Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions

TL;DR

This paper develops a regularity theory for nonautonomous Maxwell equations with perfectly conducting boundaries, establishing existence, uniqueness, and a priori estimates for solutions with variable coefficients.

Contribution

It introduces a new framework for analyzing Maxwell equations with time- and space-dependent coefficients and characteristic boundaries, providing rigorous regularity results.

Findings

Proved a priori $H^m$-estimates for solutions.

Established existence and uniqueness of solutions under regularity and compatibility conditions.

Utilized divergence conditions and regularization techniques to handle characteristic boundaries.

Abstract

In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity and permeability on domains with a perfectly conducting boundary. This leads to an initial boundary value problem for a first order hyperbolic system with characteristic boundary. We prove a priori estimates for solutions in $H^m$. Moreover, we show the existence of a unique $H^m$-solution if the coefficients and the data are accordingly regular and satisfy certain compatibility conditions. Since the boundary is characteristic for the Maxwell system, we have to exploit the divergence conditions in the Maxwell equations in order to derive the energy-type $H^m$-estimates. The combination of these estimates with several regularization techniques then yields the existence of solutions in $H^m$.