  In this article we provide a local wellposedness theory for quasilinear
Maxwell equations with absorbing boundary conditions in $\mathcal{H}^m$ for $m
\geq 3$. The Maxwell equations are equipped with instantaneous nonlinear
material laws leading to a quasilinear symmetric hyperbolic first order system.
We consider both linear and nonlinear absorbing boundary conditions. We show
existence and uniqueness of a local solution, provide a blow-up criterion in
the Lipschitz norm, and prove the continuous dependence on the data. In the
case of nonlinear boundary conditions we need a smallness assumption on the
tangential trace of the solution. The proof is based on detailed apriori
estimates and the regularity theory for the corresponding linear problem which
we also develop here.
