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...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Tsering-Xiao, Basang, Xiang, Wei
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.02.2019
Subjects
Coefficients
Mathematical analysis
Maxwell's equations
Regularity
Online AccessGet full text

Cover

More Information
Summary: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 \(\mathcal{C}^{1,\alpha}\) 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.
ISSN:2331-8422