholtz equation outside O (resp. inside Cl) with the Sommerfeld condition at infinity, then u has an expansion given by (380) in the exterior (resp. interior) of all balls B which contain (resp. is contained in) Q, and thus: R ^ R ^ s u p |x| f

(resp.R i ^R m « infjx|).

When u is due to a distribution f with compact support (see (1)), we have a similar result with (supp f) replacing Q in the expression of RM. EXAMPLE

1. Elementary solution relative to a point xQ not at the origin.

Let <&• «<fr(x - Xp) be the elementary outgoing solution of: (385)

Au + k*u»-«^.

Then we can prove that $(x - XQ) has the expansions |^2(2n+l)Jn(kr0)h!ll)(kr)PIl(aa0) (386)

ifr>r 0 ,

*(x-x 0 ) = § | (2n+ 1) jn(kr) hjJWo) Pn(aa0) if r < r0,

withr= |x|, r0= ^ l , a=sx/r, a0 = x</ro> Pn the Legendre polynomial of degree n. We can find these formulas, using that $(x-x Q ) has a Rayleigh expansion with Bessel functions for r < rQ and with Hankel functions for r> r0, and then using that the jumps of $(x-x Q ) and its normal derivative across the sphere of radius r are (in the sense of distributions on the sphere): (387) W x - j g ] s

*m\W-Vm\M*09

[

g p ^ ] s =A*< a - a o>

with

^00-^0) | ^ P r f - ^ n

which is due to the relations: (388)

«^«-^ o (r)8(«-cg f

*a-a0)= | ^t^Pn(aa0).

This follows from the relation, for all regular functions f on the unit sphere: (389)

«« 0 ) . | / § 2 ^ J ± I Pft(oa0) f(a) da

(see for instance Muller [1] p. 53). Then we can see that thefirstseries in (386) is divergent for r < rQ, and the second series for r > rQ.

147

3.11 MULTIPOLE EXPANSIONS

Indeed the generic term Sn of thefirstseries is equivalent (thanks to lemma 5), for high n, to: S n ÂŤ c [ ^ J Pn(t), t=a.cc0, Caconstant. Then the convergence is a consequence of the usual formula: (390)

(1 - 2zt + z2)- m = 2 PnW *> n

which defines a convergent series when | z| < 1, since the zeros of the polynomial ( l - 2 z t + z2)arez = eieande"iflfort = cos8. Thus the radius of convergence p of the entire series (390) is equal to 1 that is: P = limsup|P n (t)| 1/n =l, V t â&#x201A;Ź [ - l , l ] . *-" * EXAMPLE 2. Entire solutions of the Helmholtz equation. Plane waves. Expansions of the form (380) (with Bessel functions) with an infinite radius of convergence define entire solutions of the Helmholtz equation. An example is given by plane waves, where the expansions are uniformly convergent on compact sets (391) e t o o X = e*"*0" = 2 in(2n + 1)Pn(a0.a)j (kr), witha0 eS2. n

Replacing in (391) the Bessel functions by Hankel functions, and using:

Jn(kr)-^(l4V) + h?(kr)), we might think that we obtain a decomposition of the plane wave into the sum of an outgoing and an incoming waves, but this is false since the expansions 2 in(2n+l)Pn(a0.a)hSc)(kr),K=l,2, ^ ^ / ^ ( P ^ a . a ^ d a ^ ^ J j are nowhere convergent! Then before studying the behavior at infinity of Rayleigh expansions, we will answer the question: what is the source of a wave un given by (377)? 11.3. The source of a wave u n

First note that un(m) = h j J W ^ a ) satisfies (A + k2) un * 0 in ZT(R\O). Furthermore the Hankel function of order n being equivalent in the neighborhood of 0 (up to a constant in r) to r~n, it can be extended (using finite parts, see Schwartz [1]) to a distribution on R3; thus (392)

- P n " ( A + k2)ull (in^XR3))

148

3 STATIONARY SCATTERING PROBLEMS

is a distribution with support {0}. Thus p is of the form p = Qn(D) 5, with Q a polynomial. Since u satisfies the outgoing condition at infinity, it is given by the convolution product (393)

u n «*»Q n (D)8 = Qn(D)*.

Thus we have to find O so that: (393)' ( ^ ( D ^ ^ ^ D J ^ ^ h i V ) K^a). We can see that this implies that Qn is a homogeneous polynomial of order n, so that its restriction to the unit sphere is proportional to Kfl:

094) Qn(it)=Qnam«)=vn m x w We can also find this result by Fourier transformation (see for instance SteinWeiss [1] p. 158, thanks to the commutation of the Fourier transformation with rotations !). Then vn is easily obtained from the behavior at infinity of u n and of the Hankel function (see Muller [1] p. 74, Petiau [1]); we have: u ^ O ^ V ^ a ) *^e-^^^a)^^)^),

r-oo,

with a e S , therefore: (395)

3n(ka) = Q^ika)=£ (-

if^K^a),

tbm v =

n lf [V] ' andthus:

(395)'

Q ^ l ^ f

( ^ ]

K^a).

We can also find another expression for pn using distributions in polar coordinates up to 0; for this we must specify the space of distributions. Space of distributions in polar coordinates in Rn\{0} and Rn. The mapping h: x —» (r,a) is a diffeomorphism from Rn\{0} onto R+ x S n ~ l (with R+ = (0, oo)) which allows us to identify distributions on Rn\{0} with distributions on R + x S n " l . But here we have precisely distributions on Rn and not only on Rn\{0}. Let u be a regular function on Rn. Then we denote: (396)

U(r,a) = u(w) = u(x),i.e., U - u o l T 1 .

Lemma 7. Let u be a regular (C°°(Rn)) function. Then U satisfies:

3.11 MULTIPOLE EXPANSIONS

(397)

|¥(0,a)=

149

rf« X D\i(0),

VX = (X1,..,Xn)eNn,

irftft |X| 8 ^ + ,,. + ^ , XlsXi!...^!. Thus —jr(0,a) & a sphericalharmonic oforder p anrf U(Ota) = u(0), & independent ofa, PROOF.

We have, with standard notation:

(398)

f g (r,a) =, 2 DjU(x).^ = 2 OjDjiKx),

with)^ = raj, Djssg^, hence i) . Then we easily verify that: (399)

(2«jD/u= j

2 a,

a: & ...R u,

ii-.Jp1

hence (397).

Then we see that, if u is a C°°(Rn) function, U is also a Coo([0,oo)xSn""1) function. We define its extension to R x S n ~ l (also denoted by U) by (400)

U(-r,a) = U(r,-a),

Vr>0, Va€S n " ! .

Lemma 8. Let u be a regular (C°°(Rn)) function. Then its extension U to R x S n ~ l defined by (400) (and (396)) is such that: U€C 00 (RxS n " 1 ); furtheimore u € D(Rn) (resp. S(Rn)) implies U € D(R x Sn~l) (resp. S(R x Sn~*)). PROOF.

We only have to verify that lim 9?U(r,a) = lim 8{?U(r,a), which follows

from (397) and (400). Lemma 9. Lef u te iVi D(R3) or 5(R3). I7ie/i U(r,a) Aas an expansion in spherical harmonics (401)

U(r,a)« 2 Un ^ K ^ a ) ,

n € N, - n < j < n,

n,j

such that each U n j, satisfies: (401)'i) and also

Unj(0) = 0, . . . , - ^ ( 0 ) = 0, p = 0 t o n - l ,

apu •

(401)'ii)

- ^ ( 0 ) = 0 if(-l) n *(-l) p .

ISO

3 STATIONARY SCATTERING PROBLEMS

PROOF. (401)'i) is a simple consequence of Lemma 7, since we have: -57!l<0)-/a2S<0)K^a)d»-1?

ffDPu(0)/s2a*Knij(a)da = 0.

We also have K ^ - a M - l f K^a), with (400) which implies (401)'ii). Then (401)' with U n j € C^R*) implies that Unj(r) - i^gft 2 ), with Cnj € C~(R). Notation 1. Let DQ(R x Sn~ *) and SQ(R x S n ~') te *Ae subspaces of/unctions U /nD(RxSII"1)iwrfS(RxSn-l)wirA(400)aiirf: gpTj

(401)"

—jr (0,a) & a spherical harmonic oforder p.

We can identify the spaces of regular functions D(Rn) and 5(Rn) with the spaces D0(R x Sn~ *) and SQ(R x Sn~1), and then their dual spaces for the duality: <\J9&mf

jUfaaWw)^1

drda, resp. <U,0>o=s/R+xSll.1U(r,a)#(r,a)dida

with subspaces of distributions on R x S n ~ l with support in [0, oo). Notation 2. DQ90(RxSn"l)

We denote by DQf(RxSR~l) f

andSo 0(RxS "*))

and SQf(RxSn'1)

(respectively

these subspaces.

A distribution U on R x S n ~ l induces a distribution u on Rn by <u,$> » <U,$>, with $(i%a) = #(ra), but the mapping p: U -♦ u € Z>0(Rx Sn~l) or SQ(R x S n " l ), has a kernel. In the space D* (RxS n ~ ! ) of distributions with support in [0,oo) (with pairing <,>), this kernel is the space of distributions U so that: U(r,a) = 2 SW(r)Tv(a), with <Tv(a), Kp(a)> = 0 for every spherical harmonic Kp of order p, v * p + n - 1. Note that if T is a distribution on Rn which is concentrated on (0}, it is a finite sum of Dirac derivatives, thus for n = 3: T-J^dJK^a)-!

^c^pS^xXwithc^^/^KptaJda,

in 2>0Q(R x S2) or D'(R3), with Kp a spherical harmonic of order p. Thus8(r)=8(0)(r)=4ii8(x), 5 ( 1 ) ( r ) c ^ J | ( x ) . Lemma 10. Let u^ro)« h ^ W ^ a ) in R3. Then un satisfies: (402)

(A + k ^ u ^ - p

151

3.11 MULT1POLEEXPANSIONS

with p ^ - v / ^ K ^ c O i n Z ^ R x S 2 ) , P B - - f / ^ W K ^ a ) inZ^<RxS2), where n

PROOF.

(403) with:

j (2n+Q! ^ 1 ^

„ '

„ i (2n+Q! „ " kn!(n+2)! ( - 2k) '

Let ♦ € D(R3). Then we can define using Lemma 9: L » tf <(A + k2) un> ♦>=<un, (A + k2) ♦> = Um £(♦),

where * n (r)=/ 2 *(ra) K^a) da. Then integrate twice by part. We obtain: (405)

£(♦)— [ A i V ) ^ ! ) - i2 ( ^ W ^ r ) ] ^ = a-(n-1)[|(r»XV))^(r)-hi1)(kr)|<r»+V))]r=a-

Using Lemma 9 and the recurrence formula for Hankel function:

(406)

| (i^hjfti)) = r^h^fr),

we get that the first term in (405) is equivalent to a2 when a —> 0. For the second term, using equivalence (see Abramovitz-Stegun [1] p. 437) (407)

h?>(W-

F(2n+I)

(2n)!

(407)

h, (ka) -

(ka)n+l2nr(n+1)

» (ka) n + l 2 » n , .

and also Lemma 9, we have (408)

^^—^--^-^-(O),

which gives us Lemma 10. At the first order we verify that h^ (kr)» ^ O(r). We note that for k = 0, we have: (409)

A(r-(n^1)Kn(a)) = . p n ,

with On - " T

8 ( n >

« ^(a) inD^toS 2 ), p„ = - ^ 8 ( n + 2 ) ( r ) !£„(«) inZ);(R*S2). ®

152

3 STATIONARY SCATTERING PROBLEMS

11.4. Multipole expansions and analytical functional From Lemma 10 we understand the expression "multipole expansion" for the Hankel functions expansion (376) of the solution u of the Helmholtz equation (1): each term u of the expansion is due to a distribution p of order n concentrated at the origin. The sum p m =Ip n cannot be a distribution, but we can define it as an analyticfunctional: Definition 6. Analytic functional. An analytic functional is a dual form of the space A(Cn) of entire analytic functions on Cn (or holomorphic functions on Cn) equipped with the topology of uniform convergence on compact sets. We denote by A'(Cn) the space of analyticfunctional. That pm is an analytic functional results from the Paley-Wiener Theorem for analytical functional, see below and for instance Hormander [2], [3], Treves [1] Ex. 22.7. We recall some notions Definition 7. An analytic functional y in Cn is said to be carried by a compact set K(orKis a carrier for p) if for every neighborhood <aofK there is a constant C such that (410)

|p(0|:£C w sup|f|,

Vf€A(C n ).

Then we define the Fourier-Laplace (or Fourier-Borel) transform of an analytic functional ji in Cn by (411)

0(0 = FLVL(C) = <n2, e" * <2 ^>,

with C € C n , <z,0 = 2 z,C..

For all compact set K in Cn, we define the convex function on C n (412)

H K (t)=sup Re<z,C>. zeK

WhenKistheballK = BainRn,thenHK(C) = a|C , |, wfthr-(C'i,...,C'n)-ReC. Definition 8. Let a a (a^.. .,an) with a. > 0. We denote by Exp (a) the space of the entire analytic functions of exponential type a, i.e., the space of the entire functions on Cn such that there is a constant A(f) so that: |f(z)|exp(-.a 1 |z 1 |....-a n |z n |)^A(f). We denote by Exp the vector space of all entire functions on Cn which are of exponential type, i. e., the union ofall the spaces Exp (a), with a. > 0. The "Paley-Wiener" theorem for analytical functional is:

3.11 MULTIPOLE EXPANSIONS

153

Theorem 10. If\i is an analyticfunctional in C n carried by the compact set K then its Fourier-Laplace transform is an entire analytic function of exponential type, and more precisely: for every 5 > 0, there is a constant C5 such that (413)

|£(t)| <C 5 exp(H K (-it) + S|t|),

VC€C n .

Conversely ifK is a convex compact set and (an entire function satisfying (413) for every 6 > 0, there exists an analytical functional \x carried by K whose FourierLaplace transform is f. Then we consider the expansion

(4i4)

r « - 2 <yw-1 <yMy|«>- ? t ( n r 1 ) " ^ * *

where we assume that (cn), satisfies (379), with c n = /

| K^oc) | da.

Theorem 11 • The expansion (414) defines a harmonic function on R3 satisfying (415)

|p a n (y)|=o(re r V 2 )

forr= |y|-oo,

y€R

which can be extended to an entire analytic function (on C3 of exponential type. Thus fis the Fourier-Laplace transform of an analytical functional p*1 so that i) p M is the source of a wave u defined by (380) or also by (416)

u(x) = <pan(z),^(x-z)>

withx€R 3 ,

(x^R^,

ii) u is an outgoing wave, and its behavior at infinity is given by (417)

u(ra) s O(r) p an(ka), forr-> «>.

Furthermore the radius of convergence of the series (380) is given by: (418)Rw==RMd=Ifmnsup[^^

Remark 26. Note that (416) is a way to write the Helmholtz equation (A + k2)u » pan, by the convolution product of 0 with pan. But as a function of z, 0(x - z) is analytical only if z * x. Thus (416) has to be feasible, for instance if pan can be identified to an analytical functional on an open set Q in C3 which does not contain x. Anyway we can define (416) by taking the limit of the series Z <pn,<D(x - .)> and thus we obtain the series (380).

154

3 STATIONARY SCATTERING PROBLEMS

Remaik 27. Let v be a hannonic function of exponential type (a), that is, for every 8 there is a constant C s such that: (419)

|v(y)| sC 8 exp «a+5)), r= |y|.

an Then (415) implies: p (y) is of exponential type a, with a = r 0 /2. 0 For the proof of Theorem 11, we use the lemma: Lemma 11. Lett be a harmonic function on Rn of exponential type. Then f is the restriction to Rn of a unique analyticfunction on Cn ofexponential type. PROOF for n at 3. We use the inequality (due to the Poisson formula), see Chazarain-Piriou [1] p. 23, for every harmonic function f:

(420)

|8 a ff0)|£(3n) n r n sup |f(x)|,

n=|a|,

x€S r

with S r the sphere of radius r. Thus for z € C3, |z|<r 0

(421)

12^3^(0)^1 < 2 ( 2 a4(3n) n f$)r- t t sup |f(x)| , a! * |«|«n ~ x€S r n X(3nf±(3t 0) r«sup ni

x€S r

|f(x)|st2 n5^(^ 2 ) ,l ] sup |f(x)|. ft

x€S r

Then using the Stirling formula, we see that for large values of n: (422)

a/#g(^)n«^(9er0/r)n.

Thus for r=Cr 0 with C > 9e and with relation (419), we obtain: (423)

|«z)| <C s exp[r 0 (a+5)/C],

with |z| <r 0 .

Notice that we have, with (419) and (412), (413): (424)

a s inf H K (-ia)= inf H ImK (a). a€S 2

o€S 2

ITius when Im K={Im z, z € KJ is a convex set, the ball Ba satisfies B a £ Im K. ® of Theorem 11. The proof of the first part, that is, that the expansion (414) with (379) or (380) defines a harmonic function satisfying (41S) is fairly easy (see Muller [1] p . 89). Thanks to Lemma 11 this implies that f is of exponential type on C , with (424) and a = rQ/2. The Paley-Wiener theorem 10 implies that pm is an analytical functional. PROOF

3.11 MULTIPOLE EXPANSIONS

155

Then we can write (416) (see Remark 25). The behavior at infinity of u can be proved in a way similar to (15), Proposition 1. Thus u is an outgoing wave. There are many other ways to prove it (see for instance Muller [1] p. 90 using a Laplace transformation). Then we know (see Mffller [1] Chap. 3.5, Thms. 16 and 17 thenThm. 18)that for r > Rce the expansion of u defines an entire harmonic function so that R ^ Rce and conversely for r greater than RM (see (418)), the series (380) is convergent, thus R M > R • Furthermore we can characterize the radius of convergence Rce from the coefficients (cn) in (414), since we have: (425) thus: (426)

R^ = inf {r, so that c n = o [(kr/2)n+l T{n+\1/2)$ for n-^oo}, |kR a =^1i f limsup(r(n

i)c n ) 1 / n ;

P is the radius of convergence of the entire series 2 F(n + 1^2) Cn1^° <8> Note that it would be interesting also to use analytic functionals of the variable r in C , with values in L2(S2). Conclusion. Let X be the space of outgoing waves defined by: (427) X = {u€D'(R3), p-(A + k 2 )u€£'(R 3 ),/ s 2 |(^-ikuKra)| 2 da = o(l/r 2 )}. With B1a = R 3 /Bafl , we also define X = U Xa fl, where: a>0

(427)' X a = {u€Z>'(B;),(A + k )u = 0inB;, / g 2 | ( | j - ikuXra)|2da = o(l/r 2 )}. Thus for every u in X, there is an a > 0 such that u |

t €X . a Let Y be the space of traces on the sphere Sk = kS2 of harmonic functions on R 3 B

of exponential type: (428)

Y = {g€C°°(Sk), 3fonR 3 withAf=0and(419),fl

=g}. k

Y is also the space of traces on the sphere Sk = kS2 of the Fourier transforms of distributions with compact support on R3. Then we have: Theorem 12. Trace theorem at infinity. The mapping: y : u —► lim u(r<x)/(r) is surfective from X onto Y. Note that it is not injective! Its kernel is the space Xfl£'(R 3 ).

156

3 STATIONARY SCATTERING PROBLEMS

Definition 9. We denote by A'H(C ) the space ofanalytical functionate p a n so that the restriction ofp^toR

is harmonic ofexponential type.

Theorem 13. The mapping p811 - • p ** |

is (continuous) one-to-onefrom A'H(C3)

ontoY. We have to compare to the mapping on distributions with compact support: p €£'(R 3 ) — pi € Y , whichhas the kernel (A + k2)£*(R3)! Thenforeverype^XR^wedefmep^cA'H^withpL

■p a n L .

Let 2£(R ) be the space of distributions on R with support in the ball B a . The mapping p € £;(R3) -> u | , € Xa (with u » * * p), has the kernel (A + k^JS^R3), and we can identify the quotient mapping with the mapping p" 1 € A*H(C ) —• u, Note also that the radius of convergence of the multipole expansion of an outgoing wave u is given by the type of the Fourier-Laplace transform of the "source" of u (see Theorem 11). In the case of scattering of an incident wave by an obstacle, a conjecture of Bardos is that the radius of convergence corresponds to the distance of the origin to an envelop of the normals. 11.5. Multipole expansions for the electromagnetic field We consider Maxwell equations (22), with currents J and M having compact supports; the electromagnetic field (E,H) satisfies Helmholtz equations (28) with (m j ) given by (27). Then we can apply the results we obtained above on Helmholtz equation: let (j^.m^) be the analytical functional such that (429)

j M ( k a ) = j(ka), A ^(ka) =rfi(ka),

Va€S2,

with (j ***, ifi *") entire analytic vector functions on C3, of exponential type. Note that, thanks to (33), they also satisfy: (430) a. j ^(ka) = 0,

a. A ^(ka) = 0, V a € S 2 (and a A J ^(ka) = Z m ^(ka)).

Definition 10. We denote A' ^ t k ( C 3 ) ^ {j*11 € A'H(C3)3, a.j ^(ka) = 0, a € S2}. Thus (E, H) is obtained from (jan,man) by (431)

E(x) = <j an ,*(x- .>,

H(x) = <m an ,*(x- .)>, withr= |x| >R.

3.11 MULTIPOLE EXPANSIONS

157

Note thatrelations(430) imply (from Rellich Theorem) div E = 0, div H « 0 for r > R. Then from the expansion of j a n and m "" into spherical harmonics,

(432)

j "(pa)-5 2 (^) n K "(a), m an(p«) = f 1 f^fK

£(«),

we obtain the multipole expansion of (E, H): (433)

E(ro) = 2 h^'W) K °(<x), H(ra) = 2 h l V I i(a),

with the radii of convergence given by (see (418)): (434)R^ = l m s u p ^ l o g / s 2 |j" n (ia)| 2 da, RjJ = Itasup^log/ g 2 I n W ^ d a . E

Obviously, from Maxwell equations, we have R^ = R^ . Remark 28. First we have to note that the vector product or the scalar product by a of a (vector) spherical harmonic of order n is not a spherical harmonic. Furthermore a A Kn(a) and a A Km(a) (resp. a. Kn(a) and a. Km(a)) n * m, are not orthogonal. 0 Remark 29. Note also that we have the relations between j and m thanks to (27) (435)

i) icoy m + curl j «(A + k )M, ii) curl m + iwej s - (A + k2)J .

When M = 0 we have: icop m + curl j = 0, but we do not have: iwp m**1 + curl j 3 1 1 = 0 ! The differential operators (here the curl operator) do not operate in A' H t (C 3 )!

EXAMPLE 3. The (oscillating) dipole. We define the dipole oscillating at the angular frequency G> (and centred at the origin) as the charge density p(x,t) and the current J(x,t) given by: (436)

p(x,t) = e-iwtp(x), P(x) = p.grad6, J(x,t) = e"itt,t J(x), J(x) = io>p.6,

with p € R the dipole momentum; we have <p(x), Xi>=:<p.grad5, x ^ s - <6, div(pXi)> = - p.. With the usual notations we have:

158

(437)

3 STATIONARY SCATTERING PROBLEMS J(y) = iop,

j (y) = i^ [J - k"2 y (y. J)] = - co2 y [p - k~2 y (y.p)],

thus j (ka) = - a>2y [p - a(a.p)], which is also j ""(ka) from the definition. Then we easily find (438)

)m(y) =

-(*2l^){\p?-y(y.p)]-lptf-It*)},

and then: A(ka) = ikaAJ(ka) = -ka>aAp, giving: A " t y ^ - w y A p. We can write (438) also in the form: (438)'

j *%>«) = - (<*V k 2 ) {p2 K2(a) + K0(a)},

K2(a) and K0(a) spherical Harmonics of order 2 and 0 resp., lc 2 (a)=i p - a (a.p), K0(a) - 3 P k2. We have a. K2(a) = - | a.p = - a. K0(a); thus K2(a) is not tangent to the sphere S2, and a. K2(a) is not orthogonal to a. K0(a) in L2(S2). We also verify that: iwpA^yJ + i y A j ^ y ) * © and yJ"Ky)*0!

Conclusion. In the electromagnetic case we can conclude in a very similar way as in the Helmholtz case. Let X6"1 be the space of electromagnetic fields (E,H) on R3, satisfying: i) the Maxwell equations (22) with (J,M) distributions with compact support. ii) the Silver-Muller conditions (43) at infinity. ~em

~em

We also define X c m = U X a , where X a is the space of electromagnetic fields (E,H) on B; satisfying: i) the Maxwell equations: curlH + ia>£E = 0, -curlE + iam H = 0 inB a ii) the Silver-Muller conditions (43) at infinity. Thus for every (E, H) in X cm , there is an a > 0 such that (E, H) B| R , € X a ' a Then let Y*"1 be the trace space on S k Ycm = {g € C°°(Sk)3, a.g(ka) = 0, V a 6 S2, 3 f: R3 — C 3 harmonic of exponential type, i.e., with (419) and f | =* g}. 1

(The condition on f is equivalent to lim sup r- log/ 2 I f(ra)I da < oo.) r—» oo w

We have the following results:

159

3.12 SCATTERING BY A BALL

Theorem 14. Trace theorem at infinity. The mapping: y^: (E,H)-»rlim> E(iaV<!>(T) is surjective from Xm (or Xm)

toY™.

But it is not injective ! Like in the Helmholtz case, the mapping (with (27), (29)) (J,M)€Fa(R3)3x£;(R3)3 -*(E,H)| B , €X<m, or(J,M) — ^ E(ra)/(r) € Y*"1 is not injective, and we can identify itsaquotient mapping with the function: j* 11 € A'H.ktC3)— (E,H) (with (E,H) defined by (431)). We recall that m an is obtained from j* 11 thanks to (430). Furthermore the radius of convergence of the multipole expansion of(E,YL) is obtained from j*"1 by (434). Remark 30. A natural question in scattering problems is to "control" the total P (see (283)) by the total cross section o s (see (279)). This is important also for vanational methods in scattering. The question is equivalent to the following: does there exists a constant C > 0 such that (439)

|g(a 0 )|<CCf s2 |g(a)| 2 da) 1/2 , V g € Y ?

(Y being defined by (428) with k = 1 to simplify writing). The answer is negative. We can see it by taking the functions: gn(a) = Pn(a(0, with a given unit vector fl and Pfl the Legendre polynomial of degree n. We have (see Muller [1])

(440) /s2|gn(a)|2d« = 2nf J (Pn(t))2dt =

£j, Ml

butg (P) = P n (l)=l; for n— oo, we see that we cannot have 1 <C(4*r/(n+l))

1 2 . SCATTERING BY A DIELECTRIC BALL

The scattering of an incident wave plane by a dielectric ball BR is often used as a reference to numerical implementation. Results are well known due to Rayleigh expansions. 12.1. Scattering with Helmholtz equation 12.1.1. Calderon operators and Calderon projectors on a sphere First we can obtain the Calderon operators C\ Ce and the Calderon projectors P\ Pe on the sphere SR (for a wavenumber k) thanks to the Rayleigh series (380) quite easily. The derivatives of these expressions with respect to r, for r = R, are for the (exterior) outgoing wave, and the interior wave: (441)

g (Rot) = 2 k hJfldOK^ resp. g (Ra) = 2 k j ^ k R ^ a ) ,

with h^ = h^'; the prime denotes the derivative. Thus ifj (kR) * 0, V n:

160

3 STATIONARY SCATTERING PROBLEMS

h' j' (442) (;%(<«) = 8^ K^a), d ^ a ) = 0*n K^a), 6^ = kt^(kR), ej, = k f . n " Jn Thus the Calderon operators are simple multiplication in the space of spherical Harmonics of order n. This is a consequence of the fact that the rotation group commutes with the Calderon operators. Now the exterior Calderon operator has the extra property (with k > 0): Re - (Ccu0,u0)>0, andeven

Re - (Ccu0,u0)> g- Hu0ll^2(s }-

Using (442), and the decomposition u0 = 2 u0n into spherical Harmonics, we have Re(Ceu0,u0) = 2 ReO£ | Uoil | 2 , Re9« =-L I £(|h n (kr)| 2 ), and | heeler)| is a decreasing function of r, see Abramowitz-Stegun [1] p.439. Then the Calderon projectors are obtained in the space of spherical Harmonics of order n also by solving the transmission problem: (443)

i) Jn(kR) K^a) - h^kR) K„(a)

= pn(a),

ii) kj;(kR) K^a) - k h;(kR) KJ.a) = P;(a), with given Pn(«)» P'n(a)- Thanks to the Wronskian (444) W(Jn,hnXkR) = [hAJn - h„ j^(kR) = i/(kR)2, we obtain,

I KnW J

Lkh; -hnJU;(«)J

with j for j (kR), hn for hn(kR). This gives the Calderon projectors Pj;, Pjj by: (446) P^ikR 2 and:

f u g ; - K>n\ kk 2 ig; -kjg n J

pi=ikR^f-khiJn L -k2h;j;

ps + pi-i. pi-^-s., withsn=ikR2

Kin\ ug; J

^ - 2k2ig; k(igny J

12.1.2. Scattering of an incident wave by a ball Let k be the wavenumber of the incident wave Up kj be the wavenumber of the scattered wave in the ball. We assume that u{ is an entire solution of Helmholtz equation. The usual transmission conditions on the sphere SR are (447) (447)

u* u e -u u - u -up

^ 3u!.!^l an-dn-an ■

161

3.12 SCATTERING BY A BALL

Using the Rayleigh expansion (380) and (442), we have (ifjn(kjR) * 0)

(448)

e1nIuj,-eX=e,nuI>n,

i4-nS-u,.n,

with (see 442):

9en = k^(kR),

(449)

^-kfcdcR), J

ejf-kjJ^R). J

If k, is real, e" is also real, thus fn - e" * 0 since the Wronskian W(hn, h n ) * 0 . Then the solution of (448) is given by (450)

u« = R„u In , uj,=T n Ui n , Rn = - ^ - r r ,

«£-«

=^^,T e

n-°n

-R

= l.

Then for u,(ro) = I j (kr)Kj,(a), Uj n = Jn(kR)Kj,(a), the waves u1 and u e are ukw) = | f n j n (k,r)4(a),

with fn = T„Jn(kR)/Jn(k,R),

(451) ue(ra) = 2 R n h^flcr)^*), n

with Rn = R„ j (kRJ^kR), for r 2: R, "

with K^(a) = i (2n + l)P n (a.a 0 ) for a plane wave. The series for u e is convergent in L2(Sr) for all r> R. Its radius of convergence R ce is given by (418), with: ^ = /s2|Rnl^2n+l)2|Pn(«-«0)|2d« = 4n(2n+l)|Rn|2. Then we can verify that R ce = 0: the Rayleigh series converge for all r > 0 ! 12.2. Scattering with Maxwell equations 12.2.1. Debye potential of the electromagnetic field Let (E,H) be an electromagnetic field in a ball B a or its complementary, satisfying Maxwell equations (452)

curlH + io>eE = 0,

- curlE + io>y H = 0 in B a or B' a = R 3 \B a ,

with constant e,m with (locally) finite energy, and the Silver-Muller conditions at infinity. Then (see Schulenberger [1], Aydin-Hizal [1]), there exist two scalar (outgoing) solutions <pv <p2 of the Helmholtz equation for k 2 = aAy, such that (453) i) E = -curl (xc|> ) + ^ curl curl (xq> ),

ii) H = <^j curl curl (x<|> ) + curl (x<!>J.

We get <|) and <> | up to a function of r = | x| from E or H by solving the equations 2

- io)£x.E = L <>| , 2

-x.curlE = -

2 K«)JLIX.H = L

<> | ,

with L 0 = x.curl curl (x<!>); L is the angular momentum operator (we can identify L2 with the Laplace-Beltrami operator - A 2 ); in L2(Q), O = B a or R 3 ^ , it is a

162

3 STATIONARY SCATTERING PROBLEMS

positive selfadjoint operator (see Dautray-Lions [1] chap. 9B.1.1). The functions c|>j and <|>2 are called the Debve potentials of the electromagnetic field (E,H). Furthermore the fields (E(1),H(1J) and (E (2) ,H (2) ) defined by: (454)

U) H(l) = 4 curl curl (x^),

E (1) = - curl (x^), E( } = (5eCurlcurl(x<|)A

H (2) = curl(x<|>2),

are respectively called TE (transverse electric) fields and TM (transverse magnetic) fields. Thanks to relation (28) chap. 2, we have: n.E (1) = 0, and n.H (2) * 0, when n is the normal to the sphere I = S f . Then taking the tangential trace of these fields on the sphere I , we obtain: i ) n A E | 2 = grad2(-r(|)1) + curl 2 (^(I + D)<|)2), (455) ii) n A H | 2 = curl2 ( ^ (I + D) ^) + grad2 (n|>2), with (456)

*=2xk^=r?F,

or also for the projections on the sphere £: i) * 2 E = curl2 (r^) + grad2 ( ^ (I + D) <|>2), (455)' I ii) * 2 H m grad2 (^r (I + D) y{) - curl2 (r<|>2). This is due to the relations (28), (30), (30)' chap. 2, with curl (xc|>) = - x A grad <|>. The Debye potentials <> | and c|>2 on £ are obtained from n A E | by: (457)

rA2 ^ = - div2 (n A E1 2 ),

A2 (I + D)<|>2 = - io>c curl2 (n A E1 2 ).

Formulas (455) or (455)' correspond to the Hodge decomposition of currents on the sphere I (see section 6.2, Appendix). Note that the sphere is simply connected, and thus the cohomoiogy space Hl(L) is reduced to 0. Let Il c and IT be the orthogonal projectors in the space L 2 t (l) of square integrable tangent fields on £, so that: (458) f = fc + fg = grad2 o 2 + curl2 *x with fg = IIgf = grad2 o 2 , fc = n / = curl2 *x, and *,, <t>2 are obtained using the inverse G of the (opposite of the) Laplace-Beltrami operator on the sphere in LQ(I) = {u € L2(2), / 2 u dZ= 0}, by: 02 = - G curl2 f, 0 2 = - G div2 f. On the unit sphere, G is the integral operator (see Schulenberger [1]): Gf(a)=/ 2 g(a,(J) m d^,

with g(a,0) = - ^ l o g ^ ^ , a, f$ € 2 = S 2 .

3.12 SCATTERING BY A BALL

163

Note that the Debye potentials are related to the radial components Ef, Hf by (459)

A§2 <p{ = iwji rH r ,

A§2 <> | 2 = icoc rE r

This is easily seen from the radial component of Maxwell equations, with (28), (30) chap. 2. Furthermore rEf = x.E and rH = x.H satisfy the Helmholtz equation and converge to 0 at infinity (see (50)). Thus they can be used as potentials for the electromagnetic field, but it is not very easy to prove that they satisfy the outgoing condition, and the Debye potentials are more regular. For the converse of (459), we only have to eliminate constants functions on the sphere. 12.2.2. The Calderon operators on the sphere To obtain the Calderon operators on the sphere, we have to inverse the relations between (<!>,,(|>2) and E 2 . Now D in (455) is also (up to factor r) the exterior or interior Calderon operator on the sphere S of the scalar case. In the exterior case, B er = (C* + y) is an isomorphism from HS(I) onto Hs~ (I). In the interior case, we have to eliminate irregular frequencies, see 4.7, 4.8). Now in the exterior case, we obtain from (455), (457), (460) n A H 2 = Ce(n A E2) = - -X-curl2A£lBercurl2E2 - icoe g r a d ^ 1 B~* div2 E 2 . We will have a similar formula for waveguides, see (68) chap.4.2. Now we give explicit formulas thanks to Rayleigh series. Using the orthonormal basis (Kn .(a) of spherical Harmonics (see (373)), then: gradjKnj,

cur^K^, n= 1, 2,..., - n < j < n ,

is an orthogonal family of L2t(I), giving an orthonormal basis by normalization. Now we use differential operators on the unit sphere S2, instead of I, thanks to (461)

grad2(r<|>(x)) = grad^ ¥(r,a),

c7rl2 (r<|>(x)) = clfrl^ ¥(r,a),

with <|>00 = ¥(r,a). Thanks to the exterior (resp. interior) Rayleigh series ofty{,ct>2 (462)

c|>(ra) = 2 h j J W ^ t o 1

n>0

(resp 2 jn(kr)4(«)), J= 1,2, n>0

in (455)', we obtain the exterior Rayleigh series: n E

= 2 K c"ris2 Ki + | cob hn grads2 KJ;,

(463) "2H = 2 4 r K « rad ,2 * i - 2 hn c"uri , K^,

164

3 STATIONARY SCATTERING PROBLEMS

with the notations:

(464)

1^ = hjJW

hn = (I + D)hjJW

We obtain the exterior Calderon operatorfor the TE and TM waves C™'e and c ™ ' e (465)

C™'c = - i § n « A n g ,

c^-hwCe^aAlIe,

with a 6 S2 the normal to the sphere, 9 £ * h n /l^, n g , Il c given by (458). The interior Rayleigh expansion is obtained by substituting j n for h n . The interior Calderon operators for TE and TM waves are given by: (466)

C™'1 — 4 r

n«Ang,

C^-iwrte^aAH.,

with the notations: (467)

j n =j n (kr),

j>(I

D)Jn<kr),

K = ln\.

12.2.3. Scattering of an incident wave by a dielectric ball Now let (EJ,HJ) be an incident wave on a ball Ba, with a dielectric medium of permittivity e{ and permeability ]i{. We assume that the incident wave is an entire solution of Maxwell equations in free space (for instance a plane wave). Then the incident wave is given thanks to the Debye potentials <|>n and (|>I2, with the Rayleigh expansions: (468)

(|)Xj(ra) = 2 Jn(kr)Kj;j(a),

j = 1,2.

Let (<!>*,<|£) and (<|>*,<|>*) be the Debye potentials of the electromagnetic field inside the ball (E1,!!1), and of the reflectedfield(Ee,Hc). Let: (469)

^w)»2j n (kir)l4 j (a),

<|><(ra) = 2 h£V)Kj J <«). j - 1,2,

be their Rayleigh expansions with unknown harmonic functions KjjJ, K^' J . Then writing the transmission conditions on the sphere X, with r = a, forTE and TM waves we have: i> [*!], = ♦}-*?»♦[.]» 1.2 J

(470)

*»

| ii) [ i (I + D ) ^ =55^ (I + D ^ ; - 4r (I + D)4>« = ^ (I + D)*\ | iii) [ ^ (I + D>|)2]2=s^- (I + D ^ - A (I + D ^ = i (I + D ^ .

Now using the Rayleigh expansions of these Debye potentials, and the notations (471)

ji=J n (k,r), j i = (I + D)Jn(kir), e » = a i ) / j ; ,

3.12 SCATTERING BY A BALL

165

we obtain the system of equations: ;1 w-i,l

|- w-e,l

•

V I,1

(472) PjJnJSi

-jinn**

- P

^ '

We have the same system for the superscript 2, but with t instead of p. Their solutions are given by

(473)

ii^-1?^.

hX'^R^K^

j=1 2

and thus: JT ^ej „<j) (474)

" n V = ^ J n C j-1,2,

OV-'Ji Kl - *? V'J n 41), (.,r »Ji K2- liV'J, 4'2)(

with (1)

(475)

fj'Bn-P

M U _9in„(l) nRul M)_

flk

and also:

(475)

^ V e i ' - g - ' i s ' r--«»i7[r- '

with similar formulas for RJ, and T„ , butwithe, ej in place ofji, jij. Note that: (476)

li?-R®-l,

ij?-**-!,

j-1,2.

We finally obtain the reflected electromagnetic field (Ee,He), thanks to (463), by its components: ZE'TE,II = Rn

(477)

(

"2El)TE>n >

"ZH'TE.II = ^n

(nZHI*rE,n '

and thefield(ESH1) inside the ball by: (478)

* Z E TE.n = I n

(

*ZEl)TEn >

* 2 E T M , n = ^f<*Z E I>TM, n '

Z H T E , n = hi ( n Z H I>TE.n ' ^TM.n = ^ W V r M . n '

166

3 STATIONARY SCATTERING PROBLEMS

EXAMPLE. If the incident wave is a plane wave, given by: (479)

E^Ejoe 11 ™ 0 *", H,(x) = HI0e*""0", x = r<x,a€S2,

with given OQ € S2, E I0 and HI0 € C3, with aoEio = ^-Hio - °» ZH I0 ■ "o A EI0> a n d Z the impedance of free space. Taking aQ »(0,0,1), EJ0 ■ (1,0,0), we obtain the Debye potentials of this plane wave by the Rayleigh series (468), with the harmonic functions (373) (see Jones [1] p. 446):

(480)

K^-CJK^-K^],

l4 2 («)--Kt K n.l + K ».-^

with Cn = (2K) 1/2 2n(n+l) ' T W s g i v e s u s t h c s c a t t e r c d

field

*&&&& to (477).

3 . 1 3 . ADDENDUM. COMPACTNESS PROPERTIES IN SCATTERING PROBLEMS.

The scattering problems of scalar waves on bounded (regular) obstacles depend on the Fredholm alternative; this is a consequence of the compactness of the operators K, J and L (see (70)) in the framework H"" 1/2 (D (or Hl72(D, or C°(0), the range of these operators being in H1/2(r). Since we can often prove fairly easily uniqueness at most of the solution in scattering problems, then this also implies its existence! In electromagnetism, scattering problems also depend on the Fredholm alternative, and the sum: Cc + C1 = - R" lT (see (241)) is compact in H~ I/2(div,H, as a consequence of the proposition: Proposition 32. IfT is of class Cl'lt the operator T defined by (100)' is compact in H " i/2 (div,D, with Im T in Hl/2(div,D ifl is ofclass C2>l. PROOF, i) For all u f in H " 1/2(div,D, we verify that Tur is given by: (481) Tur = gradro + vr, 0€H 1/2 (O, v r €L 2 (0 3 ,withv r €H t I/2 (OifrisC 2,1 . By developing the double vector product in (100)' and using (70), we have: (482)

Tuj<x) = - Jur(x)-nx(nx.Jur<x))-22; n^grad^Lurjfr), j

which is of the form (481) with 0 = - 2^. Lur, thanks to the properties of K and J. ii) Now we verify that divr Tur € H 1/2 (0. Let (E, H) be defined by (223) with Jr = u r Using the notation {v} = v| + v| , we have like for (84) (see also (28), (30) chap.2) r

(483) divrTur = - divr ({n A HJr) = - io>6 {n.E}r Then using (95)' and (70), we have: (484)

{n.E}r = ikZ{n.Lu^r + i|j(div r u r ) = 2ikZn.Lur + i|j(div r u r ).

Since Lur and J (divr ur) are in H 1/2 (0, this implies the proposition.

£>

CHAPTER 4

WAVEGUIDE PROBLEMS

1. WAVEGUIDES WITH HELMHOLTZ EQUATIONS

Here we study the propagation of scalar stationary waves in a "waveguide" which is a semi-infinite cylinder. The method that we develop, will allow us to deal with cases of more general geometry and later, with the electromagnetic case. Let Q+ = Cl x R+ be an infinite open set which is a semi-infinite cylinder in R3 with a regular bounded cross-section Q in R2, of boundary T. Let T+ = T x R+ be the lateral boundary of Q+. We first consider the Dirichlet (resp. Neumann) problem:findu such that

(1)

i)Au + k2u = 0inQ + , ii)u| = 0 ( r e s p . g | =0), iii) u(x,0) = u°(x), x € Q. c R2,

with a real wavenumber k and a given u° (for instance in L2(Q)). We denote by Aj, the Laplacian in the transverse variables xT = ( x ^ ) (in R2), by x3 the variable along the axis of the cylinder with origin at the end of the cylinder. Then we can write (l)i) in the form (l)i)' | T + V 1 + k2u = 0 in n + . 167

168

4 WAVEGUIDE PROBLEMS D

Let AT (resp. AT) be the Laplacian with Dirichlet (resp. Neumann) condition. Denoting by A either - Aj or - AT (both self adjoint operators), we can rewrite (1) in the form: :.3 2 U

(2)

i ) ^ - | - A u + k2u = 0, ii)u(.,0) = n° u(

Dx2 ADx Let (^.♦JJ) be a spectral decomposition of A (that is ((Xjj') ,$~) for A = - A^, and ((^) 2 ,^)) for A = - Aj, with on an orthonormal basis in L2(Q). We decompose u° and u(.,x3) on this basis:

(3)

u°=2XV

u(->x 3 )=!u n (x 3 )V

and then we have tofindthe unknown coefficients un(x3) so that: (4) ii)un(0) = uS.

Denoting by ocn and fl two arbitrary constants, the general solution of (4)i) is: a)ifX2>k2, un(x3) = a n exp(x 3 Jx^.k 2 ) + P n exp(.X3 i Jx2-k 2 ). If we require that the solution be tempered with respect to x3 , thus an = 0, then: (5)

un(x3) = exp ( - x 3 Jx 2 -k 2 ) ift

b)ifX2«k2, un(x3) = a n x 3 + pn. If we require that the solution be x3-bounded, then ocn = 0, un(x3) = u c)ifX2<k2, un(x3) = o n exp(-ix 3 Jx2.k 2 ) + P ii exp(ix 3 Jx2.k 2 ). This is called the propagating part of the wave u. Then (4)ii) implies ocn + (Jn = u^. Thus an, Pn and un are not determined in a unique way: we say that problem (1) is ill-posed, i.e., the given u° does not determine a solution u in a unique way. We transform problem (1) into a well-posed problem by adding a physical condition concerning the sense of wave propagation: if we suppose that the stationary problem (1) comes from an evolution problem with a solution of the type u(x,t) = e"~lwtu(x), we say that u is a wave propagating towards the positive (resp. negative) axis x3 if the propagating part ofu is a superposition of waves of the type: V3 (6) Un(X3> = e vn («*p. e n W\n)f P n > 0 .

4.1 WAVEGUIDES WITHHELMHOLTZ

169

For a time evolution u(x,t) = e,wtu(x), we make the converse choice. We define:

(7) end=efe;=(x2_k2),/2if^>k2, 9 n d #-ie-=-i(k 2 -^)^ifx^k 2 . Thus with the physical hypothesis of positive axis x3 wave propagation, we have to take a = 0 in case c), and then we obtain a (unique) solution of (1) in the form

(8)

uUx^Je"^^^

Theorem 1. Problem (1) with u° given in \}(Q) and with the condition: (9) u is an Xy bounded wave, propagating towards the positive axis xy has a unique solution u in C°([0,+oo[,L2(Q)), and it is obtained thanks to a holomorphic contraction semigroup of class C° in L2(Q), (G(x3)), x3 > 0, by (8). For these notions of semigroup, we refer to Dautray-Lions [1], Pazy [1]. We recall that a contraction semigroup in a Banach space X satisfies: IIG(t)u°ll<llu°ll,

Vu°€X, t = x3.

We note G(x3) = GD(x3) for the Dirichlet boundary condition, G(xJ = GN(x3) for the Neumann boundary condition. Its infinitesimal generator C (C D or CN) is defined (on the basis of the cylinder) in L2(Q) by: (10)

J£-(.,0) = Cu° (withC = C D orC N ), D(C) = {u° € L2(a>, u(.,x3) = G(x3)u° € c\[0,+oo),L2(Q))l

Definition 1. The infinitesimal generator C of the semigroup (G(x3)) defined by (10) is called the Calderon (or also the capacity) operator of the guide at the wavenumber k. We can characterize this operator thanks to the spectral decomposition of the Laplacian by: (11)

O n = -e n * n ,

Vn, and D(CD) = Hj(Q)f D(CN) = H1(Q).

Thus C 2 = A - k2. Hence C is a square root of the self adjoint operator (A - k2I). Furthermore it has the following properties (for all v in D(C)): (12)Re(Cv,v)=

-0^|v n | 2 <O, Im(Cv,v)=

2 n.A^k

£ |v n | 2 > 0.

2 2

n,A^<k

170

4 WAVEGUIDE PROBLEMS

Thus L (Cl) has an orthogonal decomposition into: L2(C1) = H£®H£, with: H£ = {2cn0nwithX^<k }, offinitedimension (the space of "propagating modes"), and C has the corresponding decomposition: (13) C = - C + + iCT, + with C and C" positive selfadjoint operators on H£ and H£. Then C = C D is a continuous operator from HQ(Cl) into L (12) and by duality from L2(Q) into H" l(Q) then by interpolation from H ^ Q ) into (H^Q))'. More generally, Theorem 1 is valid if we replace space L2(Q) by D(AS) for any real s. This will allow us to deal with many examples. Example 1. Junctions and Cascades A typical junction consists of a bounded domain Q in R3 with p guides arriving at it with different directions (see for instance Jones[l]). Let (F), j = 1 to p, be bases for these guides. We can choose Q and these bases so thai I\ for j = 1 to p be part of the boundary T of the domain Cl. Let T0 be the part T/UT. of the boundary (the wall of the junction). We assume for instance: i) that the guides walls and To are hard, i.e., with Dirichlet boundary conditions. We could also assume them soft, that is, with Neumann boundary conditions; the terms hard or soft originate from problems in acoustics where pressure is the unknown; ii) that a given incident wave u{ = (uA is coming from the guides to the junction; iii) that the wavenumbers are k in tne junction and k. in each guide (with the same frequency o>). Then the reflected or transmitted wave in the jth guide (u^) must satisfy (with the capacity operator C. of the jth guide): 14

aUlj i*r-cjV

Then using the continuity relations of the wave across the boundary I\: (15)

u = ur + u„

§H=aH"+a5-oneveiyrj,

we obtain from (14), (15) that the wave u must satisfy: (16)

S - q u - f 0 onl} with f y ^ - C j U y .

Thus we have tofindthe wave u in the junction, satisfying: (17)

i)Au + k2u = 0inQ, ii)u| =0and(16)onrj, j = 1 top.

This problem can be written in a variational form; we define the space:

4.1 WA VEGUIDES WITH HELMHOLTZ

(18)

H^PHueHH

171

u|_ =0}, iQ

and the sesquilinear form a(u, v) on this space by: (19)

a(u,v)=/_ (grad u . grad v - k2uv) dx - 2 / r Cu v dT, "

{

which is continuous on Hr (ft), and Hr (ft) coercive with respect to L (ft), thanks to the above properties of the capacity operators. The variational form of problem (17) is:findu in Hr (ft) satisfying l o (20) a(u,v) = 2/ r fijvdT, V V € H } ( Q ) . j

For a regular domain ft, the natural injection Hr (ft) —* L (ft) is compact. Thus problem (17) depends on the Fredholm alternative: it has a unique solution except for wavenumbers k such that (17) has a nontrivial solution, that is, for the eigenmodes of the junction. Using the decomposition corresponding to (12) for the jth guide, we denote by: Cj = - Cj" + i Cj" the decomposition of Cj, by Vj" the space of propagating modes for the jth guide (above denoted by H£), and by P* the corresponding projection of uJ (the trace of u on Tj) onto the space Vj". Then u is an eigenmode of the junction if (20) is satisfied with fr = 0. Taking v = u in (20), and the real and imaginary parts, we easily obtain that u must satisfy: i)Au + k2u = 0inft, (21)

ii)u| r =0, g J . q U = 0onr j ,j=:ltop,and Pjuj = 0,(oruj €Vp.

Remark 1. The case of small junctions (or of weak wavenumber). If the domain of the junction is small, we can apply the Poincare inequality: (22)

Hul < C llgrad ull, V u € Hi (ft),

with C = \Q 1 , \ 0 being the smallest eigenvalue X of: i) Au + Xu = 0 in ft, (23)

Then taking the real part of (20) with v = u and f = 0, we obtain with (12): (24)

4(|gradu| 2 -k 2 |u|)dx<0;

and using (22) we have:

172

(25)

4 WAVEGUIDE PROBLEMS (l-k2C2)/aIgradu|2dxsO,

VU*HÂŁQ;

thus for (1 - l^C2) > 0 (that is for X0 > k), we have u = 0. This implies that the wave problem (17) in a small junction always has a unique solution. Remark 2. Dissipative junctions. If the wavenumber k is complex with Im k2 > 0, we also have uniqueness of the solution of problem (20). This is obtained taking the imaginary part of (20) with v = u and f = 0 and using (12): (26)

Im (- k2) iluB2 = 2 Im (Cu,u) 2> 0, j

thus giving u = 0, and we have the same conclusion as in Remark 1. Remark 3. Cascades (see for instance Jones [1] p. 257). The case when the cross-section of the guide changes (with discontinuities or not) over a finite distance, is a special case ofjunction (with two different guides arriving to it). <8>

Remark 4. We can generalize to "inhomogeneous" waveguides that is when there are different media with different wavenumbers in the cross section of the guide (with the usual transmission conditions at the interfaces of the fibers). Remark 5. We can also generalize to "inhomogeneous" junctions with a wavenumber k dependent on x. The problem can be solved numerically by using finite elements inside the junction and spectral decompositions at the boundary for the capacity operators (for instance). 0

2 . WAVEGUIDES IN ELECTROMAGNETISM

We consider a semi-infinite cylinder Q+ = Q. x R+ in R3 with x3 axis, occupied by a conservative medium with permittivity and permeability t and ji (with positive real c, p), bounded by a perfect conductor. We assume that the cross-section Q is an open regular bounded and connected set (it can be simply connected or not). We first consider the following problem: find the electromagnetic field (E,H) in the waveguide Q+, at angular frequency co, satisfying (with the same notations as in section 1) |i)curlH + icDâ&#x201A;ŹE=:0, (27)

ii) - curl E + i<oji H = 0 inQ + , ft

iii)nAE|

=0,

iv) E T (0) = E^ onQ,

where E T is a given transverse electricfield(for instance in H0(curlT,Q)).

4.2

WAVEGUIDESINELECTROMAGNETISM

173

We denote by (EpH-r) the transverse components of the electromagnetic field (E,H) and we use the index T for transverse derivations: let t be a function on Cl, v T = (Vj,v2) be a vector function on Q., we denote: (28)

gradT <> t = ( ^ - , ^ - ) = (3^, a2o),

curlT ÂŤ> = (32o, - 3^)

curl T Vj= 3 ^ 2 - 3 2 Vj.

Let e 3 be the unit vector along x3, and let S be the operator: (29)

Sv T = S(v1,v2) = (-v 2 ,v,) = e 3 Av T .

With these notations, the Maxwell equations are: i) - 3 3 H T + gradT H3 + me SE T = 0, (30)

ii) 3 3 E T - gradT E 3 + ia>y SH T = 0,

with: (31)

i) curlT H T + ia>eE3 = 0, ii) - curlT E T + io)MH3 = 0,

and since n A E = (n2, - n j , njE2 - n2Ej) = - E 3 Sn + e 3 n T A E T , the boundary conditions are: |i)n T AE T | = 0andE 3 | = 0 , (32) ii)ET(0) = ET Like in the Helmholtz case we can see that this problem is ill-posed and it becomes a well-posed problem adding to it the physical hypothesis of propagation of the wave (E,H): (33)

(E, H) is an unbounded wave, propagating towards the positive axis x 3 .

