Regularity of solutions to time-harmonic Maxwell's system with various lower than Lipschitz coefficients

TL;DR

This paper investigates the regularity of solutions to time-harmonic Maxwell's equations under various low regularity conditions on the coefficients, introducing new estimates and extending previous results.

Contribution

The paper develops new regularity estimates for Maxwell's equations with coefficients less regular than Lipschitz, including $ ext{H}^1$, $ ext{W}^{1,p}$, and $ ext{C}^{1,eta}$ estimates, extending existing theory.

Findings

Established $ ext{H}^1$ estimates for $ ext{L}^ ext{infty}$ bounded coefficients.

Derived $ ext{W}^{1,p}$ estimates for coefficients that are continuous.

Extended results to $ ext{C}^{1,eta}$ coefficients close to the identity.

Abstract

In this paper, we study the regularity of the solutions of Maxwell's equations in a bounded domain. We consider several different types of low regularity assumptions to the coefficients which are all less than Lipschitz. We first develop a new approach by giving $\mathcal{H}^1$ estimate when the coefficients are $\mathcal{L}^{\infty}$ bounded; and then we derive $\mathcal{W}^{1,p}$ estimates for every $p > 2$ when one of the leading coefficients is simply continuous; Finally, we extend the result to $\mathcal{C}^{1,\alpha}$ almost everywhere for the solution of the homogeneous Maxwell's equations when the coefficients are $\mathcal{W}^{1,p}, \, p>3$ and close to the identity matrix in the sense of $\mathcal{L}^{\infty}$ norm. The last two estimates are new, and the techniques and methods developed here can also be applied to other problems with similar difficulties.