Duke Mathematical Journal

Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds

Marius Mitrea

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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.

Article information

Source
Duke Math. J. Volume 125, Number 3 (2004), 467-547.

Dates
First available: 18 November 2004

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

Subjects
Primary: 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14] 35Q60: PDEs in connection with optics and electromagnetic theory 58A14: Hodge theory [See also 14C30, 14Fxx, 32J25, 32S35] 58J32: Boundary value problems on manifolds
Secondary: 31B10, 35J25, 42B20, 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Citation

Mitrea, Marius. Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds. Duke Mathematical Journal 125 (2004), no. 3, 467--547. doi:10.1215/S0012-7094-04-12322-1. http://projecteuclid.org/euclid.dmj/1100793678.


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