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Mixed boundary-value problems for Maxwell's equations


Author: Marius Mitrea
Journal: Trans. Amer. Math. Soc. 362 (2010), 117-143
MSC (2000): Primary 35J55, 78A30, 42B20; Secondary 35F15, 35C15, 78M15
DOI: https://doi.org/10.1090/S0002-9947-09-04561-9
Published electronically: August 13, 2009
MathSciNet review: 2550146
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Abstract: We study the Maxwell system with mixed boundary conditions in a Lipschitz domain $ \Omega$ in $ \mathbb{R}^3$. It is assumed that two disjoint, relatively open subsets $ \Sigma^e$, $ \Sigma^h$ of $ \partial\Omega$ such that $ \overline{\Sigma^e}\cap\overline{\Sigma^h}=\partial\Sigma^e=\partial\Sigma^h$ have been fixed, and one prescribes the tangential components of the electric and magnetic fields on $ \Sigma^e$ and $ \Sigma^h$, respectively. Under suitable geometric assumptions on $ \partial\Omega$, $ \Sigma^e$ and $ \Sigma^h$, we prove that this boundary value problem is well-posed when $ L^p$-estimates for the nontangential maximal function are sought, with $ p$ near $ 2$. A higher-dimensional version of this result is established as well, in the language of differential forms. This extends earlier work by R. Brown and by the author and collaborators.


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Additional Information

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri at Columbia, Columbia, Missouri 65211
Email: mitream@missouri.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04561-9
Keywords: Maxwell's equations, Lipschitz domains, mixed boundary conditions, Rellich estimates.
Received by editor(s): June 21, 2005
Received by editor(s) in revised form: April 12, 2007
Published electronically: August 13, 2009
Additional Notes: The author was supported in part by the NSF and the University of Missouri Office of Research
Article copyright: © Copyright 2009 American Mathematical Society

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