Recent developments in harmonic analysis have facilitated a deeper understanding of large classes of partial differential equations, modeling a variety of physical problems, including those arising in elasticity, hydrodynamics and electromagnetism. This proposal is concerned with regularity aspects of solutions to these equations. Most notably, the PI intends to develop new tools, as well as broaden the scope of more traditional methods, in order to be able to study how the smoothness of the data influences the smoothness of the solution, in the presence of boundary irregularities. The natural borderline is the case when the domains in question are Lipschitz, i.e., satisfy a uniform (interior/exterior) cone condition. Informally speaking, Lipschitz domains make up the most general class where a rich function theory can be developed, comparable in power and scope with that associated with the upper-half space.
The scales of Besov and Triebel-Lizorkin spaces offer a natural functional framework within which the issue of smoothness can be analyzed. They encompass both the Sobolev and Hardy classes and, otherwise, exhibit features which are generally harder to take advantage of (or are even completely lost) when working on the more classical Lebesgue scale. One of the main questions addressed in this proposal is the regularity of Green potentials on Lipschitz domains (with Dirichlet and Neumann boundary conditions) on Besov and Triebel-Lizorkin spaces. The PI proposes to investigate this issue both in the elliptic and parabolic setting, via singular integral methods.
