A central problem at the interface between harmonic analysis and partial differential equations (PDEs), on the one hand, and functional analysis and operator theory, on the other hand, is the study of elliptic boundary value problems on non-smooth domains. One typical approach for solving many classical PDEs of mathematical physics in Lipschitz domains in the Euclidean setting is the method of layer potentials.
The PI plans to address a number of important and long-standing questions regarding fundamental properties of boundary layer potential operators on Lipschitz domains, which still remain open despite the tremendous progress made in the last three decades. Some of the specific themes of this proposed research program are The Spectral Radius Conjecture for boundary layers on rough domains;Displacement prescribed problems of elasticity in multi-connected regions with rough boundaries; and Regularity properties of Green functions and Poisson kernels.
The study of boundary value problems (BVPs) with minimal smoothness assumptions (on the coefficients or on the domain under discussion) has been the driving force behind fundamental developments in the area of Partial Differential Equations and has deep connections with many areas of Analysis. The interest in such problems resides both in their theoretical importance as well as in their numerous applications to engineering.
