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Embedding transmission problems for Maxwell's equations into elliptic theory

Yuri A. Godin, Boris Vainberg

Abstract

We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary conditions, after which the problem becomes elliptic. The results are applied to general problems for Maxwell's equations in bounded and unbounded domains, as well as to the transmission problem with inhomogeneities on the right-hand side of the equations and at all boundaries. Relations between the inhomogeneities of the elliptic problem are established that provide a one-to-one correspondence between the solutions of Maxwell's problem and the solutions of the elliptic boundary value problem.

Embedding transmission problems for Maxwell's equations into elliptic theory

Abstract

We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary conditions, after which the problem becomes elliptic. The results are applied to general problems for Maxwell's equations in bounded and unbounded domains, as well as to the transmission problem with inhomogeneities on the right-hand side of the equations and at all boundaries. Relations between the inhomogeneities of the elliptic problem are established that provide a one-to-one correspondence between the solutions of Maxwell's problem and the solutions of the elliptic boundary value problem.

Paper Structure

This paper contains 4 sections, 7 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reduction of Maxwell's equations to an elliptic problem
  3. The relationship between Maxwell's equations and the elliptic problem
  4. The general nonhomogeneous transmission problem

Key Result

Theorem 2.1

Let $\varepsilon = \varepsilon ({\bm x})$ and $\mu = \mu ({\bm x})$ be either complex-valued functions with a positive real part or real-valued positive definite matrix functions, and let (w1i) hold. Then the boundary value problem (L1)-(Lc) is elliptic. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Geometry of the transmission problem. The domain $\Omega$ contains an inclusion $\Omega_{-}$, bounded by $\Gamma = \partial \Omega_{-}$, and $\Omega_{+} = \Omega \setminus \bar{\Omega}_-$.

Theorems & Definitions (15)

  • Remark 1
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • Remark 2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 4.1
  • Remark 3
  • ...and 5 more