Generalized Electro-Magneto Statics in Nonsmooth Exterior Domains

TL;DR

This paper develops a comprehensive solution theory for a generalized electro-magneto static Maxwell system in exterior domains with anisotropic coefficients, including spherical harmonics expansion and handling inhomogeneous boundary data.

Contribution

It introduces a new solution framework for Maxwell systems with anisotropic coefficients in nonsmooth exterior domains, including spherical harmonics expansion and boundary data handling.

Findings

Established a generalized spherical harmonics expansion for Maxwell equations

Developed a weighted static solution theory for anisotropic Maxwell systems

Extended the theory to inhomogeneous boundary conditions

Abstract

We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions which are up to a finite sum of special generalized spherical harmonics in another appropriately weighted Sobolev space. As a byproduct we prove a generalized spherical harmonics expansion suited for Maxwell equations. In particular, our solution theory will allow us to give meaning to higher powers of a special static solution operator. Finally we show, how this weighted static solution theory can be extended to handle inhomogeneous boundary data as well. This paper is the second one in a series of three papers, which will completely reveal the low frequency behavior of solutions of the time-harmonic Maxwell equations.