Local wellposedness of Maxwell systems with retarded material laws in low regularity

We develop a complete local wellposedness theory for a Maxwell system on R³ and a large class of nonlinear material laws which are nonlocal in time. Such constitutive relations are typical for nonlinear optics. The problem was treated before in the Sobolev space H⁸ for s > 3/2 by means of energy methods. Using a recently shown Strichartz estimate, we can lower this level of regularity to s > 1. In this context ’charge-type’ terms would spoil the analysis. We avoid them by the Helmholtz projection for the divergence operator with coefficients, which requires mapping properties of the projection also in $H^{α,q}$ with q ≠ 2.