We first formally replace E 3 and H 3 in (30) by their values in (31); thus: (34)

i) - 3 3 H T +A j E T = 0,

ii) 3 3 E T +A 2 H T = 0,

with: (35)

i) A j E T = | - gradT jj curlT E T + icoe SE T ii) A2Hj = jjj gradT j curlT H T + icop SHT .

We can write this system of equations (34), on E T or H T only: (36)

with:

3 | H T + ^ J ^ [ 2 H T = 0,

afEr+^jEraO,

174

(37)

4 WAVEGUIDE PROBLEMS i) A XA2 HT as (gradT p curlT pS + eS gradT j curly + o>2 epl) H T ,

1 1 ? ii) i4^! Ej = (gradj j curlT ES + pS gradj. p curlT + or epl) E j , or also using relations: (38) S gradT = - curlT, curlT S = divT, (39)

i) A XA2 HT =s (gradj p divT p - cS curlT j curlT + o>2 epl) H T , ii) A^A! ET = (gradT j divT E - pSZlt curl1T p curlT + or epl) ET

Now we make the following important hypothesis: The domain of the guide is occupied by a homogeneous and isotropic medium; thus e and p are constants in Q.Withk*= o2ep,(39)is: i) A ]i42HT = (gradj divT - curlT curlT + k I) H T , (40)

ii) A^iEj SB (gradT divT - curiT curlT + k2I) ET.

From now on we will not write subscripts T to differential operators grad, etc. The operators A1 A2 and A1 A1 are identical to the transverse Laplacian; we have verified only that ET and HT satisfy the Helmholtz equation! But these operators differ by their boundary conditions. We define them by the usual variational method. Wefirstdefine the following spaces: (41)

V = {v € L2(Q)2, div v € L2(Q), curl v € L2(Q)}, V- = {v€V,nAv| r =0},

V^fveV.n.vl^O}.

From the Appendix (see (121), (122)) we know that spaces V* and V~ are contained in H^Q). Then we define the sesquilinear form: (42)

a(u,v) a / (div u div v + curl u . curl v) dx.

Thus we can define the operators A+ and A " (see Appendix (156)") by: D(A+) (resp. D(A~)) * {u € V* (resp. V"), v-»a(u,v) is continuous on V* (resp. V"") equipped with the L topology} and thus D(A-)=s{u€H1(Q)2,Au€L2(Q)2, nAu| r = 0,divu| r = 0}, (44) D(A+) = {u € H W , AU € L2(Q)2, n.u | r = 0, curl u | r = 0}; (43)

A+ and A are positive selfadjoint operators, with compact resolvent, thus with discrete spectrum.

175

4.2 WAVEGUIDES IN ELECTROMAGNETISM

Then we verify (at least formally) that if (E, H) is a solution of the problem (27), (ET,HT)(x3) is in D(A ) x D(A~); condition (32) implies ^ - l r = 0, and from equation div E = 0, we have div E T | and (31)i), (32)i) imply curl H T |

= 0. Besides, (27)ii) implies n T .H T |

= 0,

= 0? *+

Determination of the transverse electric field ET We come back to problem (27) which, after (36), (32), and (44), amounts to determine the transverse electric field E T that satisfies: (45)

i)85E T = ( - A " - k 2 I ) E T , iOE^OJ^Ej.

But this is an ill-posed problem as we see it by spectral decomposition of the operator (A~ + k2I). It is transformed into a well-posed problem by adding the physical hypothesis (33) for E T . Using the decomposition of the self adjoint operator - ( A ~ +k 2 I) into its positive and negative parts corresponding to the space decomposition of L2(Q) (46)

L2(Q)2 = H+ ®H_ ,

- (A" + k2I) = (- A" - k2I)+ - (- A - - k2I)_ ,

we can define (from the symbolic calculus) two "square roots" of — (A~ +k I) denoted by A+ and A ~ , so that: (47)

on H+: A+ = A" = - (- A" - k2I)*/2 negative selfadjoint, on H_: A+ = - A" = - i(-A" - k 2 I) 1/2 with (-A" - k 2 I) 1/2 bounded, positive.

Thus A + and A" are adjoint normal operators with domain V" , with spectrum o(A+) c R" U -i[0, k] resp. o(A")c R" U i[0, k]. (The Hodge decomposition allows us a more explicit spectral decomposition using the spectral decomposition of the Dirichlet and the Neumann Laplacian.) Taking (33) into account, we then replace problem (45) by: (48)

i) 3 3 E T = A"ET in Q+, ii)ET(0) = ET.

The choice A+ would give the opposite direction of propagation. From now on we simply denote by A the operator A". With the above properties, the operator A is the infinitesimal generator of a contraction (and holomorphic) semigroup of class C° in L2(Q)2, denoted by (GA(x3)), x3 > 0; the solution of (48) is given by

176

4 WAVEGUIDE PROBLEMS

Ej(x3) = GA(x3) E j , x3 > 0. (49) For the transverse magnetic field HT, first assuming that it is given for x3 = 0, and denoting by (GB(x3», x3 > 0, the semigroup generated by the square root B" (like in (47)) of the operator - (A+ + k2I), we have: (49)' HT<X3) = GB(x3) H^O), X3 > 0. Then we easily determine all other unknown functions from the initial values E3(0), H3(0) and H^O) which we have to obtain from the initial value of Ep Determination ofHy First taking the scalar product of the relation (30)i) with the unit normal to T+ on the boundary T+ gives (50)

-9 3 n T .H T +Qjp-io)£n T AE T = 0 onl+,

which implies (using (27)iii) and its consequence n.H | (51)

8HT°

onr

=0):

Furthermore the initial value of H3 is obtained from (31)ii): (52)

H3(0) = H5=jjJircuriET<0)=1lrcurlE°.

Thus we have to determine H3 satisfying: (53)

I i) AH3 + k2H3 = 0 in Cl+ (with A in R3 !), ii) the boundary condition (51) on T+, iii) the initial condition (52), iv) the propagation condition (33) for H3.

We have solved this Helmholtz problem in section 1; its solution is given using (a generalization of) Theorem 1 (note that for Ej in L2(Q)2, H3 is not in L2(C1)) (54)

H3(x3) = G N (x 3 )H5.

Determination ofE3. Wefirsthave from equation div E = 0 in R , for x3 = 0: (55)

^ ( 0 ) = -divE T (0) = -divE£. 0

First assuming that E3(0) = E3 is given (in L (Q)), E3 must satisfy: (56)

2 u)E i) AE3 | r+ k=0 E(see in Q+, (with A in R3), 3 « 0(32)), iu)E3(0) = E 3 onQ, iv) the propagation condition (33) for E3.

4.2

177

WAVEGUIDESINELECTROMAGNETISM

Then, from Theorem 1, we know that this problem has a unique solution, which is given by: (57) E3(x3) = G D (x 3 )E°. We detennine the "initial" condition E 3 from (55) by: (58)

^ ( 0 ) = C D E° = -divE£.

If k is not an eigenvalue of{- A T ), C D is invertible and (58) has a unique solution in L2(Q) when E j is in L2(Q)2: (59)

E^-C^divE^.

Determination of the transverse magnetic field H T . We have to detennine the initial condition HT(0) = H° T . This is obtained from equation (30)ii) for x 3 = 0. Using (48) we have (60)

AE^ - grad E 3 + io>y SH^ = 0,

and thus, with (59) and using the impedance Z with kZ = u>\i (61)

S H ^ ^ I A E T + gradCDdivEj].

First properties. Calderon operator C of the guide. It will be useful to transform the basic relation (61) using the following Hodge decomposition (see chap.2 (114)") of u in L2(Q)2 in the form: (62)

u = grad A^ divu - curl Aj^ L^, + u,

with u € H2{C1\ where H2(Q) = {v € L2(Q)2, curl v=div v=0, n A V| = 0); A N is the isomorphism from H (Q)/R onto (H^Q))' associated with the Neumann i

Laplacian and Lu € (H (Q))' is defined for u in L (fir by: (63)

Lu(v) = (Su,gradv),

VveH^Q),

thus w=A£, L„ is the solution (up to constants) of: (63)'

(grad w.grad v) = (Su.grad v),

W e Hl(Q),

oralso: (curlw,curlv) = (u,curlv), V V C H ' ( Q ) . For u € H(curl,ft) , we have L„ = curl u - n A U | 5 r and for u € H0(curl, £2), we have A^ LjjsAj^curlu.

178

4 WAVEGUIDE PROBLEMS

From the spectral decomposition of the operators AD, AN, the space L2(Q)2 splits into H+ and H ~ (see (46)), with: (64) H. = grad(HpD©curl(Hi[)N©H2(Q), H+ = grad(V£)D®curl(V£)N, where (Hk) and (H£) arefinitedimensional spaces corresponding to the negative part of the operators (- AD - k2I) and (- AN - k2I) (see the decomposition in the Helmholtz case), and (V£) and (V£) are spaces corresponding to the negative part of these operators (in HQ(Q) or H^Q)). This decomposition gives us the relation (for u in V", with (62)): (65) with: (66)

Au = gradAp CDdivu-curlCN(AJ^ curlu) + iku, Re(Au,u)<0 and Im(Au,u) > 0, V u € V .

Yet we return to relation (61); we have: (67)

, and thus:

CD' + I ^ A D ' C ^ - A ^ C D .

Therefore (61) becomes (for ET in V" then by continuity, for ET in H0(curl, Q.)) (68) SH^=j^ [- k2 grad Ap CJ^div Ej - curl C N AN' curl E^+ik E §], with E T the projection of E^ on H2(&). A first consequence of (68) is: (69)

(SHT, E£) = (n A H£, E £ ) = J ^ [k2 <A^ C^div E$, div E£>

- <C N AN' curl E^, curl ET> + ik<E Jf, E $>].

This implies (for all Ej in V" and then in H0(curl,Q)): (69)*

Re (SH°, E%) = Re (n A H£, EJ) < 0.

Then using the term "mild" by reference to a solution of Maxwell equation in a "weak" sense (in the sense of semigroup theory, see Pazy [1]), we can prove: Theorem 2. Except when k2 is an eigenvalue of the Laplacian with Dirichlet condition, problem (27) with (33) and the initial condition (70) E!r€H0(curiT,fl) has a unique mild solution (E, H) which satisfies: 0 |ET€C ([0,+oo),Ho(curlT,Q)), 1

HTeC°([0,+oo),H(curlT,0)), E3andH3€C°([0,+oo),L2(a)).

4.2 WAVEGUIDES IN

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179

This solution is given by (49), (49)\ (68), (52), (54), (57), (59). Remark 6. When k2 is an eigenvalue of the "Dirichlet Laplacian", we verify easily that the foregoing problem has no unique solution, since the following electromagnetic field (E, H) is a solution for zero "initial" condition: E° = 0, H 3 = 0, but E3(x3) * E3(0) so that (A + k2)E3(0) = 0 with E3(0) in Kl0(Q) (an eigenvector of the Dirichlet Laplacian) and Hj(x3) = H^O) = j — curl T E 3 (0). Definition 2. The mapping C: SE° T — SH° T (or C: E°- -► H° T ) which is defined by (68) is called the Calderon (or also the admittance) operator of the guide at frequency <D. Proposition 1. The Calderon operator C (resp. C) is a continuous mapping from space H Q (div,Ii) onto H(div,Q) (resp.HQ(curl,Cl) onto H(curl,Q)) with the following properties: (72) Re(CSE^,E^.)<0, VE^€H Q (curl,Q). Remark 7. We can give the dependence of the electromagnetic field with respect to x 3 (therefore the semigroups (49) and (49)') thanks to the semigroups G^(x 3 ) and G N (x 3 ) of section 1; for instance: E ^ ) = grad G D (x 3 ) A^1 div E^ - curl G N (x 3 ) A^1 curl E^ + e***3 E $(x3) HT<X3) = gradG

(x 3 ) A^1 d i v H j - curlG D (x 3 ) A^1 curl H j + e**3 H £(x3),

for E j in V", with E jinH2(Q)and (73)

H j » n H ^ Q ) . We note that the relation:

HT<X 3 ) = C ET(X 3 ) (or S H ^ )

=C SE^)),

which is true for x 3 = 0, is valid for all x3 > 0: the Calderon operator C commutes with the semigroups of (49), (49)', simply denoted by (GA(x3)) and (GB(x3)) (73)*

GB(x3)C = CGA(x3),

Vx3>0.

Remark 8. Note that when E T is in H 0 (curl,Q) and not only in D(A), we have 3H 3 , -g^- (x3) = C N H 3 (x 3 ) in (H (Cl)Y (from semigroup theory), whereas divT H T is in H" (Q), and this does not correspond to the usual sense of div H = 0 in D'(R3). Applying the curl and div operators to (68), we obtain: (74)

i) curl H° = | C^ 1 div E ° ,

ii) div H° = ^ A C N A^1 curl E ° ,

180

4 WAVEGUIDE PROBLEMS

that is also P93 H^ + div Hj = 0, with P = AAJ^1 which is the (orthogonal) projection from(H1(Q))' onto H'l(Q). Conversely, thanks to the Hodge decomposition ("conjugated" to (62)): (62)*

v = - grad A^1 L^ - curl A^1 curl v+v,

with v € Hjtfl), where Hj(Q) = {v € L2(Q)2, curl v=div v=0, n.v| f = 0}, applied to v = Hj, and then using (74), we easily obtain (68). We can also obtain the inverse relation to Ej with respect to HT either as above for SHT, or using the Hodge decomposition (62) with the inverse to (74) (this gives us the inverse to the Calderon operator C): SE£ = -curl A^1 div E^- grad AJ^1 curl E£ + S E $ (75)

= f [cTrl A ^ C D curl H^ - k2 grad C^1 A^1 L S H 0 - ik H $]

when Hj € H(curl,Q), with (76) H^Z^SE^; H j is the orthogonal projection of H j onto H^Cl) whereas E j is that of E T onto Hi(Cl). The inverse relation to (74)ii) is given by (77)

curl E£ = ikZ ACJi1 A^1 L

We recall that w » - A^1 L

satisfies Aw = div Hj since:

SHnp

(gradw, gradv) = (H^, gradv),

VveH^Q).

Using (75), we can see that conversely, given Hj € H(curl,Q), we have ET in H(curl, O) with the boundary condition nT A EJ S 0 on T; this is due to the relations nT A curl <fr = - n.grad <fr and nT A grad 0 = t.grad 0, V * with T = (-n2, nj) tangent to T. This proves Proposition 1 for the Calderon operator. Remark 9. Transverse-Electric (TE), Transverse-Magnetic (TM) and Transverse-ElectroMagnetic (TEM) waves. The Hodge decomposition (62) for the initial transverse electric field gives the "usual" decomposition of the electromagnetic field (E,H) into: i) the "Transverse-ElectroMagneticfield"(TEM waves) (E, H) with E3 = H3 = 0 and &,<x3) = e^ 3 E $, Hj(x3) = e^ 3 H $, with E £ € H2(Q), H $ € HX{C1\

4.2

WAVEGUIDESINELECTROMAGNETISM

181

and ~ yo _= L* *7""ici£ H OC oj .,

ii) the "Transverse-Electic field" (TE waves) ( E ^ , H TC ) with E 3 = 0, therefore: c u r l H j = 0, divE^ = 0, then H3(x3) = ^ £ G N ( x 3 ) c u r l E j , and: E ^ ) = - curl GN(x3) AN 1 curl E £ ,

H ^ ) = ^ grad G N (x 3 ) C N A^1 curl E j .

Thus the Calderon operatorfor TE-waves is given by SH

T " EZ ™tX C N A N

curl E

iii) the "Transverse-Magnetic field" (TM waves) (E T M ,H T M ) with H 3 = 0, therefore: c u r l E j = 0, divHj = 0, then E3(x3) = G D (x 3 ) E 3 = - G D (x 3 ) C^ 1 div E j , and: ET(X 3 )

= grad GD(x3) A^1 div E j , H ^ ) = curl G D (x 3 ) A^1 curl

H£

with (74). Thus the Calderon operatorfor TM-waves is given by: S H j = =Y grad A^1 C^ 1 div E^. This decomposition of the electromagnetic field is due to the splitting of the domains V4" and V"" by the Hodge decomposition. Yet we give a more useful framework. For usual applications, we have to find an electromagnetic field with locally finite energy in the closure of Q + . This implies (see chap. 2.7.2) that in each cross-section of the cylinder we have, with notations (85), (86) chap. 2: (78)

E1(x3)\Q€

H; 1 / 2 (curl,a),

H^)^€X(curl,Q),

Vx3 > 0.

Thus we have to solve problem (27) with a given initial boundary condition E° T with (78). These spaces are not very easy to handle, so we extensively use the interpolation theory and the following Hodge decomposition (compare to the Hodge decomposition above): Lemma 1. We have equivalence between i) and ii): i)

u € H; 1/2 (div,Q) (see (85) chap. 2)

ii) u = grad o + curl <>| + u,

with <> t € D(A^/4), (J> € D(Ap 4 ), and u € H^Q)

and also, using the S operator, we have equivalence between: i)u€H;1/2(curl,Q), ii) u = grad 0 + curl <>| + u\

with 0 e

D(A{> /4 ),

<> | € DtA^ 4 ), and u' € H2(C1).

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4 WAVEGUIDE PROBLEMS

Here A D and A N denote the generalizations of the Dirichlet Laplacian and the Neumann Laplacian to the interpolated spaces and the dual spaces. ForregularQ, we will use the domains D(A D ) - H 2 (Q)n H£(Q) , D ^ 2 ) - H*(Q) and by interpolation: LXA^4) = H^2(Q), D ^ ' 4 ) - H* /2 (Q)» H 3 / 2 (Q)n Hl0(O). We also have D(A N ) = {v e H2(Q), §J| - OJ, DCAjf2) = H ! (Q) and by interpolation: EKA},/4) = H 1/2 (Q), D(Ai!,/4)={u € H 3/2 (Q), n.grad u € H^ 2 (K r )}, VT being a neighborhood of T in £2. (SKETCH OF) PROOF of Lemma 1. i) implies ii). First * is determined (up to a constant) by solving: (79) A ^ = div u € H" 1/2(G) = D(A N 1/4 ), 3/4%

"■"*

which gives 0 in D(A N ). Then since the grad and curl operators are continuous from H 3/2 (Q) into H l/2 (Q) , by duality the div and curl operators arc continuous | is obtained by solving equation from H" 1/2 (Q) 2 into H" (Q). Thus <> (80) AD<|> = curl u € H"3/2(Q), 1/2

which has a unique solution in HQQ (Cl). ii) implies i) Thanks to the characterization of the interpolated domains, ♦ € D(A N *) implies v « grad 0 € H^ 1/2(div,Q) Lemma 1 implies for (locally) finite energy electromagnetic field in R3:

Theorem 3 . Except when k2 is an eigenvalue of the Laplacian with Dirichlet condition, the problem (27) with (33) and the initial condition: (81)

ET€H; 1/2 (curl T ,Q)

has a unique (mild) solution (E,H) which satisfies (with def.(S6) chap.2); (82)

I E T € C°([0,+oo), H;1/2(curlT,Q)),

H T € C°([0,+<x>), X(curlT,Q)),

E3€C°([0,+oo),(Hy 2 (a))') and H 3 €C°([0,+oo), H- 1/2 (Q)). Furthermore (E, H) has a locallyfiniteenergy up to the boundary in Q + : (83)

E and H € H loc (curl,5 + ), i.e., CE, CH € H(curl,n+), V C € Z)(R3).

Then we generalize Definition 2 and Proposition 1 on the Calderon operator to the present framework.

4,2 WAVEGUIDES IN

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183

Proposition 2. The Calderon operator C (resp. C) of the guide atfrequency a is a continuous mapping mapp from H^ 2(div,Q) onto X(div,Q) (resp. HjJ (curl,Q) onto continuous X(curl,n», with: (72)*

Re (CSEJ,ET) ^ 0 ,

V E^ € H; 1/2 (curl,Q).

Note that the electromagnetic field in Theorem 3 and the Calderon operator are obtained by the same formulas as above. Theorem 2 is a regularity result with respect to Theorem 3 for the electromagnetic field. Then we have another regularity result in the H 1 framework: Theorem 4 (Regularity), When k 2 is not an eigenvalue of the Laplacian with Dirichlet condition, the solution (E,H) of problem (27) with (33) and the "initial" condition: (84) E?.€Ho /2 (curl T ,n) is such that: (85)

H j € H 1/2 (curl T ,Q)riH; 1/2 (div T ,Q)), E° € H"02(Q) and H°3 € H 1 / 2 (n),

and thus (E,H) satisfies continuity properties from x- (in R+) into these spaces. Furthermore (E,H) has H 1 regularity up to the boundary, that is: (86)

E and H € H ^ Q J 3 , i.e. CE, CH € H 1 ^ ) 3 , V C € D(R3).

Proposition 3 . The Calderon operator Cisa continuous mapping from the space Ho/2(curlT,Q) onto H 1/2 (curl T ,Q)nH; l/2 (div T ,Q). Example 2. Junctions and Cascades in electromagnetism We consider the situation of Example 1 with a junction and p guides incoming to it. We assume that: i) the (side) boundaries of the guides and of the junction TQ are perfectly conducting; ii) a given regular incident electromagnetic wave (EJ,HJ) with angular frequency d> is coming from the guides to the junction; iii) the permittivity and the permeability of the jth guide and of the junction are respectively (e.,^) and (e,jj). Thus the reflected wave (E.,H .) in the jth guide must satisfy (from Propositions 2 and 3, using the Calderon operator C. of the jth guide) on top T. of the guide (87)

nAHr|=C(nAEr|). j

Then using the continuity relations of the wave across the boundary T.:

184

(88)

4 WAVEGUIDE PROBLEMS [n A E] =0,

[n A H] = 0, that is:

nAE = nAE r + nAEj, nAH = nAHr + nAHj, onlj, j = l top, we obtain that the electromagneticfield(E, H) has to satisfy on T-: (89)

nAHl -C j (nAEL) = fIj, with f„ «ii A H, | r - C O i A E , ! ) . j

Thus we have to find the electromagnetic field (E,H) with finite energy in the junction satisfying: i) Maxwell equations in Cl: (90)

|i)curlH + i<oeE = 0, ii) - curl E + iop H = 0 in Q,

ii) the boundary condition on ro, then on Ij (91)

nAE| r = 0, with(89)only

We write this problem in a variational form. We define: (92)

Hr (curl,Q) =f {u € H(curl,Q), n A u| _ = 0},

and the sesquilinear form on Hr (curl,Q): (93) a(E,E)=SQ(-XcuTlE

. curll - icoeE . E)dx- 2 / r C , ( n A E | r ) . E| r d^.

Then problem (90), (91) is equivalent tofindE in Hr (curl,Q) satisfying: (94)

a(E,E) = 2 / j

F„ . E dx,

V E € Hr (curl,Q).

i) First if the medium of the junction is dissipative, that is if (95)

lme>0, and lmji>0,

problem (94) has a unique solution; this is easily proved as follows. Taking the real part of (94) with E = E, thanks to (72)', there is a constant C 0 so that: (9i)

Rea(E,E) > C 0 ! E £

(curla),

VE €H^cur!,^).

Thus the sesquilinear form a(E,E) is coercive on Hr (curl,Q). o Thanks to the Lax-Milgram lemma, problem (94) has a unique solution. ii) / / the medium of the junction is conservative (that is when c and \x are real positive) we will see that problem (94) depends on the Fredholm alternative.

4.2 WAVEGUIDES IN ELECTROMAGNETISM

185

Note that the natural injection Hr (curl,Q) â&#x20AC;&#x201D;> L2(Q)3 is not compact. o

But for given regular incident electromagnetic field, fy is (for instance) in H (curl,rj)H H~ / (div,^); E must satisfy a variational problem in Hr (curl,Q) with div E = 0 and its boundary value must be in H1/2(curl,0; thus E is in Hl(Cl) . Since the natural mapping from H*(Q) into L2(Q) is compact (for regular Q), this problem (94) depends on the Fredholm alternative: (94) has a unique solution except for the eigenfrequencies of the junction (and also of the guides corresponding to the eigenvalues of the Dirichlet Laplacian). Remark 10. In these junction problems we are essentially interested in obtaining the electromagnetic field and above all the propagating modes which are outgoing from the junction through the guides. If the medium of the junction is homogeneous, we can write the problem on the boundary of the junction thanks to the Calderon projectors P. or Pe (see Def. 3 chap. 3) or the (interior) Calderon operator C1 (see Def. 5 and (211) chap. 3); thus we have to find the boundary value of the electromagneticfieldsatisfying: (97)

P e (nAE| r ,-nAH| r ) = 0, or nAH| r = C W E | r ) ,

with (89) and (91). Taking the value of the magnetic field on T, j = 1 to p, gives us a system of equations on the boundary value of the electric field. This is very simply written thanks to the Calderon operator (but less suitable for a numerical solution): (98) 2 (CJj,-Cj5jj.)Mj, = f Ij ,j = ltop,withM j = nAE| r â&#x201A;ŹH; 1/2 (div,r j ). j ' = 1 to p

If the medium in the junction is inhomogeneous, then we have to connect (for instance) afiniteelement method in the domain Q with a spectral method on the boundary of Q. Such a method has been used and numerically implemented in order to evaluate the permittivity and permeability of a sample in a wave guide. Obviously there is a great variety of examples and of geometries for this study. <8>

Remark 11. We haven't studied the more difficult case of waveguides which are inhomogeneous in the fiber, that is, the cross-section of the guide up to infinity is inhomogeneous. It seems necessary to use microlocal analysis for such a case. 0 Remark 12. In Examples 1 (with Helmholtz) and 2 (with Maxwell), the Calderon operators (C.) depend on the angular frequency <o, and even are holomorphic functions in a neighborhood of the real axis (except for oo = 0 in the Maxwell case), see for instance Sanchez Palencia[l] p.350). Thus using a lifting, we can write problem (20) (or (94)) in the form: (99)

A(tt>)u = FI(a,) with given F^,

186

4 WAVEGUIDE PROBLEMS

where (A^) is a holomorphic (in a>) family of closed operators in a Hilbert space H, with compact resolvents. Therefore we can apply theorem 1.10 (p.371) in Kato [1]. Since outside the real axis (for o>) the operator A(a)) has a kernel reduced to {0}, we deduce that the problem (99) depends also on the Fredholm alternative, with only a finite number of singular values to on everyfiniteinterval J (in the Maxwell case we have to assume that 0 is not in J).

CHAPTER 5

STATIONARY SCATTERING PROBLEMS ON UNBOUNDED OBSTACLES

1. PLANE GEOMETRY

The study of stationary waves scattered by infinite obstacles follows naturally from scattering by finite obstacles at high frequency. For a plane geometry, we have an ill-posed problem (as in the waveguides) if we don't take into account the sense of wave propagation. Here we extensively use the Fourier transform method, which replaces the discrete spectral decomposition in waveguides. First we study Helmholtz problems, then electromagnetism with Maxwell equations. 1.1. Plane geometry with Helmholtz equation First we consider a model Helmholtz problem with a Dirichlet condition (which corresponds to scattering by a soft wall in acoustics) as the starting point to understand the basic problem. 1.1.1. Helmholtz problem in a half-space with Dirichlet condition Let R* = ( X a ( x ^ ) â&#x201A;Ź R n ~ l x R, xn>0} = Rn"1 xR + with (generally) n = 3, and let T = R n " l x {0} be its boundary. We consider the following Dirichlet problem:findu satisfying (1)

|i)Au + k2u = 0 inR? jii)u| r = u0 onT, 187

188

5 UNBOUNDED OBSTACLES

with a given wavenumber k > 0 and given IK on T, for instance in L2(D. We study this problem thanks to a (transverse) Fourier transformation. Let fv denote the Fourier transform of a tempered distribution v, with: (2)

v«) = .Fvtt)=/ n.ivMe-^dx,

forveSCR""1).

Then with the notation u(Xj) » uCx,^, we write (1) as: ,2A

:\d U

(3)

i ) ^ 3 + (k 2 -S 2 )fi = 0

forx n >0,

ii)fi(.,0) = fi0. Thus we are reduced to solving a differential equation in terms of the variable xn only. We get, with constant functions (with respect to xfl) a and 0 to specify i) for|t|>k, (4)

iKUn^attJe0* V K 9 e ~ 0 + \

with8+= tf - k2,

ii) for|t|<k, (5)

fitf^)

= <x«) e*-** + m e~l*-\

with 9. = ^k 2 -* 2 .

Wanting a solution which is tempered with respect to xn (and £) at infinity we have to choose (6) <x(£) = 0 for |£| >k, thus: «t) = fi0(9 for |t| >k, therefore: (7)

ii&xj^e-****^)

for\i\>k.

When |£| <k, (3)ii)with(5)gives: (8) a(£) + p«) = fl0(£) for|t|<k, and this does not allow us to uniquely determine a solution of (1): problem (1) is ill-posed. Similar to waveguides, we transform it into a well-posed problem by adding a physical condition of wave propagation. For a time evolution given by u(X,t)sse~lwtu(X), we say that u is a wave propagating towards the positive (resp. negative) axis xR if the propagating part ofu (i.e., (5)) is a superposition of waves of the type (9)

exp (i8jgvtf) (resp.expHe.x^vtf)),

with9_>0.

Then with the physical choice of waves propagating towards the positive axis xn, we get a unique solution for (3) (and then for (1» of the form: (10) fl(e,xn) = e"8(e)Xnfl0«), with:

189

5.1 PLANE GEOMETRY (11) 9(0 = e+ = (£ 2 -k 2 ) 1/2 when|£|>k,

8(£) = .ie. = -i(k 2 -£ 2 ) 1/2 when |t| <k.

Theorem 1. The problem (1) with uQ given in L2(Rn~l) and with the condition (12) u is a tempered wave propagating towards the positive axis xn, has a unique solution u in C°([0,+oo),L2(Rn~l))y and it is obtained thanks to a contraction (and holomorphic) semigroup of class C° in L^R11""1), (G(xn)), xn > 0, by its Fourier transform (10), with (11) (13)

= G(xn)fi0«) = e" **** fi0«), £ € Rn"l.

fitf^)

PROOF.

The operator G(xn) is a contraction in L2(Rn"~l) since

(14)

sup logK-erosgi-i, Vx n >0.

Furthermore, we have: (15)

HuCxJ - U0l|2 = (2*)- n / R n-l Ifl&Xn) -U0«)|2d£ = (2«)* n / nD n.liao«)| 2 M-e- e X n | 2 dt 'R

Since UQCL^R "" ), for all 6i> 0, there exists a>0 such that: / . . and there exists n > 0 such that 0 < xn < n implies

|u 0 | dE<Ej,

(16) |l-e" 0 X n | 2 <£ 1 , V£with|£|<a, thus for 0<x n <n, (17)

W . ^ - U / ^ M O + IUOI2),

which implies that the semigroup is continuous. Proposition 1. The infinitesimal generator A of the semigroup (G(xfl))t xfl > 0, in L2(Rn~l) has the domain D(A) = H^R11"l) and is given by its Fourier transform: (18)

Au0(£) = - 9«)u0(S)

with 9 defined by (11).

PROOF .

The domain D(A) of the generator is given by its Fourier transform

(19)

I D(A) = {fl0 € LAR11-1), Afl0 6 L2(Rn-!)}, -|flo€LV-I),/,5,>k(l2-k2)|ao|2«<«>}f

from which we clearly have D(A) = H*(Rn~*). <8>

190

5 UNBOUNDED OBSTACLES

We also have:

(20)

A^O-tf-k2^©,

therefore: (21) A2 = -(A + k2),

withEKA^H^R11-1).

Thus A is a square root of the self adjoint (but not positive) operator - (A + k2). Like in the waveguide case, we can decompose the space L (R n ~ l ) into: (22) L^R^^H^eH^, so that A splits into: (23) A » - A+ + i A", A+, A" positive self adjoint operators resp. in H£ and H£, and A satisfies: (24)

Re (Au,u) <: 0,

Im (Au,u) > 0, V u € H^R11*1);

A is a normal operator: D(A) ■ D(A ) and A commutes with A , with spectrum (25)

o(A) = (-oo,0]U[0,ik].

Definition 1. The infinitesimal generator A of the semigroup (G(xn)), x n >0, defined by (IS) is called the Calderon (or the capacity) operator of the half-space, We recall that A is the mapping u0 —»g~ | , u the solution of (1) with (12). Remark 1. We have to verify that choosing waves propagating towards the positive axis xn as above, agrees with choosing outgoing waves defined by the Sommerfeld condition. It suffices to verify that the elementary outgoing wave ka centered at point X =(0,0, - a ) , "a" being positive, satisfies (12). Using a translation, we can take a = 0. Then using the Fourier transform of the elementary outgoing wave * k given by (159)' chap. 3, and taking the inverse Fourier transform with respect to variable xfl only, we obtain (for n = 3) (26) Thus: (27)

F^Fx^^^Fî^x^^î^mâsp

(" «(© W>-

^S(£,x3) = Ok(£,x3 + a) = ^ e x p ( . 0 ( t ) | x 3 + a|),

which is of the form (13) forXn > 0 (but for a = 0, u0 = 1/(29) is not in L2). From formulas (13) and (26), we see that the semigroup is also given by:

5.1 PLANE GEOMETRY

(28)

191

G(x3)fl0tt) = - 2 g ^ ax 3 )a 0 «),

that is also with the convolution product * (a priori for u0 with compact support) (29)

G(x 3 )u 0 - - 2-gf (.,x 3 ).u 0 . ikR

The semigroup (G(t)), t = x3 > 0, is also given with <&fc(x,t) = | j g - , R = (x2 + t 2 ) i / 2 by (30)

G(t)u0(x) = - 2/ R 2^(x-x , ,t)u 0 (x , )dx'.

Thus the operator A is obtained by the convolution in R Au0 = ax*u0, with a(x) = - 2 ^ ( x , 0 ) = - 2 F P ( i | ( ^ ) ) , r = | x | , i . e . <a,0> = lim/ e °°V5 , (r)dr, with 5Xr)«2~jJV(r,B)<», V*€Z)(R 2 ). Note that 5'(0) = 0 from (397) chap.3. When k = 0, the semigroup (G(t)) is given by (31)

G(t)u 0 (x)=^/ R 2 ( ( x _ x j

+ t 2 ) 3/ 2

«o(x*) dx',

that is, the Cauchy-Poisson semigroup which is holomorphic in ail LP(R2), p * oo (see Butzer-Berens f 1] p. 248). This seems not to be true for k * 0. Remark 2. We can find formula (29) using the "image" method: by symmetry we define in the whole space R3 (32) Then (33)

U(x,x3) = u(x,x3) when x3 > 0,

U(x,x3) = - u(x, - x3) when x3 < 0.

AU + k2U = div (2u0n6r) in R3, with n = e3 * (0,0,1).

Thus, for u0 with compact support, U is an outgoing wave given by (34)

1* 3 ) x *u 0 , U = .O k ^div(2u 0 n6 r ) = -gradO k ^(2u 0 n5 r ) = - 2 3^(.,x

that is (29); thus U is a double layer potential.

Remark 3. Some generalizations to other frameworks. First note that A is not a bijective mapping from H*(R2) onto L2(R2). This suggests that it is not the best framework to consider (from the point of view of applications also). We first define spaces with a special weight for tangent (or longitudinal) waves, i.e., for £ on the sphere Sk of radius k: (35)

H£/2(R2)={U€S>(R2),

fi€L/OC(R2),

fl€L2(R2,

|e| d«)},

Hk 1/2(R2) = {u €S\R 2 ), fi € L ^ R 2 ) , fi € L2(R2, |6| - 1 d£)}.

192

5 UNBOUNDED OBSTACLES

Note that if pk(() = d(£,Sk) is the distance from £ to the sphere Sk, then in the neighborhood of Sk: (36)

|e k (E)| 2 s |(|6| 2 -k 2 )| = | ( | £ | . k ) | ( | £ | + k ) « 2 k 6 ( t , S k ) = 2kpk(£).

An example is given by the trace on the plane T of a wave due to a point source not in the plane: thanks to (26), for a * 0,4>£| € H£ / 2 (R 2 ). Since 9 and 8"l are in L^R 2 ), then u € L2(R2, |9|d£) implies by Cauchy-Schwarz inequality

(37)

/K|fi|d^(/K|fi|2|9|d£)1/2(/K | 9 r W 2 ,

for aU compact sets KinR 2 , thus u € L^R 2 ). When u € L2(R2, | 9 | - 1 dt), we have the same result. The condition u € L/OC(R2) in (35) is only used to eliminate measures concentrated on the sphere Sk of radius k. Furthermore, using the inequality: |9|<C k (l + |S| ) 1/2 , Ck=max(i,k), we have (38) » u l t ^ ^ d j f / R 2 l « I 2 | « l « ^ C j - ^ l f i | 2 < 1 + IM 2 > 1/2 *tt=C^luB^,,^^. We deduce the following space inclusions (39) H1/2(R2)— Hk/2(R2)and Hk1/2(R2)-> H"1/2(R2), by duality, Hk 1/2(R2) being identified with the dual space of Hk/2(R2). Then we have by Fourier transformation, duality and interpolation: 1/2

1/2

Proposition 2, The mapping u -»Au is an isome tryfrom Hk (R ) onto Hk 1/2

(R ),

l/2

and is a continuous mapping from H (R ) into H" (R ). Moreoverfor u0 in Hk/2(R2), the solution u of(\\ (12), given by (13) is in H/ 0C (R I), that is,uis oflocallyfinite energy up to the boundary. The framework (35) does not allow us to directly tackle all useful examples such as plane waves. So we have to generalize. The framework of tempered distributions (in R2) is not adequate since exp(-0x 3 ) is not a multiplier in 5"(R2). But we can use other functional spaces, for instance Besov space B used by Hormander [1], T.l chap. XIV p. 227 and its dual space B*. For most applications in view, it seems interesting to work with the space Ek defined by: (40) Ek = (u € S*(R), JFxu is a Radon measure concentrated on the sphere Sk c R }.

5.7 PLANE GEOMETRY

193

From the Paley-Wiener Theorem, the elements of Ek are entire functions in R3, which satisfy the Helmholtz equation in the whole space. The set of elements of £ k propagating towards the positive axis x3 is (A\\ E£ = l u € Efr ^ x u *s a ^ a c , o n measure concentrated on the half sphere S£}, with S£={(5,£3) e Sk, with £3 > 0}. Of course, there are other functional spaces which may be useful, like: (42)

X = {u€L2(R2), fi€C0(R2)}, withC0(Rz) = {v€C(Rz), v(S)-> Owhen |E| — 00}

and its dual space X' also defined by (43)

X' = {f € ^(R 2 ), f € Ml(R2) + L2(R2)}, with Ml(R) the set of bounded Radon measures.

We can prove that X' is contained in the space: (44)

Cb(R2) + L2(R2), with Cb(R ) the space of continuous bounded functions on R

Furthermore, we can prove that the spaces X and X' are stable under the semigroup (G(t)), t > 0, which is a semigroup of class C° in these spaces; X' allows us to tackle the scattering of an incident plane waves by an infinite plane wall. But these spaces are not very easy to handle from the point of view of functional analysis. Plane waves are often used by the spectral theory as generalized eigenvectors (of the operator A). Remark 4. The Helmholtz problem (1) with Im k different from 0 (0 < arg k < n) is well-posed if we only require that u be a x3-tempered solution. It is also obtained by a semigroup of class C°, which is holomorphic, and given by its Fourier transform: (45) fl«,X3) = G(X3)u0a) = exp(-ex3)u0(£), e«) = tt2-k2)1/2 with Re 0(£) £ 0. Here 9*0, thus A is an isomorphism from H1/2(R2) onto H"1/2(R2). The limit of (45) when Im k tends to 0 with Re k positive (resp. negative) gives the wave propagating towards the positive (resp. negative) real axis x3, following the limiting absorption principle (see chap. 3). 0 1.1.2. Transmission Helmholtz problems in R3. Calderon projectors First we consider the following Helmholtz problem in the whole space: find u in R 3 satisfying (for a given real wavenumber k)

194

5 UNBOUNDED OBSTACLES i) Au + l^u = 0

(46)

forx3 * 0, i.e., in R* and R3. ,

ii)[u] r d ^u| r - u | r =p,

[§-] r =P* onr, 3

iii) the restrictions of u to each half-space R+ and R_ are "outgoing" waves, i.e., resp. propagating towards x3 > 0 and x3 < 0, with given p and p'onTsR x {0}, with hypotheses to be specified later. We know (for instance from chap. 3) that this problem has at most one solution. Using the above section, we are led to find the boundary values of the solution on r only. We can use different methods. Here we use an integral method, similar to chap. 3. Under usual (regularity) conditions on p and p' (see chap. 3), i) and ii) are equivalent to (47)

Au + k2u - - (p' 8r + div (pn 8r)) in Z)'(R3).

With the "outgoing" condition iii) (at least when p and p' have compact supports), problem (1) has a unique solution u given by the convolution product: (48)

u = * < (p , S r + div(pn6r)),

with O = * k the elementary outgoing solution (see (8) chap.3), or also (49)

u(.,x3) = <K.,x3)x, p' + J£. („x3)x, p.

If we differentiate with respect to x3, we obtain: < 50 >

^<- x 3)=^^ x 3) x .P'^(-.x 3 );P.

Then, when x3 —»0+ and x3 -* 0_, using (29), we have:

['-£&)•'-[',)• (*&«)-'-(•)•

with P+ and P_ given by (52)

P+-±(I-S),

P . = i ( I + S),

Thus, we have (53)

P+ + P_=:I, and S2 = I.

We can verify that (54)

B^-AA-',

- ( : - ; >

5.1 PLANE GEOMETRY

195

is an integral operator given by the convolution product:

Bpw-«(..a>.p'-J^ 4r^T p ' ( x ' ) d x '-

(55)

Proposition 3. The operator Sis an isomorphism satisfying (53), in the space (56)

Y k d ^ H£/2(R2)x H£1/2(R2),

and thus P and P _ are continuous complementary projectors in this space, giving the boundary values of the problem (46) by (51) when (p, p') are given in Yfc. Definition 2. 77te operators P+ a/irf P_ are called Calderon projectors {for the Helmholtz problem) of the half space at wavenumber k. If we compare the half-space problem to that of a bounded domain (see chap. 3 Def. 2 and (71)), we see that we have, with notations of chap. 3: K = 0, J = 0, 2L = -A~ 1 ,

2R = - A .

1.1.3. Transmission problems with two different media We assume that the whole space is occupied by two different media, one on each side of a plane. Let k{ and k2 be (real for instance) wavenumbers in each medium. Then we consider the following transmission problem: find u in R3 satisfying i) Au + k2u = 0 forx 3 *0, withk = kj inR 3 andk = k2 inR 3 , (57) |ii)M r d = e f u| r _-u| r + = p,

[|y-] r = p >onr,

iii) the restrictions of u to each half-space R+ and R_ are "outgoing" waves, i.e., resp. propagating towards x3 > 0 and x3 < 0, with given p and p' on T = R x {0}, with hypotheses to be specified later. This problem is a scattering problem when we have a given incident wave Uj (in the first medium for instance) either produced by a source of compact support in R3+, or by a plane wave. Thus we have to find the reflected wave Uj in R + and the diffracted (or transmitted) wave u2 in R3 _, satisfying: 8u, 3uT 3u2 (58)

u 1 + u I = u2,

^-+5E-=^-

onr.

Thus taking u = ux in R +, u = u2 in R _, we see that u has to satisfy (57) with:

(59)

=Ul|r,

p'=gl| r .

196

5 UNBOUNDED OBSTACLES

Let A. and 0., j = 1, 2 resp. be the Calderon operator A and the function 0 defined by (18) and (11) for k.. Then problem (57) is reduced to finding the boundary values of u, u, and u2 on I", and these are obtained thanks to the Calderon operators of each half spaces from (57)ii) (with i) and Hi)) by: DOJOJ + M I - ? '

(60)

|ii)u 2 -fi, = p\

The solution of (60) is given by: A

< >

p'+e.p

*2-W'

p '--et2 _ p

^'T*2 *

When p and p' are given by (59) with an incident wave propagating towards the negative axis x3, we have (62)

p' = -A 1 (uil r ) = -A,(p).

We obtain the boundary values of the transmitted wave and the reflected wave by (63) with

u 2 =fu,,

u, = Ruj onT,

1 "^" 2 A

1 ^* 7.

R and T are called the reflection and transmission coefficients. They satisfy (65)

T-R=l.

Proposition 4. We assume that the incident wave u{ satisfies (66)

u,| r €H k /2 (R 2 ), t h u s ( U l , ^ ) | r € Y k i .

On the boundary T of the two media the reflected and the transmitted waves satisfy (67) u, | r € H k2(R2) and u 2 1 r e H k 2(R2)n H{/2(R2) (and even u21 f 6 H1/2(R2)) 3u, andthus ( U l , ^ i ) | r 6 Y k l ,

9u7 (u2,-g^eY^m^.

PROOF. We have (68)

/ R 2|fi2l 2 |e 2 l d 6 - / ^ | « , | 2 | t | 2 | e 2 | * - / ^ | « I | 2 | e 1 | t d t .

5.1 PLANE GEOMETRY

with T=

197

419,6? I 2 ' 2 . We easily see that T< 2, and thus:/ 2 |u 2 [ |S 2 |dt<oo.

Since | f | <2, we also have/ 2 |fi 2 | 2 |9il &< °°- Furthermore:

/^l^l^i.UlV^d^/^lii^iejad^ with g . 4 I W H ^ 1 / 2 . R

l e l+ 9 2l

But a is bounded on R2, thus we have u 2 | €H1/2(R ), and Proposition 3 follows. Generalizing to spaces such as that deflned by (41), or (43) allows us to deal with plane waves, giving the famous Snell-Descartes laws. The incident wave Uj is given by (69)

Uj(X)«iiI0e l*' ,

withuIO€C,a = (a T ,a 3 )€S 2 ,a 3 <0,

whose Fourier transform is the measure (69)' fi,(£,x3) = (2n)2 6({ - k ^ ) e*1"3*3 uI0, with |a T | 2 + a] = 1. Thus p and p' are given by their Fourier transforms: (70) £«) = {1,(5,0) = (2*)2 8« - klttT) uI0, $'«) = &T (£ ' 0) = (2n) m±i°^ ^ - kiotT) uioHere formulas (60) are also valid and imply that fl, and u2 are Dirac measures concentrated on { = kjaT. Then 8j and 02 are given by (71)

e ^ o ^ a i l ^ a j , e2(k!aT) =

(k24-^)1/2 |-i(k 2 -k 2 <4) 1 / 2

ftkl\^r\>k29 ifk!|a T |<k 2 .

Thus with the (relative) index n = k2/kj, we have (63) with: m\ (

2 g

l 3l I«3l+n9 12 '

A R=

I a 3l-n0 1 2 |«3|+n812'

' and (73) 912 = i02/k2 = i((4/ n 2 )- D 1/2 if K l >n,

e12 = (l -{al/T)2))1'2 if |a T | < n .

The inverse Fourier transforms of the reflected and the transmitted waves are, for the reflected wave A

(74)

ik,0.X

Uj(x,x3) = R uI0e

with 0 = (aT, - a3),

and the transmitted wave (74)' ^ u2(x,x3) = fu I0 e ik2 ^ X withg = (PT,P3), where P may be complex; we have

198

5 UNBOUNDED OBSTACLES

i)fork,<k2, i.e., n>l, Ms 2 ,withP T =Vn. i*3 = - 0 -(<W))m,

thus |P T | < K | , |P 3 | >|a 3 |,

or with respect to incident and refracted angles of wave propagations *j and *2, I 0 T | » sin *2 = I arl/n * (sin ♦,)/!!,

*2 < *p

ii)fork|>k2, i.e., n<l, a)k!|a T |<lc 2 , Ms2,pT=Vi» *us I M < K M P 3 l < l a 3 l ^ 2 < ^ b)kl\*T\>k2> PT = aT/n, but P3 is complex (the transmitted wave is exponentially damped) P 3 3 S -i(( a 2/ n 2 )-l) 1 / 2 ,and P 2 = ^ + Pf=l, but Pis not in S2. 0

1.1.4. Some examples of applications of the Calderon operator There are very many examples, where we have to use the Calderon operator A in order to obtain a well-posed problem, taking into account the sense of wave propagation, even for geometries which are not simple half-planes. Using the Calderon operator A often gives a boundary condition, which allows us to reduce the domain of the wave problem to a slab or to a bounded domain. Thus, we can treat multilayer scattering problems (that is, for slabs with different homogeneous media in the free space), screen scattering problems, where the screen can be thick or thin, infinite (for instance a half plane) or not, with apertures of arbitrary shape.... Here we give an example with a thin soft screen with bounded apertures. Let T0 be a "flat domain" that is an open subset of R2x{0}, modelling a thin screen; we assume that its complementary set Tl^R2x[0}\To is a "regular" bounded set (connected or not), modelling apertures in the screen. We assume that the free space is outside the screen and that a given incident stationary wave Uj with wavenumber k is propagating in R3+ towards the screen. Let \x{ and u2 resp. be the reflected wave in R3+, the wave in R3_ transmitted through the apertures. Let u be defined in R^R^O}) by u = Uj in R3+, u = u2 in R 3 _. Then u must satisfy: i)Au + k 2 u«0inR\r,withr=R 2 x{0}, 3o>.

(75)

3ui.

9uT

»I»lr1,,,r»«lrr,Iilrr,lilrI'ftrr2lrriSlr1-2lr1-0Bri Hi) U||

=-U||

>u2| =0 (soft screen conditions)

iv) u t (resp. u2) is propagating towards the positive (resp. negative) axis x3.

199

5.1 PLANE GEOMETRY

Using the Calderon operator A, this problem is reduced to finding u on both sides of r satisfying (76) Uj-Uj^UjOnTj, with (75)iii) on T0, and (Auj + Au^L ~--§5"l r onTj. Let w=u 2 + u1 + Uj on I\ Then, from (76), w = 2u2, and w must satisfy (77)

Aw|r=f,|r,

withw=0onr o ,

3Ur

withfjL ^-"gjrlr - A u i l r =-2AU|| (78)

llR2 J W € H i / 2 ( R 2 ) -

Now let V={v € H£ / 2 (R 2 ), v| find w in V such that (77)'

(Aw,v) = (f,,v),

given on Tj. We assume that Uj satisfies

thusf^HJ^R2). = 0}. Then we can write (77) in the variational form: W€V.

Lemma 1. The sesquilinear form a(u,v) defined by: (79)

u, v € H^/2(R2),

a(u,v) = (Au,v), 1/2

is "coercive" on H k (R ) and more precisely (80)

|a(u,u)|-lud l /a / .a 4 .

Vu€l4 /2 <R 2 ).

PROOF. From definition (79), we have (81)

a(u,u) = - f

2 0|G|

d*,

and thus we obtain (80) from (81)' |a(u,u)|=/ R 2 |8||fi| 2 d£. Lemma 2. H£/2(R2)(resp. H^m(R2)) is contained in H/0/2(R2)(resp.Hf0lc/2(R2)). PROOF. Working on the Fourier transform of these spaces, we decompose the corresponding L2 spaces into the sum of two spaces, one WQ with functions equal to zero in a neighborhood of the sphere S. and the other Wj with functions equal to zero outside this neighborhood. Now we see that the inverse Fourier transform of functions in WQ are in the Sobolev space H1/2(R2) (resp. into H"~1/2(R2)), and that the functions in Wj having compact support are transformed into regular functions. <8>

200

5 UNBOUNDED OBSTACLES

Proposition 5. Let u} be an incident wave satisfying (78). Problem (77)' (thus (77)) has a unique solution w (thus u2 the transmitted wave) such that

(82)

wcH^OV,

giving the unique solution of (75) (with bounded apertures Tx in a screen in R3), that is the transmitted wave and the reflected wave thanks to the semigroups (G(x 3 ))am/(G(-x 3 ))M31). PROOF. First, using (39) and the lemma 2, we can identify H^O^) with the closed subspace V of H£ / 2 (R 2 ) by the mapping u —> 5, u being the extension by 0 of u 1/2

outside Tx. Furthermore, we have fj € V = (H0 0(r1)y. Then using Lemma 1, we can conclude thanks to the Lax-Milgram lemma (see for instance Lions [4]). 1.2. Plane geometry with Maxwell equations 1.2.1. A typical problem in a half-space We first consider the following problem: find the electromagnetic field (E,H) in free space, at angular frequency a>, satisfying i)curlH + io)£E = 0, (83)

ii)-curlE + i(i)jiH = 0

inR3 = R^xR£ ,

iii) n A E | = n A E j (orwrE = Ej), with given E T , with nT the orthogonal projection on the boundary r of R+. Like in the Helmholtz case, we can see by Fourier transformation with respect to x (under usual hypotheses on E°~) that this problem is an ill-posed problem, and it becomes well-posed with the following hypothesis (84) (E, H) is a wave propagating towards the positive axis x3 . Since the components of E and H satisfy the Helmholtz equation, they are obtained thanks to the semigroup (G(x3)) of section 1.1 due to (84), from their boundary values on I\ With usual notations (see also chap. 4) we write (85)ET<X3) = G(X 3 )E£,

E3(x3) = G(x3)El HT<X3) = G(X 3 )H£, H 3 (X 3 ) = G(x3)H°3,

thus we only have to find E3, H3, Hj from the transverse component E^ of E. We can do it in many complementary ways. We first write Maxwell equations ID the form (30),(31) chap.4, decomposing the electromagnetic field into its transverse and longitudinal (along x3 axis) components. Using (31) and (38) in (30) gives us (34) with (35) chap. 4, then using the inverse of the infinitesimal generator A of the semigroup (G(x3)), we obtain (with <oji ■ kZ, and o>c = k/Z):

5.1 PLANE GEOMETRY

201

i) H T = 2 £ A " l (k2 + gmh d«vT) SE T ,

(86)

ii) Ej. = - Y A* l (k2 + gradf divT) SHT. Thus we get the following relations (87)

1)HT = C.ET,

or nAHT=C(nAE£)(i.e.SHT = C(SE£)),

ii) E j = - Z 2 C H £ , or n A E £ ■ - Z 2 C (n A H J ) , (i.e. SE^ = - Z 2 C ( S E £ ) ) ,

and thus: (88) C 2 = - Z ~ 2 I , C 2 = - Z - 2 I , and C = -SCS, C = -SCS, with C (and C) given by (86), that is, in Fourier transform, H j = C E T»

<89>

zk9^

^.^J^zke^.^.^) . ^ J .

and C =-

— zke^(k2.$J)

.^J

Now we follow another method. Taking into account that E and H are divergence free (and satisfy Helmholtz equation), we can substitute to (30) chap.4: (90)

divE = 0,

divH = 0,

that is we only consider the equations: (91)

9E 3 »a^

+ div

9H3 a—+divTHT=o,

TET = 0 '

ii) curlT H T + io)6 E 3 = 0,

- curlT E T + io>y H 3 = 0.

