Elliptic regularity theory applied to time harmonic anisotropic
Maxwell's equations with less than Lipschitz complex coefficients

The focus of this paper is the study of the regularity properties of the time
harmonic Maxwell's equations with anisotropic complex coefficients, in a
bounded domain with $C^{1,1}$ boundary. We assume that at least one of the
material parameters is $W^{1,3+\delta}$ for some $\delta>0$. Using regularity
theory for second order elliptic partial differential equations, we derive
$W^{1,p}$ estimates and H\"older estimates for electric and magnetic fields up
to the boundary. We also derive interior estimates in bi-anisotropic media.