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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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.

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

First available in Project Euclid: 18 November 2004

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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


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

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