Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds
Marius Mitrea
Source: Duke Math. J. Volume 125, Number 3 (2004), 467-547.
Abstract
We solve three basic potential theoretic problems: Hodge decompositions for vector fields, Poisson problems for the Hodge Laplacian, and inhomogeneous Maxwell equations in arbitrary Lipschitz subdomains of a smooth, compact, three-dimensional, Riemannian manifold. In each case we derive sharp estimates on Sobolev-Besov scales and establish integral representation formulas for the solution. The proofs rely on tools from harmonic analysis and algebraic topology, such as Calderón-Zygmund theory and de~Rham theory.
Primary Subjects: 31C12, 35Q60, 58A14, 58J32
Secondary Subjects: 31B10,, 35J25,, 42B20,, 46E35
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1100793678
Digital Object Identifier: doi:10.1215/S0012-7094-04-12322-1
Zentralblatt MATH identifier:
02141301
Mathematical Reviews number (MathSciNet):
MR2166752
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