Local wellposedness of quasilinear Maxwell equations with conservative interface conditions

We establish a comprehensive local wellposedness theory for the quasilinear Maxwell system with interfaces in the space of piecewise H$^{m}$-
functions for $m$ $\geq\ $3. The system is equipped with instantaneous and piecewise regular material laws and perfectly conducting interfaces and boundaries. We also provide a blow-up criterion in the Lipschitz norm and prove the continuous dependence on the data. The proof relies on precise a priori estimates and the regularity theory for the corresponding linear problem also shown here.