In the following we drop subscript T to differential operators (when it is not confusing). From these equations, we easily get (at least formally) (92)

(93) divH^-AH^, and thus: (94) that is:

(94)'

E^-A^divEj,

H^-j^curlE^,

curlH^-^E^,

div H£ = j ^ A curl Ej,

curl H ^ = | A" l div E j ,

(diViil]4(° ^ curlH^ J

V. kA"'

^A1fdiV^l 0 J ^ curlE^ J

202

5 UNBOUNDED

OBSTACLES

Then we get H T using the Hodge decomposition: (95)

H £ = grad A"' div H^ - curl A" ! curl H^.

We define the following Riesz operators (this is useful to specify spaces) Rg* = grad (- A)- 1/2 ♦, i.e., Rg ♦«)=-j|r♦«), (96) R^cTrK-A)- 1 ' 2 *, i.e., Rc$tt) = - j | | $ « ) , and their adjoint operators R;u = -div(-A)- 1/2 u, i. e .,R>tt) = - « j H p , (96)' 1 ^ u = curl (-A)".1/2. " u,

i.e., R,.fl«)=

iUfl(t) —^-.

These operators satisfy: (97) R g = SR c , and also: (98)

Rc = - S R g )

| R g $ « ) | = !♦(«)!,

R^ = RgS,

Rg = - R ^ ,

|Rc$tt)M$tt)|,

and thus (with L norm for instance): IIRg0ll = 11*11, IIRc0ll = ll0ll, (99) i.e., R and R c are isometric operators in L 2 . Then the (orthogonal) projectors P a n d r c on the images of the grad and the (vector) curl operators associated to the Hodge decomposition are given by: (100)

P gu = R g R g u = grad A"l div u , Pcv=RCR* v = - curl A"" *curl v,

or with their Fourier transforms: (100)

pgU«)—np^-,

pcv(0—jjp^-,

which are associated with the decomposition: u({) = Pg u(£) + P c u(S). Using these operators, we can write (94) in the form (ioi)

R;H£=-J^AR^E£,

and also (since R* Pg = 0, Rg Pc = 0):

R^H£=-|A-1R;E£,

5.1 PLANE GEOMETRY

203

(102) PCH°=-§RCA-,RX=ZA-1PCS4 and thus the Calderon operator C is also given by: H ^ C E ^ - ^ R g A R ^ + kRcA^RjE^ (103) =

Z[-EAP« + kA " lp J S 4-

Now we define the following functional space: (104) H k (R 2 )^ {ue.SXR2 ) 2 ,fi€L^R2 )2 , R j u s H k 1/2(R2), R^ u eHk/2(R2)}. This is a Hilbert space with the norm: (105)

» ul W = Ml e l UAU«>I2+TOT U-a^)!2]—^ dt),/2 |0| vl/2

i«r

^"^"H^RV'V'H^'V) We also define the space Hk(R2) = SHk(R2): (104)'

H^RV^UCSXRV.U^L^R2)2,

Rju €Hk/2(R2), R^u eH k 1/2 (R 2 )j.

This is a Hilbert space with its natural norm, and S is an isomorphism from Hk(R ) onto Hk(R ). Now we assume that: (106)

Ej€Hk(R2).

Theorem 2. With hypothesis (106), problem (83) with (84) has a unique solution (E,H) with locally finite energy (up to the boundary, i.e., E and H are in Hloc(curl,R2 x [0, oo)) and the boundary value of the magneticfieldalso satisfies (107)

Hj€Hk(R2).

For the other components of the electromagneticfield,we define, like (35): (108)

X k ^ f {u€y(R 2 ),fi€l^ 0C (R 2 ), / R2 |fi| 2 (|0|/|£| 2 )d£<oo}.

Note that if v satisfies/ 2 |v| 2 (|8|/|S| 2 )d£< oo, thus v € L^R 2 ), since from Cauchy-Schwarz inequality fK |v| d£<(fK |v| 2 (|0|/|«| 2 )d£) l/2 .(/ K (|5| 2 /|9|)d£) 1/2 , for all compact sets K in R2. Furthermore since |0|/|£| 2 ~ 1/|£| when |£| —> oo,

204

(108)'

5 UNBOUNDED OBSTACLES Xk c H f ^ R 2 ) .

Nowwith hypothesis (106), we have (-A)-1/2E3 and (-A)" 172 ^ € H£ / 2 (R 2 ), from (92), (93), that is: E^andH^cX,,. (109) E^ We also have: (109)' HkJR2)cHf0lf(dfcr>R2),

H^cHfJ^curlR2).

proof of (109)'. Let u € Hk(R2), then RgR^u € H£ / 2 (R 2 ) 2 , and R ^ u € Hk 1/2(R2)2, thus: u = RgRgU + R ^ u € Hf0lc/2(R2)2, thanks to lemma 2. Furthermore R^u€Hk/2(R2) implies divu€X k , andby(108)\ divucHfJ^R 2 ). This prove the inclusions in (109)', thanks to operator S. Then, like in Definition 4 chap. 3 (with (199), (200)), we define the "usual" surface operator (which depends on Z and k): Definition 3. The operator C (or C) defined by (87), or (89), or (103) is called the Calderon operator (or the surface admittance operator) for the half space (andfor Z, k, and the chosen sense of propagation). As a consequence of above properties, we have: Proposition 6. The Calderon operator C (resp. C) is an isomorphism in Hk(R2) (resp. SHk(R2)). Both commute with the semigroup (G(x3)), satisjy (8S), and (110)

Re (CSE£,E£) < 0, V E^ € Hk(R2).

This inequality is a consequence of the formula:

(111)

/R2nAH$J$dx=/R2£[-£|UE^§M^<tt,

which is easily deduced from (103), and the definition (11). Remark 5. Similar to waveguides the electromagnetic field is decomposed into transverse-electric waves and transverse-magnetic waves, thanks to the Hodge decomposition; using the decomposition Hk(R2) = Hk#g(R2)©Hkc(R2), with (112)

Hkg(R2) = {u€Hk(R2), curlu = 0}, H^R 2 ) = {u€Hk(R2), divu = 0},

we have, with hypothesis (106):

205

5.1 PLANE GEOMETRY E ^ . € H k c ( R 2 ) ^ E2 = 0 ~ H^.€H M (R 2 )forTEwaves

(113) E

T € Hk,g<R2) ** H 3 ■ ° ~

T € H k,c( R2 ) for ™

waves

and thus the Calderon operator C is an isomorphism from H k C(R ) (resp. H^ (R )) onto H k g(R2) (resp. H^C(R2)) given by OforTEwaves: H £ = C E £ = J = ^ S A E J (114) ii)forTMwaves: HJ = CEJ^S

~* ET-

Note that there are no transverse electromagnetic (TEM) waves since the whole space is simply connected. Remark 6. Calderon projectors for the half-space We can easily verify that the Calderon operator for the inferior half space C. (or C_, for the same medium, the same angular frequency) is given by (115)

q*-C(orC|-~C).

Then we consider the usual transmission problem (or with given electric and magnetic currents, see (86) chap.3): find (E,H) with locally finite energy in each half space (up to the boundary) so that i) curl H + icoe E = 0, ii)-curlE + i<iniH = 0 inR + andR_, 2 2 (116) iii)[nAE] r = M , - [ n A H ] r = J, r = R x{0}, J and M given in Hk(R ), iv) (E, H) is a wave propagating towards the positive (resp.negative) axis in R* (resp. R*). Thanks to the Calderon operators, this problem is reduced to finding the boundary values of (E,H) satisfying (116)iii) with: (117)

nAH|r =C(nAE|r),

n A H | r =C(nAE| r ).

Thus we get a linear system of equations which is very easy to solve explicitly, using (88) and (115). We can write the solution in the usual form

(,W with (119)

f n A E | > j

(M^

-1-"<J-HJP+-£(I-S),

P . = | ( I + S),

f n AE|

f MA

L—-'riJ-'-L ,J-

206

5 UNBOUNDED OBSTACLES

where S is given thanks to the Calderon operator C = Z" *C£ by:

-(c - ^ H z V i }

[Of course we can also use other forms, for instance:

L"AH|rJ *l J ' l"H'r.J "I J with: (119)'

P + = i ( I - S ) , P_=±(I + S), andS = -S.]

The operators P+ and P . are the Calderon projectors in Hk(R ) x Hk(R2) for the half space (with c, p, O>). Of course we also have S2 = I. 1.2.2. Scattering problems with two different media As in section 1.1.3, we assume that the whole space is occupied by two different media on each side of the plane T (x3 = 0), one with permittivity and permeability (Cjjjij) in the domain R3+ , the other with permittivity and permeability (E 2 ,IO in the domain R 3 _. We assume that there is an incident electromagnetic field (EpHj) in R3+, at angular frequency a>, for instance produced by charges and currents with compact support in R . We have to find the reflected and transmitted (or refracted) electromagnetic fields, resp. (E , H ) in R3 and (E t ,H t )inR 3 _. Let (E,H) be the electromagnetic field defined by: (E,H) » (Ef,Hr) in R3+ and (E t ,H t ) in R 3 _. Let (e,*i) be (t^)

in R3+, (e2,y2) in R 3 _. We first assume that

(£,,Pj), (*2»P2) are positive real numbers (the two media are conservative). Then (E, H) has to satisfy (116), with M and J given by: (121)

M = nAE,| r ,

J= -nAH,|r

Thanks to the Calderon operators in each half-space, Ct and C = - C2., we are led to find the boundary values of the field, which satisfy (116)iii), or also (122)

Hj2 - H j | » H j | ,

E*j<2 - EJJ = EJJ .

Applying the adjoint Riesz operators (see (96)') to (122), then using (101), we obtain two simple scalar systems. We define: (123) u f - R ^ E q J - l , 2, uf = RjE TI , u? = R*ETj, j = 1,2, uf-R^En,

207

5.1 PLANE GEOMETRY

and thus the Fourier transformation of (101) gives, keeping (e,|i) of the media: R

(124)

gHTl=Sirei^l»

g H T 2 = "®Jir82fi2'

g H f I = -Bpr6ifif

| R J ftT1 = iwEj 0]"1 flf, R % ft^ s - i(oe2 flj1 flf. R S H j , = - I**! S^1 fl f. We have to solve: (125)

k 0 2 lfi 2+ £ i°r lfl ? s£ i e r lfi ?

g Ti u2

A 8

ft i, - ui i i -=u

and Ifl

AC,

1 fl A C . 1 fl AC

p^«2 u 2 + f i J 0 I u l = p ^ ^ l u I

(125)

The solution of the system (125)c is given by: ft c - T ft c

- P ii c

(126) ^P^+11,82*

"^ei+>1i02

Thus for an incident field such that: (127)

E I T and H IT € H k (R2), £

1/2

we have u, € H ^ (R ), and from Proposition 4, we get (128)

uf e H*k/2(R2) and u | e

H ^ R V H ' A R

Then we define (129)

Vj-af'flf.j-i.i

^-e^flf,

so that (125) becomes: g

(130)

|i)e 2 v 2 + €,v 1 = e1Vj |ii) 8 2 ^ - 0 , ^ = 8 ^ , .

The solution of the system (130) is given by v, = T v , , (131) T=

v.-Rv,,

2E,9

l°l

«1«2 + «2V

e,92 + e29,

)

(and even u£ e H 1/2 (R 2 )).

208

5 UNBOUNDED OBSTACLES

With the same hypothesis (127), we have Vj € H*k/2(R2) and from Proposition 4: (132) V! €H 1 k / 2 (R 2 )andv 2 €Hj c / 2 (R 2 )nHj/ 2 (R 2 )(andevenv 2 €H w Thus we have: (133) and

ufeH-^R2), u

s I

uf€H^ /2 (R 2 ),

guI»

s R

I»

(134)

$ ?2£ tefil £ £ £182 * c 2 e i +€ e g=S £ 0 +€ e ^Oj ei9 2 2 r l 2 2i ' (126) and (134) are Fresnel's formulas for electric field. Note that

(135)

Tg-Rg=l,

Tc-Rc=l.

Thus we get Theorem 3 . With hypothesis (127), the scattering problem (116) of an incident wave (E^Hj) by a (conservative) medium in a half-space (with boundary T) has a unique solution (E,H) of locally finite energy up to the boundary (on each side of T), which is obtained thanks to two semigroups (G(x 3 )) x 3 >0, and (G 2 (-x 3 )) x3 < 0 (see (85) and (29)) associated to the wavenumbers kIf k2 in each medium, and thanks to the transverse components of the field on T, so that: (136)

E ^ | and Hjj | r € H* (R2), j ■ 1,2 on each side of the plane T.

Moreover the transverse components of the solution on T split into: Ejj = E jj + E j j , Hjj = Hjj + Hjj (137) |E T j = R g R g E T j,

E^SR^EJJ,

H^sRgRgHjj, HTj = RcRcHjj,

with(R g E T j , R g H T j ) , (R;E T j , RCHV)given

by(123), (126)and(134).

Remark 7. The decomposition (137) of the transverse field also corresponds to the decomposition of the total field (E,H) into TE-waves and TM-waves. When E T » E£ on each side off, then div E ^ = 0, thus E 3 j = 0 on T and in R 3 . Thus H3j = | cop. J - curl E ^ satisfies, with u f given by (126) with respect to u f: <138>

3nlr-Bfe:l«l«f

•

thusH3j|r€Xk..

5.1 PLANE GEOMETRY

209

When Ej = E | on each side of r, then curl ETj = 0, thus H3j = 0 on T and in R Pj satisfies, satisi Thus E3j = - Aj"1 div E^ with u * given by (134) with respect to u f: (139)

E 3j | r = m e rf f iAf *,

thusE^^cX,..

We call Tg, Tc, and R , Rc the transmission and the reflection coefficients of the electric field. We also define the transmission and reflection coefficients for the magneticfieldTp Tc, andR^, R; which satisfy (135) also: let (123)'

vf = R;HTj,j = l,2, v f ^ B n , V? = R;H TJ ,j= 1,2, vf-R^l,,.

Then we get Fresnel's formulas for magnetic field: ♦ S - * c H v? = R ; v f > v f = f ^ v f , O f s R ; v f ,

with:

(140) A

A £ ^ 8 I A

A P I " ?

T> _ -A—L x

T> — *—- T

D'__D

D ' — — 1?

Remark 8. We can extend these results and Theorem 3 to the case where the second medium is dissipative (with complex e, \i)9 and also where the two media are dissipative. The main point is to prove that the terms: e,92+ £29, and ^1^2+ ^2®i a r e n e v e r nu^- TWS *s t*le c a s e f° r instance when the first medium is conservative, and when Re E2, Re p2 (with Im E2, Im \i2) are all positive numbers. Furthermore for a dissipative medium, the wavenumber k has an imaginary part k2" * 0; This implies that the wave is exponentially decreasing from the boundary. We define the skin depth 6 as the distance in which the wave is attenuated by 1/e,. that is using the notations of (45): (141)

|ua,5)|/|fl«,0)|=exp(-(Ree 2 )6)<l/e,

VUR2.

Since Re 92 > Im k2 = k2, we can take 6 = l/k2. For a conductor with y2 = ]xQ, we have e2 = c2 + (io/<o) = e2 + ie0e £, with o » a>c2, so that k2 ^ <«>oy2/2, thus the skin depth is 6 = (2/o>on0)I/2 = (2/<*\t0z"T)m = (2/tyU2\/2n, with the wavelength X. Remark 9. In order to directly tackle the case of an incident plane wave, we have to use a more general framework (see (43) for instance). Here also we are reduced to the scalar case of section 1.1.3, by decomposition of thefieldinto TE and TM waves. Thus we obtain the Snell-Descartes laws for electromagnetism. Remark 10. Using the Calderon operator C (or C) defined for the half space allows us to tackle many other examples like in the scalar case, section 1.1.4. For instance we can tackle the case of an incident electromagnetic field through a (fiat) perfectly conducting screen, thick or thin, with apertures.

5 UNBOUNDED OBSTACLES

210

The case of a "thin perfectly conducting screen11, that is a surface T (a part of a plane) on which we take n A E = 0 is reduced to the scalar case, since the electromagnetic field (E,H) also splits into TE and TM waves. 1.3. The slab 1.3.1. The scalar case with Helmholtz equation First we consider the Helmholtz problem in a slab of thickness T and with a real wavenumber k, for the Dirichlet condition: (141)'

OAu + l^usO inQ t £R 3 ,withO<x 3 <T, ii)u(.,0) = u0,

U(.,T) = UT,

with given uQ, u f . It is natural to solve (141) using a Fourier transformation with respect to x = (x^x^ (if uQ, u t are tempered distributions). With usual notations we get i ) ~ ^ + (k 2 -e 2 )fl = 0,

0<X3<T,

(142) |ii)u(.,0) = u0,

U(.,T) = UT.

We note t = x3. Then (at least formally) the solution of (142) is given by

sh((x-t)4?^?) * „ s h ^ - k 2 )

i)u«,t) = u 0 «)(143) ii) u(E,t) = u 0 «)

sh(T^-k2)

sin((x-t)4 2 -£ 2 ) 2

sin(t4 -5 )

sh(t^2-k2)

when |E| > k ,

A/,xsin(tJ?^?) when | ( | <k. ■+uta)

sin(t4l?-?)

or using 8 (£) defined by (11) t\Aw (143)

att) *\= Aflsh(x-t)6 fl(., 0__r- +

flsht8 Tii-g.

Let XQ and xt be functions defined by (144)

* " » - * & * •

< * » ■ & ■

When \i\ >k, x 0 «,t)< 1, x t (U)^ 1. But when |£| <k, x 0 (t,t)andx t tt,t)-> «> when •rfl—*nn, n integer. Let \ be the wavelength, X = 2n/k, and nT = max n with n integerroc< tk, thus n<2xA. We also define: (145)

k„ = Jk 2 - (nii/x)2- k J1 - (nA/2t)2, ^ ^ ^

ST= U Sk , 0<n<nT *n

211

5.1 PLANE GEOMETRY

with Sj, the sphere of radius k^ the values a* with an = roi/x, n € N, n > 0 being the eigenvalues of - d /dt 2 on (0,x), with Dirichlet condition. Thus we have to eliminate the "irregular values" £ on these spheres, i.e., (€S T . Recall that the (generalized) eigenvectors (or eigenmodes of the slab) are: v M (x,t) = e Ux sin(tnn/x), with£2 + (nn/x)2 = k2, n € N, n>0. When k2 - £2 is equal to a 2 , the solution of (142) must satisfy the condition (146)

f0 (u"v -uv") dt = [u'v - uv']* = u(0)v'(0) - U(T)V'(T) = 0,

with v(t) = sin (nnt/x), and thus: (147) ( - l ) n u T - u 0 = 0. But since £ is a continuous parameter, we don't have to impose condition (147) in this form. We have to give a sense to (143) only. Thus when | £| < k we write: Hi 5 u 0 (E)cos(T^k 2 -£ 2 )-u T (£) r-y 5 2 2 (148) fi(£,t) = fl0«) cos (t^|k - £ ) - - ^ %-— — sin (t^k2 - t 2 ). sin (T ,|k 2 -£ 2 ) Thus we see that the hypotheses u 0 , uT € L (R ), and: (149)

r n

'(-1)nfio^>|2dt<<w,foraUn,o<n^nT, sin 2 (t*jk 2 -r)

with Vn a neighbourhood of the sphere Sj, , will imply that u(., t) € L (R ) , 0 < t < x. We can also take space H £ / 2 ( R 2 ) instead of L2(R2) with (149). We note that: i) if the slab is thin enough, that is if x < X/2, problem (141) has a unique solution (for instance in L2(R2 x (0,x)) when uQ and UT are in L2(R2)); ii) if the wavenumber k is complex, with Im k > 0, we have the same conclusion. Then we can get the values of the normal derivative of u at the boundary of the slab (at least formally) thanks to (143)'. Their Fourier transforms are in matrix form:

d50) (150)

r fi,( -' T) i =c cr fl( - T) i U'(.,o)J- *U(.,o)J'

with

(15D (i5i)

£__flLfchTe <^-shTflL i A

We easily see that CT satisfies:

-n -cfnej-

212 (152)

5 UNBOUNDED OBSTACLES (CT)2 = 92I,

and thus C t has (at least formally) the inverse: (CT)-1 = 8"2 C t . The operator C t defined by (ISO) is the Calderon operator of the slab for k. For other developments on the slab, and for guided modes of the slab, see Petit [2]. It is interesting to consider the Cauchy problem (with x } = t), also (for instance in order to treat problems with many different slabs): (153)

» * O^f+AjU + lAisO inQ t cR 3 , withO<t<t,

ii)u(.,0) = u0, §£(.,0) = u, with u0, U] given (at least temperate distributions). Using as above a Fourier transformation, we obtain the solution in the form:

For every given 8, (Ge(t)) (155) with

is a semigroup, which splits into:

G0(t) = e0tPt + e-0tP0_,

p»=|d+se), Pe_4(i-se), s e = [ o ^ J , that is, P+ and P_ are the Calderon projectors, since: s] = I, thus(P°)2 = P0, (PeJ2 = P9_, ?lP[ = ?lP{ = 0. By inverse Fourier transform, (GM) does not correspond to a semigroup in the usual L2 functional spaces, in other words: the Cauchy problem (153) is an illposed problem. But (153) is helpful because: i) we know that there is at most uniqueness of the solution of (153), ii) very often in physics, we work with given functions (or measures) uQ, u .whose Fourier transforms have bounded support, and then the inverse Fourier transform of (154) is well-defined. (Note that when the supports of the Fourier transforms of uQ and \x{ are contained in the ball of radius k, the semigroup (Ge(t)) is that of the wave evolution.) Thus if we have a heterogeneous slab composed of n homogeneous slabs with thickness T. and wavenumber k. , the values of the wave at the end of the heterogeneous slab is obtained in function of the values at the beginning by: (156>

( fiS ] =G'fO...G9fa) [ Jj ] , withx- 2xj.

213

5.1 PLANE GEOMETRY

Thus we easily obtain the solution of the scattering problem of a plane wave by a multilayer slab. 1.3.2. The slab with Maxwell equations Now we consider the Maxwell problem in a slab (with real £,p): I i)curlH + i<ocE = 0 , I iii) E^.,0) = E j ,

ii)-curlE + ia>jiH = 0 , inQ T ,

0<X3<T,

Eji.,1)» E\ with given E^, E j .

Since the components of (E,H) satisfy the Helmholtz equation, we obtain the transverse electric field Ej. from the above section. Thus we have to find the other components of (E,H) from E T . This is very easy from the usual decomposition into TE and TM waves, as follows: i) for TE waves, due to (30)ii) chap. 4, we have E^ = 0,

9 3 E T + i<o|i SH T = 0,

and thus using the Calderon operator CT given by (151): (158)

H-p = — Qjp- C^SE^;

ii) for TM waves, from (30)ii) chap. 4, we have H 3 = 0, and thus: (159)

a 3 H T -ia>£SE T = 0,

H T = ia)ECT"lSET.

These two results are given by: (160)

HT = ^ h i c T P g + k C ; 1 P J S E T ,

which is relation (103) with CT in place of A, where E T and H T denote the couples: E T = ( E T ( . , 0 ) , ET(.,T)),

H T = (HT<.,0), HT<.,T)).

Furthermore the components E 3 , H 3 are obtained thanks to (91), by: (161)

3=15^ c u r l T E T»

E3 = - C ; 1 d i v T E T .

There are no new difficulties with respect to the scalar case: we only have to eliminate the "irregular values" of ( corresponding to the generalized eigenvectors of the slab with Dirichlet condition, thanks to a weight, like in (149). In order to treat multilayer problem, we consider the Cauchy problem:

214

5 UNBOUNDED OBSTACLES i)curl H + itoeE = 0,

ii) - curlE + icoy H = 0, inQ,, 0 < X 3 < T

(162) iii)ET<.,0) = ElJ., HT<.,0) = HT, with given E £ , H £ . This is also (at least formally) easily solved due to the decomposition into TE and TM waves, using the projectors P and P (see (100)) or the Riesz operators R and R . We first write Maxwell equations in the form (30), (31), then (34) cnap. 4, ue., (163)

8 3 H T = 5 |r( k 2 + 8 r a d d i v ) S E T»

93%—Ad^+graddWSHr.

Then applying Pg and Pc to (163), we have: (164) 3 3 P g H T =4 r (k 2 + A)PgSET, (165)

93PcHT = i»e P^Ep,

33PgET= - 4 (k2 + A)PgSHT,

SjPgEr = - iuji P£HT,

Then using the commutation relations: (166)

SPc = PgS,

P^-SPg,

and using relations (154) with u = PgHT, then PCET, with (164) and (165), gives us: (167)

Pc£T<T) ^ _ f PgHT<T) J

ch (9T)

-'w*B~l sh (9T) S ^ f p c ET ^

^ -5ir8sh(et)S

ch(et

(i68) r P^T(T> 1 = r i

^ PCHT<T) J ^ io* 0-1 sh (6t) S We write (167),( 168) globally in the form: (169)

f*F

l.<v/fr

L *¥*) J

< 0 , ,

ch«h) J (^ Pg ft° J

4esh(9t)s

I f p^ ].

ch (&r) J ^

P<;ftT

\ H^O) J

Then we have the same conclusion as in the Helmholtz case. The formulas (167), (168) (or (169)) allow us to treat very easily the multilayer case like in (156). There are many applications of these formulas to scattering (Salisbury screen, Jaumann screen). 2 . PERIODIC GEOMETRY, 2-D GRATINGS

Periodic structures appear in numerous applications, especially in scattering by gratings (see for instance Petit [1]). As usual we first develop the scalar case, then electromagnetism.

215

5.2 PERIODIC GEOMETRY 2.1. Periodic geometry with Helmholtz equation We first define the notion of a quasiperiodic function: Definition 4. Let K = (Kj,K 2 )

wit

h K p K2 real numbers and L = (L p L 2 ) with Lx

and L 2 positive real numbers, be given. Let u be a function (or a distribution) in R 2 which satisfies "Bloch condition1': (170)

u(Xj + L p x 2 + l ^ ) » exp ifl^L, + K2L2) u(x p x 2 ), x = (xvx2) € R2,

(170)'

u(x + L) = exp (iK.L) u(x).

Then u is called a quasiperiodic function (or distribution) of type K-L, or a K-L (quasiperiodic) function (or distribution). When K = (0,0) or when K.L € 27iN, we obtain the L-periodic functions (distributions). Conversely given a quasiperiodic function u we obtain a periodic function v by: (171)

v(x) = exp(-iK.x)u(x).

This allows us to obtain the main properties of the quasiperiodic functions from those of the periodic functions. Let P denote the "elementary cell", that is the rectangle P = [0,Lj]x[0,L 2 ] with identified opposite faces; P is identified with the 2-D torus T 2 . Let IPI denote its area, IPI = Ljl^. We also define: Definition 5. The space of K-L "regular11 (C°°) quasiperiodic functions in R 2 (with the topology of uniform convergence of functions and all their derivatives) is denoted by DKL(R2)

(or DKL(?));

then its dual space, i.e.,

the space of

quasiperiodic distributions is denoted by D'K L(R ), and the space of quasiperiodic functions in R2 which are square integrable on the period is denoted by L2K L(R2) or simply L 2 .. We also denote by H* _ (R2) or H* . the Sobolev space of order s {real) of quasiperiodic functions (distributions), i.e., for s = man integer:

(172)

H™ L (R 2 ) = {u € 4 L ( R 2 ) , | J € 4 L ( R 2 ) , V J = <J„ J2) € N2, with|J|d#J,+J2<m}.

An orthonormal basis (up to a constant) of L2K is given by the functions: (173)

0J(x) = exp(iK.x)exp2ni( I i-x 1 +j^x 2 ),

withJ = ( J 1 , J 2 ) e Z 2 .

216

5 UNBOUNDED OBSTACLES

We also denote: (173)'

♦J(x)=expitJx,with£J = K+65 and t j ^ n ^ , A

J = (J,,J 2 )eZ 2 .

By expansion of quasiperiodic distributions on this basis, we have: (172)'

HsKL(R2) = {v€2?'KfL(R2), v = 2 v^j, 2 |K + ^| 2 s |v J | 2 < oo}, S €R.

We consider the standard problem in the half-space: find u satisfying (174)

OAu + k^OinR*, ii) u | _ = u0 given on the boundary r of R*,

with k a given wavenumber (k € R*) and u0 a K-L quasiperiodic function. The Helmholtz equation and the domain being invariant by (transverse) translations, we look for K-L quasiperiodic solutions. Thus we develop the unknown function u (and the given function uQ) on the above basis: (175)

u(x,x3) = 2 uj(x3) ♦jW,

u0 = 2 u0J Oj,

and problem (174) is reduced tofinding(Uj(x3)) satisfying (with Sj = K + £j): (176)

i ) ^ + (k 2 -£ 2 )u j3S 0, x 3 > 0 , *3

|u)uj(0) = uOJ.

When |£j| > k, there is only one bounded solution of (176): (177)

uj(x3) = u0J exp (- 9jx3),

with 9j =flj= «J - k2)1/2.

When | Ej | < k, the general solution of (176) is (with <*j + Pj = u0J): (178)

Uj(x3) = aj exp (- i9Jx3) + ^ exp (i9Jx3), with 9J - (k2 - l])U1.

Thus we cannot determine in a unique way the solution of (174), which is an illposed problem. In order to obtain a well-posed problem, we have to impose a sense to the wave propagation, like for waveguides, such as (179) u is a wave propagating towards the positive axis xy i.e., the propagating part ofu (178) is a superposition of waves Uj(x3»J(x) = e',j30J(x) withfij>0. Then taking:

5.2 PERIODIC GEOMETRY (180)

217

8j = « 2 - k2)172 when S2 > k2, 9j = - Kk2 - tym when tj ^ k2,

we obtain a unique solution of (174) with (178) by the Rayleigh series: (181)

u(x,x3) = 2 u0J e " 0 J \ ( x )

with J € Z2.

We have a situation quite similar to that of the waveguide, replacing the boundary condition on the waveguide by quasiperiodic conditions. Thus we have due to the Fourier expansion, or to the semigroup method Theorem 4. Problem (174) with a given quasiperiodic function uQ in L2K L(R2) and with condition (179) has a unique solution u in C°([0,oo),L2KL(R2)), given by (181) with a contraction holomorphicsemigroup (G(x3)), x3 > 0, in L2K L(R2). The infinitesimal generator A of the semigroup in L K L(R ) is defined by (182)

3u Au0 = |y-(.,0), EKA) = {u 0 €4 L (R 2 ), u€C\[09oo),L2KtL(R2))},

and using the spectral decomposition of the quasiperiodic Laplacian: (182)'

Au0 = - 2 6j uOJ0j, with u0 = 2 UQJOJ,

D(A) = KlK L(R2).

Definition 6. The infinitesimal generator A of the semigroup (G(x3)) is called the Calderon operator for the quasiperiodicfunctions and the wavenumber k. This operator has the same properties as that of the waveguides; A is a normal operator, a square root of the self adjoint operator - (A + lrl) in L2K L(R2). Let: (183)

U£ = {J€Z 2 sothat |£j| <k}, U£={j€Z 2 sothat |£j| >k}, U^={j€Z 2 sothat|£j|=k},

and let: (183)' H£, H£, H£ the Hilbert spaces generated by *j , J € U£, U£, u£(resp.). Thus: ker A = H k , and Im A is its orthogonal space. Furthermore A satisfies the following inequalities for all v in D(A): (184)

218

5 UNBOUNDED OBSTACLES

According to (183), we have the decomposition of L^ L(R2) and of A into: (185)

L| L (R 2 ) = /4©H^,with^=H^©Hj, andA = - A + + iA~,

with A+ and A* positive selfadjoint operators in the orthogonal spaces H^ and H£, where H£ is of finite dimension and corresponds to the propagating modes. Remark 11. The Lg L(R ) framework is not "optimal" for all applications we have in view. More generally Theorem 4 is true in the quasiperiodic Sobolev spaces fig L(R2), with real s, and in this framework the Calderon operator A is continuous from Hf^R 2 ) into H^ L(R2). The case s m -1/2 corresponds to waves u with locally finite energy. Example 1. Scattering of a plane wave by aperiodic soft wall. Let r be a (regular) periodic surface in R (with period L = (Lj,L2)) which is a modelling of a periodic soft wall (or a 2D-grating) with free space on one side. We assume that T is contained in the half-space x3 < 0. Let Uj be an incident plane wave in the free space Q given by (186)

u,(X) = uI(x,x3) = u I 0 e t o X = u I0 e^ aTX ^ 3 * X3) , a€S 2 , a 3 <0.

Therefore u. satisfies the K-L quasiperiodic condition with K^koj, (thus | K | < k) and Uj(. ,0) = U^AQ (with (173)). We have to find the wave uf reflected by this structure. This is a K-L quasiperiodic function satisfying the following on the plane TQ (x3 » 0): (187)

9ur "gjf-Aur = 0 onT0.

The Calderon operator A reduces the problem in the infinite domain O to the bounded domain C1Q with disjoint boundaries TQ and T and L-periodic with respect to x. Let u = Uj + uf be the total wave. The transmission conditions on TQ are: »„ 3uP 3U| <187>' W3'W3+W3> «=«r+uionr 0 . Thus u has to satisfy on the boundary r0: (187)"

J|- - Au = fIf

with f, = ^ - Au,.

Therefore we have to find the total wave u, which is a K-L quasiperiodic function in QQ with finite energy (in O Q ) satisfying OAu+l^usOinQo, (188) ii) u | = 0, and u satisfies (187)" on T0.

5.2 PERIODIC GEOMETRY

219

We can write this problem in a variational framework. Let V be defined by: (188)* VsfueH^Qo), u l r = 0» u(.,x3) is a K-Lquasiperiodic function in QQ VX 3 ) Then let a(u, v) be the sesquilinear form on V: (189)

a(u,v)=/

(grad u.grad v - k2u v) dx - / r Au. v dT0.

The problem (188) amounts to finding u in V satisfying (for all v in V) (190)

a(u,v)=/ r f,vdT0.

Since the inclusion of V in L2(Q0) is compact, this problem depends on the Fredholm alternative. But we prove below (with fairly general hypotheses) that there is at most one solution, which implies its existence. ® Remark 12. We can treat more general scattering problems, for instance with a "hard" wall (with the Neumann condition) or a wall coated with some medium which may be heterogeneous (but periodic). If there are dissipative inclusions in this medium, that is domains where the wavenumber is not real, then these problems depend on the Fredholm alternative, and we can prove that there is at most one solution, thus we have a unique solution also. Example 2. Scattering by a slab with periodic inclusions, or by a grid We consider a system of periodic inclusions in a slab placed in free space, and with an incident plane wave arriving from one side. The inclusions can be hard, or soft, or made of a dissipative material. We can consider the case of a grid with thickness d, or the scattering by a dielectric medium also. We have to solve the standard problem in a (fictitious) slab: i) Au + k 2u = 0

(191)

in QT = R2 x (-T,0), or P X (-T,0),

i i ) | | - A u = fI onr0, f, given by (187)", t=x 3 iii)^«f AjU = 0 onTjsr^, iv) u is a K-L quasiperiodic function with finite energy in P x (- T, 0),

where T0, Tx are the boundaries of Qt; Aj is the Calderon operator for the domain with x3 < - T where the wavenumber is kj (here kj is a real number). In the simple dielectric grating case, we have (192) Q ^ ^ u f ^ , withiUkinty)* k^kjinQj. Like in Example 1, we can write this problem in a variational framework. We define (with usual notations) the space on the periodic domain Q :

220

5 UNBOUNDED OBSTACLES

(193)

V « HkL<fig - H^L(P x (-t,0)),

and the sesquilinear form a(u, v) on V: (194) a(u,v)*/ (gradu.gradv-^u.^dx./j. Au.vdT 0 -/ r A^.vdTj. (If we have hard inclusions occupying a domain O in QQ, we replace V by the subspace of functions which are null in O). Then problem (191) is written: (195)

a(u,v)=f f^dTo,

VvcV.

Since the natural inclusion of V in L 2 (Q T ) is compact, this problem depends on the Fredholm alternative also. Under supplementary hypotheses, we can prove that there is at most one solution, which implies its existence. Proposition 7. We assume that one of the following hypotheses is satisfied: i) the grating is thin: the depth d(T is contained in a slab Q d of depth d) is small with respect to the wavelength X (d < A/4); ii) the grating is given by a Lipschitz periodic function x3 » g(x), except on vertical parts; Hi) the grating is coated by a (heterogeneous) medium with a dissipative part. Then the scattering problems (188) and (191) have a unique solution u with locally finite energy up to the boundary. Thanks to the variational method, we can prove that problems (190) and (195) with f» = 0 implies u = 0. Here we only give the proof in two cases. PROOF FOR (188), with hypothesis i). i) Taking first the real part of (190) with v s u gives, using (184), PROOF.

(196)

/ 0 (|gradu| 2 -k 2 |u| 2 )dxsRea(u,u) = 0.

Now assuming that T is above a plane T_d we can extend u by 0 up to r_d, and this extension u is in H^ L(Qd), Qd = Px (-d,0), thus: (197)

|gradu|2dx * ^ = inf/

|gradw|2dx,

the inf being taken on {w € HJ^ L(Qd), w=0 on r_d, llwil = l},and it is reached at the solution in this space, of |i)Aw+icfw=0 inQd, (198)

ii)w,

r_ d = °' £l ro -0-

Thus we have K{ £ n/2d. Since k = 2n/X, when d and X satisfy:

221

5.2 PERIODIC GEOMETRY (199)

K§-k 2 >(7t/2d) 2 -(27iA) 2 >0,

i.e., d<A/4,

we have from (196): (200)

|u| 2 dx > J

|gradu|2dx > / ^ K 2 |u| 2 dx,

which implies u = 0 from (199). We note that this is a version of the Poincare inequality. PROOF for (188) with hypothesis ii). This is adapted from Cadilhac's proof (see Petit [1]). First taking the imaginary part of (190) with v = u gives (201)

u 0 1S f u| r €#£

(see (185), u is not propagating). Then applying the Green formula to u and v (another solution of the Helmholtz equation satisfying (201) also), we have (202)

/ no (Auv-uAv)dx = - / r | v d r + / r o ( ^ v - u f ) = 0.

But the integral on T0 is zero thanks to (201). Thus

(203)

/ r givdr=o.

Then taking v = ^ in (203), and since ^ = e3.grad u = n3 g~, with n3 > Owe obtain gj = 0 at last on a part of I\ and thus (with u = 0 on T) we have u = 0 on Q0. <$ Now given a regular periodic surface T in R3, we can define Sobolev spaces of quasiperiodic functions H5K L (0 for real s, like in (172) or (172)\ We take the situations of examples 1 and 2, but with inhomogeneous boundary conditions on obstacles. Let Q be a regular connected periodic domain with one of the following hypotheses, according to example 1 or 2: i) Q £ R x (-T,+OO), with boundary r Q R2 X (-T,0) (grating with one side) ii) Q' = K\h QR X (-T, 0); CT may be connected (grid) or not (inclusions). Proposition 8. We assume that one of the hypotheses of Proposition 7 is satisfied. Then the problem: find a K-L quasiperiodic function u with locallyfiniteenergy (up to the boundary) satisfying (204)

i) Au + k u = 0 in Q, with real k, il/2, ii)u| r = u0^ve/ii/iH^£(r) 3 Iiii) iii)uuisisaawave wavepropagating propagatingtox3>0 in R+, (also to x3 < 0 in R ),

has a unique solution. Furthermore we have

222

(205)

5 UNBOUNDED OBSTACLES

g|r€HiV&0.

This is a consequence of Proposition 7 and Theorem 4. Then Proposition 8 implies that the mapping A=A f defined by: (206)

UQ—1|| , with u the solution of (204),

is continuous from fl^ffF) into Hj£ ^(0Hie operator A of Definition 6 is a particular case of this mapping when Q is the half space. If T separate R3 into two connected components Qj and 0 2 (the grating case), we have to consider two operators Af+ and Af _. Definition 7. ffte operator Aj. defined by (206)frco/fetf dte Calderon operator of the 2-D periodic surface T. This operator has the following main property: (207) Im(A r v,v)£0, V v € H ^ ( r ) . The proof is an easy consequence of the property (184) for the half space, thanks to the Green formula. Furthermore (in the case of a grating for instance) if we define the sesquilinear form (208)

a^u0, v0) = - ± [(AfUQ, v0) - (u0, A^)]

on H^£(r),

which satisfies: a^UQ.UoJ-Im^^iio) > 0, we obtain due to the Green formula that (209)

aj<u0,v0) = | P| 2 iOj Uj. Vj, with J € U£ only,

u. being the J component of the trace ofu on the plane I\ in the basis (173). Taking the conjugate of vQ in (209) leads to the reciprocity relations (see Cadilhac in Petit [1]). But note that if u is K-L quasiperiodic, its conjugate is not K-L quasiperiodic, but it is quasiperiodic for ( - K). Remark 13. In the case of 1-D gratings, that is if we assume that T is a periodic line in R2, as in Petit [1], all this is true under very little change: in particular we have to change double Fourier series with subscripts J in Z to simple Fourier series with subscripts n in Z. This corresponds to problems in R3 with cylindrical gratings (with axis x for instance) with given data which are independent of x2. But if the data depend on x2, without x periodicity, we have to make a Fourier transform with respect to x2 and we obtain a situation which is in-between the plane case and the periodic case.

223

5.2 PERIODIC GEOMETRY

We have to change the mathematical framework: we define spaces of K1-L1 quasiperiodic functions with respect to x(, using an orthonormal basis (up to a constant) ($n) of KJ-LJ quasiperiodic square integrable functions O^XjJsexpUjX!, with lx = K{ + In (n/Lj), n = Jj€Z, with a Fourier transformation with respect to x2 and then we have to substitute the following space for the Sobolev space (172)' with s = 1/2: (210)H$

L (R

) = {u = u(x1,x2)= 2 u „ ( x 2 ) V x l ) ' 2 / R | 8 n « ) | |fin(S)|2d£<oo}

where 9n «tf2 +t 2 - k2)1/2, £n = K{ + 2K (n/Lj), n € Z, and wherefln(£)is the Fourier transform of un(x2). We have a space in-between H^/2(R2) (see (37)) and H ^ L ( R 2 ) (see (172)'), which has the H ^ R 2 ) regularity. 2.2. An integral method for gratings Similar to bounded obstacle we can solve the scattering problems with a periodic obstacle from an integral method. Here we use the space D' (P x R) of distributions which are K-L quasiperiodic with respect to x. We win assume (with notations (173) and (183)) that K, L, k satisfy: (211)

|£j| * k , V J € Z 2 , i . e „ u £ = 0 , oralso: (Z^ + k2)u = 0 in P, (with u € L^ L(R2)) implies u = 0.

First we have to define an "elementary outgoing solution" of the Helmholtz equation in Z)' KL (PxR). Let 6^(x) be the Dirac (quasi) periodic distribution. We look forO*p)(x,x3) the "outgoing" quasiperiodic solution of: (212)

i) (A + k2) (P) (x,x3) = - 6(p) (x) «(x3), ii) <& p is a wave propagating from x3 = 0 towards x3 > 0 (resp. x3 < 0).

We develop <D(p) on the basis (173): (213)

<*>(p)(x,x3) = | P | - 1 2 ^P)(x3>j(x),

J€Z 2 , | P | - L ^ ;

applying (212)i) to *., we see that * j (x3) must satisfy, with t = x3:

a2$(p) (214)

-g-f- - 02 * « = - 6(t),

with 0j2 = 52 - k2,

224

5 UNBOUNDED OBSTACLES

and thus it has jumps across t = x3 = 0: (P) »?* (215) [ * f ] 0 = 0, I-^-]0 = -l, with the outgoing condition. Since 6j * 0, Oj is given by: (216)

• ? ) ( x 3 ) - a j « P ( - e J lx3lX

with 8j given by (180). Thus * (p ' is given by its (double) Fourier series: (217)

*(p)(x,x3)= i P f 1 2 ^ e x p ^ O j |x 3 |)^(x).

We can split *

(218)

into two parts with J in U£ and J in U£:

• ( P ) . * ? ) + *J >) ,

the first term corresponding to the propagating modes, and the second to the diffusion (or evanescent modes). We easily see that: (219)

* (p) €L? oc (PxR), i.e.f • < p ) | p x h b J €L 2 (Px[a f b]) t

Va,b€R,

since 0(_p) is afinitesum of terms in Lj^P x R), thus *** is in Lj*0C(P x R), and <f^p) is in L2(P x R), since J ^ |*i P ) | 2 dxdx3 « |P|" ! 2 2 (29j)-3 < oo, the sum being on J € U£. $W is the elementary outgoing solution of Helmholtz equation in D'K L(P x R). Remark 14. We can see that the two notions of outgoing waves mentioned in chaps. 3 and 5 agree: if we compare ^ with the usual elementary outgoing solution 0 (see (8) chap. 3), from (29) and (214) we have: (220)

•? ) (x 3 )«$tt J ,x 3 ) f and thus *(p)(x,x3) = | P | - ! 2 *(£j,x3) ^(x),

with O the Fourier transform of O.

® oo

Remark 15. Let x be a smooth function (C ) with x(x,x3) = 1 in a ball Br of R , and x(x,x3) = 0 in the complementary of the ball B 2r , with 4r < min (LpL^). Then 4>x is a distribution with compact support in R3, and we can define its corresponding periodic distribution by translations (*x)^ = I Tj*X- We have defined a "parametrix" which is equal to exp(-iK.x)*^ up to a regular function. Thus 0 ( p ) has a 1/r singularity at the origin in PxR and it is C°° outside. #

5.2 PERIODIC GEOMETRY

225

Proposition 9. Let T be a regular (Lipschitz) 2-D periodic surface in R 3 which is contained in a (fictitious) slab of finite thickness d, R 2 x(-d,0). With hypothesis (211), the problem: find u a K-L quasiperiodic function with locallyfiniteenergy up to T, satisfying i)Au + k2u = - f inR 3 withf«p , 6 r + div(pn6r) (221) | w l t h « i v e n <P» *>'>€ H K,&n x H K, L^> ii) u satisfies the "outgoing wave condition", i.e., the propagation condition (179) towards x3 > 0 or < 0 out of the slab, has a unique solution u which is given by the Kirchhoffformula (222) u « * W • f ■ * ( p ) * <pf «r + div (pn ty), o/irf the jumps ofu and its normal derivative across T are given by (223)

[u]r = P ,

iglr-P'-

This proposition is similar to Proposition 3 chap. 3 for a bounded surface T (which is the boundary of a bounded domain). We have to note that the convolution product in (231) is that of D9 (P x R), that is with respect to x (for quasiperiodic distributions) and to x3; this makes sense since p and p' have compact supports with respect to x3. Then we can define single and double layer potentials like in (67) chap. 3 by (224)

L ^ V - * ® . ^ ) . i^P)P = <&(p)*div(pn6r),

with the "usual" properties (see (69) chap. 3). We also define the Calderon projectors P[ p) , P<p) (or P ( p ) , Pip) for gratings), like in (65) chap. 3 in:

(225)

Y(KP2 = < l(T) x Hj^ftO.

They will satisfy (72) chap. 3 with an operator S = S(p) so that S2 = I, given by the matrix (71) chap. 3 with four integral operators L(p), K(p), J (p) and R(p) which are obtained thanks to <&(p) by Fourier expansions. For their particular expressions in 1-D gratings, we refer to Maystre-Vincent and Petit in Petit [1]. This allows us to solve scattering problems with gratings like in chap. 3. Here we do not develop this topic. We only point out that the difficulties are quite similar to those of the bounded obstacles, in particular with respect to convergence of the Rayleigh series: for gratings we know that the solution of the usual scattering problem has a Rayleigh expansion (181), which is convergent in all (reasonable) senses, above 0 (and below - d also if we substitute + for- in (181)). For the questions of convergence of these expansions, see Cadihlac-Petit [1] (quite similar to those of chap. 3). #

226

5 UNBOUNDED OBSTACLES

Remark 16. Singular cases in periodic geometry. We often assume the wavenumber k, K and period L are such that (211) is satisfied, that is the kernel of the operator A is reduced to 0. Here we study what happens if it is not true. First we consider the Neumann problem in a half-space: find u with locally finite energy (up to the boundary) satisfying OAu + lAi^OinR*, (226)

iii) the propagation condition (178), with Uj given in ker A. It is an ill-posed problem, since the solution of (226) is a priori given by u(.,x3) = x3Uj + C, C constant, and is not bounded ! We have to eliminate this case, and thus to take: (227)

u 1 €(kerA) 1 =ImA, i.e.,/ p u 1 5 J dx=0,

VJ€lj£.

This condition written for the half-space allows us to obtain well-posed problems in singular cases. First in problems with given jumps (p,p') of u and its normal derivative (see (221)) on T = P x {0}, p' must also satisfy (227), and then the solution u of (221) is determined up to an element of ker A. As a consequence we have to change the definition of the Green function (212), since the Dirac (quasi)periodic distribution is not orthogonal to ker A. Instead of (212)i) we have: (228) (A + k2>D(P)(x,X3) = F, with F = - 5(p)(x) 8(x3) + \P\~l

♦J(0)*J(X)8(X 3 ).

J€U<>

Then O ^ must satisfy (224) for J out of u£, and: (229)

[^ p ) ] 0 = 0, [ - ^ ] 0 = 0,forJinu£,t = x3.

We can take *SP) = 0 for J in u£, so that * ( p ) , ^ - are orthogonal to ker A. Now if we go back to the jump problem (221) we have: i) in the simple case where r = P x {0}, we have a unique solution in the space(s) orthogonal to ker A, when the given data satisfy: (230)

(w>) € H ^ L X H K , L

wtth

/ p 5 J * = 0, /p*5j dx = 0, V J € u£.

ii) in the general case of a grating T, we can prove (using the Green formula successively in domains above and below the grating with u and +, then with x30j), that the orthogonality of p and p' to ker A must be replaced by the following conditions:

227

5.2 PERIODIC GEOMETRY

(231)

/r(p^J.p^i)dr=0,

/ r [ ( p ^ J . p ^ ) x 3 . p n 3 5 J ] d r = 0,

and then the boundary values of u and its normal derivative on each side of the grating will also satisfy these conditions (231). Thus the Calderon projectors P^ and P_ act in the subspace VK L defined by: VK L = {(p,P,)€Hjc^(r)xHK1/^(n, satisfying(231)}. 2.3. Periodic geometry with Maxwell equations First we consider like in section 1.2.1 (or in chap. 4 section 2) the following problem: find the electromagnetic field (E,H) in free space at angular frequency (o satisfying (232)

i) curl H + iwe E = 0 ii) - curl E + iwp H = 0 in the half-space R+, iiOE^.jOJsEj (ornAE = nAET)

with E° T a given K-L quasiperiodic transverse electric field. Similar to the Helmholtz case, using Fourier series, we see that (232) is an illposed problem (in K-L quasiperiodic spaces), and it becomes (generally) wellposed thanks to the usual physical assumption: (233)

(E, H) is a wave propagating towards the positive x3 axis.

Since each component of E and H satisfies the Helmholtz equation, they will be determined by the semigroup (G(x3)) of section 2.1 from their boundary values on the boundary T (x3 = 0) at least formally by: (234) ET<X3) = G(x3)E£, E3(X3) = G(x3)E5, H ^ ) = G(X 3 )H£, H 3 (X 3 ) = G(x3)H°, and thus we only have tofindE3, H3 and HT from ET. At first we specify the framework according to the usual given data. 2.3.1. Mathematical framework A "regular" assumption is: (235)

E^€HKL(P)2.

Since (G(x3)) is also a continuous semigroup in this space, this implies: Ej(x3) is in H^L(P)2, for all x3 > 0. Then (91) (for instance) and (92) imply: (236)

H° € L| L (P), H 3 (X 3 ) € l£ L (P), E°3 € H ^ P ) , and E3(x3) € H ^ P ) ,

thus (thanks to (30) chap.4): Hj(x3) and Hj € L£ L (P) 2 . We see that hypothesis (235) implies that the regularity of the electric field E(x3) is H1, whereas that of the magnetic field H(x3) is L2.

228

5 UNBOUNDED OBSTACLES

A less regular assumption is: (237) E^€HKL2(curl,P). This space corresponds to the "usual" regularity (see H.(R2) (104)). Using (double) Fourier series, it is defined by: (238)

H ^ c u i U P ) = {u = 2UJ0J , 2 j f j - (|uj| 2 + |5j A Uj|2) < oo},

whereas: HkL(P)2 = {u = 2 ^ j , 2 ( | $ J . U J | 2 + U J A U J | 2 > < °°lThen (237) implies for the other components: (239)

H^H^curl.P),

E^HJ^P),

H° € H J ^ P ) ,

and ET(x3), E3(x3), H1<x3)9 H3(x3), will be in these spaces for all x3 > 0. With assumption (237), the electromagneticfield(E,H) solution of (232) with (233) is of locallyfiniteenergy up to the boundary. More generally the hypothesis (for any real s): (240)

E^€HKL(curl,P)

implies H T € H^L(curl,P); then E and H have the same regularity. 2.3.2. The Calderon operator We first assume that the operator A (see (182)) is invertible, i.e., (211) is satisfied. Then we have the usual relations between the transverse components of E and H given by (86) with A = A(p). Using Fourier series decomposition, we have (241)

H ^ ^ t f j t f j A E n ) + k 2 SB n ].

Definition 8. The mapping E°T — H°T (resp. SE°T — SH°T) defined by (86) (with invertible A given by (182)'), or in Fourier series by (241), is called the Calderon operator (or admittance operator) C or

(resp. C or C^)for the half space in

K-L quasiperiodic spaces (at angularfrequency o>). Proposition 10. The Calderon operator C (resp. C)isa continuous operator in: HK L(curl,P)(resp. H^/£(div,P)) and in H^ L(curl,P)(resp. H^ L(div,P)), s €R. Moreover it satisfies: (242) C 2 = - Z ~ 2 I , C 2 = - Z - 2 I , and C=-SCS, C = -SCS,

5.2 PERIODIC GEOMETRY

229

and also: (243) with:

Re(CSE^.,E^.) = Re(SCE!j.,ElJ.)<0,

(244) (CSE°,E°)=ZSH TJ .E TJ =X — j - ^ [ - e, ^ I ^ A E ^ 2 ^ ^.E^2] £iving only afinitesum (J € U£) in tfte real part (243). Furthermore from (241), we see that the Calderon operator C (or C) is naturally decomposed by Fourier series into C = ZCj (resp. C = I C}) with (see (89)):

and E^, H^ are given by (92), or with the Fourier series: (246)

E3j = i9j tj.ETJ,

^3J = £z^j A ^TJ,

and thus we can write formulas (94), (94)' between the div and curl:

Thus we obtain another expression of the Calderon operator (see (103)), corresponding to the Hodge decomposition of the transversal fields, or also of the total electromagnetic field into its TE (transverse electric, with E3 = 0) and TM (transverse magnetic, with H3 = 0) components, which is given with the Fourier series decomposition by: (248)HTJ = a j . H T J ) ^ +

ttJAHTJ)i=-^[^ajAETJ)fJ+^«J.ETJ)Sy

thanks to the relation Stj.u = tj A U , for all vectors u. This gives (244) directly. The Hodge decomposition of the space (237) into: (249)

HKL2(curl,P) = HKLtg(P)0HKLfC(P) (like in (112), %L,g( p )( res P%L,c( p )) = ( u€H KL 2 ( curl ' P )» curlu = 0, (resp. divu = 0)}

gives the TE and TM decomposition of (E,H) according to (113) and the decomposition of the Calderon operator C according to (114), given (from Fourier decomposition) by (250)

HT J = J ^ 0 J S E ^ J

for TE waves, H£J = - § OJ1 SE^forTM waves.

