Local Wellposedness of Nonlinear Maxwell Equations

In this work we establish a local wellposedness theory of macroscopic Maxwell equations with instantaneous material laws on domains with perfectly conducting boundary. These equations give rise to a quasilinear initial boundary value problem with characteristic boundary. We provide a priori estimates and a differentiability theorem in arbitrary regularity for the corresponding linear nonautonomous hyperbolic system of partial differential equations. A fixed point argument then yields a unique solution of the nonlinear problem in $H^m$ with $m\ge 3$. We further show a blow-up criterion in the Lipschitz-norm and the continuous dependance on the data.