Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coefficients

TL;DR

This paper applies elliptic regularity theory to analyze the boundary and interior regularity of solutions to time harmonic Maxwell's equations with less than Lipschitz complex coefficients in anisotropic media.

Contribution

It establishes new $W^{1,p}$ and Hölder regularity estimates for electromagnetic fields under minimal regularity assumptions on material parameters.

Findings

Derived boundary $W^{1,p}$ estimates for electric and magnetic fields.

Established interior regularity results in bi-anisotropic media.

Extended elliptic regularity techniques to Maxwell's equations with less regular coefficients.

Abstract

The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with $C^{1,1}$ boundary. We assume that at least one of the material parameters is $W^{1,3+\delta}$ for some $\delta>0$. Using regularity theory for second order elliptic partial differential equations, we derive $W^{1,p}$ estimates and H\"older estimates for electric and magnetic fields up to the boundary. We also derive interior estimates in bi-anisotropic media.