230

5 UNBOUNDED OBSTACLES

Now we assume that the operator A is not invertible. With notations (183), (185), we assume: i) E^j a grad ♦j, with ^ € H£ (thus A*j » 0, and H3 = 0). This implies Ejfa)« E^Jf and 3 3 E5 = - div ETJ » - Afy «lc2 * j , whose solution is the (unphysical) field E3(x3) = k20j x3 + constant. ii) ETJ = curl 4>j, withfy€ H k , thus div EJJ = 0, giving: E3 does not depend on x3 and E^ is not determined by E|J. (E^ is of the form: E^ » 2 Cj #j with +j € H£ and Cj constant). Furthermore we have ETJ(x3) = EJJ, and using (30) chap. 4, we have HT »jsp S grad E3 = - ^* curl E3 which is not determined by E£J. Thus in the two cases, we have an ill-posed problem. To eliminate these cases we assume that: (251)

ET is orthogonal to ker A x ker A.

With this assumption, we can develop the theory as in the regular case; for instance we can define the Calderon operator C in the space (and it is an isomorphism in this space): (252)

H£lL/2(curl,P)n (ker A ) 1 x(ker A ) 1 .

Remark 17. In these spaces of KL-quasiperiodic functions, we easily verify (for instance thanks to a Fourier expansion) that: (253) u € L^L(P), curl u » 0, div u « 0 implies u = 0, except in the trivial case where K.L/2n is an integer (then any constant vector u is a solution). This means that the cohomology space H (P) is reduced to {0} (thus there are no TEM waves), except for perioaic functions where dimiST (P)«2. 2.3.3. Some scattering problems Thanks to sections 2.3.1, 2.3.2, we can treat the scattering of an incident plane wave (EpHg) in free space by a P-periodic structure, like in Examples 1,2. From the geometrical point of view, we can tackle two types of scattering problems with a P-periodic structure: i) one-sided problems with a perfectly conducting grating, with a connected boundary T so that the domain of the scattering problem is on one side of this grating. The grating can be coated with some dielectric medium or not. ii) two-sided problems in one of the following cases: dielectric grating (in a domain which contains a half space), periodic dielectric medium contained in a slab, periodic dielectric (or perfectly conducting) inclusions or grid in a slab of finite thickness.

5.2 PERIODIC GEOMETRY

231

From the above theory, we reduce these scattering problems in an infinite domain to problems in afinitedomain with quasiperiodic conditions and with one or two Calderon operators at the boundary. Now from the mathematical point of view, for the solution of these problems we have the following cases: i) We are reduced to a problem in a bounded domain Cl occupied by a lossy medium, that is, with a permittivity c and a permeability ji, with B" = Im e > 0, ji" = Im ji > 0 (more generally the medium can be anisotropic, inhomogeneous). In this case we can apply a variational method in the usual space of electromagnetic fields withfiniteenergy in Cl, with a coercive sesquilinear form on a subspace V of the natural Hilbert space H (curl,&) of K-L quasiperiodic electricalfieldswithfiniteenergy. Thus from the Lax-Milgram lemma, there is a unique solution in Cl, and then for the scattering problem. ii) We are reduced to a problem in a bounded domain Cl occupied in part only by a lossy medium, so that we don't have a coercive sesquilinear form any more. But we can prove that the problem depends on the Fredholm alternative; this is generally due to the fact that the natural injection of V in L2KL(Q)3 is compact, but it is not true for H (curl,ÂŁ2) and thus we have to substitute the Sobolev space of quasiperiodicfieldswith free divergence in H^Q) (see chap.2 section 11.2), to H (curl,&). Then we have one of the following situations: either we can prove that the problem has at most one solution: for instance this is the case where there is really a part with a lossy medium in Cl and a part with free space (or a conservative medium), and very likely in each of the cases of the Proposition 7). This implies (thanks to the Fredholm alternative) that there exists a unique solution in Cl and then in R3 to the scattering problem; or we cannot have uniqueness; the periodic structure has eigenmodes. From a numerical point of view, we often have to couple a spectral method (in order to take account of the Calderon operator) with afiniteelement method for instance to describe an inhomogeneous periodic medium, like in waveguides. But there are numerous other methods for gratings (see Petit [1]). We can also replace the boundary condition with the Calderon operator by "absorbing conditions", or we can use an integral method also (see Nedelec-Starling [1]) briefly described below. Example 3. "One sided problem". We assume that we have a lossy dielectric (with permittivity tx and permeability p.) in a P-periodic domain Cl contained in R2 x ( - T,0). This domain Cl is bounded on one side T by a perfectly conducting medium, on the other side I\ by free space. We will assume (in order to simplify) that T0 is the plane R2 x {0}. We have tofindthe electromagnetic field (E,H) in Cl due to the incidentfield(Eâ&#x20AC;&#x17E; FL) with angular frequency o>:

232

5 UNBOUNDED OBSTACLES ik(a T .x + a*x.*)

(254)

u,(x,x3) = uI0e

,u I 0 €C ,a = (aT,a3)€S , a 3 < 0 ,

u, * E, or Hj, E I0 s - jsz a A H IO , H,0 = ^ a A E IO , a.EI0 = 0, a.HI0 = 0 . Now we write the boundary conditions on rQ, like in the Helmholtz case, see (187) Example 1: the reflected field (Ef,Hf) satisfies thanks to C the Calderon operator of the half-space: (255)

nAH^CfciAEponlV

Then the electromagneticfield(E, H) in Cl satisfies the continuity relations on T0: (256)

nAHsnAHy + nAHp nAE = nAEr + nAEj.

Thus (E,H) must satisfy the boundary conditions on r0: (257)

nAH-C(nAE) = f,,

withf^nAfy-QnAE^onlV

Thus we have tofind(E,H) in Cl satisfying: I i) curl H + mtx E = 0, ii) - curl E + iaijij H = 0 in Cl, (258)

iii) n A E | r = 0, and (257) on T0. I iv) E and H € HKL(curl,0),

i.e., (E,H) is a K-L quasiperiodicfieldwithfiniteenergy; we save notation Clfor the elementary periodic domain. We can write this problem in a variational framework, with the sesquilinear form: (259) a(E,E)=/ n (- T±- curl E. curl E - icoe, E. E) dx - / r C (n A E|_ ). E dT0 in the Hilbert space (with the natural norm): (260)

V=VKL=:{E€HKL(curl,Q), withl| r = 0}.

Thus problem (258) is equivalent tofindingE in V so that: (261)

a(E,E)=/ r f,.EdT0,

VEeV.

Thanks to Proposition 10 on the Calderon operator C (that is (243) and that C is continuous in the natural trace space) and since the medium is lossy (thus with Im £j > 0, Im \i{ > 0), we obtain that the sesquilinear form a is a V-coercive form, i.e., there exists a constant CQ > 0 such that: (262)

Re a(E,E) > C 0 IIEIIy,

V E € V.

5.2 PERIODIC GEOMETRY

233

Thus we can apply the Lax-Milgram lemma: therefore problems (261), (258) and also the scattering problem, have unique solution (E,H). Now if we substitute free space for the lossy dielectric in Q, we obtain a problem which depends on the Fredholm alternative (in a subspace of the Sobolev space Hl(Q)3). Thus except for a countable set of singular values of o>, the scattering problem has one solution. This allows us to define a Calderon operator C f for this profile, with the properties of C for the half-space (Proposition 10). For numerical applications, an integral method is preferred. <8>

Example 4. "A two-sided problem". We consider the situation of Example 3 but the perfect conducting medium is replaced by free space (or by another dielectric medium). Now the scattering problem is in the whole space, with the electromagnetic field (E,H) propagating towards the negative axis x3 under Cl. Then we can reduce this problem to the domain £2 with the Calderon operator C ! r for the inferior domain (but generally we cannot use it for a numerical method), or to the domain Q = R 2 X ( - T , 0 ) with the Calderon operator C, relative to the half-space R 2 x ^ - < » , - T ) . In this case, we have to find (E,H) satisfying (258)i),ii),iv) and (257) and the condition (see (115)): (263)

nAH + CjfaAE^O onTx = R2x{-T}, withn = e3.

Now we define the sesquilinear form in V = HKL(curl,QT): a(E,E)- J ( - Xcurl E. curl E - me E. E) dx (264) ./roC(e3AE|r).Edr0-/riC1(e3AE|r).Edr1, with £ = 6j and \i = ]i{ in £t, t = c0, M = M0 in free space. This scattering problem is written in the variational form: find E in V such that (265)

a(E,E)=/ r ft.EdTo,

VEeV.

Since e and JJ in free space are positive real numbers, this new sesquilinear form is not V coercive. We will replace V by the following space (266)

WKL(QT) = {E€4L(QT)3, E^eH^Q) 3 ,

EI^HJU/Q')3,

[nAE] r = 0,

[n.E2E]r = 0},

where Q* is the complementary set of Q in QT. We note that the (transverse) traces of WKL(QT) on the boundaries T0 and T{ of Clx are in H^jfao)2 and H^CFi) 2 , with natural inclusions H^(r 0 ) 2 Q H j ^ d i v J o ) Q H ^ L V Q ) 2 and for Tx also.

234

J UNBOUNDED OBSTACLES

Thus the boundary terms in (264) which correspond to the Calderon operators C and Cj are continuous in this new space. Since the natural mapping of WKL(QT) in L2KL(ftx)3 is compact, the problem (265) depends on the Fredholm alternative in WRL(QJ. But uniqueness at most of the solution is an easy consequence of the dissipativity in the dielectric: M taking the real part of (265) with E = E when fj = 0 implies that E = 0 in Q, we have E a H * 0 in QT; the scattering problem (265) has a unique solution in WKL(QT).0 Remark 18. Integral method in periodic structures. We first consider, like in section 1.3.2. chap. 3, the "standard problem" with given electric and magnetic currents J p M- on a "regular11 periodic surface T (contained in R 2 X ( - T , 0 [ ) in free space:find(E,H) of locallyfiniteenergy up to T such that I i) curl H + ia>c E = Jr, Jr € HJ^2 (div,D, (267)

ii)-curiE + ia)|iH = M r inR 3 , Mr € HK L /2 (div,0; iii) E and H satisfy an "outgoing" wave condition above T (see (233)).

In order to simplify, we assume that the regularity condition (211) is satisfied (if not we have to assume that JL, ML satisfy supplementary conditions similar to (231) in the Helmholtz case, in order to satisfy the condition (251)). Then (267) has a solution (E,H) which has all the required properties and which is given by the convolution product (44) chap.3, but with the quasiperiodic elementary solution **p) given by (217) instead of * (and with the convolution product in D\ (P x R), which is allowed by thefinitethickness of the grating). Then (E,H) is obtained thanks to (quasiperiodic) electric and magnetic layers L*& and Pm*p) (like in (96) chap.3). By restriction to I\ on each side of T, we obtain the operators 1<p), R(p) (similar to T. R) see (100), (100)*chap.3) and thus the Calderon projectors P ® and P _ *p' on each side off and then the operator S*p* (with (104), (105), (106)) in the space H^l2(diy9T) x H^2(diy,T). Of course we should give this operator using Fourier expansions. This allows us to apply the integral method in electromagnetism with various periodic situations: perfectly conducting gratings or inclusions or grid, or also with a dielectric medium (with constant permittivity and permeability). <8>

Remark 19. ID-Gratings. Now we assume that T is a cylindrical periodic surface (with ^ axis for instance). If the incident plane wave (EJ,HJ) given by (254) is such that its propagation direction a is orthogonal to the x2 axis (i.e. a2 s 0), (EpH,) is independent of x^ then we can see: i) that the solution of the scattering problem is independent of x« also, ii) that this problem splits into two simple "scalar" problems (with Helmholtz equation) according to the polarization of the incident wave that is Ef parallel to x2 (P polarization) or orthogonal (S polarization), see for instance Petit [1]. We can easily apply the theory developed above for 2D-gratings (changing double Fourier series into the usual Fourier series).

5.3 CONICAL GEOMETRY

235

In more general cases, with an incident wave which depends on x~, it may be useful to work with a trace space on a plane (x3 = 0) which is in-between the plane situation (104) and the periodic situation (238) thanks to a Fourier expansion (with respect to xt) and a Fourier transformation. More precisely we can use definition (104), changing the Lebesgue measure dx » dfcjd^ into the sum of measures vn = 5 ^ - S^d^, and space H^/2(R2) into the space (210) in Remark 14. This (natural) trace space give the usual expected properties for an electromagnetic field with locally finite energy. Remark 20. On 2D-gratings with a small period. When K< | k | (which is the case for an incident plane wave), we always have the propagating mode J = (0,0). Since the propagating modes correspond to numbers J in Z so that | Sj| < k, there is no other such mode (with notations (173)') when |$J| > 2k, J*(0,0). But | $J| > 2tiL^!, with LM = max (L{,L2). Thus when 2nL^ > 2k, i.e., X = 2ir/k > 2LM the unique propagating mode is for J »(0,0). This simple remark, useful for applications with composite materials, could have been made before. 3. CONICAL GEOMETRY Here we consider stationary wave problems in domains Q in Rn such that Q =Q

r0,S = {x"(r>a)» r = M » a=x/r, r>r 0 , a € S }

with S a "regular" open subset of the sphere S n ~ l , and r0 € [0,00). Thus in polar coordinates these domains are of the form (rQ,oo)xS. We essentially work on the Helmholtz equation (this will reduce Maxwell problems to problems on the boundary). We use a method of separation of variables, which leads us to take into account Sommerfeld conditions, in a somewhat different way from the case of bounded obstacles. It will be interesting to compare the results of this powerful method with the usual "limiting absorption principle" (for which we must have sources of compact support) and to the other geometries. The fact that we have wellposed problems in the present framework implies the limiting absorption principle! First we consider the Helmholtz equation in a domain Cl of the form (0,00) x S: (268)

A v + k 2 v = - g inO,

with given g in Q, with a real wavenumber k, and with boundary conditions. Using polar coordinates, (268) is:

236 (269)

5 UNBOUNDED OBSTACLES r ^ ^ ^ r ^ - ^ + ^ A . V + ^ V ^ - G inR+xS,

with Aa the Laplace-Beltrami operator on S, and V(r,a)« v(x), G(r,a)» g(x). Now if we multiply (269) by r2, and we change variables and the unknown function (270)

p«kr, u(p,a) = r(n-3)/2V(r,a),

flpfa)-i^1,/20(rfa)f

we obtain the equation: (271)

^ u - B u = f inR+xS,

with A and B given by (keeping notation r for p from now on): (272)

i t u a - d 2 ! ! ? - ! 2 ! ! - ! ! / * and £ « Aa - ((n - 2V2)2I.

3.1. Properties of some unbounded operators associated to A First we study the main properties of the operator A. Obviously the main difficulties are at infinity and at 0 since A is a "degenerated" operator at 0 (that is the coefficient of the derivative of higher degree is zero at the boundary) with weight r2 at infinity. We choose the framework L2(R+). This choice is not at all obvious since, through the change (270), it does not correspond to the usual L2(Q) space but to the space L^C^dx/r2). But the properties obtained in such a space justify its use. We first define unbounded operators in L2(R+), with different domains associated to the differential operator/*. The first properties of these operators will be given with the usual notions on unbounded operators, see for instance Richtmyer [1], Dunford-Schwartz [1]. We define the "minimal" and "maximal" realizations of A, denoted by Am and AM by their different domains: (273)

EKiV) = D((0, oo)),

D(AM)«{u € L2(R+), An € L2(R+)}.

Proposition 11. The minimal operator A is symmetric with deficiency indices (1,1). Its adjoint operator is the maximal operator AM which has no boundary condition at 0, but two linearly independent boundary conditions at infinity B{ and B2, given by (274)

B,(u) = Dm e 2k (r u(r) e"*)', B2(u) = lim e' 2 * (r u(r) e*)',

In Weyl's terminology, we say that A is of limit point-type at 0, and A is of limitcircle type at oo, see Dunford-Schwartz [1] p. 1306. Definition 9. The boundary condition B} (resp. B2) is said to be the outgoing (resp. incoming) Sommerfeld condition (with respect to a time evolution e " lft*).

237

5.3 CONICAL GEOMETRY

PROOF of Proposition 11. The minimal operator Am is symmetric, i.e., (275) Wu,v) S5 /~[r 2 (u'v'-uv)-|uvJdr=(u,>4v),

Vu, v €l>((0,oo));

Am has (AJJJ)* « AM as adjoint; its deficiency indices are: n ± = dim ker (AM ± il). To determine n+, we note that there are two independent solutions of the equation (A + il) u = 0: u(r) = JA(r)/4?, X2 = i, thus X = ± exp (w/4), with Jx the Bessel function of order X. Its behavior at 0, then at «> is:

(276) |Jv(r)| «<«^> tev Jl^ToT W h e n r ^ 0 , and ,Jv(r>l

2rCr I/2whenr

^00-

Thus /!° |Jv(r)| dr/r< oo onlyifRev>0, therefore n+= 1. Similarlyn. = 1. Furthermore we obviously have D(AM) £ Hf0C(R+), and thus for all u, v in D(AM): (277) <-^u,v>£>R=/f{(r2u,), + r2u+|u}. v dr=<u, -Av>t

+[rV.v - u.v ')£.

Then to obtain the limits of the last term in (277), we have to specify the behaviors of the elements of D(AM) at 0 and at infinity. We extensively use the Hardy inequality (see Dautray-Lions [1] chap. 8.2.7 lemma 3) for all Tfiniteor not (278) /J|w|2dr^4/J|(iw)'|2dr, Vw with (iw)' €L2(0,T). a) Behavior at 0 of u in D(AM). We have u and (rV)' in L2(0,T) for all T>0. From (278) r.u' is in L2(0,T), thus (r.u)' is in L2(0,T), i.e., w=r.u is in H ^ T ) ) . Thus w(0) exists and since u = w/r is in L2(0,T), this implies w(0) = 0. Furthermore thanks to Cauchy-Schwarz inequality, (279)

|w(T)| = |/Jw*(r)dr| S T 1 ' 2 ^ |w>(r)| W ^ T 1 ' 2 ^ .

Arguing similarly with w=r 2 .^, we obtain: (280)

u(r) = o(r I / 2 ),

u'(r) = o(r 3/2 ) whenr~>0,

and u € D(AM) implies: (281) u € L2(0,T), (r2.^)' € L2(0,T), i.e., u € L2(0,1), r.u € H^T), r*.u € H2(0,T). Now in (277), [r^u'. v - u. v *)](*) —> 0 with c, thus u has no boundary condition at 0. b) Behavior at infinity ofu in D(AM). Let: (282)

w*(r) = (e~k ru)' e2ir,

wj(r) = (e* ru)' e~2ir.

238

5 UNBOUNDED OBSTACLES

Then (for instance) w(r) ■ wjj(r) satisfies: rw"(r)=(- Axx - 1 u) e _ir € L2(0,oo). Therefore for all T > 0, W e L'CT, oo) and w<r) has a limit /=B2(u) when r -» oo with (283) |w(r) - /| » \fi W(s)ds| 2S(/^V 2 ds) l/2 cC |sw'(s)| 2 ds) 1/2 ir- 1/2 C r with Cr —»0 when r —» » . Then for all u and v in D(AM), we have: (284) and

a ^ v ) * ^ - i [ru'.rv- - ru.rv"] =|[wjjwj- wjw|] ■OM-)^!

t(AMu,v)-(u,AMv)] = - i

Iim

r.mR [ r V v - uv*)]

(285) = 1™R^«, aR(u,v)=i[B2(u)B2(v)- BjOOB^v)]. This implies that the two mappings u —♦ BK(u), K = 1,2 are continuous linear forms on DCA^) with the graph norm. Furthermore let u be in DtA^; using (283) with B2 (and Bj) we have: (286) |(ru)' + iru - e ^ u ) ! *o(r 1 / 2 ),

|(ru)' - iru - e^B^u)! «o(r I / 2 ),

therefore giving when r —♦ oo: (286)' l u - ^ B ^ + ^ B ^ u ) ! =o(r"3/2), l u ' - ^ B ^ - ^ B ^ u ) ! =o(r 3/2 ). Example 5. We take u » Jv(r)/ Vr, with Re v > 0. From the usual properties of Bessel functions (see Abramowitz-Stegun [1] p. 364), u is in D(AM), with (287)

B1(u) = ^ e x p [ i | ( v - | ) ] ,

B 2 (u)=N|exp[-i§(v-|)]. 0

Definition 10. We define the operators A , tc«1,2 by restriction ofhu to (288)

D(AK)<lif{u€D(AM), BK(u) = 0}, *=1,2.

Proposition 12. The operators A{ and A2 are adjoint to each other. Moreover the operators - iA} and iA2 are maximal dissipative operators and thus they are the infinitesimal generators of contraction semigroups of class C°. From relation (285), we easily see that (Aj)* = A^ (A^* « Ax. Furthermore, taking v = u in (285), we have

PROOF.

(289)

a(u,u) = i[(AMu,u) - (u,AMu)J =|[|B 2 (u)| 2 - |B^u)!2].

5.3 CONICAL GEOMETRY

239

Thus i)Re(iA,u,u)=± IBJOI)!2 > 0,

VuinlXA,),

(290) ii) Re(iA2u,u) = - \ jB^u)!2<,0,

Vu in D(A2).

Thus the operators iAf and ( - iA2) are accretive operators (and - iAt and iA2 are dissipative operators). Since (LA)* = -iA 2 , Proposition 12 is a consequence of Dautray-Lions [1] Thm. 8 chap. 17A.3. ® Then we define the operator Ag when 0 < 6 < 2 by: EKAg) = {u € D(AM), BL„u = 0}, AgU =Au, (291)

BI^u = exp [i | (| - 8)] B,(u) - exp [- i \ (\ - 8)] B2(u).

We can prove that it has the properties (see Dunford-Schwartz [1] 13, 9.15): Proposition 13. The operator \is a self adjoint operator whose spectrum o(Ag) has a continuous spectrum R+ and a set of discrete simple eigenvalues: (292)

o(Ag) = R+U{- (2n + 8)2, n e N},

with associated eigenvectors J^+gOMr. The resolvent R(- X, Aj) = (XI + Ag)~ of Ag when - X is not in ofAg) is given (with Re ^X > 0) by: (293) with:

R(- X,Ag)f(t)=/" Ke(t,s;X) f(s) ds,

(294)

Ke(t,s;X) =

f € L2(R+),

rt2(X)(J^(s)/^XJytVAlt), s<t,

^ ( ^ ( t v ^ x ^ s ) / ^ ) . s>t, (295) J^sin5(X + 8)Jx + sin5(X-8)J_X( PROOF.

(296)

The fact that A^ is a selfadjoint operator is a simple consequence of B1(0 = B 2 (f),i^(f) s B l (f),

thus BL,(f) * - BL^f).

Furthermore thanks to (287), we have: (297)

( < [

Bl* (JA(r)ATr) = 2i ^ sin § (X - 6).

Thus when X is in R*f J^(r)/i/r is an eigenvector of A^ (for - X) if:

240

(298)

J UNBOUNDED OBSTACLES sin|(4X-9)*0, i.e., X = (9 + 2n)2, n€N.

This gives the discrete spectrum. The continuous spectrum is obtained by finding the generalized eigenfunctions so that (A + Xl)f =* 0, with BL0f« 0. We obtain: «r) » J*^r)A[T. The general theory of differential operators (see Dunford-Schwartz [1] p. 1326) yields the resolvent of the operator Ag as given in (293), (294). ® Proposition 14. The spectrum o(A|c) ofAK is continuous with o(A|c) = R+, K= 1,2. The resolvent R(- X, A^)«(XI+AIC)""l of A^ when X € C\R~ is an integral operator: R(-X,AK)f(t)=i^KK(t,s,X)f](s)ds,

Vf€L2(R+)

with kernel Kgfcs, A) given by (when Re (4X) > 0): (-l) K - ! |(J^(s)/^)(H ( ^(t)/# (299)

whens<t,

K^A)* ( - i f 1 § (J^ty-TO (H ( $s)/^) when s > t,

where H* \ H* orerteuszia/ Hankel functions oforder v. PROOF.

(300)

The closure of D(R+) for the graph norm is the space D(Am) defined by EKAm)*{u€ D(AM), BK(u)-0, K . 1, 2}. "

Its range by (XI - A\ X € C\R+, is closed of codimension 1. Let: (301)

D ± c i f ker(A M ?iI), withdimD ± =l.

Then D(AM) is the orthogonal sum for the scalar product <u, v> = (Au, Av) + (u, v): (302)

D(AM) = D(Am)®D+®D_,

and there are spaces D\ K= 1,2 of dimension 1 contained in D + §D - such that: (303)

D(Alc) = D(Am)©DK,

K=1,2.

In order to prove that (XI - AK)D(AK)» L2(R*), we only have to verify that: (304)

(XI - \)nK = 0, uK € D(\),

implies u = 0.

But the solutions of (304) are linear combinations of (J ~(s)A[5) and (J_ -(s)A[5») and the conditions u in L (R*), B^u = 0 imply u = 0. Thus we have proved that o(AK) £ R*. Furthermore the asymptotic behavior of the Hankel functions gives:

241

5.3 CONICAL GEOMETRY

BK((Hf(s)A[5) = 0. Thus for all X in C\R~, the equation: (A + XI) * = 0 has a unique (up to a multiplicative constant) generalized solution satisfying one of the conditions i) 0 is a square integrable function in a neighborhood of 0 only: $(r,X) = J -(r)/4F. ii) $ satisfies the condition at infinity Bt(0) « 0: <Kr,X) = H\2(r)/>f?. This gives (299) according to the general theory, when K * 1; K = 2 is similar. Proposition 15. The resolvents R( - X, AJ, K = 1,2 and R( - X, A^) are only different through a (degenerated) operator of rank 1 when X is not in ( - 00,0] a/irf w/te/1 X * (2n + 0), with n an integer: (305)

PA = R(-X,A^ - R(-X, A 2 ),

P* = R(-X,A,) - R(-X,AJ,

with kernel K(t,s,X) and K0(t,s,X):

I K(t,s,X) = i* (J^(s)/4s).(J^t)/4t), (306)

K*(t«K -exphi(n/2X4X-8)] K (t,s,X) = . pr-———K(t,s,X). sin[(n/2)(NX-8)] I7i^ norms of the operators Fk and PA are given with 4X = y0 + i jij by

(307) i p ^ . ^ - i p j } ! ,

llPx» = IiPAHexp(^,71/2) [ch 2 (| ji^ - cos 2 (|(n 0 - 0))]"1/2;

thus when X = X0 € R+, vQ = <|XjJ:

008)

IIPAOII=^,

,0 „

K^w0wwm^m-

Proposition 15 is a simple consequence of (299). It implies the basic result: Proposition 16. The resolvents of the operators A satisfy: (309)

IIR(-X 0 ,AJ<K/4^,

K = (K+1)/2, VX 0 >0.

This result is the best in the sense that there cannot exist (K,a), K > 0, a > 1/2 so that (310) PROOF.

(311)

IRÂ^KAjf. From (305), (308) we have for instance for Aj : IIR(-X0, A,)ll < IIRÎI + IIPÎI.

242

5 UNBOUNDED OBSTACLES

Since - A^ is a selfadjoint operator, the norm of its resolvent is given by: (312)

»R^|= l / d i s t f X ^ - A ^ 1/ inf |X 0 -(2n + 8) 2 |.

Let [PQ] be the integer part of i*0= ^ . When [JJ^ iseven, taking 9=|i 0 - [ ^ +1 gives |sin|(JI 0 - 8)| m 1, |X0 - (0 + 2n) 2 | > 2j*0. When [vj is odd, we obtain the same result choosing 8 = n0 -

[JIQ].

Finally we cannot have inequality (310) since by

difference, we would have IIP, II s 2K XjJ°, in contradiction with (308). 0 Remark 21. The so-called "Kantorovich-Lebedev" transformation (in the Hankel form) is used to diagonalize the operator ^. But we can expect that it is not so easy, because the operators A are not selfadjoint, nor normal, nor even spectral operators ! The definition of this transformation (with ji > 0) is

013)

$Koi) = iimA r mJW^w m ds, K. i, 2,

with H; the Hankel function. The mapping 0 —»(+|,+2)is a n isomorphism from L2(R+) onto H\ x H^ with (314)

/*>{g€Lk c (R + ), / R + Ig(p)| 2 e C T , 1 (thf )p * < < » } ,

«.<-lf,

with (unusual) properties of diagonalization of A . For developments on this transformation (which can be used to solve Helmholtz problems) see Cessenat [1]. [2], [3]. ® 3.2* Solution of Helmholtz problems Now from section 3.1 we can solve Helmholtz problems. First in the domain Q = R*xS, and in space L2(R+xS), (i.e., in L2(Q, dx/r2)) we define the following framework: (315)

DM(A - 5 ) = {u€L 2 (R*xS), (A - £ ) u €L 2 (R + xS)}.

This (maximal) space has traces on the boundary of Q: let u be in DM(A - B\ then: i) B\x € L2(R+, H"2(S)), thus^u € L2(R+, H'2(S)) also, and thus BKu exists in H~2(S). ii) An € L2(S, H,;2(R +)), (A maps H2C(R+) «{u e H^(R+), supp u is bounded} into L2(R*)). Thus B\x € L2(S, Hj~2(R +)), which implies that the traces yQn and YjU of U on the boundary R* x 8S make sense (at least in H^aS.Hj^R +)), for s = - 1/2 and s = - 3/2 respectively).

5.3 CONICAL GEOMETRY

243

Definition 11. We define the operators AK - B D , AK - B N , K = 1,2 by restriction of the differential operator A -Bto the following domains: I EKAc - BD) = {u € DM(A - B), BKu = 0, Y0u = u| R+ {316)

=0J,

a,, |EKAK-BN)={U€DMU-B),BKU=O,Y1U=^|R+X9S=OK

that is with the Dirichlet or the Neumann boundary conditions and with the outgoing or the incoming Sommerfeld conditions. Theorem 5. The Dirichlet (or Neumann) problem with the outgoing (or incoming) Sommerfeld condition at infinity: (317) ( ^ - B ^ u r r f ( r e s p . ^ - B ^ u ^ O , f€L2(R+xS), has a unique solution u in D(AK - BD) (resp. D(AK - BN)), except when n = 2, with the Neumann condition; in this case, the result is valid if we change (for u and f) L2(R+xS)i>if0 (318)

L2(R+xS) = {f€L2(R+xS), /sf(r,9)d9 = 0}.

We extensively use the following property: the Laplace-Beltrami operators Aap or AaN with the Dirichlet or Neumann condition are selfadjoint operators with compact resolvents, so that the operators BD and BN defined by (272) are strictly negative operators i.e. with pure point spectrum contained in ( - «>,0), except when n = 2 with the Neumann condition (in this case the supplementary condition (318) eliminates the eigenvalue 0, which correspond to constant eigenvectors). Thus the operators AK and B D or BN are commuting operators, with disjoint spectrum (thanks to Proposition 14). Under fairly general conditions (see Grisvard [3]), we can solve problems similar to (271), thanks to the Cauchy integral in the complex plane: (319)

u=~4/ Y (A K + zr1(B + z)-1fdz,

B=sBDorBN,

where y is a contour in the complex plane including the eigenvalues of - B (with horizontal asymptotes when Re z tends to +oo); a priori the use of (319) is purely formal, since the estimate on the resolvent of AK at infinity (see (309)) is not sufficient to apply this formula. We will justify it below. PROOF of Theorem 5. Let (\,*n) be a spectral resolution of - B, (with B = B D or BN) where the X are the eigenvalues (accounting their multiplicity) and ($n) a corresponding ortnonormal basis of eigenvectors, giving the decompositions

244

(320)

5 UNBOUNDED OBSTACLES

I«(«) = 1 ^(r)* (a), ««*)= 2 fn(r)*n(a), a € S, r> 0, " , , |with: br-2hlir<eD> MT-2igr<»,

so that (271) is reduced tofind(un) satisfying (321)

(iWK-f,,.

Then since - Xn is not in the spectrum of Ay, (321) has a unique solution given by (322)

u n = (AK + V ) - 1 f n = R(-Xn,Alt)fn.

Moreover inequality (309) implies: (323)

hJ^H^I/^.

Thus the condition f € L2(R* x S) implies 2 Bunll2 < oo. The solution u of (317), is given by (324) u(.,a) = 2 R(-* n A> f n *n<a>> and satisfies u € L2(R* x S). Thus u € DM(A - B) and also u € L2(R+, D(B1/2)) since (325)

2X n liu n !l 2 <oo,

thanks to (323). Furthermore each u n is in D(AK) and thus we have BKU = 0 in a weak sense, but we don't have u in L2(S,D(AK)). Note also that (323) implies: (323)'

kl£K%;1/2m,

withK^^+D^^^infX^

The solution u of (317) is then given by the Cauchy integral: (326)

u = | u n = | (AK + X n ) - 4 n = ^

ton/^(AK+z)-1(B

z)- 1 fdz,

where y is a family of growing contours including the first v eigenvalues in the complex plane. This is a consequence of the usual formulae (with Q n the orthogonal projection on the eigenspace of - B for X and for large values of n) +z l (B n T

>~ =-2x^Q >

t\+\rt-A/

^(A.^zr^dz.

Remark 22. Rayleigh series. Using formula (299) of the resolvent, the solution u of (317) (see (324)) is written in the form (for K = 1): u(ra)=§ [ | ♦n(a)(J^_(r)/^^)J7, (H ( ^s)/,|5)f n (s)ds (327) + 2 ♦ n (a)(H ( ^-(r)/^)^(J^_(sy^)f n (s)ds],

5.3 CONICAL GEOMETRY

245

with f n (r)=/ s f(ra) «n(a) da. This gives the usual Rayleigh series, see (380) chap. 3: i) when supp f is contained in the ball Ba, then for r > a, (328)

u(rot)=11 cn *n(a) (H( .LWtf),

with c n = / ~ (J r^sy^s) fn(s) ds; ii) when supp f is contained in R"\ Ba, then for r < a, (328)'

u(ra)=f | c n ^(a) (J^r)A|f),

with c n = / ~ ( H ^ s y * ) fn(s) ds. In R , ^ = n + k, n € N, with multiplicity equal to 2n + 1, the eigenvectors of B are the spherical harmonics. Remark 23. In order to prove the following proposition, we note that the solution u in D(Aj) of (A{ + \I)u = f, with fin L2(R+), is obtained (with (305)) by: u as R( - X,Aj)f = R( - X,A^f + Pxf; thus its traces at infinity are BjU = 0, and (329) B2u = B2Pxf - in B2(J^(r)A[r) L^f,

with L^f=/~ (J^tWtWt) dt.

Hence from (287): (330)

B2u = i^exp[-f(<fX-±)]L^f,

we have (thanks to the Cauchy-Schwaiz inequality): (331) using: (331)'

|B 2 u| 2 s2n|L 4X fi 2 < : |llfll 2 , J R+ |J v (t)| 2 dt/t=l/2v, v>0.

We note Y = D(A!-B) + D(A 2 -B),Y = YDwhenB = B D ,Y = YNwhen B = B N . We recall that D(B1/4) = H^2(S) when B = BD, D(B1/4) = H1/2(S) when B = B N . Proposition 17. Trace at infinity and Green formula. The mapping BK: U —► BKU is a continuous mapping from Y onto D(B1/4), K= 1,2. Furthermore all u, v in Y satisfying the Dirichlet or the Neumann condition, also satisfy the Green formula (332) aB(u,v) d=?f i {(A - 5)u,v) - (u, (A - B)v)=| / [B 2 u.I^ - BjU.I^] da.

246

5 UNBOUNDED OBSTACLES

PROOF. First we prove that u in D(Aj - B) implies B-u in D(B1/4). Using the decomposition (320) with (322), we have with (331) and f= (Aj - B)u: (333)

»B2ull2D(Bl/4)as 2 jfc | B 2 u n | 2 * * | lffn«2^w»fl2< °°-

Then given g in D(B1/4), we construct now a lifting of g into EKAj - B). Let g = 2 &n+n be an expansion of g with 2 ^ defined by: (334) with

| g | < °° • Then the function u

u-tAj-B)-1?,

F - ^ | ^* n «PP5<^-2>]<V r ) / * ) V is such that F is in L2(R+ x S) (from (331)') and u satisfies (with (329) and (331)'):

The first part of Proposition 17 follows easily. Now we prove the Green formula (332) using the spectral decomposition of B and (285), since we have: aB(u,v) = i 2 [((A + Xn)un,vn) - (u n ,U + Xn)vn)J (335) <8>

Proposition 18. The operators - i(Aj - B D ), KAj - BD) are maximal dissipative operators, adjoint to each other, and so that (336) EK(AK - BD)2)£L2(S,D(AK))nL2(RM)(BD))£D(AK - B D )£L 2 (R + ,EKB{) / 2 )). The same properties are valid if we replace the Dirichlet condition by the Neumann condition, but for n = 2, we have to substitute space L2(R*X2(S)0) for L2(R* x S), where L2(S)Q is the orthogonal space to constants in L2(S). PROOF. The Green formula (332) implies that: (337)

Re [ i((A! - BD)u,u)] > 0, V u € EKAj - BD).

Thus - i(A. - BD) is dissipative. Furthermore since this operator has a bounded inverse, it is a maximal dissipative operator (from Dautray-Lions [1] thm. 8 chap. 17A.4). We easily prove (336) and that i(A.-B D ), - i ( A 2 - B D ) are operators which are adjoint to each other from Green formula (332). 0

5.3 CONICAL GEOMETRY

247

We can prove (see Cessenat [1]) that if u is in D(AK~BD) (or in D(A K -B N )), then (338)

§£ € L?0C(R+ xS),

thus EKA, - B) £ H^tfO, oo)x S).

With this result (and (323)'), we can obtain an estimate on the norm of the solution u of (317) in Hjoc((0,oo) x S), like the estimate (181) of the limiting absorption Remark 24. Asymptotic behavior in DMC4 - B). Sommerfeld conditions. We can specify the behavior of functions u in DMU - B ) at infinity like in the proof of Proposition 12. Using the decomposition (320), we define (where ' is used for derivation with respect to r): (339)

w^(r,a)«(e* ru(r,a))'e-2ir, wj(r) - (e* run(r))'e-2ir,

thus w^(r,a) = 2 ^(r^Ja). Then we can prove that the formulas (286) are also valid, n " 3/2 the o(r"* ) refers to the norm H" *(S). This immediately implies: if u is in D(Aj - BD), i.e. B ^ = 0, then u(r,oc) = Q(t"l)t u'(r,a) - iu(r) = o(r~3/2), that is, u satisfies the outgoing Sommerfeld condition. We can also specify the behavior of functions u in DM(^4 - BD) at the origin like in (280), using the spectral decomposition of B = BD (then the o refers to the D(B"1/2)norm). Remark 25. Sources of compact support. Rellich lemma. i) If we have a source g (see (268)) with compact support and such that f = t^g (see (270)) is in L2(R* x s \ when n = 3, then we know (see (15) Prop. 1 chap. 3) that the asymptotic behavior of the outgoing solution u = v is given by: (340)

u(ra) = §^B2u(a) + o(r 3/2 ), with B 2 u(a)«i- g(a), a € S 2 , k = l .

ii) Proof of the Rellich Lemma 1 (chap. 3). Let u satisfy outside a ball Ba (341)

(A + k2)u = 0, i.e., (A - 5)u = 0, and/ 2 |u(rcc)|2da = o(r 2 ),

sothatu(ra)€L2(Bl). Furthermore (341) implies that B ^ B ^ O . Letu a =uL . Then using the spectral decomposition (320), we obtain that un satisfies: (342)

(A^ + y u ^ O ,

r>a, ic=l,2,

un(a) = u£.

Thus un is given by the two formulas (with K = 1 and 2):

248

5 UNBOUNDED OBSTACLES u„(r) = < (H ( ^-Wtf) (rf li(a)/4a)- l ,

V r 2: a,

which necessarily implies u*=un(r) = 0, for all n, thus u = 0 ! We emphasize that the Rellich lemma can be applied to any domain whose boundary is a conical surface when r is large enough. ® Then we solve completely inhomogeneous problems thanks to Theorem 5 and a duality method. First we define the following trace spaces with K » 1,2 |X5 = {u0=y0u = u | R + x 9 s , U E E K A . - B N ) } (343)

|XHul=*lu=lilR+x3S' ■•DCA.-BDH equipped with the quotient topology, and also with (315): (344)

X ^ l u o ^ u , u€D M W -B)}9

X ^ u ^ u , u€D M M - * ) } .

We can prove that XQ = XQ , x} = x\ (thus denoted by XQ, XJ), and

(345)

x 0 =(x 1 y,x 1 =(x 0 y, with x 0 cx 1 c(x 1 yc(x 0 )'.

Usual regularity results on elliptic problems in regular domains (see GilbargTrudinger [1] for instance) imply:

XocHf^xas^^cH^xas), JCocHf^Vxas), x^Hf^Vxas). Theorem 6. Let (f, g, uQ), resp. (f, g, Uj), be given such that: (346)

f€L2(R*xS), g€EKB1/4y, u 0 €Xo,

resp.u^Xp

Then there exists a unique solution u in T>M(A - B) of:

(347)

|i)04-5)u=:f inR + xS, ii) B2u = g (or BjU = g) I Hi) Y0u - u0

(resp. yp = Uj).

The only exception is the Neumann problem with n = 2. In this case the theorem is valid //(f,g,Uj) satisfy the conditions (with S identified with ( - 80,80)) g = 0,

jo"(ui+ + Uj-) + f € ImA! (or ImA2), with

(348) ^=^(.,+80), ^.=^(.,.90), f=^/^°f(.,e)d8,g=2^/^0g(e)de.

5.3 CONICAL GEOMETRY

249

The proof is based on the weak form of the problem (347): find u in L2(R+ x S) such that (349) with

(u, (4 - B)v) = LfgfU()(v),

V v € DfA, - B D ),

f,g,u0<v> = <f'v> - 5<g,B2v> - <u0,Ylv>.

Thanks to Theorem 6, we can define the Calderon operator C relative to Cl: when n * 2, this is the isomorphism defined, with u the solution of (347) for f = 0, g = 0,by: (350)

u 0 € Xo(resp. XQ) — YJU € Xx (resp. X^.

When n = 2, we have to replace spaces Xx and Xx by X \ and X | with (351)

XKresp.XlJ^fu^^^JeX^resp.X!),

u 1+ + u ^ e l m A ^ .

With the usual regularity properties of S (thus of Cl = R+ x S) we can extend the functions u in D^(AK - BD) or Dft(AK - BN) to Rn by symmetry and truncation. This implies that the spaces of traces (343) and (344) are the same for Cl and its complement Cl9. This allows us to solve the "transmission" problem (or with given sources on the boundary T of a cone Cl):findu in L2(R+ x S n ~ { ) such that i)04-B)u=rO inQandQ'^R+xS', S' = S n " \ s , (352)

ii) B ^ u l ^ O ,

B|(u| a > )s0 (i.e., u is outgoing),

with given jumps p and p' of u and its normal derivative across T. Theorem 7. Let p € X^ p* € XJ . Then (352) has unique solution u such that (353)

u|0€D&4-*),

u^eD^C*-2?)

(i/n ss 2 the condition onp'is: p'+ + p*_ € Im Ai with notations used in (351)). Furthermore we have the regularity result on the traces of u and its normal derivative on each side of the boundary T ofCl (354) (y0±u,yl±u)€X0xXl when (p, p')€XQXX X (when n = 2,we have to substitute Xj for X^. Let P ± be the mappings (pyf>9) —»(yQ ±u, Yj ± u), u the solution of (352), we verify that P * are continuous projectors in spaces XQ x Xj and in XQ x X, (or X 0 x X | w h e n n = 2).

250

5 UNBOUNDED OBSTACLES

These maps are the Calderon projectors in these spaces; they satisfy the usual relation: P + + P~ =1. These operators can be made explicit thanks to the spectral decomposition of the Laplace-Beltrami operator on Sn~*, and we can use them to solve scattering problems with an incident stationary field in free space on a cone occupied by a medium with wavenumber k.. We can generalize in many ways (see Cessenat [1]), for instance to domains Q = (a,oo)xS, a>0, to dissipative media (associated with the limiting absorption principle). Application to Maxwell equations. We can use the Calderon projectors above for Helmholtz equation to transform a boundary problem with Maxwell equations in a conical domain, into an (integrodifferential) problem on the boundary T of this domain. We can consider also the scattering of an incident wave by a dielectric cone. In order to do that, we have to transform relations between the electricfieldE and its normal derivative on T into relations between the tangential components of E and H on T, thanks to the differential formulas (310) in the Appendix. Using the same notations, the Maxwell equations at the boundary Tare: | i ) - ^ ( n A E ) + curl r E Il -2R m nAE + i(i)|iHr = 0, (355)

ii)^(nAH)-curl r H f l + 2RlllnAH + i<i)£Er = 0, iii) - curlr (Er) +toyHn = 0, iv) curlr (Hr) + io>£ En = 0,

with E r =t r E the tangential component of E. We recall the divergence formulas: (356)

d i v r E r + ^ + 2R m E n = 0, div r H r +^£ + 2R m H n = 0.

We emphasize that g- (n A E) and n A ^j E r (for instance) are different! Using an orthogonal system of coordinates (x^x 2 ^), the Riemannian metric is: g = ds2 = gjdxf + g2dx| + ds2 . If X = X ^ + X282 is a tangentfieldto T, then: (357) with:

^-(nAX) = n A ^ + f x - 2 R m n A X ,

« - ^ - x a 5 8 | + X | 5 % 1 , go=(gig2)"l/2 " , d 2 R --^4 , o 1 ' Of course, we would have to specify the functional framework for such Maxwell scattering problem in correspondence to the above scalar problems with Helmholtz equation. The functional spaces are based on weighted norms. We don't develop these (technical) spaces here, but results are especially interesting in the simplest case where the conical domain is the whole space! &

CHAPTER 6

EVOLUTION PROBLEMS Evolution problems in electromagnetism and more generally non-stationary wave problems are of many different types: First we consider Cauchy problems, i.e., evolution problems with given initial conditions (or also mixed Cauchy problems, i.e., with boundary conditions). These problems are fairly easy to solve and quite "standard" in free space, or if we assume an "ideal" linear isotropic media with constitutive relations D = eE, B = pH. In the general case the constitutive relations have to be written with a convolution in time, see (22) chap. 1. This implies that the usual Cauchy problem is ill-posed: we have to know all the past of the material to solve evolution problems, which are called Cauchy problems with delay or with thick initial conditions, see Dautray-Lions [1] chap. 18. Here also scattering problems lead us to define incoming and outgoing waves, associated with Cauchy problems under fewer initial conditions. Then we consider "causal" evolution problems, which are well-posed problems without any initial condition. Many evolution problems concerning waves and electromagnetism are of this type. Maxwell equations are an example of symmetric systems of Friedrichs. We do not develop this general theory, but only refer to Friedrichs [1], Lax-Phillips [3], Chazarain-Piriou [1], Colton-Kress [2]. We do not either use microlocal analysis (and the trace theorem of Hormander) in this book. 1. CAUCHY PROBLEMS

First we consider the following Cauchy problems. Let H be a Hilbert space; given UQ and F, an /f-valued function, find an H-valued function U such that: | i ) ^ + ^ U = F, with t>0 or with tâ&#x201A;ŹR, ii)U(0) = U0. 251

252

6 EVOLUTION PROBLEMS

Problem (1) for t > 0 only, is called a one-sided (or unilateral) problem. Problem (1) for real t is called a two-sided (or bilateral) Cauchyproblem. First we assume that -A is the infinitesimal generator of a semigroup (G(t)) of class C° in H. From usual theory (see Pazy [1], Dautray-Lions [1] chap. 17) we know that if F € C°([0, oo) ,H) (or if F € Lr([0,T) ,H) for all T) with UQ € H, the one sided problem (1) has a unique solution U € C°([0, oo) yH) (in a weak sense) given by: (2)

U(t) = G(t)U0+/0G(t - s)F(s) ds.

Then assuming that —A is the infinitesimal generator of a group (G(t)), and F is given for all real t, the two-sided Cauchy problem (1) has a unique solution with the same regularity properties. 1.1. Scalar wave Cauchy problems We recall some well-known results on Cauchy problems for waves. As a reference on this subject see Hormander [1], Chazarain-Piriou [1], DautrayLions [1]. The usual standard Cauchy problem for scalar waves is: given a (complex or real) Hilbert space H, find an H-valued function u so that: a2,, i ) ~ j + Au = f inR^orR^, (3)

ii)u(0) = u°, 3u, gW-u1, with given initial conditions u°, u1, and with a given source f. The operator A is a positive (or even bounded from below) selfadjoint operator defined using a variational framework (V, H, a(u,v)). The usual space with finite energy waves corresponds to u°€V, u*€H, and f€C°(Rt,H) (or more generally f€L l loc (R t ,H)).Thentaking# = VxH, and: <4, U

= ( ^ , U < , , . ( ; « ) . * . > = ( * > ) . A.[<

->).

- « . » .

problem (3) is written in the form (1), with A the infinitesimal generator of a group (G(t», and thus (3) has a unique solution U€C°(R,/0. giving u € C°(R, V), with u' e C°(R,H); u is given by (5) u(t) = cos(tAlV+A-y2sia(tAuW

+f0A~U2sin((t-s)Am)f(s)<1S'

We obtain very easily this formula of symbolic calculus, defining z and w by: (6)

z(t) - iA1/2 u(t) + u'(t), w(t) = -iA 1/2 u(t) + u'(t), Zo-iA^uO + u1,

w0 = -iA , / 2 u° + u1.

253

6.1 CAUCHY PROBLEMS

Then z and w satisfy: (7)

1/2 z* iA1/2 z(t) + f, z' == iA z(t)

v/ = - iA ,/2 w(t) + f,

and the solution of (7) is given as in (2) by: _/*%

(8)

iA1/2t 1

Izft^e * w(t) = e _lA

Zo+io l

iA1/2(t-s)~ e

T won +J„e J0r

'f(s)ds, L1/2„

'f(s)ds.

We obtain (5) by taking the difference z(t) - w(t). We note that the operator: (9)

M(t)=A-1/2sin(tA1/2)

is a bounded operator, and that (5) is also given by: (5)'

u(t)=|j M(t)u° + M(t)u! + f0 M(t - s)f(s) ds.

Furthermore, multiplying (3)i) by u' (the time derivative of u) and integrating, we check that u satisfies the energy balance (for f=0 this is the energy conservation law) (10) \[H§jJ(t)ll2 + HA'^t)!2] =|IBu'U2 + IIA^VB2] + Re/ Q (f,g)ds. The first usual example is the Cauchy problem in free space occupying Rn associated with the wave equation or d'Alembert equation: 7

(ID

i ) D u d J f 4 | ^ . A u = f in^xRt, ii)u(0) = u°, fjW-u 1 ,

c being the velocity of light in free space, u°, u1 being given initial conditions, and f being a given source. The usual framework is the space offiniteenergy solutions. Using the Beppo Levi space: (12)

W1(Rn) = {v€Z)>(Rn), v€L?oc(Rn), /Rn(|gradv|2)dx<oo}

(see also Appendix, Def. 7 section 5.2), we assume that the data satisfy: (13)

u^W^R11), u!€L2(Rn), f€C°(Rt,L2(Rn)).

254

6 EVOLUTION PROBLEMS

Then taking V = W^R11), H = L2(Rn), H=W^R") x L2(Rn)f A » - c 2 A (but V is not contained in H), we see that problem (7) is of the form (3); thus it has a unique solution u obtained in the form (5), thanks to its Fourier transform (we take c = 1 in order to simplify the formulas): /nx */*** ^sin(|6|(t-s))j; / I H A A 0 m sin(|£|t) A i (14) u(t,t) = cos(|t|t)u (|) +—jt| u (*>+/o~—TH f&s)ds. We can write this formula using elementary solutions of the d'Alembert equation as will be seen below. We haw used the semigroup theory to solve this Cauchy problem, but there are many other ways to solve it (see Dautray-Lions [1] for instance). We can solve mixed Cauchy problems also, i.e., problems in a domain Cl with a Dirichlet or a Neumann condition. Let Q be the exterior of a bounded domain with boundary T. For a Dirichlet condition, we have to substitute for W*(Rn) the Beppo Levi space (see Def. 8 in the Appendix) which is also defined in the regular case by: (15)

wi(Q) = {u€D'(Q), u€l4 c (Q), / Q |gradu| 2 dx<oo,

u|r=0}.

We first give a general property of waves satisfying (3) when t goes to infinity. We define the kinetic energy Ec(t) and the potential energy E (t) of u by: Ec(t) = i l | j (t)ll2t

Ep(t) 4 IIAl/2u(t)ll2.

Then using the notion of spectral measure (see Kato [1]) and (5), we can prove the following property when there is no source (for instance when A = - A in the exterior of a bounded domain, see Goldstein [1], Lax-Phillips [1]) Proposition 1. Equipartition of energy. IfOis not an eigenvalue of A, then the solution u of'(3) with f = 0 satisfies equipartition in the mean: Moreover if the spectral measure dE(X) is absolutely continuous with respect to the Lebesgue measure dX, u satisfies: lim Ec(t)= lim EJt).

1.2. Cauchy problems In electromagnetism 1.2.1. Cauchy problems in free space R3 We consider the Cauchy problem: find the electromagnetic field (E,B) (or (E,H)) in free space Q = R3, satisfying the Maxwell equations (1) with (5) and (3) chap. 1, for positive t (one-sided problem), or for all real t (two-sided problem):

6.1 CAUCHY PROBLEMS

255

i ) - f ^ + curiH = J, §£ + curlE = 0, (16)

ii) divD = p, divB = 0, iii)D = £oE, B = »»0H, iv) E(0) = E0, B(0) = B0, (or H(0) = H0, D(0) = D0),

with given charge density p and current density J satisfying (3) chap. 1, and with given initial conditions for E, B (or D, H). It is more common to work with (E,B) in free space than with (E,H) for evolution problems, so we will use (E,B) only. As indicated in chap.l, these equations are partially redundant. We assume that: (17)

E0€L2(R3)3,

B0€L2(R3)3,

andJeL20c(Rt,L2(R3)3.

