## Duke Mathematical Journal

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

Marius Mitrea

#### 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 in Project Euclid: 18 November 2004

https://projecteuclid.org/euclid.dmj/1100793678

Digital Object Identifier
doi:10.1215/S0012-7094-04-12322-1

Mathematical Reviews number (MathSciNet)
MR2166752

Zentralblatt MATH identifier
1073.31006

#### Citation

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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