Regularity of solutions to time-harmonic Maxwell's system with various lower than Lipschitz coefficients. (March 2020) - Tsering-xiao, Basang; Xiang, Wei
Abstract: 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 less than Lipschitz. First, we develop a new approach by giving a H 1 estimate when the coefficients are L ∞ bounded. Second, we derive W 1, p estimates for every p ≥ 2 when one of the leading coefficients is simply continuous. Last, we extend the result to C 1, α almost everywhere for the solution of the homogeneous Maxwell's equations when the coefficients are W 1, p, p > 3 and similar to the identity matrix in the sense of L ∞ norm. The last two estimates are new, and the techniques and methods that are developed in this paper can also be applied to other problems with similar difficulties.