Regularity of solutions to time-harmonic Maxwell's system with various lower than Lipschitz coefficients
In this paper, we study the regularity of the solutions of Maxwell's equations in a bounded domain. We consider several different types of low regularity assumptions to the coefficients which are all less than Lipschitz. We first develop a new approach by giving $\mathcal{H}^1$ estimate when the coefficients are $\mathcal{L}^{\infty}$ bounded; and then we derive $\mathcal{W}^{1,p}$ estimates for every $p > 2$ when one of the leading coefficients is simply continuous; Finally, we extend the result to $\mathcal{C}^{1,α}$ almost everywhere for the solution of the homogeneous Maxwell's equations when the coefficients are $\mathcal{W}^{1,p}, \, p>3$ and close to the identity matrix in the sense of $\mathcal{L}^{\infty}$ norm. The last two estimates are new, and the techniques and methods developed here can also be applied to other problems with similar difficulties.