  In this article we develop the local wellposedness theory for quasilinear
Maxwell equations in $H^m$ for all $m \geq 3$ on domains with perfectly
conducting boundary conditions. The macroscopic Maxwell equations with
instantaneous material laws for the polarization and the magnetization lead to
a quasilinear first order hyperbolic system whose wellposedness in $H^3$ is not
covered by the available results in this case. We prove the existence and
uniqueness of local solutions in $H^m$ with $m \geq 3$ of the corresponding
initial boundary value problem if the material laws and the data are
accordingly regular and compatible. We further characterize finite time blowup
in terms of the Lipschitz norm and we show that the solutions depend
continuously on their data. Finally, we establish the finite propagation speed
of the solutions.