We define H = L2(R3)3 x L2(R3)3, and with e0ti0 = 1/c2,

Then the Cauchy problem (16), without taking ii) into account, is written in the form of (1). Moreover we can prove easily by Fourier transformation that -A is the infinitesimal generator of a unitary group (G(t)) in H, with domain D04) = H(curl,R3)xH(curl,R3). Then we know that the two-sided Cauchy problem (16) with (17) has a unique solution (E,B) in C°(Rt,/f) with finite energy (with conservation of energy when J = 0, see (7) chap. 1): (19)

W(t) = i/ R3 (D(t).E(t) + B(t).H(t))dx = W ( 0 ) - / 0 d s / R 3 E . J d x ,

but does not necessarily satisfy (16)ii). If the data satisfy, in addition to (3) chap. 1: (20)

div B0 = 0 and div E 0 = p/£0 with p € L ^ R ^ L2(R3)),

then we can prove that the solution (E,B) of the Cauchy problem also satisfies: div B(t) = 0, and div E(t) = p(t)/e0 for all t. 1.2.2. Cauchy problems in a domain fit Now let Q be a free space domain in R3 bounded by a perfect conductor, with boundary T. We have to solve a mixed Cauchy problem: find (E,H) satisfying the Maxwell problem (16) in O, with the boundary conditions (21)

n A E | r x R + = 0,

n . B | r x R + = 0.

We assume that the data satisfy (17) with Cl instead of R3. Then with H= L2(Q)3 x L2(Q)3, we can prove that operator 4, defined in (18), with domain D(A) = Ho(curl,Q) x H(curl,Q) is such that:

6 EVOLUTION PROBLEMS

256

A*=-A, with D(A*) m D(A) (U is a selfadjoint operator if we complexify the space). As a consequence of the Stone theorem, A is the infinitesimal generator of a unitary group (G(t)) in H> at least when fi satisfies one of the following conditions: i) Q is a bounded domain, or the complement of a bounded domain, ii) Q is a cylinder (see Duvaut-Lions [1], Dautray-Lions [1] chap. 9A and 17B.4). Of course we can take other frameworks which are subspaces of the above space H for these evolution problems, in order to take into account usual conditions of free divergence for B for instance, or regularity properties. This allows us to indicate in which sense the boundary conditions are satisfied (thanks to the notion of weak and strong solution). Thus we can use any variational framework ofchap.2.11.1. It is interesting to compare the Maxwell problem to the wave problem, for instance with respect to H (or B) only. We eliminate E by a usual trick to obtain: 2

9H i K1 ^ - A H = curlJ, c dtr

(22)

ii) H(0)=* H0,

divH=0,

fH(0) = - j^curl E 0 ,

iii)n.H| rxR+ =0,

nA curlH| rxR + =0.

The boundary conditions imply (see (147) chap. 2) that (-A) is the positive selfadjoint operator (A*-I). For E, we obtain (in the regular case) a similar result but with the operator (A~ -1). Of course we can generalize these results with respect to the Cauchy problem in many ways: for instance, thanks to a variational method described in DuvautLions [1], to the case of several linear (isotropic or not) media with different permittivities and permeabilities (but frequency independent in time Fourier transform). Now we recall some properties of the wave equation and above evolution problems. These properties are common to hyperbolic equations: the wave equation is a typical hyperbolic equation. 1.3. Some hyperbolic properties of wave evolution The main property of the d'Alembert equation is the existence of an elementary solution <&(x, t)(or 4> (x,t)) with support in the future cone orforward light cone: (23)

C+ m {(x,t) € R n x R, with |x| £ ct}.

This solution which is thus characterized (taking c = 1) by: (24)

D * d ^ ~ ? - A=6(x, t) m 6(x) 8(t),

with supp * £ C + ,

is unique: if <DQ is another elementary solution with the same property, then the convolution of * n and * is defined and we have

(25)

CAUCHYPROBLEMS

257

□(<& ♦ 4>o)»(□*) ^ o = % = * * <□*<>) = • •

This existence is typical of hyperbolic equations (see Dautray-Lions [1] chap. 5). There are many ways to calculate this elementary solution, each method having its own interest (one of them is the Radon transform method). We can also obtain simply by its space Fourier transform (compare to (14)): (26)

A A A sin(Ult) <IKU) = Y(t)R(U), withR(U)* j ^ 1 ' ,

where Y(t) is the Heaviside function. Here we give the very simple result when n = 3: (27)

«(xft).i{«(r-t)-^Y(t)«(i2.t2),

r=|x|.

The operator M(t) given by (9) in this case is simply given by the convolution: (28)

M(t)v = R(.,t) ; v, with R(x,t)=^(signt)8(r 2 -1 2 ), t * 0 .

In the general case, results differ essentially with the evenness of space: when n is even the support of <& is the whole future cone, whereas when n is odd the support of O is the surface of this cone only. There is also a unique elementary solution 4> _ (., t) with support in the past cone or backward light cone, given by: <&_(. ,t) = - Y ( - t ) R ( . , t ) , and thus R(.,t) = 0 + (.,t) -<!>_(., t) is a solution of the homogeneous wave equation. When Q is the complement of a bounded domain £1\ we obtain as a consequence of the properties of 4>: Proposition 2 (Finite velocity of propagation). Let R > 0 be such that Q' is contained in B R . If the initial conditions (u°, u 1 ) are zero outside B R , the solution uof: i ) ~ ^ - A u =0 (29)

inQxR^,

ii)u(0) = u°, IfCO)-!! 1 , iii)u

lrxR+ = 0

is zero outside the ball B

(or

(uV)€W£(0)xL2(Q),

SHlrxR+

= 0)

. t ,. Furthermore if the initial conditions (u°, u 1 ) are

zero inside the ball BR(xo) centered at xQ then the solution u of (29) is zero inside the ball BR _ ct(xQ) when t < R/c. Then we have the property of local propagation of energy: Let E(u,t,p) be the energy of the wave u at time t in a ball B (B being arbitrary):

258

(30)

6 EVOLUTION PROBLEMS E ( u , t , p ) = i / n n B (li|*<x,t)| 2 + | gradu(x,t)|2)dx,

then we have (31)

E(u,T+ t,p) £ E(u,T,p + ct) for all T € R and t > 0.

1.4. Radon transforms The Radon transfonn method is a basic tool to obtain many properties in scattering and wave evolution. We briefly give the main definitions and properties, and refer to Gelfand-Graev-Vilenkin [1] and to Helgason [1] for more developments. First using regular functions we have: Definition 1. The Radon transform of a function f in 5(Rn) is the function on RxSn~l defined by; (32)

10l8 9 a).7(s 9 a)./ U a s f(i)dS 9

s€R, a € S n - \

dx being the surface element ofthe plane x. a = s, with ds dx = dx. Note that we can take a in Rn\{0} instead of S11"1. Then (32) would give a function satisfying the homogeneous condition: (33)

f(Xs,Xa)=|X|"1f(s,a),

V X * 0 , (s,a)€RxR?.

We first have the basic property: (34)

/ R e-^f(s,a)ds=/ R e-^ds^^^

where f is the Fourier transform off. Thus if we define (like in chap. 3, (396) (400)): (35)

F(r,a) = f(ro), r ;> 0, and F(r,a) = f((-rX-«))» r * 0 , V a € S n " \

F(r, a) is a function, defined on R x Sn"l, which is in space S0(R x S n " ) when f is in S(Rn) with the notation 1 of chap. 3.11.3. Then we define the following space: (36)

S~(R x Sn"l) is the space of inverse Fourier transforms of 50(R x S n " l ).

Proposition 3. The Radon transformation is an isomorphism from 5(Rn) onto S^RxS11"1); the Fourier transform of f and the Radon transform of f are related to each other through the Fourier transform (34) on s. The elements of S~(R x S n ~ l ) are characterized by |i)g(s,a) = g(-s, -a),

V(s,a)€RxS n - 1 ,

H) J R g(s,«) s* ds = Pk(a) a spherical harmonic of order k, k € N.

259

6.1 CAUCHY PROBLEMS

We note the properties of the Radon transformation with respect to translations, derivations and convolution: i) fa(s,a) = f(s + a.a,o) (38)

with fa(x) = f(x + a),

ii)(g|) ( s ^ c c ^ ^ a ) , andthus: (AfT(s,<x) = y ( s , a ) , iii) h = f * g with f and g in5(Rn) implies: h(s,oc)=/R f(s\a) g(s - s\a) ds\

The inverse Radon transformation is obtained using the inverse Fourier transformation: (39)

i W - ^ / ^ ^ i M - ^ / ^ ^ i ^ ^ ^ d n f c u

Then g(r,cc) = e1^" f(ra) has a natural extension to R x Sn~l by (36). Thus we have tw< cases according to the evenness of n: i) n is odd: (40)

W - 5 ^ / . ^ , e t a a F(r,«)r»-'drd«.

Then by the inverse Fourier transformation with respect to r, with m «(n - l)/2,

<4,)

» - ^ r V i $ S * - . I * " ^ ' - A > ° V- *"• •»""■

ii)niseve/i: (42)

fi(x) = 2(^jH/ RxS n.l e ^ ftr,«)(sign r)r^drda.

Then thanks to the Fourier transform of finite parts (see Schwartz [1] chap. 7.7): (42)' FK-lf^-^Pfs-^Msignr)^-1, we obtain

lA-n

«™ ( - l A n - l ) !

(43)

*).

/ R j t i B . 1 P f ^ — - ^ dsd«.

-JM..

We can define the Radon transformation on tempered distributions, either by using a Fourier transformation, or by duality. Let g be in 5(R x Sn~ *), and: (44)

g(x)=/ n -i g(x-a>«) da-

Then g —♦ g is a continuous mapping from 5(R x Sn~ ) onto S(Rn) so that (44)'

<f,g> = <f,g>,

Vf€5'(R n ).

260

6 EVOLUTION PROBLEMS

That this mapping is onto is a consequence of inverse formulae (41) and (43). Then / R nf(x)g(x)dx=/ s n . 1 da/ R ds/ a X a s s f(x)g(s,a)dx=/ R x s I l - _ 1 f(s,a)g(s,a)dsda.

Thus (44)' defines the Radon transformation as a continuous mapping from SWimoSXRxS 1 1 - 1 ). Another basic property of the Radon transformation (and its advantage with respect to Fourier transformation) is that it preserves the boundness ofsupports. If suppf is bounded then it is obvious that supp jRf is also bounded. For the converse we refer to Helgason [1]. We also have the obvious property for odd n (45)

f(s,a) = 0for|s| s a , a > 0 , VacS"" 1 , impliesfi(x) = 0for |x| <a.

Furthermore we have a "Plancherel" formula for Radon transformation; applying the Plancherel formula twice to Fourier transforms leads to: i) when n is odd (and m = (n - i)/2): (46)

/Rnf(x)g(x)dx=^T/RxSn.1p(s,«).^(s,a)dsd«.

ii)whenniseven; (-l) n/2 (n-l)! (47)fRllf(x)g(x)dx=sV" ' ( 2 | y '* J ^ ^ u i Pf «Pi.«>g<h.«><»l - s ^ d s ^ d a . When n is odd, we define the following space: (48) L^(RxSn - l ) = {v€L2(RxSn"1), v(-s, -a)«(-l)Fv(s, a), withm = ( n - l)/2}. Then from the "Plancherel" formula we have: Proposition 4. The mapping: f € S(Rn) —* -=ps *<** a continuous unitary (up to a constant factor) extension from L2(Rn) onto the Hilbert space L^(R x Sn"!). Definition 2. The unitary (up to a constant) extension of Proposition 4 (with n odd) is denoted by Rd and is called the "differentiated Radon transformation". Thus it satisfies, with R f = |~M (49)

/R„J«x)|2dx=^^

6.1 CAUCHYPROBLEMS

261

Given two functions f0 and fj in L2(Rn), we define the function £ on R x S n ~' by: (50)

n(r,a)-?4(R) ifr>0,

h(r,a) = f ,((-rX-a)) ifr<0.

Let h = h(s, a) be its inverse Fourier transform (with respect to r). Then we note (51)

h=« 2 (f 0 ,f,).

Proposition 5.

3mh

The mapping: (f^fj) —»g-gr & * unitary (up to a constant factor)

mappingfrom L2(Rn)xL2(Rn) onto L2(R x Sn~'), with n odd, m - (n - 1 )/2: (52)

/ R n[|f 0 (x)| 2 + l f l( x )| 2 l d x =7T7rT/ R x S n-l I 0 ( s , « ) | 2 d s d « .

Definition 3. The unitary (up to a constant factor) mapping of Proposition 5 is denoted by JLd and is called a "two-sided differentiated" Radon transformation. We see that R^ and R^ are natural extensions of R? and R since /?2d(f,f) = and RJS,t)=Rf. Furthermore we have the following basic properties:

(53)

*2«o.fi>-|i*2Mi>.

*2((-A>1/2f' -<-A) l / 2 f)—it£.

/#(-A) 1/2 f 0 , - (-A)1/2)f,)= - i ^ ^ f , ) —

i^r«2(f0,f,)-

Definition 4. We denote by Rw the mapping (called "wave Radon transformation") definedfor (f,g) in Wl(Rn) x L2(Rn), with n odd, by (54)

*w(f,g) = / # - ^ ^ ^

Proposition 6. The mapping Rw is a unitary (up to a constant factor) transformation from W'(Rn) x L2(Rn) onto L2(R x S n l ):

(54)

' &&'***-* i*w««-«-

and satisfies, with A given by (4), and A = - A: (55)

Rw(A(f,g))^Rw((,g).

^^'KhK^Kw

262

6 EVOLUTION PROBLEMS

This Proposition is fairly easy to prove. Here we give its inverse only. Let h be in L2(Rx S n ~'); we have to find f and g so that h = rtw(f,g), that is with (54): <56>

" ^ 5 5 T + ^ w = h.

Using the different behaviors of each term with respect to symmetry, we have: ^ ( s , o ) = | ( h ( s , a) + (-l) m h(-s,-a)), ^ J | ( s , a ) = - ^(h(s,a) + (-l) m+1 h(-s,-a)). Then applying the inversion formula (41), we obtain: (57) «<x> = ^

/ s n.l^S(x.«,«)d«, «x) = i ^

F r

/ s n _ 1 ! s = T -(x.a,«)d«.

1.5. First applications of the Radon transform method Now the (usual) Radon transform of the wave problem (11) with c = 1, f = 0, and with (38), is the one dimensional wave problem: i)^|(s,a,t)-0(s,a,t) =O

inRsxSn-1xRt,

(58) ii) u(s,a,0) = u °(s,a), H (s,a,0) = u ! (s,a). at

Its solution is given by the d'Alembert formula: (59)

u(s,cc,t)=| (u °(s + t,a) + u ° ( s - t,a)) + ^f£

u V , « ) <k\

and we obtain the solution u of (11) with f=0 applying the inverse Radon transformation (41) or (44). Now let x,

a +a ,

be the characteristic function of the

interval [-a,+a] when a>0. When a<0, we define Xi_a+ai = "~Xr+a _ a i- Then the operator M(t) (see (9)) is easily obtained thanks to the convolution: (60)

M(t)v(x) = R ( . , t ) ) ? v = | ^ 1 <X|_ti+t,. v),

thus (using (38)): (61)

R(.,t)=±R-l(X[_tM]).

Thus to obtain the elementary solution O with support in the future light cone which is related to R by 0(.,t) = Y(t)R(.,t) (see (26)), we have to calculate the inverse Radon transform of Xi_t+ti thanks to (41) or (44). We obtain the "Herglotz Petrovski" formulas for these elementary solutions:

263

6.1 CAUCHY PROBLEMS 0(x, t) = Y(t) (- l) m+1 — ^ / (2it) "

n -!

6(n ~2)(x.« - t) da, with odd n, m = (n - l)/2

(62) (x,t) = Y ( t ) ( - l ) n / 2 y ^ J J / s n - 1 Pf

withevenn.

We can obtain R(t) when n is odd (at least formally): we have 6 t W=/ s n.l «*•« - ^ d a - a ^ j / t 8(|x|s - t)(l - s2)"1"1 ds,

(63)

withx = inf(t/|x|, 1), o n - 1 = |S n ~ 2 | the area of the unit spheres11"2. Let: J(x)= x1 ( 1 - l ) " 1 - ^ | x | > l , I l |x| Then we have (64) hence: (65)

J(x) = 0 i f | x | < l .

*tW = o n . 1 | j ( f ) ; R(t) = - c n ( - A ) m - 1 1 | J(f), with c n = - on.1/2(27i)n-1.

We can also obtain usual formulas of the elementary solution (see Lax-Phillips [1] p. 125, Methee [1], Zieman [1]). Thus when n is odd, the support of this elementary solution is the surface of the future light cone. This implies: Proposition 7. Huyghens principle (with a velocity c). For odd dimension n of space (n > 1), if the supports of the initial data are contained in the ball B R , then the support of the solution u of( 11) with f=0w such that: supp u(t) = {x € Rn, 3 e € Sn~l, x + cte € supp u 0 U supp ux] Q KR t , (66) | K R t = {x€R n ,c|t| - R < | x | < c | t | + R } if |t| >R/c, K R t = {x€Rn, | x | < c | t | + R } i f | t | < R / c . PROOF.

Thanks to (5)' and (60) the solution u of (11) (with c = 1) is given by:

(67)

u(x,t) = | R(t)x* u° + R(t)x* u1,

and the property: supp (R(t) * v) £ supp R(t) + supp v, implies (66). We can verify (66) on the Radon transform of the solution using again (45).

Furthermore we can prove that (for an odd dimension of space) when f = 0 and when the initial conditions are regular, the solution u of (11) and all its derivatives converge to 0 at each point x when t goes to infinity (see for instance Goldstein [1] for generalization to hyperbolic operators). Although the energy of the wave is globally constant in time, it decreases locally with time.

264

6 EVOLUTION PROBLEMS

2 . SCATTERING PROBLEMS - INCOMING AND OUTGOING WAVES

We now study problems which are relevant to scattering, following Lax-Phillips theory. We assume that space dimension is odd. 2.1. Another application of the Radon transformation We consider the Cauchy problem: find u satisfying (68)

.•^u i)^-j±-Au = 0

inlTxRf,

ii) u(0) = u°, §f (0)=u 1 , with (u°, u1) given in H, and we apply the wave Radon transformation Rw. Then with (6) let: (69) k - * W ) ,

k(.,t)=/?w(u(t),v(t)) = ^(w(t),2(t)), withv(t)=^(t).

Using (54) we have: k s at

(70)

< - ' > - - f ^ T <«•«•«)+iim ft.0*1).

Thanks to (7) and (S3), (or directly with (55) on (1)), (68) is transformed into:

i)fjf(s,«,t) = -f k -(s,a,t), (71) ii)§£(s,a,0) = k(s,a), in L2(R x S n l ) . The solution is given by: (72)

k(s,a,t) = k(s-t,a).

Applying the inverse formulas (57), we obtain the solution of (68) by

(73) m

(-D | v ( x , t )9=uf, („ x. , , t ) = ^ j r r / rs n . 1 ^ ^( xk ., a - t , a ) d « ,

and thus for n = 3: (74) u(x,t) = ^ 2 / s 2 «*.« - t,a) da,

| f (x,t) = - ±f$2

g (x.a - t,a) da.

Furthermore the conservation of energy is given by (10) and (35):

6.2 SCA TTERING PROBLEMS

(75)

265

W(t)=A [ KOH2 + IK-A)1 /2u(t)B2 ] = —"THTT / ^ n - 1 |k(s,a)| 2 dsda. 4(2it)

Kx^

Thus R is a unitary transformation, which transforms the evolution group (G(t)) into translations; J?w(u0,u*) is called the translation representation of (u0,u*) by Lax-Phillips [1]. Definition 5. Outgoing waves, Incoming waves (with the velocity c = I) . A wave u is called outgoing (resp. incoming) if it satisfies (76)

u(x,t) = 0 when |x| < t, i.e., u is zero on the forward light cone C+ (resp. u(x,t) = 0 when |x| > t, i.e., u is zero on the backward light cone C_ ).

The corresponding initial conditions (n°f\xl) are called outgoing (resp. incoming), and we denote by D + (resp. D")the set of these initial conditions. Definition 6. p-outgoing waves, p-incoming waves. Let p be a real number. A wave u which satisfies (77) u(x,t) = 0, V(x,t)with|x| <t + p(resp. with |x| < p - t ) is called a p-outgoing (resp. p-incoming) wave; the corresponding initial conditions are called p-outgoing (resp. p-incoming). Hie set of these initial conditions is denoted by D* (resp. D~). Thus D$ = D + , DQ = D". A wave u is called eventually outgoing (resp. eventually incoming) if there is a number p such that u is p-outgoing (resp. p-incoming) and the corresponding initial conditions are called eventually outgoing (resp. eventually incoming). Note that changing t into - t changes outgoing waves into incoming waves and conversely. We can prove the equivalences (see Lax-Phillips [1] Thm. 2.3 chap. 4.2) with the translation representation: (78)

(u^u1) € D + (resp.D~) ** k = /*w(u°,ul) satisfies k(s,a) = 0 if s < 0 (resp. s > 0), (u^u1) € D* (resp. D~) «■♦ ksatisfies k(s,a) = 0 if s - p).

Proposition 8. i) The spaces D*, D " are orthogonal supplementary spaces in H. ii) The outgoing waves (resp. incoming) satisfy equipartition of energy for all t > 0 (resp. t < 0). iii) The evolution of outgoing waves for t > 0 is given by a semigroup (g+(t)) in WJ(Rn), and if (u^u 1 ) is an outgoing initial condition, then we have (79)

/?du1(s,a) = -(signs)^i? d u 0 (s,a),

or u ! ( s , a ) - - (signs)^u°(s,a),

with a similar propertyfor incoming waves (change - into + in (79)).

266

6 EVOLUTION PROBLEMS

PROOF, i) is a straightforward consequence of the above equivalence (78). ii) When (u0,u*) is in D + , we have

(80)

(u°, - u1) € D* and for all t > 0 (u(t),u'(t)) € D* and (u(t), -u'(t)) € D".

Since D+ and D~ are orthogonal spaces we have: (81)

3/ R n|gradu(x,t)| 2 dx-A/ R n|g(x,t)| 2 dx = 0.

iii) From (70) (or from (56) with h = k(.,t), f = u(.,t), g = v(.,t)) we see that k(s,a,t) = 0 when t<0 implies: (82)

^%,a,t)=^(s,a,t) s -(signs)u ( m + 1 ) (s,a,t),

that is (79) when t s 0. Then if we consider L2(R+,L2(Sn~ *)) as a subspace of L2(R,L2(Sn~ *)), the translation group in L2(R,L2(Sn~*)) given by (72) induces on L2(R+,L2(Sn" *)) an isometric semigroup for t > 0 only, given by: (83)

k(s,oc,t) = k(s - t,<x) if s > t, k(s,<x,t) = 0 if s < t,

whose infinitesimal generator is (84)

Ao = - ^ ,

D(A0) = Hi(R+,L2(Sn-1)).

The translation semigoup gives, in Lm(R x Sn~ ) (see (48)), a semigroup (g(t)): |g(t)v(s,a) = v j s + t,a) + v + (s-t,a)if|s|>t, g(t)v(s,a)*0if | s | <t, |withv.=:v| R ., v+ = v| R+ , with infinitesimal generator A = - (sign s) g- . Then thanks to the inverse Radon transformation (J?d)~l, we obtain the evolution semigroup (g+(t)) (on u°) in W*(Rn). Thus we have proved that D + and D~ are graphs of continuous mappings C + and C " from W*(Rn) into L2(Rn). Definition 7. The orthogonal projectors P+and?" in H » W*(Rn) x L2(Rn) on D + and D~ are respectively called the outgoing and the incoming Calderon projectors forfree waves. The operators C* and C ~ are called the outgoing and the incoming Calderon operators forfree waves, and they are defined also (thanks to Rd) by: (86) C+ = - (*V[(signs)^]tf d ,

C- = ( * V [ ( s i g n s ) ^ ] * d .

We can identify C with the infinitesimal generator A (thanks to R ).

6.2 SCA TTERING PROBLEMS

267

These Calderon operators are unitary operators from W^R") onto L2(Rn) (this follows from equipartition of energy) with: Re (Cu,u) = 0, u in W^R"), C = C* or C ~. Proposition 9. The problem:findan outgoing (resp. incoming) wave u with finite energy satisfying (87)

i)5-^-Au = 0 in R n xR^(resp.R n xR t -) ) ii) u(0) = u°, with the only data u° in w'(R n ),

has a unique solution, and thus is a well-posed Cauchy problem. Now going back to the p-outgoing or p-incoming waves we have the properties: (88)

DjcD;, and Dp-.cDp" ifp'<p.

Thus (D*)

_ is a decreasing family of spaces, whereas (D~)

Pp€R

„ is increasing.

" p t K

Spaces D* and D~p are orthogonal supplementary spaces: H = D* ®D"p. Moreover p-outgoing (resp. p-incoming) waves satisfy equipartition of energy from time t = - p (resp. p). The evolution of p-outgoing waves is given from this time thanks to an isometric semigroup (g+ (t)), t > - p, by: g+p(t) = g+(t + p). We have similar results for p-incoming waves but with p > t. The main properties of these spaces for scattering (which are very easy to verify thanks to the wave Radon transformation) are: DGWDJCD;

(89)

ift>o, G(t)D;cD; ift<o,

ii) OG(t)D!«0, p t>0

HG(t)D-=0, p

t<0

iii) u a t ) DpS U G(t)D;=if, p teR

t€R

that is for iii): the set of eventually outgoing (resp. eventually incoming) initial conditions is dense in H. 2.2. Incoming and outgoing waves in electromagnetism Now we consider the Maxwell Cauchy problem (16) in free space, without charges nor currents:find(E,B) satisfying i ) - ^ | ^ + curlB = 0, | ^ + curlE = 0 i n R ^ R ^ c (100) ii)divE = 0, divB = 0, iii) E(.,0) = E0, B(.,0) = B0, (with free divergence) and E0, B0 in L2(R3)3.

6 EVOLUTION PROBLEMS

268

We write the Radon transform of this problem. First we easily have: (101) *<curlvXs,«) = §(aAvXs,a), *(divvXs,«)=^(a.vXs,«), (s,a)eRxS 2 . Thus (100) becomes (with c = 1): i)-^+^(«AB)=0, (100)'

ii)a.E = 0, iii)E(.,0) = E0,

^+^(«AE)=0

inR s xS 2 xR t ,

o.B=0, B(.,0) = B0.

Let: (102) n(.,t) = -E(.,t) + oAB(.,t). Thus n satisfies: (103)

i)gr=-^-. with <x.n = 0, ii)n(.,0)=n0 withgivenn0 = - E 0 + aAB0.

Note that for c * l , then n = -^E+aABsatisfies: gjf^-eg^Here it is better to use the transformation Rd instead of R; with the notation v = R\ I 0 and B0 are in L2(R x S2) (see (48)), thus n0 is in L2(R x S2) but not in Lf(R x S2): (104)

n0(-s, -a) m - E0(-s, -a) + (-a) A B 0 (-S, -a) = (E0(s,a) + a A B0(s,a)).

The evolution of n is given by a group of translations; if we define the outgoing electromagnetic waves as previously by n(s,a) = 0 when s < 0, we obtain that an electromagnetic wave is outgoing if it satisfies the relation: (105)

l(s,a,t) = - (sign s) a A B(s,a,t), for t > 0, (s,a) € Rx S2,

and thus the initial conditions must satisfy: (106)

E0(s,a) = -(signs)aAB0(s,a),

(s,a)€RxS 2 ,

which is equivalent (with (101)ii) to: (106)*

B0(s,a) = (sign s) a A E0(s,a).

Of course incoming electromagnetic waves will correspond to initial conditions with + instead of - in (106). Thus if (E,B) is an outgoing wave, then (E, - B) is an incoming wave for - 1 . Hence using orthogonality of these waves, we have: Proposition 10* Proposition 8 is valid in the electromagnetic case, by replacing only condition (79) by (106), concerning initial conditions of outgoing waves.

269

6.2 SCA TTERING PROBLEMS

We can also define outgoing and incoming Calderon projectors P+ and P~ in the space H of free divergence fields of L2(R3)3 x L2(R3)3 on the spaces D+ and D~" of outgoing and incoming initial conditions, then outgoing and incoming Calderon operators C+ and C " in space of free divergence fields of L2(R3)3) by: B0 = C+E0 given by (106)' (for C+). Furthermore C+ and C~ are unitary operators, with: C 2 = - I (C2= - c~2I whenc*l) and Re(CE0,E0) = 0, C = C + andC~. The Calderon projectors P+ and P~" are also given by:

(107)

P+ = I(I-S),

P-=1(I + S), with S=£ _° c+ ^ + J.

Like for scalar waves, we have: Proposition 11. The Cauchy problem: find an outgoing (resp. incoming) electromagnetic field (E,B) in R 3 xR + (resp. R 3 x R " ) with finite energy, satisfying (100)i),ii) and only the initial condition: (100)iii)'

E(.,0) = E0 with divE0 = 0,

E0 € L2(R3)3,

or with given initial condition on B only, has a unique solution (E,B) in C(R+, H) (resp.C(R~, H)), and thus it is a well posed problem. 2.3. Incoming and outgoing waves with a bounded obstacle; Lax-Phillips theory Let Q be the open complement of a bounded obstacle Cl\ and let B be a ball so that CT is contained in it. Let (G (t)) be the evolution group of waves in H = W!o(Q) x L2(Q) with a Dirichlet boundary condition. We first note that if u is a wave in Rn x R + which satisfies (77), that is u is zero in the truncated forward (backward) cone C + (resp.C ~), then u(t) satisfies the Dirichlet boundary condition when t > p (resp. t < - p). From this trivial remark we can consider D + (resp. D ~) as a p-outgoing (resp. p-incoming) space for the mixed problem. Then we apply the evolution group (Gfl(t)) to these spaces. We can prove (see Lax-Phillips[l]) that the essential properties (89) are also valid when substituting GQ(t) and HQ for G(t) and H. The only difficulty is (89)iii) which is proved to be equivalent to: (108) lim E(u,t,K) = 0, oronly liminf E(u,t,K) = 0, V K compact set, K c Q . Now let F = (f°,fl) € U G(t)D*. Then there is T> 0 so that F e G(-T)D£, and thus there is F + € HQ so that Gft(t)F+ = G(t)F, V t > T. This defines a mapping W+: F —* F+ which can be extended continuously from H into HQ. We define a mapping W _ in a similar way.

270

6 EVOLUTION PROBLEMS

Definition 8. Wave operators. The mappings W+, W_ from H into Ha defined by: (109) W ± F =

lim GQ(-t)G<t)F, VFinff.i.e., W ± = lim GQ(-t)JG(t),

wftJt J r/ie natural imbedding of HQ into H, are called the outgoing and the incoming wave operators. We can also define wave operators from L2(Rn) into L2(Q) using the groups generated by the operators iA1/2, A being the Laplacian in Rn or in Q, like in Wilcox [1]. Now these wave operators have the essential properties: Proposition 12. The wave operators are unitary operators from H onto Ha, with (110)

G a (t)W ± =W ± G(t),

Vt€R.

From this we can define the "scattering operator** SinH by: (111)

S = W;*W..

Then from Proposition 12, S is a unitary operator in /f, such that SG(t) = G(t)S, for all real t. These operators are a basic tool in scattering problems by a soft bounded obstacle. Obviously we can generalize to other boundary conditions, for example to hard obstacles with a Neumann condition. That the scattering operator S contains all the scattering information is seen from solving the inverse scattering problem (see Lax-Phillips [1] p. 173): Let Ox and 02 be two bounded soft obstacles; let Sf and S2 be the associated scattering operators in H. Then S% » S2 implies Ox = O . Furthermore we can easily deduce spectral properties from the wave operators on the selfadjoint operator A = - A with Dirichlet boundary condition: A has an absolutely continuous spectrum with no eigenvalue. Here we are essentially interested in outgoing and incoming waves. Thanks to the wave Radon transform Rw we define mappings R* and R"" (called outgoing and incoming representations) by:

(112) __

tf-rt^w;1,

R'^R^Z1* ^

n i

They transform the evolution group in Ha into translations in L (R,L (S )), with: R+(Dp = L\p,+<*)X2(Sn-1)), ir(D;) = L2(( -oo, . p)9L2(Sn-{)). We can also define "outgoing and incoming initial conditions" DQ+ and DQ*" as the inverse images of L2((0,oo),L2(Sn"!)) and L2((-co,0)fLHsn~l)l respectively by R+ and R~, or DQ+ « W+D* and DQ~ = W^D", and the Calderon oroiectors P + . P ~ on these soaces bv:

271

6.2 SCATTERING PROBLEMS

/ J\ ~ 1 P ^ - W ^ C W ^ ) """1 ,1 oa«n^dDP Q- -_=WW _Dp~-/ («W but DQ+ and DQ " are not orthogonal, nor necessarily graphs of operators ! Then the problem: find an outgoing wave u with finite energy satisfying:

i ) ^ 4 - A u = 0 inQxR*, (113)

at2

ii)u| r x R + = 0, iii) u(.,0) m u° with u° given in wl(Q), and supp u° c B' = R M L

has a unique solution u in C(R£, W^Q)), with ^ in C(R^,L2(Q)). Thus it is a well-posed problem (as a trivial consequence of Proposition 9), but we cannot eliminate the condition on the support of u°! Application to stationary waves We can also define notions of outgoing or incoming waves in the stationary case in the following way. First we consider the stationary problem in a form similar to (1) in free space, in the whole space: (114)

4W + XW = F inR n ,

where A is given by (4) with A= - A, with given X in C and F = (f1,f2) given distributions in Rn with bounded supports. Using Definition 6, we have: Definition 9. A solution of (114) which is also an eventually outgoing (resp. incoming) initial condition for the evolution problem (68) is called a X-outgoing (resp. X-incoming) stationary wave. When F = (0,g), then W = (w,Xw), with w satisfying the Helmholtz equation: x\r - Aw = g, and w is called a X-outgoing (resp. X-incoming) stationary wave when W is X-outgoing (resp. X-incoming). Note that a X-outgoing stationary wave w is a ( - X)-incoming stationary wave also, and conversely. We can easily obtain results on stationary wave thanks to the wave Radon transformation Rw. We only give one example of application. Applying transformation Rw to (114) (this is possible up to now if F is in H, but we can generalize Rw to distributions, see Lax-Phillips) gives (115)

g + Xk = f inRjjeS11"1,

with k = /*wW and f=/*wF, f having a compact support. The solution with support bounded to the left (resp. to the right) which will give the outgoing (resp. incoming) solution of (114) by its inverse transform, is (116) k 0Ut (s,«)=/ S co e A(s, - s) fl[s',a)ds', kta(s,«) = - r e A ( s ' - s ) f ( s ' , « ) d s \

272

6 EVOLUTION PROBLEMS

WhenFisinff, thenfisinL^RxS 11 ' 1 ), and we see that if (Re X)>0, onlykout is in L2(R xS""1) whereas if (Re X) < 0, only k^ is in L2(R xS n " ! ), the other solution being not even tempered. (When X is real, both solutions are tempered, butnotinL^RxS 11 " 1 )) Then applying this to F = (0,-8) gives the elementary X-outgoing (or Xincoming) elementary solution of the Helmholtz equation * r When n = 3 we have the very simple result: (117)

*x(x)ss^S^withr=|x|.

When F = (0, -f)> f being a distribution with bounded support, the X-outgoing (resp. X-incoming) stationary wave w is obtained by convolution of f with $ x , and then we can verify that Definition 9 agrees with Definition 1 chap. 3 thanks to Theorem 1 chap. 3. Now we consider a domain Q exterior to a bounded soft obstacle. Then we can define X-outgoing (resp. X-incoming) stationary waves as in Definition 9. Here we have to use a more general framework taking waves into account which are not offiniteenergy, thus not in HQ = W!0(Q) x L2(Q). So we define: (118)

#loc(Q) = {U = (u°,ul), such that CU€ Ha,

VC€D(Rn)}.

Waves which are not of a finite energy may be of two very different types according to different situations: i) scattering states; they appear in usual stationary scattering problems as Xoutgoing (or X-incoming) stationary waves. They are generalized eigenvectors, w+(x,£) (resp. w_(x,t)) with £ in Rn, i.e., satisfying: i)(A + S2)w±(x,S)=0 inQ, (119)

ii)w ± (.,t)€H* loc (5), i.e., Cw±(.,t)€Hi(G), VC€2>(Rn), iii) wr+(x,£)=w+(x,S) - w0(x,£) is an i| 11 -outgoing wave, (resp. wr_(x,£) = wJx,{) - w0(x,S) is an i |S|-incoming wave),

with w0(x,6) as e a \ Recall that iii) means that wr ± (.,x) satisfies the outgoing (resp. incoming) Sommeifeld condition. This problem (119) has a unique solution thanks to the Rellich lemma (Lemma 1 chap. 3), wr being the usual reflected wave by the obstacle for the incident wave wQ. Then if we define for all fin L2(Q) with bounded support: (120) f+«)=/Qf(x)w+(x,?)dx,

resp.fJO=/Qf(x)wJx,«)dx,

the mapping f —> f+ (resp. f J has a continuous extension r (resp. F") from L (Q)

6.2 SCATTERING PROBLEMS

273

onto L2(Rn). Furthermore F* and F~ are unitary (up to a constant), and are natural generalizations of the Fourier transformation which diagonalize the Laplacian AD with Dirichlet conditions. They are called the incoming (resp. outgoing) spectral representation oftP. The operators F~ lF* and F~ lF~ with F the Fourier transformation, correspond to (inverse) wave operators W_ and W+ in L2, see Wilcox [1]. Note that w+(x,£), w_(x,£) are tempered distributions. ii) resonance states. Let v/bea non trivial solution with locallyfiniteenergy of: |i)-Aw + X2w=0 inO, withX€C, ii)w| r = 0, thusw€Hi loc (Q), (121)

iii) w is a \-outgoing stationary wave, i. e. w has the asymptotic behavior w(x)=^j- 9(a) + CHr-2")forr = | x | -♦ oo, 9(a) a regularfunction of a = x/r.

Definition 10. A wave w satisfying (121) is called a resonance state and \ a resonance. (We emphasize that this terminology is not universal: sometimes, see Sanchez Palencia [1], resonances are called scattering states.) From Rellich lemma, a resonance cannot be real. Furthermore it must satisfy Re \ < 0 (see Lax-Phillips [1] p. 161). Thus a resonance state cannot be tempered; its physical nature is quite different from that of a scattering state. There is at most a countable set of resonances, as a consequence of the Rellich compactness result: Proposition 13.

The unit ball of D(A) in Hft, {F € HQ, \\A Fll + IIFII <l}isa

precompact set in Hloc(Q) equipped with the family ofsemi-norms for all compact setKinRn (122)

IIUIIK = ll(u 1 ,u 2 )ll K = s [|/ K n Q [|gradu 1 | 2 +|u 2 | 2 ]dx] 1 / 2 .

A complementary result on resonance states is the following (Lax-Phillips [1]) Proposition 14. The problem:findwwith locallyfiniteenergy satisfying i) X2 w - Aw = f, with f € L (Cl) with compact support, (123) ii)w| r = 0, thusw€H£ loc (Q), iii) w (or W = (w,Xw)) is a \-outgoing stationary wave, has a unique solution when \ is not a resonance. In order to study resonance we define the following spaces and projectors: (124) D P = D * © D ; , Kp the orthogonal of Dp in HQ9 thus# a = D <BK

274

6 EVOLUTION PROBLEMS

Q* « I - P* (resp. Q" = I - PJ), the orthogonal projector in Ha on the orthogonal space to D* (resp. D~), and Q = Q+Q" = I - P p , the orthogonal projector on Kp. A basic tool to study resonance is the family: (125)

Zp(t) = QjG n (t)QJ,

t£0.

We can prove that (Zp(t)) has the properties: i)D p c kerZp(t), V t > 0 , Zp(t)KpcKp, V t * 0 , ii) (Zp(t)) is a contraction semigroup of class C° in K p , with: Zp(0) = Q , and Zp(t)U0-> 0 when t —+oo. Let Bp be its infinitesimal generator in Kp. Then the main point is: Proposition 15 (Lax-Phillips). The operator Zp(2p) (XI - Bp)~1 is compact in K p , forall\>0. This implies that the spectrum of Bp is a sequence of eigenvalues in half plane Re X < 0, with finite multiplicity without accumulation point at finite distance, and that the set of eigenvalues ofBP is the set ofresonances. Moreover if W*. is an eigenvector of Bp for the eigenvalue X. then we obtain a resonance state W. by: W.(x)»lim W* (x) when p -♦ oo, xfixedin Cl. Then resonances are useful to study evolution of waves when t goes to infinity: Proposition 16. Limiting Amplitude Principle, Let g be in L2(Q) with bounded support, and X a complex number which is not a resonance. Then the solution u with finite energy of the problem: i ) ^ | - A u = eAtg inQxRt, (126)

ii)u(.,t)| r = 0,

Vt€R,

iii)u(.,0) = u°,

^ ( . , 0 ) = u!,

with given (\x°tul) in W*(Q) x L2(Q), satisfies: (127) U(t) = (u(t), §J (t)) converge in Hloc(Q) to V(t) = (eAtv, XeAtv) when t -> +oo, where v is the \-outgoing stationary wave solution of: (128)

X ^ - A v s g inQ.

275

6.2 SCA TTERING PROBLEMS

Another result is the local decrease with time of wave with respect to energy. The first result is due to Morawetz [1]: Proposition 17. Let Q' be a bounded star-shaped domain. Then the solution u withfiniteenergy of: i ) ^ - A u = 0 inQxRj, (129) ii)u(.,t)| r = 0, V t â&#x201A;Ź R ,

Cl^R^Q',

I iii) u(.,0) = uÂ°, | * (.,0) = u1, with given U0 = (u^u1) in wi(Q)xL2(Q), satisfies (with C a constant and with t > 2p): (130) E(u I t,B p nQ) = i / B

2 na[|gradu(t)| +|l?

W h d x * jf

IIUQII^.

When the obstacle is star-shaped, and even when there is no captive ray (for this notion, see for instance Bardos [1]), the energy of the wave is exponentially decreasing on every compact set when t goes to inflnity (for an odd dimension of space). Moreover we have (Lax-Phillips [2]): Proposition 17*. Let Cly be a bounded star-shaped domain. Then for every LL given in HQ with bounded support, there are resonances (\X j = 1,...,N, with associated resonance states w., constant numbers C, j = i,...,N, so that the solution u of (129) satisfies for any compact set K: (131)

E((u(t)-

2 UltoN

QJ e J w.\uK)*C(K)e J

N+1

' .

Thus resonance states allow us to have an asymptotic behavior of the wave when t goes to inflnity. This formula would be generalized to more general domains. But we know (see Ralston [1]) that if there are arbitrarily long rays following optical geometry, contained in a ball of finite radius, energy may be arbitrarily slowly decreasing in compact sets. For a soft obstacle which has a cavity with a tiny hole (of diameter 6) resonances converge to the eigenvalues of the (closed) cavity when 8 goes to 0 (see Beale [1]). There are other similar examples with waves due to a drum in Sanchez Palencia [1]. Obviously all these results can be generalized, to hard obstacles with a Neumann condition, and to systems of equations, in elasticity and in electromagnetism. For Maxwell equations we can use potentials (scalar and vector potentials, and Debye potentials also) to obtain results thanks to those for the wave equation. We shall not develop these questions here, and we refer to Schmidt [1], Dassios [1], Bardos [1].

276

6 EVOLUTION PROBLEMS

3 . CAUSAL PROBLEMS

A great number of evolution problems in scalar waves and in electromagnetism have no initial condition but are of the type: Causal problem with a support condition. Given a distribution f in Rn x Rt with support bounded in the past: (132)

suppf c Q**{(x,t),x€R n ,t^a},

find a distribution u in Rn x Rt satisfying: (133)

|i)Ou=f inD,(RnxRt), ii)suppu c Q*,(orsuppu C supp f+C+with (23)), i.e., with support bounded in the past, with the same bound a.

Thanks to the elementary solution <& with support in the future light cone (see section 1.3) we have: Theorem 1. The causal problem (133) has a unique solution u, which is given by convolution: (134)

u=**,f. x,t

PROOF. We have to prove that the convolution product has a sense with the above hypotheses on the supports of f and $, that is their supports satisfy:

(135)

A n (K - B) is a bounded set, for all compact set K in Rn x Rt>

with A * supp f, B = C+ » supp <&. Let (x,t) be in A, (y,t) in C+, and (z,X) in K, with t = X-t,x = z-y.Wehavea^t, thustsX-a, and |y| ^ T S X - a , thus x and tare bounded, giving (135). Now if w is a solution of (133)i) with support bounded in the past only, then applying the convolution by $ to (133)i) gives: u=*»f=$,(Dw)=(D*)*w=i*w=w, which proves uniqueness of the solution with a weaker hypothesis on the support. ® Remark 1. Obviously we can transform a one-sided Cauchy problem (11) (for t > 0 only), into a causal problem, by usual extension by 0 for t < 0, so that it becomes: find U a distribution in R n xR t with support in Q* = RnxR+t satisfying: (136) DU - F + u18(t) + u° 8*(t),

277

6.3 CAUSAL PROBLEMS

F being the extension of f by 0 in the past. Equivalence between the two problems is realized if we assume usual regularity conditions on the data. Remark 2. For n = 3, the solution u of (133) is given at least formally by the "retarded potential": (137)

ufrO-^J^-jp^

We have solved the causal problem (133) by convolution. Under similar hypotheses we can apply a Laplace transformation method which gives also Theorem 1. For instance, let X be a Banach space of functions on Rn, let S'(RfX) be the space of tempered X-valued distributions. We define (138)

L+(R,,X) = {f € Z T ^ X ) , 3 E € R, e" * f € S>(Rt,X)}, D\(X) = {f € JD'(Rt>X) with suppt f bounded in the past}, L^(Rt,X) = L+(Rt,X)nDV(X)

Taking for instance fin L^CR^X) with X = L2(Rn), then we prove that (133) has a unique solution in the same space (see Dautray-Lions [1] chap. 16.1) which is obtained thanks to the resolvent of the operator A = - A. Furthermore the Laplace method allows us to define well-posed causal problems without condition on supports: we substitute for the condition of bounded support in the past a L2-condition with a weight in time, such as L2 (X) (also denoted by e^L^R, X)), with y > 0, defined by: (139)

L27(X) = {f: R ^ X measurable with Ilf1l2x=/Re"27t||fll^dt< oo}.

A typical example (which is relevant for wave equation and for Maxwell equations) is the following: let A be an operator in a Hilbert space X, which is the infinitesimal generator of a unitary group (G(t)) in X. Let f be a given function in L2 (X), with y > 0; find u in L (X) satisfying: (140)

| f + Au = f.

Proposition 19. The problem (140) has a unique solution u in L2 (X). PROOF.

The Laplace transform of a function 0 in L2 (X) being defined by:

(141)

$(p) = / R e-pt«t)dt,

the Laplace transform of (140) is: (142)

(pI + A)fi(p)=f(p).

= £ + in , t > Y,

278

6 EVOLUTION PROBLEMS

When U T > 0 , this equation has a unique solution, given by: (143)

u(p) = R(p)?(p) = (pI+A)-1?{p).

Thanks to the Stone theorem, we have: (144) IR(p)l*{, and thus: dn =/ = f_ DR(o)f(vriL dn** \4 HI** llf„ , , (145) /f_ Hufolll Ifi(p4dn R IR(p)f(p)lxdn which implies the Proposition. Furthermore if f has a bounded support in the past, we obtain that u also has this property, with the same bound (thanks to a Paley-Wiener theorem). 3.1. Some examples with the wave equation 3.1.1. A typical example of causal problem We first consider the simple problem which will be very useful for solving other problems with the d'Alembert equation:finda function v on R satisfying, with a given function f on R, and real X > 0 (146)i) vM+X2v=f, wherev H =|~, with either: (146)ii) supp v c supp f + R+, if supp f is bounded in the past, or (146)ii)' v€Lj(R) = eYtL2(R), iff€l^(R). In both cases, the solution for f = 0, v0(t) s A e ^ + B e"^1 must be zero. Thus (146) has the unique solution: (147)

v(t) s / o o S m M x t " S ) f(s)ds = (Gx.fXt),

with Gx(t) = Y(t) ^ ^ , Y(t) being the Heaviside function, and also: (147)'

e - v V t ) = / M e ^ - s ) . 5 ^ i > e - ^ s ) d s = G y , A< (e-%)),

with: GT>A(t) = Y(t)e- Yt 5l n : ^; thus GyX 6 L!(R), with / R |GyX(t)| d t < ^ . Therefore when f e L^R), we have v € LJ(R) and: (148)

My^Vly

withIMIy=/Re"2Yt |v| 2 dt.

6.3 CAUSAL PROBLEMS

279

3.1.2. An example with boundary conditions Now we consider the evolution problem: given a (regular) bounded domain Q in Rn, and given fin l^X), find u in Lj(X), X = L2(Q), so that (149)

i ) ^ "2 - A u = f inQxR., n 9t ii)u| 2 = 0, I = TxR.

Using the spectral decomposition (A2.,<>.) of the Laplacian with Dirichlet condition in L2(Q), we have: (150)

iKO-ZujCQftj, andRt) =2^(1)*. with I/ R e- 2Yt |fj(t)| 2 dt< oo,

and Uj has to satisfy: Uj e L^R), with: (151)

32u-^u.-f., j

â&#x201A;Ź

From section 3.1.1, we know that this problem has the unique solution: (152)

, sin X.(t - s) UjCt)-/^ 1 fj(s)ds = (GAj,fXt),

and Uj satisfies (thanks to (148)): (153)

1 I I ^ S ^

This implies that u is in L;(X), X = L2(Q), with: (154)

1 llull^Cllfl^ C = ^ - .

Moreover since ((-AD)1/2u). = Xj Uj, we obtain with Y = HQ(Q), and C = y max (^-, 1) (155)

$ C

" , '

Furthermore differentiating (152) with respect to t, we have: (156)

u'j(t) = (Y(t)cos(Xjt)).fj

and thus: llu'jlly :ÂŁ | llf}^, giving: (157)

llu'll^llfl^.

280

6 EVOLUTION PROBLEMS

Thus we have obtained: (158) u € Lj(Y) with Y * H*(Q), U' € LJ(X) with X = L2(Q), thus e"Ytu € H ^ Q x R). Then we can prove also the trace result: e"Yt | ^ | € H"l/2 (I) on I ■ T x R. But this trace result is not optimal. Using the proof of Lions [3] p. 40, 41 in the framework of L2 (X) functions, we get:

<159>

| l z € e Y t L 2 ( Z ) , i.e., • ^ g l 2 « L a t e .

Remark 3, An a priori estimate. We multiply (149)i) by e~2ytu9 (the time derivative of u) and integrate by parts. We get: (160)

Y/ O x R e"^[|uM 2 +|giadurtd]cdt-/ O x R e- 2 y t fu»dxdt^0.

Then using the Cauchy-Schwaiz inequality, we obtain the estimate where IMI denotes theLj(L2(Q))normofv: (161)

Y[|Bu'll5 + IgradullJ]<^;llflJ.

This also implies (158), thanks to the Poincare inequality. ® Remark 4. We can solve in the same way the evolution problem (149) but with a (homogeneous) Neumann condition. The only new difficulty is with the eigenvalue 0. We obtainfinallythe same results (158) for the solution. Usual trace results give u | € eY H1/2(Z), but this is not an optimal result. If the unifonn Lopatinski condition (see Chazarain-Piriou [1] p. 362) were valid, we vt

could hope to have u | € e H (£). But this is not the case. From results of Lasiecka-Triggiani [1], we have u | € eYt H2/3(X), and when Q is a parallelepiped u | € eY H

(I), t > 0.

•

3.1.3. An example with transmission conditions, Integral methods Let Q be a regular bounded in Rn with boundary I\ We consider the problem: find u with given jumps (p, p') across I = T x R, so that a2,, i)^-Au =0 (162)

in(R n \r)xR t ,

» M 2 - P . [^}2 = P\ (P,P') e L ^ Y ) , Y=H I / 2 (OxH- 1 / 2 (n, in) supp u c supp (p9or) + C+,

281

6.3 CAUSAL PROBLEMS

where [v] is the jump of v across Z » T x R. Writing (162)i) in D'fR 1 ^), we see that u must satisfy: (163)

| ^ - A u = f, withf=-(p , 8 2 + div(pn62)),

and thus we obtain the solution of (162) in the form: (164)

u = $*f. x,t

But we have to specify the functional space, especially with respect to the boundary conditions. So it is better to use a Laplace transformation. Then problem (162) becomes:findu = u(p) (with Re p > $0), satisfying |i)(p 2 -A)iU0 inRV, (165)

r3fi

This problem has for each p a unique solution in: (166)

X = H1(A,RM^ = {u, u\a € Hl(AyCl), u|Q, € H!(A,Q%

Then we have a continuous lifting of the jump conditions into L^R^X) so that we are reduced to solving the problem:findU in L^(Rtf H2(Rn)) satisfying (167)

- AU = F,

F given in L^,L 2 (R n ».

This problem has a unique solution. Thus we obtain a unique solution u of (162), in L?(Rt,X) with X given by (166). Thus we can define causal Calderon projectors P? and P£ in L^R^ Y) by: (168)

Pffe>,p')-(u|ri, g y ,

Pec(p,P') = ( u | v g | F e ) .

with the usual property: P? + P£ = I. We get u in R3 x R from (164) and the retarded potentials by the KirchhofT formula (see Bamberger-Ha Duong [1]): u^LV + P^with:

282

6 EVOLUTION PROBLEMS

From these definitions, there is no jump of the trace of v = Lcp' across £, nor of the normal derivative of w=P°p across 2. From Lc and P°, we can define "usual retarded" integral operators on the surface £, and an operator Sc such that

(170)

Pf=^(I + Sc),

P°=4(I-SC),

withsCss

(2^c2jc].

and: (171)

LV«iV| 2 .

ICP-I^PL

+^PL,

3 /c»»

Of course it is interesting to specify these operators in a L2 (X) framework. We will do it, but optimal regularity results are missing. a) A regularframework. We define for real s and y > 0 (172)eYtH^(RW=f{f, e-ytf€S*(Rn+1), < y = / R n x R ( | p | 2 + £2)s|f«,p)|2cttdn< 00} with p a Y + in, ?(6,p) being the x-Fourier transform and t-Laplace transform off, Then we define for Cl a regular open set in Rn, by restriction, the space eY H^(Q x R), and (thanks to an atlas) eytH^(r x R), see Chazarain-Piriou [1] p. 412. Thus we have: (172)'

8u 2/T 2/ 2/i 2/ n l / x R)kdef eJyi*Hj(Q V [u € 1^(1/(0)), ^ ,du g € L^(I/(Q))}.

Now if we successively consider the wave problem (149) with Dirichlet boundary conditions, and Neumann conditions, we define the trace spaces: (173)

j o x ^ f u 1 ™ | 2 , u€e 7t Hj(axR) the solution of (149)}, lii) Yy = {u° = v | r v€eYtHj(QxR) the solution of (149)i) with §J | 2 = 0}.

We recall from the regularity results (Remark 4 and (159)) that: (174)

X^ceYtH2/3(2), Y^ceYtL2(I).

The elements of X x Y are traces at the boundary of waves u such that: u€D y (2?)^D/* D ) + Dy(5N), with: Dy(/?D) = {u€eYtHj(acR), D U € L J ( X ) , X = L2(Q), u| 2 = 0J, Dy(*NMu€eYtHj(QxR), DueLj(X), X = L2(Q), §f l 2 = 0}.

