Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions

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 $H^m$.