Strichartz estimates for Maxwell equations on domains with perfectly conducting boundary conditions

We consider Maxwell equations on a smooth domain with perfectly conducting boundary conditions in isotropic media in two and three dimensions. In the charge-free case we recover Strichartz estimates due to Blair–Smith–Sogge for wave equations on domains up to endpoints. For the proof we suitably extend Maxwell equations over the boundary, which introduces coefficients on the full space with codimension-$1$ Lipschitz singularity. This system can be diagonalized to half-wave equations amenable to the results of Blair–Smith–Sogge. In two dimensions, we improve the local well-posedness of the Maxwell system with Kerr nonlinearity via Strichartz estimates.