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.