6.3 CAUSAL PROBLEMS

283

Let f be given in L^(X), X » L2(Q). Let u f (resp. vf) be the solution of the Dirichlet (Neumann) problem for f. Let U = U f « vf - uf. Then U satisfies: (175)

I i ) D U = 0,

!u)U|2 = v|2 = u<>, g , x . . g ,

I*t 9 (resp.9,) be the mapping f€ I^(X) ^ with: D*(B) = {U€Dy(B),niJ = 0}.

( ^

We can prove that ker 9 = ker 9I# Then taking the quotient space Ly(X)/ker 9, we obtain an isomorphism from L^XJ/ker 9 onto D°y(2?) and onto a subspace Gt of X^xY* Then we define an isomorphism C l = Clc from X^ onto Yly by: (176)

Ci(u°) = ^ l

(with(175)).

This operator C1 is the interior causal Calderon operator. Now we can operate in a similar way for the exterior domain. If the trace space X1 (resp., Y l ) is the same, for the interior domain as well as the exterior domain (which seems to be a very natural conjecture for regular domains), that is X1 = X e =X (resp. Y1 =Y* =Y ), we also define a subspace G e of X x Y and an operator C c for the exterior domain which is the exterior causal Calderon operator with G(Ce) = Gc; C e is also an isomorphism from X onto Y . Note that G1 and G c are closed subspaces in X xY with intersection reduced to {0}. A natural conjecture is that X x Y = G(C i )^G(C e ). b) A more general framework of "finite energy". Now we define: | w ^ = {u€e 7t H^(nxR), Du = 0 i n Q x R } , (177)

We can consider the space W as a natural space of waves with finite energy on both sides of I = T x R. The corresponding space of sources concentrated on 2 is (178)

Zy = {(p,p') = ([u] 2 ,[^] 2 ), u€W y }.

The mapping u —> (?,(>') is an isomorphism from Wy onto Zy, with inverse: (179)

u = <t> * ( p ^ + div (pn6j)).

284

6 EVOLUTION PROBLEMS

Then Z is a Hilbert space, which is the direct sum of two closed subspaces G., Ge with intersection reduced to {0}, which are the set of boundary values of elements in W a and Wya : Z = G.eG c . The projectors on G. and Ge (along Gc and G.) are the causal Calderon projectors in Z . Furthermore we easily see that G. and Gc are the graphs of two operators, the causal Calderon operators, which are extensions of those defined in the preceding section. A reasonable conjecture is that Zy ■ Xy x Yy with the spaces of traces (180)

Xy = {v=u| r ,u€W^,Y Y = { v s s ^ | r , u € W f } ,

and that these spaces of traces for Q are equal to the spaces of traces for Q*. We refer to Bambeiger-Ha Duong [1] for numerical implementations, and for results with the space ^(R,X) = {u,/ R |p|2s»u(p)l^dfi< oo}, with s = 3/2, and X«H 1/2 (r). We note that proceeding as in Remark 3, we obtain the positivity property: (181) Y/ axR e- 2lft ([iu'| 2 +|gradui 2 l)dxdt=/ 2 e- 2Yt u'^drdt>0. 3.1,4. An example with a waveguide Let Q+ = QxR + z (with z=x3) be a semi-infinite cylinder (modelling a soft waveguide) with a regular bounded cross-section Q (with boundary O in R2. Then we consider the wave problem with a given value u° at the end of Q+: i

(182)

>^"<i3 + ^u)s0inQ*x^'^thX!=(xi'x2>€Q'

IH) u(xAt) = u ° & t ) o n °* R * u ° € L \ ( X ) > X = L 2 ( Q ) ' y > °* iii)u(x,z,t) = G onliXRjsTxR^xRt, iv)u€L2Y(L2(Q+)).

We can solve this problem due to a spectral decomposition of the Laplacian Ax with a Dirichlet condition. Let (X2.,^) be eigenvalues and eigenmodes so that we can decompose u and u° into: (183)

u(.,z, t) = I Uj(z,t) ^ ,

u° - X u? *,.

Then (182) is reduced tofindingu- satisfying: a 11

3 U

(184) ii)ui(0,t)-u,0(t), teR.

285

6.3. CAUSAL PROBLEMS

Now using the (time) Laplace transformation, we obtain:

(185)

i ) - ^ - < P 2 + X?)fij = 0

inR£,

ii)fij(0) = fij.

The solutions of (185)i) are given with two constants Aj, Bj by: (186)

uj(z) = A j e jZ + B j e" jZ , with 0? = p 2 + X?.

Now in order to have a solution in L y(X), we have to take Aj = 0, with Re 0j > y. Thus the solution Uj is given by: (187)

fi^-e^flj0.

The image of the half plane Re p £ y by the map p —»p2 is the set: 2

i.e., the exterior of a parabola/^, which contains the set {6j =p z + Xj, Rep > y}. We have the inequalities: (188)

|u j( z)M|uj>|,

/R+|fij(z)|2dz=I^|uJ0|2^^|flJ°|2.

This implies that (184) (and thus (182)) has a unique solution which is given thanks to a contraction semigroup (G(z)) (of class C°) in L2(X) by u(.,z,.) = G(z)u°. The infinitesimal generator A of (G(z)) is defined thanks to the spectral decomposition of the (transverse) Laplacian with Dirichlet condition and thanks to the time Laplace transformation by: (189)

Atyp) = -fl,flj(p)= (p2 + X?)1/2fij(p), with Re p > y, Refl,> y,

and its domain is: (189)'

D(A) = {v€L5(X),Z/ R |9 j | 2 |^ j (p)| 2 dn<oo, withp = y + in}.

Since |p| s 10j I , we have D(A) c eYtH^(QxRt). Thus the problem (182) is equivalent tofindingu in L^(X), X = L2(Q), satisfying: (190)

i ) | - A n s O , z>0, |ii)u(0) = u° onQxRt.

The operator A is the causal Calderon operator of the waveguide.

6 EVOLUTION PROBLEMS

286

3.2. Some examples with Maxwell equations 3.2.1. Causal Maxwell problems in the whole space We first consider the basic problem: let J and M be given electric and magnetic currents in free space, which are distributions in the whole space R3 x Rt, with supports bounded in the past. Like in the stationary case, we assume that there exists magnetic currents, which will be useful to consider in order to solve other problems in electromagnetism. We also assume the existence of related electric and magnetic charges p and C- Let cQ, M0 the permittivity and the permeability of the free space resp.. We have to find the electromagnetic field (E,B) satisfying i)||-c2curlB = - ^ J , (191)

i i ) | ^ + curlE = - M

inR^xR^

iii)divE=fjrp, anddivB = C, iv) supp (E, B) C supp (J, M) + C + ,

with: (192)

^ + divJ = 0,

§£ + divM = 0.

We first note that the equations (191) iii) are consequences of i), ii), iv) (and (192)). By the usual tricks, we eliminate B, then E, to obtain the d'Alembert equations (with (11)): (193)

DE=j,

DB = P0m,

with (j, m) given by: (194)

- j = i ( - y j + gradp) + cur!M,

-m=^(^^+grad?)-curlJ,

and thus we get the solution (E,B) of (191) thanks to the convolution product with the elementary solution O (see (24) and (135)) (with c = 1 to simplify): (195)

E = *j,

B-O.G^n).

We only note that taking the divergence of (195) implies (191) iii): (196)

divE = **divj = <D*^np=f£,

divB = <J>«*i0divm = «nt = C.

Theorem 2. Let (J,M) be given currents with support bounded in the past. The causal problem (191) has a unique solution (E,B) in D'(R3 x R ) 3 x D'(R3 x R ) 3 given by (195) with (194). Remark 5. Causal vector and scalar potentials. When M = 0, we define the causal 1

1 o^

vector and scalar potentials: A = n0<D*J, ♦ = FT # • p; they satisfy: -5 S + div A = 0; aA ° and from (195) (E, B) is given by: B = curl A, E = - ^ - grad 0. <8>

6.3 CAUSAL PROBLEMS

287

3.2.2* Causal problems with currents on a surface. Transmission problems Now let (JL,Mr) be given electric and magnetic currents concentrated on a regular surface T in R3 which is the boundary of a bounded open set Q. Furthermore we assume that they satisfy, with notations (138): (197)

J r , M r € L°(Rt,X)

with X = H"1/2(div,r).

Then the electromagnetic field (E,B) due to these currents must satisfy (191). Theorem 2 implies its existence and uniqueness at least in the sense of distributions. Thanks to the hypothesis (197) we can prove (for instance thanks to a lifting) that thefield(E,B) has finite energy, that is: (198) (E,B)€L2(Rt,Y)xLj(Rt,Y), Y = L2(R3)3, thenE, B c L ^ H f c u r U R V ) ) . Thus E and B have traces (n A E | , n A B | ) on each side of the boundary I of Cl x R. Furthermore they satisfy (197). Using jumps formulas (25) chap. 2 we obtain that (E,B) satisfy (199) | w - c 2 c u r l B = c 2 [ n A B 1 2 S 2 ' S + curiE—[nAB] 2 6j f inR^xR,, div E * - [n.E] r div B = - [n.B]2, so that if we compare to (191), we obtain: (200)

J 2 = - ± [n A B ] r

M 2 = [n A EJ r

Moreover applying the divergence to (199), we obtain (201)

- 1 [n.E]2 * c 2 div ([n A B]2 6J) , | [n.B]2 = div ([n A EJ2 5^.

These basic relations correspond to those of the stationary case (see (84) chap.3). They are compatibility relations for traces, and they give the normal components of E and B thanks to the tangential components of B and E. Thus we have obtained with hypotheses (197), results similar to those of the stationary case (see Theorem 3 chap.3), but here (E,B) satisfies (198) and thus is of finite energy in the whole space. Furthermore (E,B) is obtained by the formulas (195) (with retarded potentials), with (j,m) given by (194), and with: (202)

p 2 = - e0 fa.E]2, Cj — [n.B]2.

These formulas (195) with surface currents are the "Stratton-Chu" formulas in evolution case (see (87) chap. 3 in the stationary case), which are simply written in a form similar to (96) chap. 3. We define like in (95) chap. 3: (203)

I ^ J = c ( 4 | f + gradp)*<f>, c

/ £ J = - (curl J)♦ <D,

288

6 EVOLUTION PROBLEMS dp

with^ + divJ = 0, thusp = -Y.divJ, Y being the Heaviside function. Thus keeping the magneticfieldas unknown, we have (203)'

H= Z-1i4Mr-i^Jr.

E=^Mr+Zl4Jr,

Now we define the causal Calderon projectors for the electromagneticfield(E, H), P? and P* in L?(R,X) x L°(R,X) (see (197)) by: (204) P?(M,J)«(nAEL,-nAH|.),

P°(M,J)*-(nAE|_ , - D A H L ),

like in (93) chap. 3. These operators satisfy the usual relations (104), (105), (106) chap. 3 with an operator Sc given by: (205)

f-1* S°=|^R

ZRC^ _rj,

with 1*1 —<Yl + Yt>l'Sil.

RcJss2nAl4j|r

We have to specify these operators in spaces with a L2 time behavior. Here also regularity results are missing. Like in the case of scalar waves we can define spaces with more or less regularity properties: here we define a general space with "finite energy" (but always with the time decreasing weight e " Yt) W?H(E,H)€l^(YQ), withYQ = L2(Q)3xL2(Q)3, (206)

o ! f - c u r i H = 0>

M0|{- + curlE = 0,inQxR},

| w y = {(E,H), (E,H)| ftxR €W?,

<E,H)| ft , xR €Wf}.

Then we define the space ofjumps across I = T x R: (207)

Zy-KMzJz). M2 = [nAE] r J2 = - [ n A H ] r with(E,H)€Wy}.

The mapping (E,H) —»(M2,J2) is an isomorphism from Wy onto Zy with inverse given by (195) with (194). Like in the scalar case Z is the direct sum of two closed subspaces G., Ge (with intersection reduced to {0}) corresponding to the set of boundary values of elements in W a and in W fl : Z ■ G.0G c . The projectors on G. and Ge (along Ge and G.) are the causal Calderon projectors Pc. and Pcc in Z (see (204)). Furthermore we easily see that G. and Gc are the graphs of two operators, - Cc. and - Cce, i.e., up to the sign, the causal Calderon operators, Cc. and Cce. Then let:

289

6.3 CAUSAL PROBLEMS (208)

X m Y tflnAE| 2 ,<E,H)€¥# f

Y m / # { n A H | 2 , (E,H)€W^}.

We easily see that Xm = Y1" . Here also a reasonable conjecture is that Z = X m x X m (we thus admit that the spaces of trace are the same for the interior as for the exterior). Then the Calderon operators Cc. and Cce are isomorphisms in X m . As usual, these operators are directly related to interior and exterior boundary problems, since Cc. and Cce are defined by: (209)

C?(n A E°) = n A H | , (resp. C\ for the exterior).

where (E,H) is the unique solution in L2 (YQ) (resp. Yft,) satisfying: i ) £ 0 ^ - c u r l H = 0, (210)

3H i i ) p 0 - ^ + curlE = 0, inQxR(resp. Q'xR), iii)nAE| =nAE° with given nAE inX m y . — 2vt

Furthermore multiplying!) by Ee

— 2vt

and ii) by He

then integrating, we get

(211) y/ a x R (e 0 E + M 0 H )e"^ dxdt = - / I n A E . H e-**"^d2>0 for the interior, and obviously a similar result for the exterior. This gives the usual positivity properties (see (201), (212) chap. 3): z

* *l.

.-2yt-

(212)

(C?m, nAm)=/ 2 (nAE).He^ 7 l dI^0 (C*m, nAm)=/ (nAE).He"

2Yt

(withm^nAE^),

dI > 0.

For a numerical implementation of these time integral methods, see for instance Pujols [1]. 3.2.3. Causal Maxwell problems with boundary conditions i) First we recall that the evolution problem in free space (for a regular domain Q bounded or not):find(E,H) in L2 (YQ), YQ = L2(Q)3 x L2(Q)3, such that (213)

i) e 0 ^ - c u r l H = J,

P 0 ^ + curlE = M inGxR,

ii) nAE| =0, with given electric and magnetic currents (J,M) in L 2 (Yft), has a unique solution thanks to Proposition 19. Furthemore it satisfies the energy relation:

290

(214)

6 EVOLUTION PROBLEMS

y fax

(c0 E2 + M0 H2) e'2Yt dx dt = - fQ% R (J.E + M.H) e'2yt dx dt.

ii) Now we solve a more general evolution problem: we assume that the boundary T is coated with a dielectric medium, in a domain Q p and the complementary domain in Q, Cl0 is occupied by free space. The dielectric medium is linear isotropic, its constitutive relations are given by (22), (or (23)) chap.l, with hypotheses H5) ((27),(28)) and H14) in chap.l. Thus with p «y + i n - - i w, <»>«ip = - n + iy, n € R, y > 0 fixed, we have: (215)

aw VRe (- ico?) > 0,

8 d i f Re <- A) > 0,

and there exists a constant C > 0 so that: (216)

aw > CE0Y,

PW * C y/(M0 | a> |

Note that (216) is also true in free space, with C= 1, and even with equalities. Now given an electric current J in L2 (Xft), XQ = L2(Q)3, we consider the evolution problem: And the electromagnetic field E, B, D, H in L2 (Xft), such that i ) ^ - c u r l H = J, (217)

ii)D = c*E,

^ + curlE = 0 inQxR,

B = p*H inQxR (witht*e06(t), y = M0S(t)inQQXR),

iii)nAE| =0. This is an evolution problem with delay. Using a time Fourier-Laplace transform method like in chap.l, with y = Rep = Ima>>0, problem (217) becomes: (218)

i)ia>6 E + curlH = J,

- ia>£H + curlE = 0 inQ,

ii)nAE| =0. We can solve this problem where o> is a complex parameter, with Im <o > 0, using a variational method. We define the sesquilinear form: (219) a(o,<|>)=/0 ( - A - curl ♦. curl ) dx, **

♦, <> | € V = H0(curl,Q),

lop A

and then the problem (218) is equivalent tofindingE(o>) in V with: (220)

a(E(a>),(|>) = - fQ J(a>). $ dx,

V <> | € V.

The sesquilinear form a($,c|>) is coercive on V for all o> with Im a> * 0, thus (220) has a unique solution for all o> with Im o> * 0.

6.3 CAUSAL PROBLEMS

291

Now we integrate with respect to n, and take the real part of (220) with <> | = E; using (216) in ft, and the Cauchy-Schwarz inequality, we obtain with Cx = Ce0y: |E(c*)| 2 dxdn</^ R BJ tt I c u r l E t ^ l ^ a ^ lE^l^dxdn^llJIIIIEII

(221) CJ^

with IIJII2=/ftxR I J(«)| 2 dx dn. Thus we have: (222) E(o>) € L2(C1 x R,,)3, and X curl E(o>) € L2(C1 x R / , with o>. that is: (223)

+ iy,

E € L^X), X = L2(Q)3, and E(t) = f / ^ E(s)ds satisfies: curl E € LJQC).

This implies: (224) thus:

H(o>) € L2(Q x R / , and X curl H(a>) € L2(Q x Fty3,

(225)

H € Lj(X), and H(t) = f / ^ H(s) ds satisfies: curlH € L^X).

Furthermore since 2s(ci>) = - X E(o>) and H(v>) * - ]jj H(w), (with | <o | > y), thus £(o>) and H(u) are in L2(C1 x R^)3, and since n A &(O>) | = 0, we have: (226)

IE € L 2 ^ , H0(curl,Q)), H € L 2 ^ , H(curl,Q)), I thus: E € Lj(H0(curl,a)), H € Lj(H(curl,Q)).

Proposition 20. The causal evolution problem (217) with delay, that is in a domain Cl occupied in part by free space and in part by a linear anisotropic medium with permittivity and permeability (E,JI) satisfying H5) and H14) chap. 1, has a unique solution (E, D, B, H) in L 2 (X Q ), with X = L2(C1)\ for a given electric current J in L2 (X), and this solution satisfies (226). y

3.2.4. A causal waveguide problem Like in section 3.1.4, let Q + = Qx Rz+ be a free space domain, which is a semiinfinite cylinder bounded on its lateral boundary by a perfect conductor (modelling a waveguide), with a regular bounded cross-section Cl (with boundary T) in R2. We assume that a tangential electric field is given at the end of the waveguide. Then we consider the evolution problem: find (E,H) in the waveguide, with (E,H) € L^(X3 x X 3 ), X = L2(Q+), satisfying: i ) - £ 0 ^ + curlH = 0, (227)

i * 0 ^ + curlE = 0 inQ + = Q + xR t ,

ii)nAE| =0 onl^TxR^xRt, iii)nAE| 2 =nAE° onZ0 = QxRt, Ex€l^(H0(curlT,Q)).

6 EVOLUTION PROBLEMS

292

Wefirstwrite Maxwell equations like in (30) chap. 4, using the same notations: € 9t SET 9SHT gradT H

3" °

* °*

|ii)d 3 E T -grad T E3 + >i()atSHT«0 inQ+xR^ (where 8 t = ^ ) with: (229)

OcurljHj-co^Ea* 0 *

^curljEr+^a^sOinQ+xRt,

and with the boundary conditions: (230)

O^AEr! =0, E 3 | =0,

From now on we drop the subscript T for differential operators. The determination of the electromagneticfieldwill be made in several steps. Determination of the transverse electricfieldET In the same way as in chap. 4, wefirstobtain that E T satisfies: OafEj + A E T - c - ^ E T s O , (231)

inQ+xR^

ii)n T AE T | =0, (divEjH^O, iii) Ej(0) s E j o n I 0 , i^ErC^X^X^L2^);

the boundary condition on the divergence is a consequence, with (230)i), of: (232)

div

x,x 3 E = 8 3 E 3 +

div

TETs0-

Let AQ be the positive self adjoint operator in L (Q) defined by: AQU * - Au, with (233) ^ i V H u c H ^ Q ) 2 , curluand divueH 1 ^), nAu| r =0, (divu)|r=0} (see (147) chap. 2 with n = 3). Then using the spectral decomposition (X2.,*.) of AQ, with c = 1, we define the operator AT from its Laplace transform DA^pJ^-Gj^p), 9jSS(p2 + X?)1/2, withRee j>y , v^Iv^. inLj(XT), (234)

u)Dy(AT) = {v€Lj(XT),XT = L2(Q)2, Z/Rlflj^jVj^diK+oo, p = y + in}. We can prove that AT is the infinitesimal generator of a contraction semigroup (G^Xj)) in L2y(X,.), X^L^Q) 2 , of class C°. Thus like in section 3.1.4, the problem (231) has a unique solution Ej. which is given thanks to (G^x^) by:

6.3 CAUSAL PROBLEMS (235)

293

E^^G^Ef

2/ V 2x Then we can verify thanks to the Laplace transformation that E T € L~(X ).

Determination ofEy so that

First assuming that E 3 is given at x 3 = 0, we have to find E 3

i ) 8 3 E 3 + AE 3 -c*- 2 3 2 E 3 :=0 ii)E 3 | (236)

inQ + xR t ,

=0,

iii) E 3 1 2 = E 3 , with given E^ in L2y(L2(Q)), iv)E 3 €L?(X),X = L2(Q+). Then E 3 is obtained thanks to the semigroup (G(x3)) of section 3.1.4 (with generator A defined by (189), (189)') by: (237)

E 3 (X 3 ) = G ( X 3 ) E 5 .

Then we have to find E 3 . It is obtained (like in chap.4) from (232) by: (238)

33E3(0) = AE 3 = - div Ej.

We have div E j € L^(H~ l(Cl)) c (D(A))\ and thus, since A is an isomorphism from D(A)ontoLj(L 2 (Q)) (239)

E° * - A* W

Ej) € L}(L2(Q», thus E 3 € LJ(X).

Determination ofHy We get H 3 from <> | = curlj. E T by time integration of (229)ii). Applying curlj. to (231)i), then by taking the scalar product of (228)i) with the normal nT, we get <J> satisfying i)8^c|) + A(|>-c"2a2<|) = 0, (240)

inQ+xR^

dip ,

I iii) <K0) = <|>° e 1^(L2(Q)), and <Kx3) is obtained by the semigroup (G N (x 3 » with the Neumann condition: (241)

<j>(x3) = GN(x3N>°.

2,i 2, This gives H 3 e L~(I/(G)) and H3 e L^(X).

6 EVOLUTION PROBLEMS

294

Determination ofHT Now we can obtain H T by time integration of (228)ii); we have to verify that H- is in L 2(X2). We can prove this result like in the stationary case, using the Hodge decomposition for H p see (68) chap .4 (we can also use an inverse Laplace transformation on the formulas of chap. 4 taken with complex 03). Causal Calderon operator of the waveguide. Now we are especially interested in having the boundary value of H- at the end of the waveguide, i.e., for x3 = 0. This is also obtained thanks to (228)ii) with x3 = 0: (242)

8tSHT(.,0,t)=ATET(.,0,t) + gradA-ldivE!J<.,0,t).

We get a formula similar to (61) chap.4 (with (29) and (38) chap. 4): (243)

HrCAt) — £ ^ [SATE^.,0,s) + curlTA"1 divTE^<.,0,s)] ds.

We can also obtain a form similar to (68), chap.4, more useful for spaces. The operator C:E£—H^.,0,.)

(orCiSEj^SH^.A.))

is the causal Calderon operator ofthe waveguide. Of course we have to specify its domain. We can do so thanks to the operators Aj and A; we do not detail this question here. The interest of this operator is (like all previous Calderon operators) that it contains all informations at hand for the domain exterior to the waveguide. Furthermore we can prove that the causal Calderon operator satisfies the positivity properties: (244)

(CSEj, E£) = - (CEj, SE£) £ 0

which we get from t 245 * - ^ x R ^ o f

E +

>*of - W e - ^ d x d t ^ ^ d i v t H A ^ e - ^ d x d t ,

and integrating by parts: (246)

y / ^ x ^ ( £ 0 E 2 + M 0 H 2 )e- 2 y t dxdt=/ 2 o x R ^nAH.Ee- 2 Y t dI 0 dt.

3.2.5. Uniqueness at most In all these causal problems we have uniqueness (at most) results for the solution of the wave equation like the Maxwell equation (this is to compare to the stationary case, chap. 2.13; these results are given for a bounded domain Q, to simplify): Theorem 3. Uniqueness at most. Given a (regular bounded) domain Cl in Rn, with boundary T, the wave problem: find u so that

6.3 CAUSAL PROBLEMS

295

i) Du=f, f given in D'+(X), in Q x 1^, (247)

ii)u! 2 = u°, i J l ^ w i t h l ^ r x R , , withu°€D%(H-i/2(r)), u1 €D\(H'm(T)) liii)u€l)\(X),X = l/(Q),

has at most one solution. Theorem 4. Uniqueness at most. The Maxwell problem: find (E,H) so that i ) £ 0 ^ - c u r l H = J, (248)

j i 0 ^ + cur!E = M i n Q x R ,

with M and J given in D%(X), X = L2(Q)3 ii) n A E | 2 = M°, n A H | 2 = J° withM°andJ°€DV(H" 1/2 (div,r)) iii) (E, H) € Z)V(X), X = L2(Q)3xL2(Q)3,

has at most one solution. The essential difference with the stationary case is that these uniqueness results are not local in time: we can substitute a part Z 0 =sr o xR, with T0 a "regular" part of T, for I . However we cannot replace R by finite time intervals, except for bounded Q, in which case these time intervals have to be large enough. This is a consequence of the finite velocity of propagation. Finding the optimal time interval for which we have uniqueness is of great interest in control problems (see for instance Lions [3]). These uniqueness results imply that the causal Calderon operators and the causal Calderon projectors have the same importance as in the stationary case: these operators on the boundary T of a domain Q contains all the informations on the wave inside the domain Cl that we consider. Of course these operators are not very easy to handle especially for numerical implementations. So they are often replaced by approximate boundary conditions. However they are very useful in obtaining a priori estimates. CONCLUSION We have solved many evolution problems, but we would have to deal with many other linear interesting problems, which are not a simple application of the above theory, for example with a moving obstacle, and a moving antenna... Furthermore there are very often parameters giving different scales, so that an asymptotic analysis is necessary, with singular perturbations.

APPENDIX

DIFFERENTIAL GEOMETRY FOR ELECTROMAGNETISM

Using differential geometry in electromagnetism is quite natural, atfirstin the modelling of Maxwell equations by differential forms, according to the right transformation laws when changing a coordinate system into another one with a different orientation (this explains the notion of "polar vector" often used in electromagnetism). The notion of cohomology allows a deep understanding of many points, notably in the study of the differential operators grad, curl, div with their kernels and their images, in an open domain Q; and then in the trace spaces on the boundary T of Q for electromagneticfieldsoffiniteenergy. We have to use the manifold structure of T, and the differential operators gradp, curlp diVp on T, with the technical difficulty of having enough regularity in order to apply usual tools generally developed for very regular manifolds. Differential geometry is also useful in numerical methods in electromagnetism with Whitney forms (see Bossavit [1]). We suppose only elementary prerequisites onfinitedimensional manifolds. We review some simple notions in order to set notations, following Morrey [1], Arnold [1], Abraham-Marsden-Ratiu [1]. We also need some elementary notions on variational framework (the notion of a V-coercive bilinear form) and we often use the Peetre lemma (see chapter 2). We emphasize that this appendix is not a course on differential geometry; we refer to Malliavin [1] or to Kobayashi-Nomizu [1] for instance for that. 296

A.l INTRODUCTION

297

1. INTRODUCTION. MATHEMATICAL FRAMEWORK

1.1. M anifold with boundary The notion of manifold is supposed to be known, with the notions of charts, of mutually Câ-compatible charts, of atlases and of C*'a-compatible atlases (we refer to Dieudonne [1], Bourbaki [1]). We recall the standard definition of an n-dimensional manifold M with boundary T of class C**", k € N, 0 £ a <* 1 (C00, analytic). Each point of M is contained in some open set Fof M, which is the homeomorphic image either of the unit ball in Rn, or of the part of it for which ^ ^ 0, the points where ^ = 0 correspond to the boundary T of M; any two coordinate systems are related by a transformation of class Câ (C°°, analytic). Let Q be an open subset of Rn; Q. may be considered as an n-dimensional manifold with boundary T; following Grisvard [1], we then define: Definition 1. We say that Q. is cm n-dimensional continuous (resp Lipschitz, continuously differentiable, of class C**", k €N, 0 £ a < 1) submanifold with boundary in Rn, if for every x€T, there are a neighborhood V of x in Rn and an injective map yfrom Vinto Rn so that: i) Q n K= {y € Q, <>| (y) < 0}, where <>| (y) denotes the n-th component ofitfy); ii) <>| together (j)"1 defined on <|>(V) is continuous (resp. Lipschitz, continuously differentiable, ofclass Câ). o The boundary r of M = Cl is then locally defined by (> | (y) = 0. The interior M = Q, is a C°° (even analytic) manifold without boundary! The boundary T of an ndimensional continuously differentiable (resp.C , C '*) manifold M, has an induced structure of (n - l)-dimensional continuously differentiable (resp. ) manifold, according to the implicit function theorem. In the case of Lipschitz regularity, this is not true (see Grisvard [1]). Example 1. Polyhedrons, which are in common use for numerical applications, are only Lipschitz manifold, so we often have to make direct proofs of the main properties, like for the Stokes theorem! Let M be a regular (differentiable) manifold, we will use the following notations in this appendix.

A - DIFFERENTIAL GEOMETRY

298

TXM the tangent space at a point x of M (the space of tangent vectors at x), TM = U TxYM the tangent bundle of M; x€ M TXM the cotangent space at x € M (i.e., the dual space of TXM, or the space of covectors atx), T M = U TAYM the cotangent bundle of M (or the phase space); x€M r s * VSXM the tensor product &TXM # <8>TXM (the space of r-contravariant s-covariant tensors), T5M the tensor bundle oftype (r9s) over M,TfM = U TLM; x€M r , r ATXM and ATXM the spaces of v-covectors and of t-vectors, r * r * AT M = U ATAVM the r-exterior bundle over M (with A the exterior or wedge x€M product). A vectorfield(resp. a (r,s) tensorfield)is a section of TM (res p. of T^M), that is, a mapping v from M into TM (resp. T^M), such that « o v a l , with n the canonical projection from TM (resp. T^M) onto M; a differential form is a section of T*M, i.e., a mapping o>: M —> T*M such that it*ow = l, with n* the canonical projection from T M onto M. r r * A (differential) r-form » is a section of AT M, and an r-vector field is a r section of ATM, the regularity of these mappings being so far not specified. ft

In the domain of any coordinate system (x 1 ,...^) we denote by Trff-fQZn (or simply by 3!,...,3 n ) the respectively associated vector fields, by dx 1 ,..., dx" ft

the associated 1-forms, so that(-2r) ,...,(^i) is a basis of TXM, (dx1)v,...,(dxn) 0J£ X

is the dual basis of TXM, with (dx*) . 3j-*y,

Vij.

An r-foim o> may be represented as follows: (1)

0)=

Wj ; dx1 A...Achr,

where (coj j) are the components of o> in that coordinate system and A denotes the exterior product. Let I be a sequence I = {ip.. ,,ir} with 1 < ij <.. .< i,. < n. We often abbreviate notations by: (iy

a) = 2 (Ojdxj. I

A A INTRODUCTION

299

We assume that M is an n-dimensional orientable manifold (i.e., there is a continuous n-form co on M such that <o(x) * 0, V x € M). Then two orientations of M may be chosen M + and M _ (so that M becomes an orientated manifold). Let o be the symmetry exchanging the orientations. Then we can define in two equivalent ways even and odd r-forms (see Schwartz [1], Bourbaki [2], Morrey [lj. The first one is intrinsic, the second uses local charts). Even r-forms are rforms which are invariant by o; odd r-forms are r-forms which are changed into their opposite by o. (We can also define even and odd r-fields on M.) If two coordinate systems with coordinates (x) and ( x ) overlap, the components of co in each system are related by: (2)

,~x

/ ~ x% <*vK » • • • » *

)

^"■ ^ »^r?i'

<Hx* 1111xn) with 6 = 1 for even r-forms, J/| J| for odd r-forms, J= r —— . 3(x,...,xn) An odd form is usually (see Schwartz [ 1]) denoted by ©; we also note ± o> or eo>. If co is a (regular) r-form on M, its exterior derivative is the (r + l)-form da>: (3)

dco^^dx'Adx1, I,a

or (3)'

9&>~

da> = 2 (ckO.*^, (*>),» 2 ( - i f " 1 — p , J

v=l

JV=J\UVK

g^v

dcox(v0,...,vr)= S^lrtD^x-ViXvo,...,^,...^,),

Vje^M,

where v{ denotes that Vj is missing, and Do>x is the derivative of co in charts. Note also d2 = 0, i.e.,d2a> = 0, V<o, (4) d(a A P) = da Afl+ ( - l) r a A d{J, for all r-forms a and all p-forms 0. 1.2. Riemannian manifold M9 with or without boundary A Riemannian structure (or metric) on a manifold M is a (covariant) tensor field of order 2, i.e., a mapping g: M —*T*MxT*M which is a positive definite bilinear form on TXM (5)

gx(v,v)>0,

VveT x M,v*0.

In a local coordinate system x 1 ,..., x!\ g (which is often written as ds2) is given with fe«) = (& (9i > 9j)) a positive definite matrix, by: (5)f

g^gy^1®^-

300

A - DIFFERENTIAL GEOMETRY

A manifold M with a Riemannian stucture is called a Riemannian manifold. IfM is of class C^", g(x) is of class C*"1,01. For all x in M, e is an inner product onTxM; wenote: (6) gx (h,k) = (h,k) = 2 g ( x ^ , for all tangent vectors h, k of TXM, given in a local coordinate system x 1 ,..., x11 by h = Ih^i, k = Ikiai. Recall that (aj) is the dual basis of (dx i ),i=l,...,n. The length of the tangent vector h € TXM is defined by: |h|=(h,h) 1/2 . g x A tangent vector is said to be unitary if |h| = 1 (we say that h is a unit vector). For all x € M, the metric g assigns to each tangent vector h € TXM the covector a)h = Gx(h) € TxMby: (7)

(8)

coh(k)3SGx(hXk)«gx(h,k),

Vh,k € TXM,

and Gx is an isomorphism from TXM onto T*XM. The cotangent space T* M is therefore also equipped with a positive definite bilinear form (thus an inner product, corresponding to a tensorfieldg* on M): -1

(9)

-1

(cop <t>2)g, =gj (*>!, a ^ g ^ G ^ G ^ ) ,

V»i, o)2 €TXM.

Thus using notations (8): (10)

K , <ok) =gj (coh, cok) = gx(h,k) = (h,k) , *x

V h, k€TxM

(i.e., G x carries the metric of the tangent space onto the cotangent space), or in a local coordinate system: (11)

g* = I glj di®8j, (glj) inverse matrix of (g..).

r This can be extended more generally to exterior algebras: on the space ATXM of r tangent r-vectors at x, and on the space AT*XM of r-covectors at x, we define, for all hj, kj in TXM, and all coj, 3 j in TXM (hi A ... Ahr, kx A ... AkJ ^deta^hj) ), *x *x («! A ... A(or, S j A ... A 3 r ) adetftcoj, Sj) ); (12) ©X

©X

r r♦ Gx defines an isomorphism (again denoted by Gx) from ATXM onto ATXM by:

A.I

(13)

INTRODUCTION

301

G x (h,A...Ah r )1i f G x (h 1 )A...AG x (h r ) = 0 ) h i A . . . A a ) v

We will often denote by (,) x the inner products (6), or (9), or (12). For r = n, we can define an odd n-form v = v* which is positive (for the chosen orientation of the orientable manifold M): (14)

^ K o j A "-^n-l^A.I.Aa^r

"l A '"'

Aa)

nWetg*Ka)j)r1/2,

for a family C0ly...9(on of n independent 1-forms, where lo^ A ... Aa>n| is afunction ofx defined according to (12): Iwj A ... Ao)nlx=(a>1 A ... Aa>n, oj A ...Aw n ) x . This definition is independent of the chosen family. Thus (14) defines a Lebesgue measure on M. In a coordinate system x 1 ,.. .,x n , v is given by: (14)' v = v g = ± dx1 A ... Adx^det(g*(dx\dx*))r 1/2 = ± dx1 A ... Adx n (detg..) 1/2 . We often denote by dM the measure defined by Vs (and also by g the determinant of the matrix (g..)). But the notation dM may be misleading, since in general it is not the exterior derivative of any form. We then define a scalar product on the space of (smooth) r-forms (with compact support if M is only locally compact) by: (15)

(o),n) = (o),n)g d = f / M ^(G x n x ) dM,

with G x defined by (8) and (13). We also have, from the symmetry of g: -l

(16)

-l

a>(Gn) = n(Ga>),

(o>,n) = (n,<t>).

In a coordinate system, with r-forms a>, n, o) = I(o. dx1, I

n = 2 n v C^K» K

we have: (17) o>x (Gxtix) = IgIK(x)(uI(x)nK(x), with: ghh

,K gJK = det

...

g 'A|

dM = (det(g..))1/2dx1...dx11.

^ g ^ The scalar product (15) is independent of the system of local charts. Thus we can define a Hilbert space L r(M) as the closure of the space of smooth differential r-forms with compact support in M. It is also possible to define it as the space of r-forms with square integrable coefficients in every coordinate system. The elements of the space

302

(18)

A- DIFFERENTIAL GEOMETRY

LJ(M)»(<D

"generalizedr-formsM, (G>,G>) <+oo}

are called square integrable r-forms. We often do not specify whether o> is an even or odd r-form. But if necessary, we shall use the notations L2r e(M) and L2„ AM). Note that from the definition (14), we have (19) (v*,v«) x =l, Vx€M, and thus (19)' (v*,^) - J ^ - I M I (finite or not). For all n-forms © on M, we have (20)

a) = (co x ,v*) x v*,

o> = (a>, v*) g .

Then we define by duality to the exterior product the inner (or interior) product «-» of a p-covector u by an r-vector k, r s p, as the (p - r)-covector iku ■ k ^ u such that: (21) <h, i k u>s<h, k*J u> = <kAh, u>, V (p-r)-vectorh, using the duality bracket between p-vectors, and p-covectors <, >. We also define the inner (or interior) product k*«£ u of a p-covector u by an r-covector k*, r £ p, by (21)' (k* A h*, u )x = (h*, k* ^ u )x, V (p - r)-covector h*, i.e., we also have: (22) < G"\ k* A h*), u > = < Gx"!h*, Gx"" !k*J u >. Thus (23)

k*«i u s G ^ k ' ^ u .

We then define the Hodge transform (or * transform) of an r-form n by: (24)

(•n)x=nx^v* = ( G - \ ) ^ v x \

Therefore the Hodge transform of an even (resp. odd) r-form is an odd (resp. even) (n - r)-form, with: (25)

(nAa,v*)f«(af.n)g,

V a € Lj_r(M), n € L*(M).

Note that the Hodge transformation has the following properties: (26)

.v*«l,

.l^v*.

One of the main properties of the Hodge transformation is: (27)

w A *n = (<t>,n)vvg=<o), G ' W v * .

303

A.l INTRODUCTION giving for all <■>, n in L2r(M), even or odd: (28)

/ M M A •n»(»,n).

This is expressed in a coordinate system, with (29) by: (30)

» 2 **i d*1' n = 2 n K d x K ,

card I s card K s r,

/ M » A . n = («,n)f = X / M 8IK «mK g 1/2 dx1.. .dx".

Taking the inner product of (27) with v*, and using (19) and the commutativity relation: (31)

(0A»ns(-l)

" *nAw,

Vr-formsa),n,

we obtain for all r-forms o>, n (32)

(a) ,n)x=(<■> A ♦ n, v \ = ( - l ) * - ^ ♦ n A a>, v*) x =(- l)*""^*, * * n)x,

hence for all r-form n: (33)

= (-D r ( n " r ) n.

Besides, we have: (34)

(oA (o A *n * n==( - 1)

(*w)An = n A *G>

(which may be verified by inner product with v?), thus (35)

(a>,n)g = (*n,*a>) g = (*a),*n) g ,

Va>,neL?(M).

The Hodge transformation is an isometry from Lr(M) onto Ln_r(M), changing even into odd differential forms, and conversely. Remark 1. Expression of the Hodge transformation in a coordinate system. For each sequence K C {1,..., n}, with card K = r, there are numbers (a KJ ), with card J = n - r, so that (36) *dxK = e 2 "KJ^* J

withe=*+l if(x1,...,x11) has the chosen orientation, E = - 1 if not. To specify (36), we define with the usual notations (see Bourbaki [1] A III 79-87) V = {1,.. .,n} \ I (37) p n , =5 ( - l)v, v ss number of couples (i,j) € I x V such that i > j , |pTV = 0 i f J n K * 0 , PIV = (-1) V i f J n K = 0 , (38)

|v s number of couples (\,\i) € J x K such that X > JJ, we have for all r-forms o>, n:

304

A - DIFFERENTIAL GEOMETRY

(39)

(40)

dx*A • dxK = ca K r dx I Adx r ,

d ^ A d ^ ^ p j p d x 1 A ... Adx".

Comparing with (30), we find: (41) (42)

«KI—«IKPir*l/2.

cardK = r, c a r t r = n - r ,

•dx K «£2ig I K P i r g 1 / 2 dx r ,

caidI = caidK=r.

Whenr=n, letN = {l,...,n}; we have P = 0 , d x 0 = 1, P N 0 = 1, a N 0 = g* 1/2 , thus *dxN = eg- 1 / 2 . A /\ Whenr=l, •dx k =c ^ g i k ( - D ^ ^ ^ d x 1 A . . . Adx i A... Adx11, wheredx* i

indicates that dx1 is missing; in a Cartesian system, we get • d x k * e ( - l ) k " W A ... Adb^A... Adx11.

1.3. Definition of the codifferential For any (regular) r-form <or, the codifferential 8wr is an ( r - l)-form defined with the exterior derivative d by: (43)

8cor = ( - l ) n ( r + 1 ) + 1 . d * o ) r .

We can easily show the following properties: (44)

58 = 0, 61=0, 5f=0 for all functions (O-forms)f, , 6 d = d6», *d6 = 6 d , , 6*d = d«5 = 0,

(45)

•(5o>r) = ( - l) r d( • cor),

r-l 8( ♦ <ar) = ( - If" * do>r.

From Stokes formula (see (81) below), 5 appears as the adjoint of the exterior derivative: (46)

(da,P) = (a,6(J),

o for all (smooth) (r - l)-fotms a and r-forms 0 with compact support in M, because: (46)'

(da,P)-(a,8P)=/ M daA*P-aA*6P=/ M daA*P + ( - l f - 1 a A d * P = / M d ( a A , 0 ) = O.

We will show the generalizations of (46) below.

305

A A INTRODUCTION

1.4. The gradient, divergence and Laplace-Beltrami operator Let f be a (smooth) function on M. Then its gradient, denoted by grad f, is the vector field defined by: (47)

gradf=G(df),

i.e., df=G(gradf) = Vadf •

Let X be a (smooth) vector field on M. Then its divergence denoted by div X is the function defined by: (48)

divX = -6GX = -6a>x.

Then we define the Laplace-Beltrami operator (also called the Laplace-de Rham operator) by: (49)

A = -(d6 + 6d).

Applying A to a (smooth) function f, we obtain: (49)'

Af=-Mf=-8GGdf=-SGgradf=divgradf;

the Laplace-Beltrami operator applied to the functions is identical to the Laplacian. o Note also that for all smooth r-forms $ with compact support in M (Aa,P) = -(6a,8P)-(da,dp) = (a,AW. or

Expression in a coordinate system (x{,.. .,xj. We have df = 2 Z I *^> thus: i ***

(50)

grad f = Gdf = X 5 Gdx1 = £ 3 gU 3j,

and for X = 2 X19}, we have: (51) »x = G(X)=2Xigdxi, (52)

.wx=±2(-iy"1Xjg1/2dx1A...Ad^A...Adxn.

We verify that (OXA ,(o x = ±(X,X) g 1/2 dx' A ... Adxn= ± ^ . x b c V 2 * 1 A ... Adxn, X

and we have Sa>x = -*d*u> x =-* ±2^r(X j g 1 / 2 )dx 1 A...Adx n , thus:

306

(53)

A- DIFFERENTIAL GEOMETRY

divX = g" 1/2 2 X (XV 2 ). j ftr"

Furthermore the Laplace-Beltrami operator is given by

(54)

Af-g-'^yfeV 2 ^.

1.5. Decomposition of the space of tangent p-vectors and tangent p-covectors into tangential and normal parts on the boundary Under the usual regularity hypotheses on the Riemannian manifold M, on its boundary T and on g, the tangent space TXM of M at a point x of T has an orthogonal decomposition with respect to gx into tangent and normal part to T (55)

T x M=T x TeR n ;

similarly the cotangent space TXM of M at x is orthogonally decomposed (with respect to g*) into: (55)*

T*M=T£©R;,

and we write the decomposition of elements with n a unit normal to TXM, a>£ = Gxn, (55)'

h = th + (nh)n, Vh€T x M, h* = th* + <h*,n><4, Vh*€T*M.

There are corresponding decompositions of the spaces of tangent r-vectors and r-covectois: (56)

Vx€l\

r ♦

teforall<D and we write for all <DxX€AT € AT xM XM (57) ci>x = taix+nci>xAa)^ = tci>x + 5(i>x. r r Then Gx (see (30)) transforms the decomposition (56) of ATXM into that of APM (58)

GtA=tGA, GnA=nGA,

VAcAT x M.

These decompositions involve decompositions of the restrictions to T of (smooth) r-fields and r-forms on an oriented manifold M (59)

0 - $ + i0A«ji-t0+iiP,

tjJ (resp. nfl) is called the tangential part (resp. normal part) of (5.

307

A A INTRODUCTION

Admissible boundary coordinate system; adapted chart. The decomposition (59) may be extended to a neighborhood of I\ using an "admissible boundary coordinate system" or a chart "adapted to the unit outgoing normal vector field". We recall the definition (Morrey [1]). An admissible boundary coordinate system on a manifold M with boundary T of class is a coordinate system (of class C; ,M) which maps its domain G^o c R^ (with o the part of 9G on xn = 0, o * 0 ) onto a boundary neighborhood N, in such a way that o is mapped onto NnT and the metric is given on o by: (60)

ds 2 = 2 g..(x,,0)dxi®dx? + (dxn)2, x ^ x 1 1 ) . lJ

i,j = i

[Also, if (W, <> | ) is a chart of r withy €W, there are a > 0 and a diffeomorphism: (t, y)€[0, <x)xW — z = Vy(t)€U c M, such that Un T = W, and: (61) Vy(0) = y, v ^ 0 ) = n, the unit normal to T. Thus we can define a chart (U,x) of M on U by: (62)

Xj(z) =

Then (^r I, )• , 9

Xj /j = l

*j(y),j = l,...,n- 1, xn(z) = t withz = Vy(t)€U. n

can be identified with the normal n.]J

If M is a manifold Q, Cl being an open set of Rn, we can use the Euclidean distance of z € Cl to the boundary T, d(z,0: t = s n = <>| (z) = d(z, 0 , and: w^ = dsn » d(|>n. This allows us to extend the decomposition (59) in a neighborhood V of I\ ® Let j be the canonical mapping I*—»M. IfjJisaC r-form on M, the pull-back j*P of 0 byj is the C k_1 r-form on T defined by (see Schwartz [1], Bourbaki [2]) (63)

0*P)x(v1,...,vr) = Px(v1,...,vr), Vx€r,v 1 ,...,v r €T x T.

This r-form j*(J can be identified with the restriction of the r-form t|J to T, and we write (64)

tr$=j*P = tPl r

Using notations (21), (23) we define an (r - l)-form n ^ on T for all r-form (J on M: (65) nrP = ( - l / -

» l r - ( - If-^W

A<oi)lr=nplr = ( - l ) 1 - 1 ^ ! , . .

Indeed for all (r - l)-forms a on M, we have: |(a, w l J (nfl A o>i)) = (o>l A a, nP A o>i) = ( - lf'l(a A «£, nfl A a>]>) (66)

= ( - l)r-'(taAo>i, npA • » ) - ( - l)T-\ta,a»)(mla, o^) = (-l) r - , (ta,np) = (-l) r - , («,nP).

308

A - DIFFERENTIAL GEOMETRY

g Let g denote the Riemannian metric induced by g on T, and v r =v r the (n - 1)form on r (corresponding to the Lebesgue measure dT on I\ see (14)); then (67) v r = a>£ J v g | r = ( - l f - V * (oralsov rss t r »o>i). It is thus possible to define the Hodge transformation on T as in (25). Lemma 1 • For all (smooth) r-forms f>onM, we have (68)

Ot^lM-ir'.nrP, |ii)nr<P = ( - l ) n - \ t r p .

PROOF. Applying the Hodge transformation to (59), we obtain (69) *0 = *t0 + ,(n0A<4) s ,t0 + *n0. We have (70) U 0 - . S 0 , n*P = «#, t r «P*t r ,(n0A(i>i). Using definitions (25), (24), we have for all (n - r)-forms a on a neighborhood VofT (71)

(a, * (nP A a>£))g = (a, (np A a>£) ^ v) = (nfl A <4 A a, v)g «(-l) r " 1 (<oiAnPAa,v) g = ( - l ) r - W A a , a > ^ v ) g .

But using (67), we obtain: r 4 (72) (a, *(nPA^)) g = ( - lxf-1/ r ^ A a , v r ) g = ( - l)xr-1/ - (cc, n(J^ vr)g; therefore we deduce the identity on T: (73)

\f-l

t r »(nPA<) = ( - l ) r - V ^ v r = ( - 1 )

T-l

giving (68)i) thanks to (70). Applying the * transformation to (68)i), we find: r 1 fr IXB r , (74) - " V r. t *r . * - < - If . r r. n ^ - ( - l ) Then taking 0 = • a, with a an (n - r)-form on M, we have: (75) nr.a-C-l/^tro, with (33), thus (68). <8>

Remark 2. We also define t f and n r on 0-forms by: t r l = l, t r f=f| r , n r l = 0 , n r f=0. Onn-forms, wehave: tro> = 0, nrco = ( - l)n-la>£«j <D = ( - l) 11 " 1 ^ vr, forw^fV*. <8>

A.l INTRODUCTION

309

Remark 3. From the properties of the exterior derivative, we have for all smooth r-forms w on M, using (63): (76) thus: (77)

trdo>=j*do> = d(j*a)) = dtra), tjdusdt,*).

We also have, in a similar manner for all smooth r-forms (5 (78)

8(nr^) = n r ^ .

[We prove this, taking the exterior derivative of (68)i) and using (45): dtr.p = (-l)r-1d.nrfc thus with (77): d t r . p = t r d . P = ( - l ) r t 8p = . n « r (79) | . . . „ _, d . n r P = ( - l ) r . 5 n r p .] r We remark that in an admissible boundary coordinate system, t r u and nru (with u= ^ Ujdx1, J c {l,...,n- 1}, card J = r) are given by: j

(80)

„.J t r u = 2uj(x',0)dx"\

J c { l , . . . , n - 1 } , cardJ = r

n ^ = I Uj.n(x',0)dx'J', r e { l

n - l | , card J* = r - 1.

Using the tangential part of a (smooth) (n - l)-form o> on M, the Stokes theorem (see Bourbaki [2] and Arnold [1]) may be written: (81)

/Mdo»=/rt^,

where the (bounded) boundary T has its usual orientation induced by that of M. All this is true for smooth differential forms. We shall generalize it below. We give here another application of the Stokes formula. Let a and £ be respectively a smooth ( r - l)-form and an r-form, with the same parity. From (45), we have: (82)

d«A « # - a A *8p = d(aA *p),

then integrating on M, and using (81): (83)

(da,«-(a,5p)=/ r t r (a A *W.

But we have from (64):

310

(84)

A - DIFFERENTIAL GEOMETRY t r (otA ,P)=J*(<XA ♦P)=j*a AJ*(»» = t r a A tj{,P).

Then with (68), we obtain Stokes fonnula for all (smooth) (r - l)-forms a, and all (smooth) r-forms f> (85)

(da,P).(a,8P) = ( - l ) r - , / r t r a A i ? n ^ = ( . l ) r - I ( t I « , n r P ) . &

Example 2. We take a = f,fl= o>x, X a vector field; thus with (47), (48) (86)

l ( d f » •x)-( | 0 indf • <">x) = («radf, X), |(f,8a>x) = (f,divX),

n r a> x =n.X,

and (85) gives the usual Stokes formula: (87)

(gradf, X) + (f, divX) = (fl r , n.X)=/ r fn.XdT.

From (85), we can deduce other useful formulas for the Laplace-Beltrami operator. We first take 0 = dy in (85), then we apply (85) replacing a by Sy and 0 by a, to get (88)

(da, dy) - (a, 5dy) = ( - l J^- l /V , n^y) |(d&y , a) - (8y, 8a) = ( - l) r (ti*y, n ^ ) .

Then by difference, we have with A = - (d5 + 8d) (89)

(da, dy) + (Sa, Sy) + (a, A y M - l ) ' " 1 ^ , nrdy) + (nI<x, tj*y)]

for all (smooth) (r - l)-forms a, y. Since we also have (89)'

(dy, da) + («y, &x) + (Aa, y) = ( - D'^KtfY, nr<Ia) + (n r y, tjAx)],

we get by difference: (90)(Aa, y)-(a, Ay) = (-l) r -^(t r y, n ^ a ) - ^ , nj4y) + ( t ^ , n r y)-(n^c, t^y)]. 1.6. Currents; generalized r-forms (or distribution r-forms) Let DT(M) (resp. D (M)) be the space of C°° even (resp. odd) r-forms with compact o co support in the interior M of a C -manifold M, these spaces being equipped with the usual inductive topology ("Schwartz topology") generalizing the scalar case. o Note that it would be better to use notation DT(M) instead of Dr(M), but usual notations are as if M were an open set.

311

A.I INTRODUCTION

The dual space is called the space of odd (resp. even) (n - p)<urrents, and often denoted by 2>;(M) (resp.D;(M)), thus: 2);(M) = (2>n_r(M))*, D\(U) =

(I^JM))9.

We identify currents with generalized (or distribution) even or odd r-forms: for all co in D f (M), we define the odd current Tw by (91)

<TW, *> S / M <» A ♦ ♦ = / M * A ♦ *>, V # € Dr(M).

The mapping T: o> —» Tw extends naturally (continuously) from the closure of DT(M) onto the space of currents I)'(M). We denote also by Dj.(M) this closure, and we call it the space of generalized (or distribution) even r-forms. Similarly we have generalized odd r-foims, and we denote by D^(M) their space; (91) is then written (92)

< T W , 0 > = <OJA

**, 1>,

V*€Dr(M).

[The exterior product between smooth (n - p)-forms and generalized p-forms is naturally defined, and then « A * 0 is a generalized n-form with compact support.] The Hodge transformation has a natural extension from D*r(M) onto |>n^r(M) (and from IT(M) onto D n l r (M)). The exterior derivative of generalized r-forms is also defined by (93) with

<dT M ,+>«<T i l ,8t>, d 2 T =0,

<6TU),0> = <T w ,d0>,

V*€D r (M),

8 2 T =0.

