Strichartz estimates for Maxwell equations in media: The fully anisotropic case

We consider Maxwell equations in media in the uniformly fully anisotropic case in three dimensions and prove Strichartz estimates for Holder-continuous material coefficients. To this end, we use the FBI transform to conjugate the problem to phase space. After reducing to a scalar estimate by means of a matrix symmetrizer, we show oscillatory integral estimates for a variable-coefficient Fourier extension operator. The characteristic surface has conical singularities for any non-vanishing time frequency. As a consequence of the Strichartz estimates, we improve the local well-posedness for certain fully anisotropic quasilinear Maxwell equations. For these we establish local well-posedness for initial data with Sobolev regularity, which could previously not be covered with energy methods.