Regularity of solutions to time-harmonic Maxwell’s system with various lower than Lipschitz coefficients

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 H1 estimate when the coefficients are L∞ bounded. Second, we derive W1,p estimates for every p≥2 when one of the leading coefficients is simply continuous. Last, we extend the result to C1,α almost everywhere for the solution of the homogeneous Maxwell’s equations when the coefficients are W1,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.