  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 $H^m$.
