Time-Harmonic Electro-Magnetic Scattering in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions

TL;DR

This paper develops a mathematical framework for solving time-harmonic electromagnetic scattering problems in exterior weak Lipschitz domains with mixed boundary conditions, using advanced Sobolev space techniques and a Fredholm alternative approach.

Contribution

It introduces a solution theory for Maxwell's equations in complex domains employing polynomially weighted Sobolev spaces and extends Weck's selection theorem to ensure existence.

Findings

Established a Fredholm alternative for Maxwell's equations in weak Lipschitz domains.

Derived a-priori estimates and polynomial decay of eigenfunctions.

Extended Maxwell compactness property to new domain classes.

Abstract

This paper treats the time-harmonic electro-magnetic scattering or radiation problem governed by Maxwell's equations in an exterior weak Lipschitz domain divided into two disjoint weak Lipschitz parts We will present a solution theory using the framework of polynomially weighted Sobolev spaces for the rotation and divergence. We will show a Fredholm alternative type result to hold using the principle of limiting absorption introduced by Eidus in the 1960's. The necessary a-priori-estimate and polynomial decay of eigenfunctions for the Maxwell equations will be obtained by transferring well known results for the Helmholtz equation using a suitable decomposition of the electro-magnetic fields. The crucial point for existence is a local version of Weck's selection theorem, also called Maxwell compactness property.