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

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.