In this talk we shall focus on the recent solution of three basic potential
theoretic problems: (I) Hodge decompositions for vector fields, (II) Poisson
problems for the Hodge-Laplacian, and (III) inhomogeneous Maxwell's
equations, in Lipschitz subdomains of a smooth, three dimensional,
Riemannian manifold. They are all considered in the context of
Sobolev-Besov spaces, i.e. when the global smoothness of both the data
and the solutions is measured on these scales.
In hindsight, the problems (I)-(III) above turn out to be closely related.
A manifestation of this is that they share a common, (asymptotically) sharp
`well-posedness region', stemming from necessary limitations on the indices
(i.e., smoothness and integrability) of the spaces allowed in the
formulation of these problems. In turn, this region depends entirely on the
geometric characteristics of the underlying domain.
The main tools we employ are those of Harmonic Analysis (such as
Calderon-Zygmund theory), and our methods are constructive, in that they
rely on boundary integral equations.
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