1.7. Application: Jump formula of exterior derivative for a p-form which is discontinuous across a surface T Let T be a regular n~ 1 surface of class Cl in Rn (for example T is the boundary of an open set O in Rn). Let a be an r-form which is continuous on each side of T, which is naturally oriented; G> has a jump a across T: a = a>. - <*2 = [w]-. The exterior derivative of o> in the generalized sense is given by (see Schwartz [lj) (94)

do) = (d(o)- TAo = (do)) -6 r <4Atj<j,

0 = [a)] ,

where (dco) is the "usual" derivative of w (this is an (r + l)-form with continuous coefficients), and where the last term is an (r+ l)-form defined (using (65) with 8p the Dirac distribution on T) by

(95)

(r A a, »«(«r«iAt,c, »«(«£ At,*, P| r )=(- 1)V»%&

312

A - DIFFERENTIAL GEOMETRY

for all smooth (r + D-forms 0 on Rn. In fact, (94) is a direct consequence of the Stokes formula (81) written for the interior part T{ and the exterior part r2> with a smooth (r + l)-form on Rn. Naturally we also have (using (85)): (96)

5o) = (6(D) + ( - lf~l i^nf*, a) ar-form, o = [a>]_.

Remark 4. We assume that a> is of class C 2 , with limits, on each side of T. We can then take the exterior derivative of (94), and we get, using (93):

(97) thus: (97)'

OsSr^Atrld^+dtSr^Atro), d(8 r c0iAt r 0)«.6 r 4At r [da>] r

2 . SOBOLEV SPACES OF r-FORMS; MAXIMA SPACES

We introduce the following space of r-forms of Sobolev type: (98)

H^(M) = {(!>€ L*(M) with all its components <D| in a coordinate system in the Sobolev space H1},

and more generally, for all real s: (99)

H^(M) = {o)€ L^(M) with all its components Q>I in a coordinate system in the Sobolev space Hs}.

We write H* or H* 0 for odd r-forms and H* c for even r-forms, if we have to specify these even or odd r-forms. Let U = (U j , . . . , UQ) be afiniteopen covering of M by coordinate patches 6^ with domains G q and ranges U q . If © and n are elements of H^(M), we define:

<ioo)

(Kn))u%?|

^ | <„<*,<«, j ^ ^ - j a x

where cop and t|j are the components of» and r\ with respect to 9q; we could also use (IS) for that definition. We can show that this space of r-forms is a real Hilbert space with inner product given by (100), which depends on the chosen coordinate covering U. Anv two such inner products are topologically equivalent; we say that the space H*f(M) is Hilbertible, see Marsden [1] p. 54. We also define the two following "maxima11 spaces:

A.2 SOBOLEV SPACES

(101)

313

H r (d,MH<o€L*(M), dcoeL^M)}, Hr(S,M) *{a) €L*(M), &o € L2_i(M)}.

We can equip these two spaces with the following inner products: (^n) H (102)

( d j M ) «Kn)

+ (dw,dn),

(w,n)H (5 M) »(«,n) + (6a>,5n),

with ( , ) on the left hand side given by (15). These are Hilbert spaces with the natural norm. Remark 5. Since the coefficients of an r-form in Hsf(M) are in the Sobolev space Hs(t/) for any local chart U of a C 00 manifold, then the Sobolev spaces of r-forms inherit from the usual Sobolev spaces the principal properties of the Sobolev spaces, essentially Sobolev inclusions and compacity properties: H*(M) -* H*'(M), s > s\with compact inclusion for M a compact manifold. We also have trace properties, such as: The trace map: o> —♦ tfa (resp. n ^ ) is continuous from Hr(M) onto H r (T) (resp. H^i(O), and from H*(M)onto H*-1/2(r)(resp. H*:{/2(r)), foranys> 1/2. [Using a partition of the unit (OL), LX.= 1, with regular a. with support in the local chart U., we can prove that the mapping is onto.] But the imbeddings: Hr(d, M) -* L2(M) and Hr(6, M) — L2(M) are not compact and there are no "Sobolev" imbeddings: Hr(d,M) (or Hr(6, M)) — L^(M), p * 2. That will be specified by the Hodge decomposition: Hr(d,M) contains the space dH*(M) which is closed in Hr(d,M) and in L2(M), and is of infinite dimension 1

(and Hr(6, M) contains the space 6H *(M) which is closed in Hr(6, M) and in Lr(M), and is of infinite dimension). Also the imbedding: Hr(d,6,M) = Hr(d,M) H H r (6,M)- L2(M) is not compact since: HT(M) = {a e Lr(M), da = 0, 6a = 0} is a closed subspace of Hr(d,6,M) and of Lr(M) and has infinite dimension. ® We also define the (Hilbertible) spaces, for real s (103) H?(d,M) = {co€Hsr(M), do>€Hsr+1}, Hsr(6,M) = {co€Hsr(M), 5a>€H*_1}. Then for T a smooth manifold, we have the following basic theorems (see Paquet [l]pp. 113, 116):

314

A- DIFFERENTIAL GEOMETRY

Theorem 1 (Trace theorem). The map t r : a> —♦ tf& = j*(a>) which is naturally defined from the space of smooth r-forms on M into the space of smooth r-forms on its boundary T C°°(AT*M)=DT(M) — C°°(AT*r). Dt(T), extends continuously into a (continuous) linear map from Hr(d,M) onto Hf m(d,T). Furthermore the space C~ (AT*M) ■ Dr(M) is dense in: (104)

H ro (d,M) = kert r .

Theorem 2 (Trace theorem). The map n r : a> —♦n^ which is naturally defined from DT(M) into Dfmml(T) extends continuously into a (continuous) linear map from Hr(8,M) onto H^l[2(8,r). Furthermore the space DT(M) is dense in: (105)

H ro (5,M)=kem r .

We also denote by tj-w and nfa the extended trace of <D for w in Hr(d,Q) or in H^S.Q). Theorem 3. Remark 6.

The spaces Hf l/2 (d,I) and Hf l,2(b,T) are dual spaces. The Hodge transformation * is continuous from Hf(d,M) onto

Hn_f(5,M) and from Hf(5,M) onto Hn_r(d,M) (with opposite parity). The Hodge transformation * extends into a continuous transformation from: H; l/2 (d,D onto H ^ J f a O (therefore also from H;l%,T) onto H'5r2(d,r)). Thus by the Hodge transformation, Theorems 1 and 2 can be deduced one from the other. ® We can prove Theorem 1 (or 2) using local charts in order to work on the halfspace (see Paquet [1]). Remark 7. i) Using Stokes formula (85) with a € Hr(d,M) and 0 € H^(M) or with 0 € Hr+1(6,M) and a € H*(M), we obtain: (106)

tja€H r * l/2 (r) fora€Hr(d,M), ntf€H;1/2(D

for0 €Hr+1(6,M).

ii) But if a € Hr(d,M) (resp. 0 € H*+1(5,M)), then da € H^ ,(d,M), since we have; da € Lj+1(M) and d(da) = d2a . 0 (resp. 60 € Hr(5,M) since 50 € LJ(M), and also 6(60) = 620 = 0), then from (106) applied to da and 60:

A. 3 HODGE DECOMPOSITION (107)

t r da € H r -^(D,

315

nr % € H ^ r ) .

Thus using relations (77) and (78) (generalized to this case) we have: (108)

V* € H; 1/2 (d,r),

%P € Hr-l/2(5,r>.

iii) Therefore we only have to prove that the mappings tp and n^ are onto. Using local charts, we are reduced to prove this result on the half-space (see Paquet [1] p. 114). <8> 3 . THE HODGE DECOMPOSITION (FOR A COMPACT MANIFOLD WITH BOUNDARY)

We remark that: i) d maps Hf(d,M) into Hf+1(d,M), and5maps Hf(8,M) into H r _ ,(8,M). ii) d maps Hf 0 (d,M) into Hf+1 0 (d,M), 8 maps Hf 0(8,M) into H f _ x 0 (5,M) with (see (104), (105)): (109)

I H r>0 (d,M)»{« €H r (d,M), tr-o-0} Hrt0(S,M) = {o>€Hl<6,M), nf» = Q}.

This results from (77). Thus we have the following sequences: i) for the "maxima spaces": H 0 ( d , M ) - H 1 ( d , M ) - . . . - H n . 1 ( d , M ) ^ Hn(d,M) H0(8,M) £ H,(8,M) - ...-H n .,(S,M) £ Hn(8,M), ii) for the "minima spaces": H 0 Q (d,M)4 H I # 0 < d , M ) - . . . - H n - I f 0 (d,M)4 Hn(d,M) H 0 o ( 8 , M ) l H I o (8,M)-...^H n - l o (5,M)i-H n > 0 (S,M), each image of d (or 8) being contained in the kernel of d (or 8) in the following step, that is: Im d(r) - d Hr(d, M) Q ker d(r+1) = {m € Hr+1(d, M), dm = 0} (110) Im S (r) » 8 Hr+1(8,M) £ ker 8(r"1} = {co € Hr(8,M), 8c* = 0} withfinitecodimension. The operators d and 8 may be viewed either as unbounded operators in L*(M), or as bounded operators from one space into another. i) First we define various realizations of operators d and 8 in LJ?(M), denoted d =d o m> d M» 8 o = sm> *M» b y t h e i r domains:

316

(HI)

A- DIFFERENTIAL GEOMETRY D(d0) = EXdJ = HjoCd.M); D(dM) = Hr{d,M);

D(80) = D(6m) = H ro («,M)

D(8M)=Hr(8,M)

(with d0w s djj,*!) = do, dMo> = du, 80« = 6m« = &», SJ^OI = &>, for o> in the relevant domain); d,,,, d M (resp. 8m, b^) are called the minimal and maximal realizations of operator d (resp. 8) in Lj. From the Stokes formula, we have

(112)

I ( $ « , W=(a.^).

VaeEKdJ.fJeEK^), V a € D(dM), 0 6 D(8m),

thus d£, 6jjj (resp. d^j, Sjjj') are adjoint operators (113)

«eV=6S,

( 6 ^ = d£>, (6&V = 8 2 ,

(d2r=52.

From the Stokes formula (or also from (113)), we have the following relations (114)

(where V (114)'

IdrnfiM)1 = kerd 0 ,

(Imd M ) X «kerS 0 ,

I (Im 8 0 ) x = ker d,

(Im d 0 ) x = ker 8,

is the orthogonal of space V) and: |lm8M=(kerd0)1, X Im8W0 = ,, - (kerd) 1WV* V*/ 0

Imd M = (ker8 0 ) x , Imd

AMI \ *0Q

=(ker8)x.

(Actually, we will prove that spaces Im 80, Im 8M, Im d 0 , Im d M are closed.) ii) If we consider 8 = 8m as a continuous operator from H ro (8,M) into L r - j(M) and d a d m as a continuous operator from H r- { 0(d, M) into L, (M), then we can identify their dual mappings, i.e., the continuous mapping from L r - l into (Hr 0(8,M))' (resp. from L^ into (H r . lo (d,M))*) with the operator d (resp. 8); we denote (115)

H;kd,M)«(H ro (8,M))*,

H;k8,M) = (H ro (d,M))\

these spaces being current spaces (or generalized r-forms) since Dr(M) is dense in H ro (d,M)andH ro (8,M). TTien we define the spaces of harmonic r-forms by (116)

H?(M) = {a€Lj(M),da = 0,8a = 0} #J(M) = {a€H*(M),da = 0,8a = 0},

A. 3 HODGE DECOMPOSITION

317

and also the so-called cohomology spaces (117)

^ = ^ ( M ) = {ae^(M), /f r -=/f;(M) = {a€i^(M), tj4t*0}.

Note that + H¥X=IT' , +H~=H? _ , hence for an orientable manifold M, spaces H*T , IQT andH~_r (resp. H~ , fT _ f and /?£_r) are isomorphic, thus: dim //£ = dim #JJ_r, #7 = dim //^_r. Wehave/ffOVl) £ H*(M) for alls, for M of class C°°, thus H?(U) Q C°°. In order to study operators d, 8, d8, 5d with the decomposition of relevant spaces and boundary problems, we have two variational framework levels in L2. 1 s t level: the usual one, associated to the Laplace-Beltrami operator, with the bilinear form (118)

ax(u,v) = (du,du) + (8u,5u) + X(u,v),

(with X > 0), in a subspace of Hr(M). 2 n level: linked to the operator d or 8 separately, with the bilinear form (119) or: (119)'

a+(u,v) = (du,du) + X(u,v), a£(u,v) = (8u,8u) + X(u,v),

in a subspace of Hr(d, M) or Hr(8, M) only. Each variational framework gives interesting specific results. 3.1. Variational frameworks for the Laplace-Beltrami operator We define the following spaces (120)

HJ 0 (M)={a € H}<M), a| r =O }=V; n V ; ,

(this is also the closure of DT(M) in H*(M)), with: (121)

IV; = H*n(M)»{co € Hj(M), if* = 0}

| Hr n(d,8,M) = Hro(d,M)nHr(8,M) (122)

= {co€L2(M), d<o€L2+l(M), SwcL^M), ^ = 0}, (Hr>t(d,8,M) = Hro(8,M)nHr(d,M) = {<o€L2(M), do>€L2+1(M), 8»€l*_,(M), nrco = 0}.

318

A - DIFFERENTIAL GEOMETRY

Later, we will show (see (177)) that (122) and (121) are equivalent. A fundamental property relative to these spaces is given by the following theorem Theorem 4 (Morrey [1]). For each finite system U of admissible coordinate systems whose range covers M, there are constants X^j and CJJ > 0 such that (123)

a(a>,co) + \ v (a>,a>) * Cv ((co,*)^,

V <o € \ £ (resp. V;)

with ((a>,Q>)) defined by (100), and (124)

a(o>,a>) = (da>,da>) + (5a>,&o).

Thus the bilinear form a(a>, 3 ) is coercive on V^ (resp. V^) relatively to L2(M). Formula (123) is called the Gaffney inequality. Consequences. The map: o> —• (d&>, &>, a>) being continuous from H,.(M) into the space L2+1(M) x L~_j(M) x Lr(M), there is also a constant C so that (125)

a(a),co) + X (o),co) 3 C ((o),*))^ ,

V « € V*.

Thus the norms (a(<o,a>) + X (Q>,<O)) and ((<o,a>)) are equivalent, and we can identify the spaces (for a proof see below) v£ = H^ t(M) with H r t(d, 5, M), and \ 7 = H^ n(M) with H r n(d, 5, M), equipped with the norm (a(<*,) + \ (w,o>))1/2. Corollary 1. We assume Mto be compact and regular (of class C1,1). Then the natural mappings a> € VJT -♦ w € Lr(M) and o> € \ £ —• o> € 1^(M) are compact. Corollary 2. The trace mapping o> —» npo) (resp. tpco) is continuous from V" (resp. \£) into H ^ ( D (resp. Hr1/2(D). From the Peetre lemma, we deduce that the cohomology spaces / J * have finite dimension. Then we denote by V^ and VJ" the spaces in V" and V£ which are L2-orthogonal to H~ and H*t respectively; we have the orthogonal decomposition: (126)

V; = Hln(M)=H;*V;,

V^*Hi t (M) = ^ ® V ? . o o Theorem 5 (Morrey). The bilinearform a(a>, 3 ) is V f (resp. V ^-coercive, that is, there is a positive constant C+ (resp. C") 50 *Aaf (127)a(o),o>)^C+((a>,a)))u, V<o€Vf, (resp. a(a>,a>) * C " ( ( c o , ^ ,

VcoeV").

319

A.3 HODGE DECOMPOSITION

We can also obtain this theorem as a consequence of Theorem 4 and the Peetre lemma. Note that (127) is a generalization of Poincare inequality. We use the following new spaces:

(128)

withnotationW& = Wnker5,forW = V;, V J , \ £ , V+. Lemma 2. With definitions (111), (109), spaces Im dM, Imd , ImS M , Im5 are closed in L2(M). PROOF. We first prove that Im dM Q ker d, from which we deduce for the orthogonal spaces (in L ) (129)

( I m d ^ 1 ^(kerd)1.

But for all a€H r (d,M), there are M k e r d, y€(ker d) 1 nH r (d,M), such that a s 0 + y, thus da = dy, and then, from (129): (130)

Imd M = d((kerd) X nHr(d,M)) = d((Imd M ) 1 nHr(d,M)).

Using (114), we have (with (128)): (131)

Imd M = d(ker6 0 nH r (d,M))dV^ = d V ^ .

Now, from inequality (127), we have: (132)

lldcoll2 > ClMI*

V a) € V £ f

H r (M)

which proves that: O

°*

(133) Im dM = d V J$ is closed in Lr+1 and d is an isomorphism from V £$ onto Im d. In a similar manner, we can prove that (134)

Imd/j f dH r o (d,M) = d((Imd 0 ) 1 nH ro (d,M))==d(ker6nH ro (d,M)).

Thus with spaces defined by (128), Imd 0 = dV; 5 = dV; 5 ,

o and we have also inequality (132) on V ~b from which we deduce: o 2 ° (135) Im d0 = d V ~s is closed in Lr+1, and d is an isomorphism from V ~5 onto Im d0. Remark 8. Anticipating on the Hodge decomposition (see below), we have:

320

(136)

A - DIFFERENTIAL GEOMETRY

V £ = Im8 0 fl Hr(d,M) = Im5 0 O V j , o I Vrt=Im6 n H ro (d,M)=:Im6n V; .

Hence we have proved: (137)

Imd MSS dH r (d,M) = d\£=d(Im5 0 n \£), |Imd 0 *dH ro (d,M)=:dV; = d ( I m 8 n v ; ) ,

(138)

I Im8 M =8H r (8,M)«8V;«5(Imd 0 n Vr-), Im8 0 = 8H r o (8,M)=8V^*8(Imdnv;),

and: d is an isomorphism from Im 80 n V^ (resp.Im 8 n V p onto Im d (resp. Im d0), 8 is an isomorphism from Im d0 D V~ (resp.Im dDVJ) onto Im 8 (resp. Im 80). Remark 9. Let V^A (resp. V~A) be the space which is orthogonal to Hro(M) in \ £ (resp. V"), for the scalar product ((a,o)) * (da,d*) + (8a,8#). These spaces are also characterized by: (139)

\£ A = {a€\£,(8d + d8)a = 0},

V;A*(0€Vr~ , (8d + d8)0«0}.

Note that the intersection with harmonic r-forms (see (116)) is: (140)

V^A n HT(M) = ITT9

V;A n HT(M) = H; .

Thus relation (d8 + 8d) a = 0 does not imply da = 0, ha = 0: the kernel of the Laplace-Beltrami operator is not the set of harmonic r-forms (in the general case). ® We deduce fairly easily from Theorem 4 and from Lemma 2 (and the Stokes formula) the following theorem of decomposition of space L2r(M) for a compact manifold M with boundary. Theorem 6 (The Hodge decomposition). Using the notations above (see (117)), we have the orthogonal decompositions of the kernel spaces (141)

321

A. 3 HODGE DECOMPOSITION (143)

and also the orthogonal decomposition of the space Lr(M): (144)

We will sum up these results in a diagram at the end of this chapter. Definition 2. An r-form a> is said to be closed (resp. coclosed) i/da> = 0 (resp. 6o) = 0) and to be exact if there is an(x- l)-form 0 so that w = d0 (resp. coexact if there is an (r+l)-form 0 so that a> = 60). (Of course, we have to specify spaces in this definition.) An exact r-form is closed and the converse is locally true (it is the Poincare lemma) but globally wrong as we can see it from (141). We can also deduce from Theorem 2 another coerciveness result. Let W be the subspace of H^M) which is lAorthogonal to # f (M), thus (with (116)): (145) H*(M) = W©#J(M). Note that we have: (145)' W = (Imd0 © Im6) n H^(M) = (Imd0 n H^(M))©(Im60 n Hj(M)). We also have with notation (124): Theorem 7 (Morrey). The bilinearform a(w, 5) is coercive on W: there is a constant C>Qsothat (146) a(a>,co) £ C (((0,(0)^, V o) € W. r>

Let Vf be the intersection of the two spaces V ; and V J (see (126)): (147)

*? = Vf n V? = {co€H*0(M), co orthogonal toH;®H+T}.

Then from Theorem 2, we also have: Theorem 7\ The bilinearform a(a>, 5) is coercive on V% : there is a constant C > 0 so that (148) a((o,a>) * C ((»,»))<,, V (o € V°T. Remark 10. The orthogonality of H^ and H~ is easily seen from the fact that H^ and H~ are contained respectively in Im 6 and Im d. ® We give the particular cases r = 0 and n. i) In the r = 0 case, we have: (149) L^(M)«L2(M) = Im6 = i^® Im6 0 ,

322

and: (150)

A - DIFFERENTIAL GEOMETRY

HQ = H0(M) = {0},

Hi = ker d = {f € L2(M), grad f = 0}

thus Hi is the space of constant functions on each connected component of M, the dimension of H% is equal to the number of connected components of M, and: Im60=8H, 0(5,M) = {f€L2(M), f*8co x *divX, o) x €H! 0(«,M)}. ii) In the r=n case, we have: (151) L*(M)-Imd«#; ® Imd 0 , (152)

^ = # n (M) = {0}, and

H^ktib.

Thus H^ = {fv*, f constant on each connected component of M), the dimension of /7~ is also equal to the number of connected components of M. Application to some boundary problems. Potentials. Proposition 1. Let (a be a given r-form in L (M), orthogonal to H^ (resp. H^). Then o o there is a unique r-form co+ (resp. or) in VJ (resp. VJ) such that (with a = co+, resp. a)-): (153) (da,dC) + (&x,5C) = (co,C), V C € V ♦ (resp. V r~). Furthermore, for M ofdass C ' ,we have: (154)

dco+eV^,, Sco+cV^

(resp.dco-€V;+1, 5a>- eV;_{).

o o Note that we also have: dco+ € V ++1, &o+ € V +_ j . Definition 3. We call co+ (resp. a>~) the plus potential (resp. minus) potential of CO.

We can prove that <o* (resp. co"") is the solution of the problem: (Sd + Sd^+sco,

(155)+

(resp.(I55)

npco+sO, npdco^sO (8d + 5d)ar = co, tpCO-srO, t j & O - s O ) .

PROOF OF PROPOSITION 1. Thefirstpart of Proposition 1 follows directly from Theorem 2 via the Lax-Milgram lemma. The second part is a regularity result, for which we refer to Morrey [1]. For other regularity results, see Morrey [1]. #

323

A. 3 HODGE DECOMPOSITION

Then we define the Laplace-Beltrami operators A+ and A" as self adjoint negative operators in L2(M) thanks to the variational framework: ( V * , L,J(M), a(w, 3 ) )

(156)

D(A * ) = {<!)€ V * , such that the map ? —»a(a>,t) is continuous on V ± equipped with the L -topology}, = {*> € V * , Ao> € L2(M), a(«,C) = - (Ao>,C),

VC € V * } ,

and A ± w = Aa> = -(d6 + 6d)a>,

VweEXA*).

Thanks to Proposition 1, we have the following characterization of their domains: (156)' or also: (156)" Since (157)

D ( A ± ) = {o>€Vr± , d o e Vr± ,

SmeV*^

D(A+Xresp. A") = {o> € Hj(M), dco €Hj +1 (M), 8O> € H * _ J , npco = 0, npdo) s 0 (resp. t ^ = 0, tj&o = 0)}. Vj Q Hro(6,M) = D(6m),

Vr- Q Hro(d,M) = D(dm),

using notations (111), we can write (158)

- A + = dMSm + 6mdM,

- A - . d ^ + iMd^

Remark 11. The operators (d + 5) + in S L 2(M) defined by: (159)

(d + a^o^dco + fci), V<o€D((d + 6) + ) ,

are self adjoint and elliptic (see Gilkey p.243), and we have: (160)

((d + S ) ^ ) 2 ^ *

inL2(M), 0<r<Sn.

We also have: Theorem 8. Let a> €H r (d,M) (resp. Hr(6,M)), be orthogonal to H^(M) (resp. H~(M))> <o+ (resp. o>-) the plus (resp. minus) potential of<*> (G>* € V^, » • € V~). Then (161) do) = d S d <o+(resp. 6o) = 6 d 6 w-), and we have: (162) do>+ m(do>)+ (resp. 8o)- = (So)-) i.e., do>+ (resp. 6a>-) is the plus (resp. minus) potential of dm. PROOF. Let o> € Hr(d, M) be orthogonal to H$(M). Since do € L2+1 and da> € Im d, we have da> orthogonal to H^. Since 6 d (do>+) = 0 and da)+ € \ £ , then from (161), da>+ is the plus potential of d(o, therefore: 6 do>+ € V^ from Proposition I.

324

A- DIFFERENTIAL GEOMETRY

We would prove the other part of the theorem in a similar way.

This theorem is a simple generalization of theorem 7.7.S of Morrey [1], for o> in the space H ! r (M). Theorem 8 leads to many consequences. Proposition 2 (Money).

Letue 1^(M) be orthogonal to Hr(M). Then there is

a unique r-form o>N € H^(M), \}-orthogonal to Hr(M) such that: (163)

(da>N,dC) + (&>N,5C) «(<■>,?),

V C € Hj(M).

Moreover: (164) do>N = da)+, 6(DN = &)-, toft/t r/ie afore notations and then: da>N € H*+1(M), &>N € H^j(M). Proposition 3.

Let a> € L*(M) te orthogonal to H~ 0/f£. 27te/i fAe/« & a unique

r-form a>D € H^ orthogonal to H^ ®H$ such that: (165)

(da>D,dC) + (8coD,SC)a:(co,C), VC€H^(M).

Moreover we have: (166) d<0D = dG>-, &DD = &>+. and then: da>D € H^+1(M), & » D € H J - P PROOF OF PROPOSITIONS 2 AND 3. The first part is a mere application of the GafTney inequality (Theorem 4) and of the Lax-Milgram lemma. The second part follows from Theorem 8. ® Definition 4. We call o>N the Neumann potential of Q> and o>D the Dirichlet potential of o>. We can prove that a>N, a>D are the unique solutions to the following problems: find coN in H*(M), wN orthogonal to /fr(M), with (167)

(6 d + d 6) wN = OJ, a> given in Lr(M) orthogonal to HT(M) |nrda>N = 0, tjADNsO,

find coD in H*(M), and orthogonal to ff^(M) and H^QA), with (168)

I (5 d + d 6) <*>D m a), u> given in L^(M) orthogonal to fl£(M)e/f~(M), tr(DDaO, n ^ ^ O .

325

A. 3 HODGE DECOMPOSITION

Then using the variational framework H ^ M ) and H1r(M) in L2f(M) with the bilinear form a(<D,C), we can define the Dinchlet Laplace-Beltrami operator AD and the Neumann Laplace-Beltrami operator AN as (unbounded) self adjoint negative operators in I?r(M), by: (169)D

I D(AD) = {(i)€ H*0(M), such that C —► a(a>,C) is continuous on Hj0 equipped with the L2-topology} I = {a) € H*0(M) AO> € L*(M), a(o,t) = ( - Ao>,t), V t € Hj0}

(169)N

D(AN) = {<«>€ H*(M), Ao> € L*(M), a(o,t) = ( - Aco,C), V C € H^}.

We also have the characterizations: (170)

^(A^^lcocH^MXdcocH^M),

SwcHj.J

I D(AN) =r {o> € Hj(M), dco € Hj+1(M), 8o> € H*_ y(M\ n^w = 0, tjAo = 0},

so that we can write with notations (111): (171)

- A D = dM5m + 8Mdm, - A N = dm6M+5mdM-

CONSEQUENCES OF THEOREM 8: The Hodge decomposition of Ur(d,M\

Hr(6,M) From Theorem 8, we easily deduce the following Hodge decomposition of the spaces Hf(d,M) and Hf(6,M) into orthogonal subspaces: (172)

An easy consequence of these decompositions is the following theorem due to Tartar and Murat: Theorem 9. "Compensated compactness" theorem. Let (wn), ( 3 n) be two sequences with <on € Hr(d,M), <3n € Hr(5,M) so that (a>n) (resp. (3 n )) converges weakly to co° (resp. S °) in Hr(d,M) (resp. H^.M)). Then: (173) <on A . S n — o>° A S ° in D^(M), or equivalently: (173)' (oo>n, 3 n ) ^ K ( 5 ° ) , V ♦ eD0(M). PROOF.

From (172), there are sequences (a,,), (Pn), (o n ), ( £ n ) such that:

(174) S

n =8 « n + ^ n .

«n «HJ +1 (M),

|J n €Imd0f~lH^(M),

326

A - DIFFERENTIAL GEOMETRY

are weakly convergent in H*(M), therefore strongly convergent in L2(M). Thus: (175)

(♦«m 3 n) = (^ a n + ^P n » 8 «ii + Pii)-

Using the formula: (176) (♦do n ,5a n ) s (d#Ada n ,a | l ) we immediately see that this sequence converges as well as each of the term of the & development of (*on, 3 n ). We now prove the equality between the spaces (see (121),(122)): (177)

Hrt(d,8,M) = Vj,

Hr n(d,8,M) = Vr" .

We first prove that if a> € Hr t(d,8, M), then o> € Vj. Let h+ be its projection onto i/JXM), &+ the plus potential of o> - h+, thus: (178) then: (179)

a>-h+ = (dS + 8d)u>+, a>+€\£

(withda>+€\£+1, Sa^eV^)

da)«d8(dco+)€l^fl, 8<DS=8d(8o>+)€^1,

da>+ is the plus potential of dco from Theorem 8, thus 8 d o>+ € V£ from Proposition 1. Besides, we have 8a>+€ V ^ n i m 80 from (138), thus 8a>+ is orthogonal to H^ (see diagram), thus 8o>+€V£, then 8o>+ is the plus potential of 8<o, and Proposition 1 implies that d8a>+ € V^; thus we have w € V£. # oA Remark 12. From the proof above, it also follows that if <o € V J, then its plus potential co+satisfies 0*=5 (o+ = (5o))"\ i.e., is the plus potential of8oi. Also if cocVf then its minus potential a>- is such that a- = dor = (da>)", i.e., is the minus potential of dco. 0 REGULARITY RESULTS (FOR REGULAR MANIFOLDS WITH BOUNDARY):

The Hodge decomposition of (regular) Sobolev spaces. We give here, without proof, the following decomposition (see Marsden [1] p. 62, see also DautrayLions [1]), for all positive integer s, with usual notations (see (121) for s = 1): (180)

with #J(M) = {a€Hj(M), da=rO, Sa = 0}, (181)

Hjt(M) = {a>€Hj(M), n ^ =* 0}, f£ n (M) = {a> €Hj(M), tj*> = 0}.

327

A. 3 HODGE DECOMPOSITION

[This follows from regularity results on elliptic operators, see Paquet [1], Money [1]. For a regular manifold M (C°°), the hypotheses: co € H*(M), dto € H*+1(M), 8to € H^flVf), tfco = 0 (or n ^ = 0), implies <o € H^^M). Or also if co satisfies: A<o = (d 8 + 8 d) <o = f € H*~ *(M), with tj*) = 0 (or nj*) = 0), then co € H^^M).] 3.2. Variational frameworksforthe operators d8 and 8d For many boundary problems, it is interesting to use other variational frameworks which do not a priori suppose H l regularity. They are linked to the d8 or the 8d operator instead of the Laplace-Beltrami operator. 3.2.1. Inhomogeneous problems a) We first consider the inhomogeneous problem: Let f € L*(M) be given, with 0 < r < n, find u € Hr(d, M) so that: (182)

8du + u = f, t r u = 0.

We define in the space V = Hro(d,M), the bilinear form: (183)

a(u,v) = (du,dv) + (u,v),

u, v € V.

Then the problem (182) can be rewritten as:findu in V such that (184)

a(u,v) = (f,v),

Vv€V.

From Theorem 1, V is a closed subspace of Hf(d,M), and the bilinear form a(u,v) is V-coercive. Therefore, there is a unique solution u of (184) (then of (182)) in V. Note that (V, L2r(M), a(u,v)) is the variational framework relative to the selfadjoint operator: (185)

D ; +1, with D£ = 8 d, and EKD^) = {u e Hro(d,M), 8d u € L*(M)}.

b) In a similar manner, we can solve the problem in Hr(8, M), 0 < r < n: (186)

d8u + u = f, f given in L*(M) n r u = 0,

with the variational framework: V = * V a Hro(8, M), and (187)

a(u,v) = (8u,8v) + (u,v), u , v € V ;

328

A - DIFFERENTIAL GEOMETRY

then (186) is equivalent to: (188)

a(u,v) = (f,v),

W€V,

which has a unique solution in V. We define the associated selfadjoint operator (189)

D ; + I,withD; = d8, andD(D;) = { u € H ^ M ) , d8u€L*(M)}.

Note that it is possible to solve (182) with f € V * ( H ^ M ) ) * (resp. (186) with: f € V'-H^ftM?), sinceD'AM) is dense in Hro and H ^ M ) . c) If we substitute Hr(d, M) for the space V in (184) (resp. 11,(8, M) for V in (188)), we solve the following problem: (190)

6du+u=f npd u = 0,

d6u+u=f resp. tjiUsO.

Indeed the variational framework (Hr(d,M), L,(M), a(u,v))(resp. Hr(8,M), L*(M), a (u,v)) defines the positive selfadjoint operator D+ +1 (resp. D" +1), with: (191) D(D+) = {u€Hr(d,M),8du€L^(M),a(u,v) = (8du + u,v), VveH^d.M)}, that is, using Stokes formula: (192)

llXD^^IucH^MXSducL^MXnrdu^O}

D + u*8du, Vu€D(D + ), and also for D"

(193)

D(D-) = {u€Hr(8,M),dSu€Lr(M), trdu = 0} |D"u = d8u,

VucEKD").

d) Then we consider the inhomogeneous problem: (194)

8du = f, tru = 0,

f given in L*(M),

or in variational form with a0(u,v) = (du,dv) (195)

a0(u,v) = (f,v),

V v € V = Hro(d,M).

In order to have a solution to this problem, it is required that f be orthogonal to the space ker d0, therefore that f € Im 8, thus 8f = 0 with f orthogonal to H~(M). Then using the L2-decomposition (with (136)):

329

A. 3 HODGE DECOMPOSITION

(196)

we have from the proof of Lemma 2 (197)

a0(u,u) = (du,du) ;> C+llulf, ^

2: C+ (u,u) 2,

H r (M)

VueV^.

L O

This is a (generalized) Poincare inequality. Thus a0(u,v) is V "^-coercive and from Lax-Milgram lemma, the problem (195) has a unique solution in the quotient space o Hro(d, M)/ker d0, or also in space V 78. We have thus solved the problem: find u in Hr(d, M) satisfying: 8d u = f, 8u = 0, t r u = 0, u orthogonal to H~(M). We can also solve (195) with V = Hr(d, M); it corresponds to the problem: (198)

8du = f , n r d u = 0, f given in L^(M),

with f orthogonal to kerd, thusf €lm8 0 , i.e., 8f=0, n r f=0, f orthogonal to ff£(M). Then using the L2-orthogonal decomposition (with (136)): (199)

and using inequality (197), we obtain a unique solution u of (198) in the quotient o space Hr(d, M)/ker d identified with V *$, thus corresponding to: 8du = f, 8u = 0, nj4u = 0, nru = 0, u orthogonal to i/£(M). In a similar manner, we can solve the problem (200)

d8u = f, nru = 0, f given in L,(M),

with f orthogonal to ker 80, then (200) has a unique solution u in H,.0(8, M)/ker 60, thus in Im d n Hro(6, M) (that is with du = 0, u orthogonal to /J^(M)); or (201)

d8u = f, t^u^O, f given in L*(M),

with f orthogonal to ker 8; then (201) has a unique solution u in Hr(8,M)/ker 8 identified with Im d H Hro(d, M), that is with d u = 0, u orthogonal to H~(M). 3.2.2. Homogeneous problems We have the following results for boundary value problems (see Paquet [1]): 1/2

Proposition 4. i) Let u0 € H, (d,0, r< n; the problem:findu in Hr(d,M) with (202) 8du + u = 0, tru = u0,

330

A- DIFFERENTIAL GEOMETRY

has a unique solution. it) Let\xx £ H~1/2(8,0, 0<r<n; the problem:findurn Hr+1(8,M)$o that (203)

d8u + u = 0,

nru = Uj,

has a unique solution. PROOF. Using a lifting of the boundary conditions (thanks to Theorems 1 and 2), we are reduced to resolve a problem of type (182) or (186). We can also prove directly, following Paquet, that u is the projection of 0 onto the convex set: (204)

E ^ f v c H ^ M ) , t r v»u 0 }, resp. E Ui = {v€Hr(8,M), n r v=u,}.

Remark 13. Note that the orthogonal space in Hf(d,M) of Hro(d,M) is (205) Z = {a€Hr(d,M),(da,d» + (a,» = 0, VP€H ro (d,M)}, therefore: (205)' Z = {a€Hr(d,M),6da + a = 0}. In a similar manner the orthogonal space of Hro(6, M) in Hr(8, M) is given by: (206)

Z ={a€Hr(6,M), (8a,80) + (a,» = O VP€H ro (8,M)} = {a€H r (8,M),d8a + a = 0}.

Thus we see that the solution u of (202) (resp. (203)) belongs to Z (resp. to Z). Proposition 5. (207)

i) Let the in H~ 1/2 (8,0, 0 < r< n; the following problem:

6du + u = 0 inM, npdu=0,

or in a vocationalform: (207)'

a(u,v) = (du,dv) + (u,v) = ( - l ) W r v ) ,

Vv€H r (d,M),

has a unique solution u in Hf(d,M), ii) Let(\>€ H^1/2(d,r), 0 < r< n; the following problem: (208)

d8u + u = 0inM, tjAiscJ),

or in a variationalform: (208)'

a(u,v) = (8u,8v) + (u,v) s (- l)r(<|>,nrv)

has a unique solution u in Hr+1(8, M).

^ve^ftM),

A. 3 HODGE DECOMPOSITION

331

Let v » du (resp. w = 8u) in (207) (resp.(208)), applying d to (207) (resp. 5 to (208)), we see that v is the solution of (203) for u{ = $, and that w is the solution of(202)foru0 = <|>. 3.2.3. Capacity or Calderon operator relative to ( - A +1) and to M Using Propositions 4 and 5, we can give: Definition 5.

The mapping:

c;:u 0 €H r - 1 / 2 (d,0-*(- ljFnjducH^foO.

0<r<n,

with u the unique solution of (202) is called the capacity operator (or the Calderon operator) relative to (- A +1) in r-forms. We also define C~: ut € H; l/2 (8,0 — ( - l)r"l tr b u € H^ 1/2 (d,0,

0 < r < n,

with u the unique solution of'(203). Proposition 6. The mapping C* (resp. C~) is an isomorphism from H~ 1/2 onto H; (6,r) (resp, from U;V2(b9T) onto H;l/2(d9T)\ with: (209)

C;C+ = l,

C ; c ; = I,

(dj)

0<r<n.

Remark 14.-4 regularity result on Propositions 4 and 5. We assume that u0 € H / (T) (resp. u lf 0, q> €Hj /2 (0), then the solution u of (202) (resp. (203), (207), (208)) satisfies u € H^(M). PROOF (in the case of problem (202)). Let U 0 € H*(M) with t r U 0 = u 0 . Then with u = U 0 + U, we have tofindU solution of:

(202)'

6dU + U = - ( S d U 0 + U0)€(Ho(d,M))\ t r U = 0.

Using the variational form of problem (202)', we obtain a unique solution U of (202)' in H*(d,M), with: 6 U = - 6 U 0 € L2(M), thus U € Hj(M), andu = U + U 0 € H*(M). Furthermore, if we suppose that T is C00, with u0 € C™(T) (resp. Uj, 0, q>), then <8> the solution u of (202) (resp. (203), (207), (208)) satisfies u € C~(M). Some consequences. Then the capacity operator is also a continuous mapping from H* (T) onto H* (T). Using interpolation between H"1 (d,0 and H^ (0, it is thus also a continuous mapping from X0 = [H 1/2 (D,H; I/2 (d,n] 1/2 onto YQ = [H^ 2 (r),H;: 1/2 (8,0] 1/2 (with u the solution of (202) in (H*(M), Hr(d,M)] for u0 in X0.

332

A- DIFFERENTIAL GEOMETRY

This also implies regularity results for problem (202)). This allows us to consider the capacity operator in the L2r(0-framework if we have: (210) X o = {u o €L?(0, d^€H r -i / J(D} f Y 0 -fu I CI^n > 8 u, € H ^ O } . Finally if r is regular (C°°), then C* and C~ are isomorphisms in C°°(T).

Remark 15. Applying the Hodge transformation to relation: (211)

( - l)rnrd« = C j u 0 , u0 = tru (withueH^M), i^eH^d^OX

we obtain, for all Vj taH^£i(*»0. withv=*u, vx = ( - if'1 *u0, 0 < r < n - 1, (212)

C^ r _ l V l =:(. l ) n - r - V * u = ( - I f - ' v . d u = ( - lf-T-l.nTdu

thus (213)

C-n_r_ir.C;,

= (- lf-l*C+Tu0,

C^f.1-(-l)P.C*?.

The capacity operator has many other properties. Let us equip the spaces H~ I/2 (d,0 and H^1/2(8,0 with the quotient norm: KL-1/2^= 0

(214) W

1/2

inf

1/2

((dv,dv) + (v, v))

( d , 0 - v € H^d.M), t r v« u 0

H-l/2„nS . o Jftk «8v,8v) + (V,V)) H r I/A(5,r) v € Hr+1(S,M), n r v*o

1/2

The infimum is at v = u, the solution of (202) for the first formula, and of (203) for the second; thus: llu0HH.1/2(dn = ((du,du) + (u,u))1/2 (215)

1/21/4 u 11/2 IWI , „ =tofdull= [(8 d u,8 d u) + (du,du)] * = ((du,du) + (u,u)) \ I Hrm % (o,r)

Therefore we have proved that the capacity operator C* (or C~) is unitary: (216)

»CX» H . 1 / 2 ( 5 n = «u 0 l H . 1 / 2 ( d n .

We also have

333

A. 4 HODGE WITHOUT BOUNDARY

Proposition 7. The capacity operator is a positive symmetric operator in the following sense (217)

(u o ,c;u o )= s 8u o « H;: i/2 (dn >0,

(218)

(u 0 ,C*u 0 ) = (C;u 0 ,u 0 ),

Vu 0 €H; 1 / 2 (d,r), u o *0,

Vu 0 , u 0 €H r - 1 / 2 (d,0.

PROOF. Using Stokes formula (85) with a = u the solution of (202) and M d u with u the solution of (202) for u 0 , we have: (219) thus (220)

(du,du ) - ( u , 6 d u ) = (-l) r <t r u,npdu>, (du,d u ) + (u, u ) « <u0,C+ u 0> = <C+u0> u 0>,

from which we deduce Proposition 7.

The capacity operator is therefore also a positive symmetric operator in LT(T)9 with C* as an unbounded operator with (221)

D(C>{u 0 €L 2 (r), C*uo€L*(0}.

Note that we can also define a capacity operator for - A (instead of - A + I) as in definition 4, here using the unique solution in Hf(d,S,M) of: 6du = 0, 5u = 0, u orthogonal to H^(M)9 tru = u0 (resp. d5u = 0, du = 0, u orthogonal to #£(M), npUsu^, but since nj<6du) = 8(npdu) = 0, the capacity operator is not an isomorphism from H- 1/2 (d,OontoH- 1/2 (6,D. 4 . THE HODGE DECOMPOSITION FOR A COMPACT MANIFOLD WITHOUT BOUNDARY

For a compact manifold T without boundary, results are much simpler. Let n be the dimension of I\ First using a variational point of view with the bilinear symmetric form a(a),a>) defined by (124) on H1 (O, we have: Theorem 10 (Morrey). For r = 0 to n, and for every coordinate covering U ofT, there are constants CJJ > 0 and Xy such that: (222)

a(a>,a>) + AuKco) > Cv ((0,0)^,

with (Ka))^ defined by (100).

V<* € HJ(T),

334

A- DIFFERENTIAL GEOMETRY

Thus the bilinear form a(a>, 5 ) is coercive on H^(0 with respect to LJ?(0Since the canonical map Hr(r> —»Lj.(0 is compact, we can apply the Peetre lemma with the maps: A co € Hj(D - i (dco,8co) € L* + ,(0 X L*.,(0,

A co € H^O - ? CO € L*(0-

Theorem 11. For r=0,...,n the space: (223)

ffr(0

= kerA,={co€Hj(0, d c o - O ^ - O }

/5 a finite dimensional vector space. The image space (224)

ImA1-{(cto»€ 1^,(0x1^,(0, V<O€HJ(0}

is closed, and there is a positive constant C so that: (225) with:

a(co,to) > Cftco.co)^, V co € V r ,

(226)

Vr * {co € H*(0, <«> is \}-orthogonal to HT(T)}.

Thus a(co,co) is Vr-coercive. Using Definition (101), (102), we derive that we can identify the space H^( T) with (227)

H r (d,8,0 = H r (d,0HH r (8,0 = {co € L*(0, d co € l £ , (0, 8co € l £ , } .

Then we have: Lemma 3. The spaces Im d = d H r (d,0 and Im 8 = 5 H r (8,0 are dosed in L,.(0. PROOF.

As for Lemma 2, we prove:

(228)

Im d = d ((ker d ) l D H r (d,0).

From Stokes formula, we have (as in (114)): (229)

(8cc,|}) = (a,dP),

Va€H r (8,M), VP€H r _,(d,M),

and (8a,df) = 0, therefore: |(Im8) 1 =kerd, (Imd) 1 =ker8, (230)

|Im8=(kerd) 1 , Imd = (ker8) 1 . Since we have:

A.4 HODGE WITHOUT BOUNDARY (231) thus:

I m d c k e r d , Im6 £ ker6,

(232)

( I m d ) 1 ^(kerd) 1 ,

335

(Im*) 1 2(ker8) 1 ,

so that, with (228) and (230): (233)

Im d = d ((Im d) X n H ^ O ) = d (ker b nH r (d,r)).

But K 0 d = f kerS H Hr(d,r) is a closed subspace of H*(D which contains HT(T). Thus it is decomposed into: (234) V0 = HT{T)(BV00, V^iH^T))1

C\V0 a closed subspace of Vr,

and we also have: (235) Imd=:dK0 = dK 00 . Now from inequality (225), we have: (236)

lldcoll2 > C IMlJ,,

V o> € V00 ,

which proves that Im d = dVQO is closed in L J + 1 ( 0 , and that d is an isomorphism ® from VOQ onto Imd. We then derive the theorem: Theorem 12. (Hodge decomposition for a compact manifold without boundary). We have the l?-decomposition of the kernel spaces: (237) with (238)

ker d = Im d $ # , ( 0 , ker & = Im 6 <&HT(T), #r(I>kerdnker6

and also the L -orthogonal decomposition ofspace L r (r): (239)

We shall sum up these results in a diagram at the end of the chapter. n The numbers b r = b r (0 = dim HT(T) and x r = 2 ( - D\(X) Definition 6. r=0

are respectively called the r-Betti number and the Euler-Poincare characteristic. A compact orientable manifold T without boundary, of dimension n, satisfies: 0 Xp = 0 if it is odd-dimensional, ii) bT(T) = bfl _ r(T) if it is connected, iii) Hn _ j(D = {0}, 1ST (D = {0} if it is simply-connected.

336

A - DIFFERENTIAL GEOMETRY

If r is contractible to a point, then HJJ) = {0} for all r different from 0 and n. Note that HQ(T) is the space of functions on T which are constant on each connected component of I\ therefore: dim HQ(T) is equal to the number of components of r. The variational framework given by H 1 (D and the bilinear form a(o>,a>) allows us also to give the following proposition: Proposition 8.

Let mbea given r-form in L 2 orthogonal to HT(T). Then there exists

a unique r-form <o0 in Vf (i.e. in Hj(0, orthogonal to HT(T))f such that

(240)

(da>0,do+(&>0,ac)=(co,o,

v c € H^O.

We can write (at least formally, or in (H*(0)') (241)

<o = (d6 + 6d)o)0 = A(«)0.

Definition 7.

The r-form o>0 is called the potential of a.

Note that Proposition 8 is a simple consequence of Theorem 11 (inequality (225)) and of the Lax-Milgram lemma. There are also regularity results of a>0 with respect to a> (see Morrey [1] p. 296). Proposition 8 allows us to give a Hodge decomposition of L2r(H in a more precise form: Theorem 13.

Letue L*(0; then there are unique forms h, a, 0 with h € HT(T)9 l

a € H, + 1 (0, 0 € H Tmml(T)such that (242)

a> = h + 5a + dp, da = 0, 60 = 0, a = da>0 , 0 = 5a>o,

where a>0 is the potential ofw - h; h, 8a and df$ are mutually orthogonal in Lr(T)* Thus (243)

L2r(T) = d H ^ D ® SH;+1(r)0i5Tr(n.

Furthermore i/o> € H^Ot then we also have 6a and df$ in H r ( 0 . For a regular (C°°) compact manifold without boundary, we have a similar Hodge decomposition of spaces H*(0, s € N, given by (see Marsden [1] p. 58) (244)

H*(D = dH^}(r)e5H^}(n <BHT(T).

The set of harmonic forms is contained in the space C~(0> and (245)

HT(T) * {a € Cr°°(0 (or H*(0), Aa = 0}.

337

A. 5 HODGE UNBOUNDED

[Note that here the relation Act = 0 implies da s 0 and 6a = 0.] Actually we can prove that (o € HT(T) is of class C*"1,11 if T is of class C^*1 (see Morrey [1] p. 296) then that a) € HT(T) is of class C°° (or even analytic) if r is so. Thus HT(T) is independent of s. Using duality, we prove that the decomposition (244) is true for any reals. Remark 16. Let us apply the trace map tp (or also np) to the "sequences" of H (d,M) (and to the Hodge decomposition) for M a compact Riemannian manifold with boundary I\ We thus obtain the diagrams: H0(d,M)

l*r H^dpO

- 1 * H^d.M)

l*r

- ^

—» H 2 ( d , M )

1/2

(d r ,n - * d —»

-i/2dr

—*

ker

1/2

(d r ,n

- ^

to ker

.i/2dr

Diagram 1 The image of the spaces Hr(d,M) by t r is in the framework of the Sobolev spaces Hj(0, s = - 1/2; if a) € # r (M), then tp© is not necessary in the space HT(T). We have (see below): dim t r Itt > dim HT(T),

dim n r ^ + 1 ^ dim HT(T).

For a comparison of the Euler-Poincare characteristics of M and T, see Gilkey [l]p.246. 5 . THE HODGE DECOMPOSITION IN L 2 FOR UNBOUNDED MANIFOLDS

We are essentially interested in the Hodge decomposition for L r(M) when M is the complement of a bounded open set £2 in Rn. The main difficulty lies in the new spaces to introduce. First the study for M being the whole space R n is interesting by itself and will allow us to treat the general case in view. 5.1. The Hodge decomposition for L2f(Rn) We will extensively use the Fourier transform of r-forms in L2f(Rn) or in S'r(Rn), that is, with coefficients (in a Cartesian system of coordinates) in L2(Rn) or in 5"(Rn), the temperate distributions.

338

A- DIFFERENTIAL

We define

F<o(E) = &(t) = 2 «,(£) dx, as the Fourier transform of co.

Lemma 4 . (246)

GEOMETRY

The Fourier transforms ofdw and 6Q> are: F(da>) = U A & ,

F(&>) = - i i ^ ,

i.e., the action ofd (resp. b) is (up to a coefficient i or - i) simply the exterior (resp. interior) product by £. PROOF. T h e first equality is a direct consequence of the definition (formula (3)) of d, t h e second equality is obtained by duality: from (46)' and (21)' a n d the Parseval equality, we have: (247)

(JT(dv),Fu) s (Fv,F5v) = (it Av,fi) = (v, - ii^fi).

The operators o(d) = it A . and o(S) = - i i^ are called the symbols of the operators d and 8 (see Gilkey [1]). From the properties of the interior product, we have

(248)

i e «A^) = ( i ^ ) ^ - « A i ^ - ^ - E A i ^ ,

thus, corresponding to (49): (249)

U i ^ + i ^ A ^ = /r((d5 + 8d)v) = ^ = F(-Av). <8>

Then w e have t o introduce some spaces (for n > 2): Definition 8 . T h e Beppo Levi spaces on R n . Let W ! ( R n ) be the closure of the space of smooth functions, with compact support or rapidly decreasing at infinity, for the norm:

(250)

MwI(Rn) = ( f R „ 2 | ^ | 2 d x ) 1 / 2 .

Let W*(R n ) be the space of r-forms with coefficients in W*(R n ). It is also the closure of the space of smooth r-forms (in Z> r (R n ) or in 5 r ( R n ) ) for the n o r m : (251)

w»(Rn)=(lldft>ll^i+

I W

i ? i>1/2'

This is a direct consequence of (3), since from Parseval equality: (252)

(2n)nB o>| * .

= K A SB2 + lli^ll 2 = (i { « A &),&)+« A i £ u,S) - (?&,&).

A. 5 HODGE UNBOUNDED

339

We can also characterize the Beppo Levi space W*(Rn) by (see Lions [2]): (253)

Wl(Rn) = {ueL q (R n ),q=HT2. ^€L 2 (R n ), i=l,...,n}, n > 2 ,

or also using a weight function (see Nedelec[l]), for example in the n = 3 case: wkR 3 ) = {u, — W

£L2(R3), r= | x | , ^ € L 2 ( R 3 ) , i= l,...,n}.

We then define spaces Wr(d,Rn), Wr(S,Rn) and Wr(d,S,Rn) by: Wr(d,Rn) = {<o e Lq(Rn), q given by (253), do* € L2(Rn)} (254)

I Wr(6,Rn) = {o) e Lq(Rn), q given by (253), 8co e L2(Rn)}, I W r (d,5,R n )=W r (d,R n )nW r (5,R n ).

Thus a) € Wr(d,Rn) orWr(5,Rn) implies: <o € L^/R^nS^R"). We could also use a weight function instead of q, in (254), as before for W*(R3). Remark that W'(Rn) for example is strictly contained in the space (which contains constant functions): {u€LgMj(Rn)nS'<Rn), ^€L 2 (R n ),

i=l,...,nj.

From (246) and the definition 7, we see that d maps w|(R n ) into L2+1(Rn), 6 maps w'(Rn)intoL2.,(Rn), and by duality d maps L^R") into W;+\ (Rn) = (WJi+1(Rn))' and 6 maps L2(Rn) into W ^ (Rn) = (W)_ ^R11))'. Note that we can also define Beppo Levi space W2(Rn) to order 2 as the closure of smooth functions (with compact support) for the norm: (255)

Hullw2 = ( 2 / R n | ^ | 2 d x ) 1 / 2 ,

and then we define spaces W2(Rn), and by duality WT2(Rn). Let P and Q be the following operators in Lr(Rn): (256)

FPu(t) = \ ifi A u),

FQu«) = \ U (i^u).

