Regularity of solutions to time-harmonic Maxwell's system with various lower than Lipschitz coefficients. (March 2020)

• Record Type: Journal Article
• Authors: Tsering-xiao, Basang; Xiang, Wei
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: Maxwell's equations -- Regularity theory -- Anisotropic; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111693
• Languages: English
• ISSNs: 0362-546X

Abstract

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.