Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maxwell's equations with mixed impedance boundary conditions

Ben Schweizer, David Wiedemann

TL;DR

This work develops a robust Fredholm framework for time-harmonic Maxwell equations on bounded Lipschitz domains with mixed, matrix-valued impedance boundary conditions that can be singular. By combining a boundary-operator formulation with a Helmholtz decomposition, the authors prove a Fredholm alternative: either a homogeneous problem has a nontrivial solution, or the inhomogeneous problem admits a weak solution for all sources away from resonance. The analysis handles polarization-dependent impedance and includes cases with singular boundary coefficients, leveraging Maxwell compactness results and a limiting absorption principle to establish existence and uniqueness results. The results unify impedance and perfect-conductor boundary models and provide a solid well-posedness theory for cavity-type problems with complex boundary behavior and polarization effects.

Abstract

We study the time-harmonic Maxwell equations on bounded Lipschitz domains with an impedance boundary condition. The impedance coefficient can be matrix valued such that, in particular, a polarization dependent impedance is modeled. We derive a Fredholm alternative for this system. As a consequence, we obtain the existence of weak solutions for arbitrary sources when the frequency is not a resonance frequency. Our analysis covers the case of singular impedance coefficients.

Maxwell's equations with mixed impedance boundary conditions

TL;DR

This work develops a robust Fredholm framework for time-harmonic Maxwell equations on bounded Lipschitz domains with mixed, matrix-valued impedance boundary conditions that can be singular. By combining a boundary-operator formulation with a Helmholtz decomposition, the authors prove a Fredholm alternative: either a homogeneous problem has a nontrivial solution, or the inhomogeneous problem admits a weak solution for all sources away from resonance. The analysis handles polarization-dependent impedance and includes cases with singular boundary coefficients, leveraging Maxwell compactness results and a limiting absorption principle to establish existence and uniqueness results. The results unify impedance and perfect-conductor boundary models and provide a solid well-posedness theory for cavity-type problems with complex boundary behavior and polarization effects.

Abstract

We study the time-harmonic Maxwell equations on bounded Lipschitz domains with an impedance boundary condition. The impedance coefficient can be matrix valued such that, in particular, a polarization dependent impedance is modeled. We derive a Fredholm alternative for this system. As a consequence, we obtain the existence of weak solutions for arbitrary sources when the frequency is not a resonance frequency. Our analysis covers the case of singular impedance coefficients.
Paper Structure (11 sections, 8 theorems, 43 equations)

This paper contains 11 sections, 8 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Function spaces and weak formulation
  3. Overview of the available literature
  4. Organization of this text
  5. Discussion of weak formulation and boundary conditions
  6. Coercivity
  7. Formulation in $H$ instead of $E$
  8. Helmholtz decomposition
  9. Verification of the Fredholm alternative
  10. Compactness property
  11. Limiting absorption principle

Key Result

Lemma 1.2

Let a boundary condition be given by a map $\Lambda \in L^\infty(\Gamma, \mathbb{C}^{3 \times 3})$. For almost every $x\in \Gamma$, with the kernel $Z = \ker(\Lambda(x))\subset \mathbb{C}^3$ and its orthogonal complement $Z^\perp\subset \mathbb{C}^3$, we assume: (i) $\nu(x) \in Z$ such that $Z^\perp In this situation, we set $\Theta(x) \coloneqq \Pi_Z$, the orthogonal projection onto $Z$. We defin

Theorems & Definitions (17)

  • Lemma 1.2: Assumptions in terms of $\Lambda$
  • Remark 1.3: Special case $\Lambda=0$
  • Theorem 1.4: Fredholm alternative
  • proof : Proof of Lemma \ref{['lem:AssumptionsLambda']}
  • Lemma 3.1: Helmholtz decomposition
  • Lemma 3.2: Reduction to divergence-free data
  • proof
  • Lemma 3.3: Equivalent formulation in $Y_\varepsilon$
  • proof
  • proof : Proof of Theorem \ref{['cor:existence']}
  • ...and 7 more