To be more precise (and not to use space W2(Rn) nor its dual) we have to write: (256)'

FPu(£) = i ^ « A u),

FQu«) = U \ (i^u),

since - A is an isomorphism from W* onto WJT1. Thus we have

340

A - DIFFERENTIAL GEOMETRY

Theorem 14. (The Hodge decomposition in L2(Rn)^. The operators P and Q are hermitian orthogonal projectors with P + Q = I, and correspond to the orthogonal Hodge decomposition : (257)

I^(Rn) = kerd eker6=Imd e i m S ^ d W ^ j e&wj +1 .

From the definition and the properties of d and 5, operators P and Q are continuous in L2(Rn). Using (256), we have: P 2 = P, Q 2 = Q, and (from (249)) P + Q = I. Therefore P and Q are supplementary projectors, and they are also hermitian since: PROOF.

(FPu,/V) = (L \ £ A u,$) = (4 U u ,* A $) = tfU X £ A V) = (Fu,FPv). Then, using (256), we have: (258)

kerP = ImQ = dW^1(Rn), kerQ = ImP = 5W*+1(Rn).

5.2. The Hodge decomposition of L2 (M), M the complement of a bounded open setinR n We also define Beppo Levi spaces in this case. Definition 9. (Beppo Levi spaces for unbounded domains). Let Q, be an open regular (Lipschitz) bounded set in Rn, and let M be the complement ofCl in R n . Then W*(M) (resp. W1 (M)) is the closure of the space of smooth functions, i.e., D(Q') (resp. D(&)) with Q* = R'NQ, relatively to the norm

Let Wy(MXresp. w]l0(M)) be the space ofr-forms o«M = 5 \ with coefficients (for a Cartesian system of coordinates) in W!(M) (resp. W10(M)). Note that W!(M) is contained in the space (which contains constant functions) {u€L?oc(M), ^ € L 2 ( M ) , Vi}, withM = 5 \ We can also define spaces W*(M) and Wr(M) by restriction to M of the elements of W ! (R n ) and W^(Rn). Then we define the spaces Wr(d,M) and Wr(5,M) as those of the restrictions to M of elements of W*(Rn) and W*(Rn), and then Wro(d,M) and Wro(6,M) as the subspaces of W*(Rn) and W,(Rn) with vanishing tangential (resp. normal) trace.

341

A. 5 HODGE UNBOUNDED

Thus spaces W*(M), W^M), Wf(d,M), W (8,M) have the same trace properties and the same trace spaces for M s Q bounded open set or its complement. From these properties, we fairly easily obtain (using extensions of r-forms to Rn) that spaces dW f-1 (d,M), 5Wr+1(6,M), dW f - 1 Q(d,M), 5Wf+1 0(5,M) are closed inL 2 r (M). Using all these spaces we can prove that the Hodge decomposition of L r(M) remains valid as in the case of a compact bounded manifold, see theorem 6, formula (141), (143), (144), with finite dimensional spaces of cohomology defined as above by: | ^ ( M ) = {(D€L2(M),da) = 0,8a) = 0, n ^ r O ) |H;(M) = {o>€Lr(M),da) = 0, 60 = 0, ^ = 0} giving then: ker d = Im d ®lfv ker & = Im h ®H~. Remark 17. (260)

As in the bounded case (see (177)), we have:

Wro(d,M)nWr(5,M) = win(M), Wro(5,M)flWr(d,M) = W^(M).

Indeed using a smooth function 0 with compact support in Rn, 0 = 1 in a neighbourhood of the boundary, we easily prove that for any u in the first left hand side of (260), 4>u is in the Sobolev space H*(M) thus is also in W^(M), then we have (1 - o)u in W^ft") thus is also in W^(M). Note that as in the bounded case: (261)

Wr(d,M)HWr(6,M) D wJ(M)strictly.

We can also prove the trace property: trWr(d, M) = trWr(d,S, M) = tj<Wr(d,S, M) H ker &), similarly for 5, but note that the mappings u —>(tru,nru) from Wr(d,8,M) and Hr(d,5,M) into H~ (d,T) x H~_ j (d,0 cannot be both surjective: if not, every element u in space Wr(d,5, M) or in Hr(d,&, M) will have an extension U to Rn in Wr(d,5, Rn) = W*(Rn), thus giving U | = u € W*(Rn), in contradiction to (261). <8> Finally remark that the cohomology spaces H~, H^ are contained respectively in W^(M) and in W^(M), since they are also in the Sobolev space Hr(M), and thus we have an orthogonal decomposition similar to (126): (262)

wJ,(M) = # r -(M)0W r", ,

W^(M) = flJ(M)eW +.

Let a((l),(o ) be as in the bounded case, the bilinear form:

342

(263)

4 - DIFFERENTIAL GEOMETRY a(a>,a>*) = (da>,da>') + (&>,&i>*).

Then the main tool for proving results similar to the bounded case is the following theorem, analogous to Theorem 5 for Beppo Levi spaces. o o Theorem 15. The bilinearform a(a>,a>') is coercive on space W ^ (resp. W J): there exists a positive constant C such that: (264)

a(«,«') £ C ((«,»'))„,

Vo) € W J (resp. o> € Wf),

w/7/z the notation (usedfor (100)): (265)

(((0,0))) = U

2 q«0

2 G

q I j»l

I-S5H <k n

M?/re/e fAe cfca/t q = 0 & unbounded, the others being bounded. Using a partition of unity (♦), q = 0,..., Q, we decompose all elements <o in W*(M) into o> = 2 o>a , a>a = 0_a>, with support in the chart Ua. PROOF.

q=0

Then we have two cases: first on each bounded chart, we apply the GafTney inequality (123) of Theorem 4, then on the unbounded chart UQ we have: Then summing (and applying Peetre Lemma to eliminate the L2 norm relative to terms in the bounded charts) we obtain the theorem. 5.3. Comparison between the cohomology of M = Q, its complement and its boundary Let M be (the closure of) a bounded open (regular) set of Rn, or its complement. Atfirstwe need the following lemma: Lemma 5. The Hodge decompositions ofthe trace spaces R~ H; ,/2(8,r) of Hr(d,M) are: H;

(266)

(d.Oof H,.(d,M),

1/2

PROOF. Let <o be in H~ (d,r). then using the potential decomposition, there exists <D0 and a harmonic r-form h such that a> = (d6 + &d)a>0 + h = - Aco + h. Then we have do> = - A do>0 € H^'ftO, therefore d« 0 e H^2(r>, «<-> = - A 6o>0 € H ^ D , and thus &>0 e Hr^2(r)> from which we obtain the lemma.

343

A. 5 HODGE UNBOUNDED

Then we have (with Definition 2): Proposition 9. Let o> € L*(M) be a closed form, with if* an exact form. Then o> is an exact form. PROOF. For M bounded or its complement, let a> € L*(M), do> = 0, tro> = dr9, 9 € H ^ ( 0 from Lemma 5. Then if M' is the complement of M, there exists 9 € H J ( M ' ) , with t r 9 = 9. Let 3 = <o on M, d 9 on M\ Then we have d 3 = 0, with 3 € Lj(Rn), then 3 is exact ( 3 = d* with <fr in W*+1) and thus a> the restriction of 3 to M is exact. ® Remark 18. Let u>t and a>2 be two closed r-forms on M and M' (respectively) in L , with tj-o)! = tr<o2. Then 3 s o)1 on M, a>2 on M' satisfies d 3 = 0, with 3 in L^(Rn) and thus 3 is exact, therefore <ax and a>2 are exact. Some consequences. With notations d{ and de for d in M (bounded) and M' respectively, we have: (267)

trker dt n trker de = Im d r ,

and there arefinitedimensional spaces if*(0 and H^(T) so that: (268)

I t r kerdi= lmdT<BH\(T), trker dc = Im d r efljff),

H\(T)QHT(T),

flfr)

QHT(T).

Thus if P r denotes the orthogonal projection (in H~m(d9T) on HT(T)\ we have: (269) P r t r ^ ( M ) = H[(T), P r t r ^(M') = H*T(T), and P r t r is an isomorphism from H^(M) and ff^(M') onto HlT(T) and /f^OProposition 10. decomposed into:

(270)

The cohomology space of the boundary TofM

if r (n=//j(nei^(n,

and thus the Betti numbers: bJ(M)tfdimflJ(M),

b^M^'dimi^M'),

br(r)=fdimtfr(r)>

and the Euler-Poincare characteristics of M, M' and T satisfy: (271)

b r (0 = bJ(M) + b+(M'),

xCD = x(M) + X(M>).

and M9 is

344

A - DIFFERENTIAL GEOMETRY

PROOF. We only have to prove that every a in Hf(T) is decomposed according to (270). Thus we have to solve with notation (94): da> m - Sj. <4 A o, 6© . 0 in Rn, with m in H*(M) and W*(M*), therefore m must satisfy: (272)

^ = 8 ( 8 ^ ^ 0 ) inRn,

and thus <* is obtained by convolution of the usual elementary solution of the Laplacian with the right member of (272). We can see (by Fourier transformation) that « has the required properties, with a jump of its traces on each side of T given by a. We use in fact the properties of the simple layer potential, see for example Dautray-Lions [1]. 6 . APPLICATION TO 3-DIMENSIONAL AND 2-DIMENSIONAL CASES

6.1. The 3-dimensional case Let M be a (regular at least for its interior) oriented Riemannian dimensional manifold: for the applications in view, M is the closure of an open set in R3. There are special features of differential geometry in 3 dimensions due to the identification by the Hodge transformation of even 2-covectors (or 2-forms) with odd covectors (or 1-forms), and also (using the Riemannian G-transform) of even 2-vectors with odd vectors (often called polar vectors in physics). Remark that in 3 dimensions we always have * * = 1. Let us define the vector product of two vectors Xx and Xj on M, (here denoted by X ^ X - , to differentiate with the exterior product), using the exterior product and Hodge transformation by: (273)

X 1 A X 2 = G"1*(COX AO) X ) = G"" 1 »(GX 1 AGX 2 ) = G * 1 * G ( X 1 A X 2 ) .

We also have, with usual notations (see (24)): (274)

Xj_A X2 - Xj A X2^ G"*vg .

Indeed we have for all vectors X: (X,X1AX2)g = (X,G-l(X1 A X 2 Jvg))g = <X, X{ AX2*iVg> (275)

= <X, A X2 A X,vg> . (Xx A X2 A X,G-!vg)g = (X,Xj A X2^G-1vg)g,

which proves (274). In a coordinate system (x1, x2, x3), we notably have forXj» dlt X2 = d2, X3 = 83 (since <d{ A 92 A d3,v^> = g1/2 from (275)): (276)

dxAd2 = g1/2G-!dx3 = 2 g1/2g3k3k. k

A.6

345

APPLICATIONS

Differential operators grad, curl, div (Representatives ofd and 8.) We successively define for all (smooth) functions f and vector fields A, B: df = V a d f > o r gradf «G-ldf,

(277) (278)

do>A = * o ) ^ A , or curl A = G"** dcoA ,

(279)

d-iWg^divB)^, or divB = *d»(0 B = _&«>B .

Definitions of grad and div are that of the general case (see (47),(48)). For the curl operator we can also write (with (45)): (280)

*do>A=s6*ci>A = a> curlA , or curlA = G" 6*a>A. i

S*ad

curl

div

U\Q) — > H(curl,Q) — » H(div,Q) — > L2(Q) 1 lG 1*G 1* H 0 ( d f Q ) - i Hx(d,Q) - i » H2(d,Q) - i > H3(d,Q) = L2(Q) 1* 1* i* I* H3(6,Q) - i H2(5,Q) - £ * Hx(b,Q) - i H0(8,Q) = L2(Q). Diagram 2 Usual relations are mere expressions of the fundamental relations: d 2 = 0, and 8 2 = 0. The Laplace-Beltrami operator may be identified with the usual Laplacian (on functions, see (49)'): Af = -6df=div gradf, AcoA = o> g r a d d i v A . c u r i c u r i A = ^AA» giving the "usual" relation on vector fields: (281)

A A a grad div A - curl curl A .

In a coordinate system, we have the expressions of grad f and div X given by (50), (53). Let us give that of curl A; from formula (51) for A: (282) we have: (283)

. . 1 coA= 2A^.dx , for A= %Ald{, 1J ij

da>A= 2 ijk

atAV.) k jp- dx Adx*, 9x

3(Alg..) 3(Aig..) l k 1 curlA=2 J-G~ *(dx Adx ) = 2 i ^ G " 1 dx k AG" 1 dx*. x ijk 9x ijk 9xK —

Applying (275) with X, = G" *dxk X 2 = G" ldx*, X = G" *dx\ we get:

346

(284)

A- DIFFERENTIAL

GEOMETRY

( G " d x , G - d x A G - d x ) = <G" (dx A dx* A dx*), v> = g" 1

/ 2

E'*',

•

e** the symbol of Levi Civita, = 1 if ikj is an even permutation of 123, - 1 if it is odd, 0 otherwise. Thus: (285)

G ~ d x ^ G " dx =g"

(286)

curlA=

J r

1/2

m)s

z *d

^ 9x - r ^ K

Ukm

For an orthogonal coordinate system, with Q. an open set of R , the Riemannian metric is: (287)

ds = 2g. dxtedx*, (alsodenotedasds = 2gj(dx ) with g» » g . ^ ) .

With: (288)

curl A m 2 (curl A ) & ,

grad f - 2 (grad f) 3;,

we thus olbtain, for A (or B) given by A= 2 A f y

(289) (

)

and thus: (290)

9(A g )

l T O

Af=divgradf=

- ^ '

2 .r^!5). 8

Some applications. In a Cartesian coordinate system, formulas (289) give the usual expressions for grad, curl, div. Then we give in two other coordinate systems, the Riemannian metric. In a cylindrical coordinate system, we have: x,=pcos 9 , x = psinv, x =z; 2

(291)

ds = dp +pW + dz , thus g = 1, g =p , g = 1. P

In a spherical (Euler) coordinate system, we have: Xj

(292)

= r sin 8 cos 9 , x = r sin 0 sin 9 , x = r cos 6, 2

ds = dr + r sin 9 d* + r d9 , thus g = 1, g = r sin 9, g = r , a

and also dx, A dx A dx = pdp A d$ A dz=i^sin 8 dr A d8 A d<t>. 2

A.6

347

APPLICATIONS

Due to (291), (292), we obtain the grad, div, rot operators in these coordinate systems by (289) and (290). Note that the orientation of (r, 0, 0) is the same as The cylindrical (and the spherical) coordinate system given by (291) (resp. (291)') does not define a chart on the whole space, but only on a part of it: the whole space without the half-plane x 2 = 0, Xj > 0, onto the open set: (0,+OO) X (0,2TT)XR for(p,0,z), resp. (0,+OO)X(0,2T0X(0,TI) for(r,0,0). Note that relations (289) are often written in the "orthonormal" system: e. = ( g . ) - m d{ (recall that l^ll*=g(ditd{) = g.). The advantage of this system rests on the possible identification with the dual basis Ej as (g.) 1/2 dx\ and the very simple formulation of the Hodge transformation:

therefore for A = 2 A1 e{, o>A = 2 A^j, we have (see Arnold [1]): i

(293)

(i)^ = *ci)A = A E 2 AE 3 + A E3AEJ +A £j A€2-

6.2. The 2-dimensional case Let T be an oriented Riemannian 2-manifold. For the applications we have in mind, T will be the boundary of a ("regular") open set Cl in R3. This situation seems fairly simple, but it is interesting to specify the differential geometric properties of T with respect to that of Q. At first we prove that the Hodge transformation on T is directly related to the vector product with the normal to T, more precisely: (294)

* (ox = t r G (n_A X) = t r <on A x , thus G"l * G(X) = n_A X

for all vectors X tangent to I\ PROOF.

Using the definition (273), t r G (n_A X) is given by:

(295) t r G (G"[ * (Gn A GX)) = t r „ (Gn A GX) = - ♦ n r (Gn

A GX) =

- * (^

GX)

(see (60)). Since (see (58)) (296)

n r (<4 A G X ) = - G X ,

we obtain (294) from (295). Note that * * <o = - co for all 1-forms o> on T. rr Differential operators on a surface representatives ofd and 6 With usual notations (see (8)) adapted to the present case: o>x = GpX, Gp being the canonical isomorphism due to the Riemannian metric gp induced by g on T, we successively define for all (smooth) function f and vector fields A, B on T:

348

A - DIFFERENTIAL GEOMETRY

(297) or: (297)' then (298) or

df^co^^f,

dcoA=scurlrAvr,

(298)'

curirf^-G'^tfvr),

gradrf=Gfl df, - 6 (fvr) = <!)-♦ , r curlrf

curlrA = *do>A, 6 Q>B s - divr B, r °

d,o>B = divrBvr.

These definitions are that of the general case (47), (48). Applying the Hodge transformation to (297), we obtain: (299)

*df=-5*f=-5fr rr = *a> g rerac a a , If=o) -♦ ,

curirf

♦da)AA = curlAA = 6«o) A A,

then using (294), we have: (300)

curlrf» n_A gradr f, curlr A = - divr (n_AA). grad r

curlr

Hl(T) —> H(curlr,0 —> L2(0 I 1G i* H 0 ( d , r ) - i H,(d,r) - i H2(d,i>L2(r> l

H2(8,D - i

H^S.D « i H0(6,n = L2(O Diagram 3

curlj.

div r

H!(T) —> H(divr,r) —»

1 Ho(d,n-^>

I* H2(5,r) - £ »

I.G H,(d,n - i

L2(T)

i* H2(d,D = L2(r)

11,(4,0 - i H0(5,O = L2(O Diagram 3'

We can easily verify that relations (297) may be obtained from (277), (278), (279), relative to Q by taking the trace on I\ that is by applying t r to relations (277), (278), (279). Then we obtain (298) with the Hodge transformation. Thus Diagrams 3 and 3* are deduced from Diagram 2, but we have to change spaces H(d,D into H " 1/2 (d,D, and H(5,D into H _ , / 2 (6,0 for all r. ® The Laplace-Beltrami operator Af on T is given by the formulas below (using (297) to (299)): (301) Aff m - (d8+8d) f = - 6d f = divr gradr f, A,(fVr) = - d6 (fvr) = (curlr curlr f)vr

A.6

APPLICATIONS

349

giving: A r = div r grad r = curl r curl r . On 1 -forms we have Aro>A = - (d5 + 6d) o)A = d(div r A) - 5 (curl r A v r ) = CDA A , with A r A = grad r div r A + curl r curl r A. <$ Remark 1 9 . Since T is a manifold without boundary, the operators d and 6 are adjoint (this follows from Stokes formula). _ Thus the operators grad r and - div r are adjoint, as well as curl r and - curl r . This follows from relations: (302) giving: (302)'

(df,o>A) = ( f > A ) ,

(do>A, f v r ) = (o)A,5 fv r ),

(grad r f,A) = (f,-div r A), (curl r A,f) = - (A,curl r 0.

In a coordinate system (x1 ,x 2 ) we have the following expressions of curl r A and curl r f, for A = 2 A ^ : (303)

curlrA = .dcoA = g -

1 / 2

( ^ - - ^ L ) ,

using * (dx1 A dx2) = g" m • v r = g" I / 2 . We recall that: r r (304)

div r B=:g- I/2 2 ^ ( 6 * g1'2)i ox

From (294), (also from (300) and (304)) we have, with(nAA) = 2 ( n A A ) 1 ^ i

(305)

(nAA) = - g "

,/2

2A g.,,

(nAA) = g-

,/2

ZA *.. i

Then using (300) we obtain (in the frame dltd2): (306)

c^irf=g-1/2(-a2f,a10.

In an orthogonal coordinate system where the Riemannian metric is (307)

ds 2 = 5 g 1 (dx 1 ) 2 + g 2 (dx 2 ) 2 ,

and the Lebesgue measure on T is dT = (g J g 2 ) 1/2 dx 1 dx 2 , we obtain with A = 2 A ^ , B = 2 B 1 ^, and with gQ = ( g ^ ) " l / 2 : gradrfs j

i.*^,

diVrB=go2^B»(glg2)

1/2

)

(308)

curlrlr f-g f= 0 ( - V 3i +»if 32).

curlrA = go ( ^ A 2 g 2 ) - ^ A \ ) )

350

A- DIFFERENTIAL GEOMETRY

We also have for the Laplace-Beltrami operator: (309)

M =

* ( i ^ %

Application to T a cylindrical surface or a sphere. Then we have: g = P2, g =1 for a cylinder, gA = A i n ^ , gfl = r2 for a sphere. Remark 20. Using both relations (289) and (308) in an admissible coordinate system in R3 (with g n = g3 = 1), we obtain the expressions of the operators grad, curl, div, with respect to the tangential and normal differential operators (310)

i)gradf=grad r f+|Jn,

divB = d i v r t r B + ^ - + 2R m B n ,

ii) curl A = curlr (trA) n + g- (nAA) - curlr An + 2Rm nAA, where n is identified with g-, where t r B is the projection of B on the tangent plane, n

1/5

B = n.B, and R-. is the mean curvature given with g = (g.gJ~

by:

We recall that A is given by A * 2 A ^ . Then the Laplace-Beltrami operator is: (310)'

Af.Arf+|3f

2Rllg.

The proof relative to the div operator follows from formula: (311)

div B = divrtrB + ( g ^ ) " " 2 ^

(B\g2)m).

For the curl operator, we have:

(312)

curiAm

*Jh*r~--5?1-) ^+go( - &AVl+^A . . /3Ana

and the following expression due to (305): n A A = (gxg2rm(

- (A2g2) 9, + (A 1 ^) 82),

from which we easily deduce (310). Note that from (310): (313)

n.curl A Âť curlr (trA).

ÂŤl*) 9A n a \

A.7 MAXWELL EQUATIONS

351

We emphasize that the decomposition of curl A given by (310) does depend on the basis, and is not "intrinsic": in other words the normal derivative in (310)ii) is not a "true" vector, i.e., is not the "covariant derivative". We will give the main formulas in cylindrical then spherical coordinates at the end of this Appendix. 7. MAXWELL EQUATIONS WITH DIFFERENTIAL FORMS

7.1 Maxwell equations with differential forms in R3 We first claim that there is no unique definition for electromagnetic quantities. We can define them either as r-fields or as differential forms. We make the following choice. The electric induction D, the electric field E and the current density J may be taken as time dependent even 1-fields on R3 (or on open sets in R3). We denote by (o^, co^, a) j the time dependent (even) 1-form corresponding to them, through the Riemannian metric g. The magnetic induction B and the magnetic field H may then be taken as time dependent odd 1-fields in R3, and will be identified with time dependent (even) 2-forms ca2B, <o2H; thus in a Cartesian system, for all vectorfieldA: a>^ = GA=2A i dx i ;

« i — « i « A ! dx2Adx3 + A2dx3 Adx1 +A 3 dx 1 Adx2.

i 3(oA

Then the evolution Maxwell equations are with o>A =-gr- = <^A/at (314)

da>£+ a> 1 = 0,

8(0^ - & D = CO],

(ordco^ - a) Q = (DJ).

7.2. Maxwell equations with differential forms in R4 We can also consider differential forms in R4 (rather than in R3 with time dependence) and it is better to do that for a deep understanding of the transformation laws of the electromagnetic field. But we have to take space R4 equipped with the bilinear form ds2 given in a Cartesian system by: (315)

g = ds 2 =

(dx k ) 2 -c 2 (dt) 2

fc*l,2.3

(with c the velocity of light in free space). This metric is not positive, therefore not Riemannian; it is called a pseudo-Riemannian metric or more precisely a Lorentz metric. Space R4 equipped with this metric is called Minkowski space. It allows us to define (as in the Riemannian case) an isomorphism, also denoted by G (or GL), mapping r-fields onto r-forms, and a unique odd 4-form, denoted by ]xQ or Vs, defined as in formula (14), so that: (316)

n c =|detg y |~

o^ Aa>2Ao>3Aa>4,

for (a)j), j = 1, 2, 3, 4, a "basis" such that (o)j,a>k) = g*(<t>j,a>k) = gjk.

352

A- DIFFERENTIAL GEOMETRY

For a Cartesian system of coordinates i»c=v8 = c dx1 A dx2 A dx3 A dt. Therefore n satisfies with (316): (317)

( i V ^ g - 2 I d e t g ^ r ' d e t g * ^ , ^ - 1.

Then there also exists a Hodge transformation * (which is an isomorphism from the r-forms space onto the ( 4 - r)-forms space) so that (318)

a A . 0 d # (a, p) u ,

(fJ A a, p ^ - (a, . (J)g,

thus with (21)' .(1 = 0 ^ c ,

. l = |i c ,

.Hc = - 1 ,

.•«r-(-lffV.

Then the codifferential operator is defined by: (319)

&» r =.d„« r ,

or .6<or = (-l) r d.o) r .

From 1-forms and 2-forms o>g, o>g, cap, o>^ defined in R , we define the following 2-forms p, 3 L in a Cartesian system of coordinates by: (320)

o>| tf «>k A dt + o»|,

S L d^" (of, A dt - <o^.

In fact (Dp, & L are fundamental entities, respectively even and oddforms, and we have to define the electomagnetic field from them. From the current J and the charge p, we first define a field j = (J, p) or a 1-form Wj by: (321)

a)j = coj1ssGj = G(2Jk9k + P^)=2J k dx k -c 2 pdt.

We can easily see that the 2-forms a> p, 3 L correspond to the 2-fields (or t( skew symmetric tensors) F (even) and L (odd) defined by (322) F = G • p - ^

L=G •L-[IM52

with

To B3 -B 2 ^ To D3 -D 2 "\ 3 3 1 (B) = 1 -B 0 B I, (D)=| -D 0 D1 I. [ B2 -B 1 0 J ^ D2 -D 1 0 J Using the inner product (21), we obtain o>g, w^ from o>p and 5 L (and then w2}, »i>)by

A. 7 MAXWELL EQUATIONS

(323)

<4-W"F.

353

"H^a/at^L.

&>3 = COp - 0)£ A d t ,

(OQ S

CD L + (Ojj A dt.

Then we can write the Maxwell equations in the differential form (324)

d<4 = 0,

da>[=4ijHc = ijii.

With the following (even) 2-form: (325) a>2£ = - c * c5^ = a)^ + c2a)|>Adt,

with L =G-l<»2~ = f j ^ *

0°]

we can also write Maxwell equations in the form (324)'

d(o| = 0,

8a)^=a)?.

From these equations, using the usual properties, we immediately see that: i) the current j must satisfy i

80)^=0, i.e., divJ + -g-p = 0; ii) from Poincare lemma there exists a vectorfield(called the vector potential) A so that o>p = dco^ and co^ is defined up to an exact form (D^» = o>^ + df. We say that the vector potential A is defined up to a gauge transformation. See also Abraham-Marsden-Ratiu [1] for some developments in the free space case. 7.3. Transformation laws. Lorentz and evolution transformations Here we recall the action of mappings on differential forms and tangent fields, previously used in (63) for example. Let M and M' be two (continuously differentiable) manifolds and let u be a diffeomorphism from M to M* (i.e. u and its inverse are continuously differentiable). Let x' = u(x). The pullback of an rform a)' on M' by u is an r-form on M, o> = u*(o>) defined by (326)

(u*o>Ox(v1,...,vr) = coV(Tux(v1),...,Tux(vr)), 9x»i where Ti^ is given in local charts by the matrix (—%) with x' = u(x). For a vector field X' on N, we also define:

(327)

(u»x') x l?av- l xw

Pullback has the properties: (328)

with respect to exterior derivation, to exterior product and to inner product.

354

A- DIFFERENTIAL GEOMETRY

Application to the Electromagnetic Field and to the Maxwell equations Let u be a difFeomorphism on R4. The pullback of Maxwell equations (324), for an electromagneticfieldgiven by F' and L\ with a current j ' is (329)

d u*4. = 0,

d u* S I - ^ . u ^ .

7.3.1. Lorentz transformations A Lorentz transformation u is a linear transformation preserving Lorentz metric, that is for all vectorfieldsX, Y in R4: (330)

g(u*X,u*Y) = g(X,Y), or also u*g = g with (326).

Then u commutes with the Hodge transformation and satisfies U*PC = Mc • Thus we see that u*cof?. and u* <* L» satisfy Maxwell equations with the current u*j'. A special Lorentz transformation u with the velocity v in R3 (the velocity of an observer S' with respect to the observer S, each observer having charts us(N) = (x,t), u~,(N) = (x',f) of the physical system, which are exchanged by u s , s = u) is defined by (x\f)« u(x,t) with (331)

x' v =ch9x v -sh9ct = WXy-vt) andn°x' = w°x, ct , = -sh9x v +ch9ct = P(-(v/c)x v +ct),

where xv=x.v/v, v= |v| <1, th9 = v/c, ch9 = 0 = (l - ^ r 1 7 2 , sh9 = pv/c, and*° is the orthogonal projection on the plane orthogonal to v. Thus (332) x^Afr.v-vfyv+i^x,

t'«0(t-x.v/c 2 ),

K°X=-4(XAV)AV-

Let E\ B\ H\ D' (and thus F\ L') be the electromagneticfieldat (x\ t'). Then at (x, t), E, B, H, D (and F, L) are obtained by the pullback of the 2-forms o>£., a> v by u and from relations (323). Since <o|. = A (dx^ A <*>& A v) + ^ nvB' a>y, we have: £a>^»A cdt' + co^sôôEtA cdt'-â)B.AVAdx'+JnvE,dxÂ cdt' + v^v^'^v Then using (331), and since dx^ A cdt' and «$ are invariant, we obtain (333)

n°E = «7i°E' + BVAV),

and

* ° H = P(JI°H'- DVAV),

BAV=P (B'AV + 4 *%&) DAV=P(D'AV-

Furthermore using (331), we have (333)

Jv= 0 (J'v+P' v),

p c = P ((v/c) rv+f>> c),

4*? H '>

355

A. 7 MAXWELL EQUATIONS

and (333)"

E\v=E.v, B'.v=B.v, D\v=D.v, H\v=H.v, J'Ay=J_AV.

Inverse formulas are obtained changing v into - v . The transformation law of the electromagneticfieldis given by: E'«PE + (1 - PXE.v^/V2 + 0vAB,

B' = 0B + (1 - PXB.vJv/v2 - 0vAE/C 2

D' = 0D + (1 - MD.vJv/v2 + P VAH/C 2 , J' = J + (0 - IXJ.VJVA2 - & p v,

H' =* 0H + (1 - PXH.vJv/v2 - 0 V A D

p' = P (p - (J.v/c2)).

We see that electric field and magnetic induction (resp. electric induction and magnetic field) are coupled through these transformations. 7.3.2. Transformation laws for solid motions and for pure Galilean motions Here we consider transformations U in R4 such that: (334)

U(x,t) = (x\f) = (u(x,t),t) with u(.,t) = ut an isometryin R3;

u represents the evolution of a solid medium which is in the connected domain Q. at time 0 and in the domain Q t = ut(Q) at time t. Since Q is a connected set, u is naturally extended to the whole space R3; ut is given for all t by (335)

UjX = xt a RjX + t t , with 1^ a rotation and t t a translation in R .

The time derivative of (335) is: dx, (336) - ^ | =£tARtx + vJ=t t A(Xt- T t) + v t = s £ tAx t + v ? ) with v£ = w[ - £t _A_Tt> v t being the time derivative of t t and £t a vector in R . If we differentiate (Rtx,Rty) = (x,y) with respect to time, we see that £t is independent of x; £t is the angular velocity (Arnold [1] p. 140). Then we define (337)

^ X . ^ A X + HJ.

Now (335) allows us to define (generalized) Galilean transformations on R4 by U: x' = utx, t' = t. We assume that we have two observers S and S': S is linked to the solid medium, S' is at rest, each having charts u (N) = (x,t), ug,(N) = (x\t') of the physical system, which are exchanged by u = u . Pure Galilean transformations are given by (with v the velocity of S with respect toS'): (335)'

x ^ u ^ x + vt, t' = t,

v€R 3 (thuswithR t = I,Tt = vt).

356

A - DIFFERENTIAL GEOMETRY

Let E \ B \ H \ D' (and thus F \ L') be the electromagnetic field at (x\ t') for S \ and E, B... be the corresponding electromagnetic field at (x,t) for S. The pullback of the 2-form <o2F, by U is given by U*<4» = U*(<4' A dt> ) + U*w|. = ut*<4« A dt' + U*co|..

(338)

Since u t is an isometry we have: ■■•«L = « «o)^ i £ ,, (339) u^cofe. and also,

u,*,.*2 t'a>§. = (o^ B ,,

(U*<4')x t = B'^M) ( ( R t d x ^ + ^ t ) A ( ( R ^ + ^ - d t ) +. and then developing, we have (340)

U*<4 = ut* <*|. - u^J, A { t x ) A dt\

With (337), we finally obtain: (341) U ^ - ^ . r ^ A A Similarly we have (341)'

W S l ^ H . ^ . ^ ,

+ uJ*. Adt-u t V D> .

Thus we obtain the transformation law E(x,t) = (u,*(E' (342)

B'A«M X ')) X

H(x,t) = (ut*(H' + D ' A 4 x ' ) x

Bfe*) - K ( B ' ) ) X .

and D(x,t) = (u;(D')) x ,

that is also' (342)'

|E(x,t) = R t - , (E'(x t ,t)- BXXt.OA.^Xt),

BCx,t)-R t - | (B , (x t ,t»,

H(x,t) = Rt-,(H'(xt,t) + D'(xt,t)A«Mxt). D(x,t) = Rt-1(D*(xt,t)). The transformation laws of current and charge are given byj = U*j', thus by

thus also by (343)' J(x,t) = Rt"1(J,(xt,t)-

fap'bvt)),

p(x,t) = p,(xt,t).

Application to the Galilean transformation (335)'. We have tj^x = v, thus (344)

I E(x, t) = E'(x + vt, t) - B'(x + vt, t) AV, H(x, t) = H'(x + vt, t) + D'(x + vt, t) AV,

D(x, t) = D*(x + vt, t), B(x, t) = B'(x + vt, t),

A. 7 MAXWELL EQUATIONS

357

and J(x,t) = J'(x + vt, t) - p'(x + vt, t)v,

p(x, t)Âť p'(x+vt, t).

We verify that these transformation laws are in agreement with those relative to Lorentz transformations for small velocities (v/cÂŤl), changing v into -v. CONCLUSION

We can define "proper electromagnetic field11 in moving bodies asfieldson the tangent space TM (identified with R3 x R3) according to (from (344) for moving particles, or from (342) for solid medium): (345)

E(x, v) ISfyx) + vAB(x),

B(x, v) V B(x),

H(x,v) 4? H(x) - VAEKX),

D(X,V)

d f

=i D(X).

Then the force (called Lorentz force) on a particle with charge q and velocity v in an electromagnetic field is: (346)

F = q E(x,v) = q (E + vAB).

Furthermore since ut is an isometry for all t, pc is invariant by U, and thus (see (329)) the transformed electromagnetic field (by any transformation (334), therefore by any Galilean transformation) satisfies the Maxwell equations. We insist here on the fact that we have transformed equations (324) and not (324)', because U does not commute with the Hodge transformation in R4. The necessity to use (324) rather than (324)' is due to the transformation law (328). The definition of E, B, D, H from the fundamental 2-forms according to (323) implies that they have different transformation properties with respect to time and space inversions. From the two fundamental forms, it is also possible (see for example Jones [1] p. 112) to define the Maxwell stress tensor in T X(M\ (f) by:

[

H<8>B + EÂŽD-;kH.B + E.D) 7 -EAH/C

B A D ^

_ 1 ^(H.B + E.D)

There is no universal agreement for such a tensor (see for example EringenMaugin [1] p. 62), but it is the key point of the relations of electromagnetism with mechanics, and to define forces in a solid medium. Then conservative laws are obtained from Maxwell equations, either using the Maxwell tensor or associated differential forms: for example we have the relation giving the usual conservative law for the Poynting vector, see (51) chap.l (348) - d(o>^ A0>{j) = (o>jj A & B + 0)^ A o> E>) + O>EA5.

358

A - DIFFERENTIAL GEOMETRY

7.3.3. Transformation of the constitutive relations Now we assume that the electromagnetic field for the observer S' satisfies the constitutive relations: (349)

D' = eE',

B' = >iH\

P = oE\

We can obtain the constitutive relations for the observer S either using Lorentz transformation laws (333) to (334) giving E, D, B, H as functions of thefield"at rest", or using (342) for a given evolution of the body, and in particular using (344) for a translation motion at a velocity v. i) Lorentz transformation. By projection on the velocity v, and on the orthogonal plane to v, we easily obtain from (333), (333)% (333)" the relations (350)

D + V A H / C 2 = C (E+VAB),

J v -vp = (o/p)Ev,

B - VAE/C2«|I(H - VAD), TI?J = OKTI<>E + VAB)

thus J - pv « op (E + VAJJ - (E. v) v/c 2 ). We can solve relations (350) with respect to B, D as functions of E, H. Then we obtain with n =

** ? , n^E (or n3H) = projection of E (or H) on an orthogonal

plane to v B = ji(H-n^*vH)+n-i v AE *»«H + n 4 v A ( E + »,VAH), (351) D = e(E-n4' l 2 E )-'>-TVAH = eH-!i-7VA(H-evAE). c*

These relations are of chirai type (see (95) chap. 1) but for small v/c, we have the same constitutive laws for D and B as in (349). Note that free space has always the same constitutive laws. ii) Galilean transformation. Using inverse relations to (344), we have from (349): (352)

B = ji(H+v/a>), D = C(E-V_AB), J + pv=o(E-VAB).

Solving these relations with respect to B, D as functions to E, H, we obtain with eaerfl-eiiv 2 )" 1 : | B « M H + 8VA ( E - J I V A H ) ,

(353)

—**—'• | D = £ E - 9 V A _ ( H + EVAJE).

These relations are also of chirai type.

A.7 MAXWELL EQUATIONS

359

7.3.4. Boundary transmission conditions for a solid moving body We consider as in section 6.3.2 the evolution of a solid medium in the (bounded) connected set Cl (in R3) at time 0 and in Qt at time t, with an evolution given by u as in (335), and thus Qt = u (Q). We assume that the boundary r of Cl is regular and given by equation *(0,x) = 0. Then the equation of the boundary of Clv Tv is: <Kt,x)=S4K0,ut-1x) = 0. This is also the equation of the boundary T of the domain Cl = {(x,t) with x € Qt} in R4. Then the (outward if 0 < 0 in Q) unit normal nt to Tt at x (resp.v to f at (x,t)) is: 1/2 (354) nt = grad0/|grad0| , v = (*\,gTad*)/[(*\r +\grzd*\*) witho't^. With Withvv =a $^ , w ce hhave: *\ - - v.grad 0. Thus the unit normals v and n to T and to Tt are related by: (355) v = (1 + (v.n)2)

. | g f ^ d 0 | {0't, grad0} = (1 + (v.n)2)

. { - v.n, n}.

Then for all smooth function 0 on R we have with i» 1, 2, 3, from Stokes formula: (356)J 7 0vidr . / g g

d x d t ^ / R d t / Q t g d x = / R d t / r t 0 n i d T t = / T WdTjdt,

thus we have with (355) for i = 1, 2, 3: .

(357) n1dTtdt = v 1 dr,

ov-l/2

dTtdt = (l+(v.n) 2 J

dl\

v°dr =-v.ndTdt.

Then we consider an electromagnetic field E, B, D, H satisfying the Maxwell equations (1) of chap.l, or (324), in the differential form with F and L. If we have no concentration of currents nor charges at the boundary of the moving medium, and if the electromagnetic field is "regular" on each side of the boundary, what are the continuity relations of the electromagnetic field across the boundary ? We can answer this question in two different ways: i) either using an elementary point of view calculating expressions such as: <curlE + | ^ , 0> = <E, curio - <B, | | > ,

Vo € D (R^xRt),

then using the Stokes formula. Surface integral terms are of the following form: / R dt/ r - n A E * d T t - / 7 v 0 B * d r » / ~ (-nAE + v.nB)*(l+(v.n) 2 r 1 / 2 df ; ii) or using "Schwartz" jump formula (94) for differential forms such as: (358)

2 v A[<4] r =0,

thus with (95) and (355) (here only Euclidean structure of R4 matters):

360 (359)

A - DIFFERENTIAL GEOMETRY (<D£

- n.v dt) A [<4 A dt + o>|] ~ » 0,

that is: (360)

K » i A 4 - n . v a i | ) A d t ] 7 + [«&A«£]~ =0,

therefore, using (273): (361)

[nAE] ? - n.v [BJ ~ * 0,

[n.B] ~ = 0.

From the relation: (362)

nA(vAA) = (n.v) A - v(n.A),

for all vectors A in R3, wefinallyget: (363)

nA[E - Y A B ] J = 0,

[n. B] ~ = 0.

Similarly with the 2-form 3 L we have: (363)*

nA[H + V A D ] ~ * 0 ,

[n.D] ~ = 0.

Summing up results, using relations (342),(342)\ we have with n' = R(n: (364)

[n'AE'] = 0,

[n'.B'] = 0,

[n'AH'] = 0,

[n\D'] = 0.

They are the usual transmission conditions for an observer moving with the velocity of the solid medium.

1 i

o I-

A 1

iff

3 1

S :

I—1

diagram 2

{ {

diagram 1

kerd«

The Hodge decomposition for a regular manifold M with boundai for r * 0, n.

HODGE DECOMPOSITION

361

•O

J u

1 ft |

362

A- DIFFERENTIAL GEOMETRY

Table of the main differential operators in an orthogonal coordinate system. Let (x1, x2, x3) be a (direct) local coordinate system, with Riemannian metric: d s ^ I g ^ d x 1 ) 2 . LetgO^Cgjg^)" 2 . Cylindrical system (p, ♦,z)see(291);g = l,g =p 2 ,g =1; thusg° = p. P

Spherical system (r, 9, ♦) see (292); g « l , g = r sin 0, gfl « r2; thus g° * A i n 9. 1) Orthonoimal basis (e lt ^ e3) for vectors, thus A s 2 Aj ej, with e^ = (g.)"1/23j gradf=

2(gj)

g ^

= g-ep+pgje^se2

dhrA-jjLl^^^AjX divA * l |

( p A p

.ffl/2e

)

ffl/2e

curlA=4y| 8 i , 1/2A

Af=divgradf^2^(^f)

I ^

= ^r+ragee+F^ea* V

s h

^|(r2Ar)+Fi^[^(sin9Ae)+^(A^]

-1/2 e

er, re 9 , rsin9e 0 sph

cyil

82,

,1/2 1/2A

e l/2 A

1 i^sinS

Ap,pA0, AJ

A,., rAe, rsin0A 0

2) Differential basis (3j, 82, 33) for vectors; thus A s 2 AJ3j.

l afa pyi af~ ^ 1 af~ afa vh af i af i 2 , gradf=2^^9j = g-a p + - I ^a^^a 2 = g - a ^ - j - a ^ — 9&> ffraHf

♦'

divA^2^(Ajg°) divA * l | ( 9

p A p

1 >

) ^

2 >

9x, ' 9 x 2 '

+ 9

S h

£ ^r(r2Ar)+iiIg[|(sin9A0) + s i n 9 ^

cyl 1

9x 3

9 p ' 9 0 ' 9z

|8l A l'8 2 A 2' «3A3 Af

»p. V 9z JL JL JL sph

3e>

JL JL

^sinfl 9 r ' 9 8 '

A,, r ^ r 2 starve A,;

Ap.P^.Aj

cyt 1 9 . 9f . 1 9 2 f , ^ f sph 1 9 _ 2 9 f .

^ - * 3 p p S f + j ? 5 ? 5?

9.:-

9f.9

9f,

?9? r ^5F + ?^ n lf9e s m 9 9e + 9*iinla» ] -

3) Definition of the covariant derivative V x by V g^PO - gff V XY+X(f)Y], withIy theChristoffelsymbols, I3,= r{ 3 = U - - ^ , v\2=1^=^S""gjp withx3 = n,and V ^

forB= 2 tfa. j-1.2

= 2 I*(x)9h. Then V„B = _ j ^ V i ^

0otherwise, ^ * ^ •

363

T A B L E O F E L E M E N T A R Y SOLUTIONS OF THE WAVE EQUATION

O H

V* . *

O o

4-#

* *

.SP <*

b»

C1.

—in —in ii c

e o u

6«

«*

~x"

k. Ut + k. MT

e b

oiS"

n «=

—

4-rf

0? uT

e o u

•a* II

ST .£ <N

^£

a w

«> c

o o

C "i/>

4-*

.a Or > a

X w

«• ~

^ II II

II >

4-#

* •2

•k

- 3 ©

*; "X <

4-«

+ 4-«

+ k. «o ll

4-*

A**

' M >

•

—

e 4-#

4-1

a* >

a §•

•4

e .3*

4-rf

<s u>

til

* 1

This page is intentionally left blank

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Index admittance 19 analytic functional 152 anisotropic medium 20 antenna 136 back, scattering 124 Beppo Levi spaces 338, 340 Betti number 335 Bloch condition 215 bundle (tangent, cotangent) 298 Calderon operators 62, causal 283, exterior 109, 113, interior 112, for a half space 191, 204, for a slab 212, for a sphere 163, for a conical domain 249, for a waveguide 169, 179, for gratings 217, 228, relative t o ( - A +1) 313 Calderon projectors 83, causal 281, interior P4, exterior Pc 83, 87, for a conical domain 250, for a halfspace 195, 206, capacitance matrix 55 cascades 170, 183 Cauchy problems 251 causal problems 276 causal Calderon operators 283 causal Calderon projectors 281 chiral medium 22 chiral obstacle 136 closed form 321 coefficients of influence 130 coercive sesquilinear form 59, 321 cohomology spaces 51 compensated compactness 325 conductivity 12 constitutive relation 7 curl 2, 28 currents 310 Debye potentials 162 differential cross-section 123 differential r-form 298 373

div (divergence) 305 double layer potential 83 Eddy current 12 electric layer 87 electric susceptibility 8 electromagnetic energy 3 elementary solution, incoming, outgoing 73, causal 256 equipartition of energy 254 Euler-Poincare characteristic 335 eventually incoming waves 265 eventually outgoing waves 265 exact r-form 321 exponential type 152 extension operator 29 extinction theorem 129 exterior (outgoing) Calderon operator 109 exterior Calderon projector 83 exterior product 298 ferromagnetic medium 23 Fourier-Laplace transform 5, 152 Fourier transform 36, 188 Fresnel's formulas 208, 209 Gaffney inequality 318 Galilean transformation 355 grad (gradient) 26, 305 Hardy inequality 237 Hertz potential 134 Hodge decomposition 52, 202, 315, 333, 337 Hodge projectors 56 Hodge transformation 302 Huyghens principle 263 impedance 19 incoming Sommerfeld condition 72, 236 incoming Calderon operator 266 incoming Calderon projector 266 incoming (stationary) waves 73 (in evolution) 265

374

p-incoming waves 265 X-incoming waves 271 inductance matrix 56 influence coefficients 127 inner (interior) product 302 interior Calderon operator 112, 113 interior Calderon projector 83, 87 interpolation 57 irregular frequency 118 junctions 170, 183 Kirchhoff formula 79, 82, 281 Knauff-Kress conditions 102 Kramers-Kronig relation 10 Laplace-Beltrami 35, 305, 317 Laplace transform 4 limiting absorption principle 99, 104, 105 limiting amplitude principle 274 Lorentz transformation 354 magnetic layer 87 magnetic susceptibility 8 magnetization 7 manifold 297 Maxwell stress tensor 357 multipole expansion 142 Ohm's law 12 optical theorem 125 outgoing Sommerfeld condition 72, 236 outgoing (stationary) waves 73, (in evolution) 265, p-outgoing waves 265, X-outgoing waves 271 Peetre lemma 49 permeability 8 permittivity 8 polar set 41 polarization 7 potential (plus, minus) 322 power (absorbed, scattered, total) 14,124 Poynting vector 3, 40, 79 pullback 47, 307, 353 pushforward 47 quasiperiodic function 215 radar cross-section 124

INDEX

radius of convergence (of a Rayleigh series) 145 Radon transformation 258, 260, 261 Rayleigh series 143, 217, 244 Rellich lemma 74, 247 resonance 273 resonance states 273 r-form, r-field 298 Riemannian manifold, structure, metric 299 saddle point method 119 scattering coefficients 127,130 scattering operator 270 scattering states 272 screen effect 129 sheet 42 Silver-Muller conditions 78 single layer potential 83 skin depth 209 Snell-Descartes laws 197 Sobolev spaces 26 Sommerfeld condition 72, 236 spherical Harmonics 143 Stratton-Chu formula 79 tangential part of a r-form 306 TE (transverse-electric) 180 TM (transverse-magnetic) 180 tensorfield298 total cross-section 124 total power 124 trace in H(div,Q) 29 trace in H(curi,Q) 29, 35 trace theorem (at infinity) 155, 245 variational frameworks 58, 317 vectorfield298 vector potential 79,286, 353 waveguides 167 wavelength 19 wavenumber 19 wave operator 270 Weston theorem 126 Whitney elements 68

Notations D, E, B, H electromagnetic field J, p electric current, electric charge a> the angular frequency, v the frequency EQ , pQ permittivity and permeability of free space c the velocity of light in free space E , p permittivity and permeability of the medium, E' , y' their real part, E" , u" their imaginary part, a conductivity of the medium k the wavenumber, k2 * arEji, X the wavelength, X Âť 2n/k in free space Z = a>}i/k = k/a>E the impedance of the medium, Y = 1/Z its admittance C + the future (or forward) light cone $ the (outgoing) elementary solution Cl an open set in Rn, T the boundary of Q, n a unit normal to T Y n v=n.v| the normal trace of thefieldv on T 7trv or t r v its tangential trace Ytv=n A v| for Cl in R3 the vector product of n with v at the boundary [v] the jump of v across I\ [yL = v| - v| , with n oriented from I\ to Te r

Differential operators: grad, div (in Rn), curl in R3 gradr, divr, curlr, curlr on a surface T (in R3) A the Laplacian l a2 â&#x2013;Ą = -^2 2â&#x20AC;&#x201D; A the d'Alembertian c at 2 Integral operators: (for Helmholtz) L, P, 3.1.3. (67), L, K, J, R 3.1.3 (70) (for Maxwell) L m , Pm9 3.1.3 (95), T, R 3.1.3 (100) Riesz operators Rg, 1^, R*, R* 5.1 (96), (96)' Calderon projectors Pif P c , (and S) 3.1.3 (72) and (104), G{, G e , their graphs Pseudodifferential operators: Ov Cv the Calderon operators Transformations: F Fourier transformation, FL Fourier-Laplace transformation, L Laplace transformation, R> Rd, l? d 2 , Rw Radon transformations Specific notations for the Appendix: g = ds 2 the Riemannian metric, G its associated isomorphism <o a r-form, dco its exterior derivative, 6a> its (exterior) coderivative ^o = ^m' ^M' *o = *m' ^M t ' l e m * n i m a * o r maximal realizations of d, 5 A.3(l 11) A*, A~, A D , A N , Laplacian operators C f + , C f ~ Calderon or capacity operator A.3 (221)

375

376

NOTATIONS

Regular functions D(G) 2.1, D(Q) 2.1, (2), C" regular functions Distributions D'(Cl), ,S"(Rn) tempered distributions, E '(Rn) with compact support 2>\(X), L+(Rt,X), L ^ X ) , X a Banach space lipschitz functions (^"(Q), (^"(Q), k € N, 0 < a£ 1 L2 functions L2(Q), L2(T), 1^(0 2. l.LJ^fi) 2.13 (179) 3.1 (57) Usual Sobolev spaces H*(Q), Hs(0, seR, Sobolev spaces on Q or I\ 2.1 Hl(G), H0(G), H , / 2 (0, H-,/2(T); H(A,Q), H1(A,Q)2.1 (3) Some less usual Sobolev spaces rl/0C(Q) 3.1 (57), H ^ r , ) 2.7 (67) H ^ r , ) ' 2.7 (68), (68)', H^2(f0) 2.7 (72), H^'2(f0) 2.7 (72) and also Hk(R2) 5.1 (104), Hk(R2) 5.1 (105), Hk/2(R2), Hk1/2(R2) 5.1 (35) Beppo Levi spaces w'CR"), W^Q), w£(Q) "Sobolev" spaces with time: e^H^QxR), eytH^(rxR), Dy(B) Spaces for div and curl: H(div,Q), H0(div,Q), H(curi,Q), H0(curl,Q), 2.1 (5), (6), (7), (8) H" I/2(div,r), H - ,/2(div,r) trace spaces 2.4 (36) HS(Q'), Hl0C(dhr,Q'), Hloc(curl,Q'), 2.2 (27),(37)Q'unbounded open set Some less standard spaces on curl and div H-s(div,Q), 2.7(79), H-5(div,r,)2.4(42), 2.7(83), rrs(curl,rj) 2.7(84) H;s(div,r,) 2.7 (85), H^div.r,) 2.7 (85), X(div,r,) 2.7 (86), X(curi,r,) 2.7 (87) Hs(curl,Q)2.10(128) Hi (£2)2.9(106), H r (curLG) 2.9 (106), H r (div.Q) 2.9 (106) 'o

cohomology spaces Hl(Q), H2(0) 2.8 (104), H(T) 2.9 J(Q), J(£2), J^Q), J^(Q)2.9(111), V(Q) Whitneyspaces 2.14 Spaces for quasiperiodic functions £>KL, £>K*L, 5.2.1 (172) H K L ( R 2 ) , H^ L(curl,P) Spaces for the Appendix Let M be a Riemannian manifold (with boundary T) Spaces of regular r-forms DT(M), D r(M), Dr(M) Spaces of generalized r-forms D't(M), D'^M), SJ(Rn) Spaces of L2 r-forms L2(M) Spaces of Sobolev r-forms Hj(M), H?(M), H*(M), Hr"(M), H^(M)

v; = H^(M),V; = r4(M) Spaces of r-forms based on the exterior differential: rL/d.M), Hr(5,M), H/dAM), H?(d,M), H*(6,M), H ^ M ) , Hro(5,M) Hm(d,5,M) = Hro(d,M)nHr(5,M), Hrt(d,5,M) = HJb,M)r\HJ.d,M).

Series on Advances in Mathematics for Applied Sciences Editorial Board N. Beltomo Editor-in-Charge Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy

C. M. Dafermos Lefschetz Center for Dynamical Systems Brown University Providence, Rl 02912 USA J. G. Heywood Department of Mathematics University of British Columbia 6224 Agricultural Road Vancouver, BC V6T 1Y4 Canada S. Lenhart Mathematics Department University of Tennessee Knoxville, TN 37996-1300 USA P. L. Lions University Paris Xl-Dauphine Place du Marechal de Lattre de Tassigny Paris Cedex 16 France S. Kawashima Department of Applied Sciences Faculty Eng.ng Kyushu University 36 Fukuoka 812 Japan V. Kuznetsov Institute of Chemical Physics Russian Academy of Sciences Kosygin Str. 4, Bid. 8 Moscow 117 977 Russia

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