Finite Energy Solutions of Maxwell's Equations and Constructive Hodge Decompositions on Nonsmooth Riemannian Manifolds |
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Authors: | Dorina Mitrea Marius Mitrea |
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Institution: | Department of Mathematics, University of Missouri–Columbia, Mathematical Sciences Building, Columbia, Missouri, 65211, f1 |
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Abstract: | We consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of Lp forms in Lipschitz domains, we show that both are well posed provided that 2−<p<2+, for some >0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented. |
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Keywords: | Hodge decompositions Lipschitz domains Maxwell's equations Lp layer potentials |
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