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...
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Published in | arXiv.org |
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Main Authors | , |
Format | Paper |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
12.02.2019
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Subjects | |
Online Access | Get full text |
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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. |
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ISSN: | 2331-8422 |