On $L^r$ Estimates for Maxwell's Equations with Complex Coefficients in Lipschitz Domains

Abstract

This paper considers the time-harmonic Maxwell's equations with anisotropic complex coefficients in a bounded Lipschitz domain. We first establish the $W^{1,r}$ estimates for divergence form equations with the coefficients being the small complex perturbations of real symmetric matrices in Lipschitz domains. As an application, we then show the $L^r$ estimates of electric and magnetic fields for $\frac{3}{2}-\varepsilon(\Omega)<r<3+\varepsilon(\Omega)$, where $\varepsilon(\Omega)>0$ depends on the Lipschitz character of the domain $\Omega$.