Mathematical Methods in Electromagnetism

The following sections are included:

• Acknowledgements

• Preface

• Contents

The following sections are included:

• EVOLUTION MAXWELL EQUATIONS

• STATIONARY MAXWELL EQUATIONS

• CONSTITUTIVE RELATIONS
  • Linear isotropic dielectric media
    • Conditions obtained from invariance by time translation and causality
    • Consequences of the hypothesis H4). Kramers-Kronig relations
    • Constitutive relation between J and E: Ohm's law
    • Stationary Maxwell equations with Contitutive relations
    • General constitutive relations with Ohm's law
    • Conditions on ε and μ for a dissipative medium in a bounded domain
    • Some other definitions
    • Summary of the main properties for a linear isotropic dielectric medium
  • Linear anisotropic dielectric media
  • Linear chiral media
  • Nonlinear constitutive relations

• CONCLUSION

The following sections are included:

• INTRODUCTION

• SPACES FOR CURL AND DIV; TRACE THEOREMS

• JUMP FORMULAS ACROSS A BOUNDED HYPERSURFACE Γ IN Rⁿ

• DIFFERENTIAL OPERATORS ON A “REGULAR” SURFACE Γ

• THE SPACES H^−1/2(div,Γ), H^−1/2(curl,Γ). TRACES FOR H(curl, Ω)

• TRACES FOR W¹(div, Ω) (FOR THE POYNTING VECTOR)

• SOME COMPLEMENTARY RESULTS. POLAR SETS

• TRACES ON A SHEET
  • Trace problems on sheets (the scalar case)
  • Trace problems on sheets (the vectorial case)

• SOME REGULARITY RESULTS

• THE HODGE DECOMPOSITION

• INTERPOLATION RESULTS

• SOME VARIATIONAL FRAMEWORKS
  • Variational frameworks based on H(curl, Ω) spaces
    • A variational framework based on H_o(curl, Ω)
    • A variational framework based on H(curl, Ω)
  • Variational frameworks based on H¹(Ω)³ and the Laplacian

• FIRST PROBLEMS WITH INHOMOGENEOUS BOUNDARY CONDITIONS
  • Problems with H^−1/2(div, Γ) boundary conditions
  • Problems with H^1/2(Γ) boundary conditions

• BOUNDARY PROBLEMS OF CAUCHY TYPE. UNIQUENESS THEOREMS

• WHITNEY ELEMENTS; NUMERICAL TREATMENT

The following sections are included:

• STATIONARY WAVES DUE TO SOURCES IN BOUNDED DOMAINS
  • The main properties for the Helmholtz equation in R³
    • Local regularity properties
    • Outgoing and incoming Sommerfeld conditions
    • Elementary outgoing (incoming) solution
    • Fundamental properties
  • The main properties for Maxwell equations in R³
    • Relation to the Helmholtz problem
    • The Silver-Müller conditions
  • Transmission problems and surface integral operators
    • Helmholtz problems with charges on “regular” surfaces
    • Maxwell problems with currents on “regular” surfaces
    • Some regularity results
    • Integral method for a sheet
    • Incoming waves

• SCATTERING PROBLEMS WITH A COMPLEX WAVENUMBER k
  • Helmholtz equation in R³ with a complex wavenumber
  • Maxwell equations with complex coefficients
  • The Calderon projectors for a complex wavenumber k

• VECTOR HELMHOLTZ EQUATION, KNAUFF-KRESS CONDITIONS
  • Knauff-Kress conditions at infinity
  • Vector Helmholtz problems with jumps conditions

• BOUNDARY PROBLEMS WITH REAL WAVENUMBER k
  • Limiting absorption principle
  • Exterior boundary problems
  • Some consequences; the exterior Calderon operator
  • Interior boundary problems
  • Some consequences; the interior Calderon operator
  • Integral equations and boundary problems
  • Some consequences for the integral operators
  • On the numerical solution of some scattering problems
    • The problem of irregular frequencies
    • A saddle point method

• SCATTERING PROBLEMS BY A DIELECTRIC OBSTACLE
  • A general variational formulation
  • Solution using an integral method
  • Behavior at infinity. Radar cross section. Optical theorem

• SCATTERING AND INFLUENCE COEFFICIENTS FOR SEVERAL OBSTACLES
  • Decomposition of the trace spaces
  • Screen effect. Extinction theorem
  • Coefficients of mutual influence of antennas

• MUTUAL INFLUENCE OF SHEETS
  • Introduction
  • Electromagnetic field due to currents on a sheet
  • Matrix elements; influence coefficients

• FIELDS DUE TO CURRENTS ON A LINE

• SCATTERING PROBLEMS BY A CHIRAL OBSTACLE

• CONCLUSION ON THE CALDERON OPERATORS FOR SCATTERING PROBLEMS

• MULTIPOLE EXPANSIONS. RAYLEIGH SERIES
  • Multipole expansions for Helmholtz in R²
  • Multipole expansions for Helmholtz in R³
  • The source of a wave u_n
  • Multipole expansions and analytical functionals
  • Multipole expansions for the electromagnetic field

• SCATTERING BY A DIELECTRIC BALL
  • Scattering with Helmholtz equation
    • Calderon operators and Calderon projectors on a sphere
    • Scattering of an incident wave by a ball
  • Scattering with Maxwell equations
    • Debye potential of the electromagnetic field
    • The Calderon operators on the sphere
    • Scattering of an incident wave by a dielectric ball

• ADDENDUM. COMPACTNESS PROPERTIES IN SCATTERING PROBLEMS

The following sections are included:

• WAVEGUIDES WITH HELMHOLTZ EQUATIONS

• WAVEGUIDES IN ELECTROMAGNETISM

The following sections are included:

• PLANE GEOMETRY
  • Plane geometry with Helmholtz equation
    • Helmholtz problem in a half-space with Dirichlet condition
    • Transmission Helmholtz problems in R³. Calderon projectors
    • Transmission problems with two different media
    • Some examples of applications of the Calderon operator
  • Plane geometry with Maxwell equations
    • A typical problem in a half-space
    • Scattering problems with two different media
  • The slab
    • The scalar case with Helmholtz equation
    • The slab with Maxwell equations

• PERIODIC GEOMETRY, 2-D GRATINGS
  • Periodic geometry with Helmholtz equation
  • An integral method for gratings
  • Periodic geometry with Maxwell equations
    • Mathematical framework
    • The Calderon operator
    • Some scattering problems

• CONICAL GEOMETRY
  • Properties of some unbounded operators associated to A
  • Solution of Helmholtz 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 = εE, B = μH. 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.

Using differential geometry in electromagnetism is quite natural, at first in 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 electromagnetic fields of finite energy. 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 on finite dimensional 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.