Calderon-Zygmund Operators on Sobolev-Besov Spaces and Boundary Problems with Minimal Smoothness Assumptions

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Following A.P. Calderon's program (sketched in his Helsinki ICM address in 1978), many basic boundary value problems in physics and engineering can be attacked by starting with a representation of the solution in the form of a potential operator, and then reducing matters to analyzing the functional analytic properties of a system of pseudodifferential operators on the boundary. The new phenomenon, occurring when non-smooth structures are present, is that these boundary operators belong to a larger class, consisting of singular integral operators. The lack of a symbolic calculus in this context makes the study of such operators fundamentally harder, particularly when one has to address such delicate issues like Fredholmness and invertibility. While in the intervening years these ideas have undergone a dramatic development, much remains to be done. The main goal of the current proposal is to investigate the specific nature of the singular integral operators arising in connection with basic problems in mathematical physics, such as Maxwell's equations, vector Poisson problems, Hodge decompositions, and Stokes's system of hydrodynamics, in light of recent advances in the fields of harmonic analysis and partial differential equations. For maximum applicability, it is important to consider these problems in the context of general Sobolev-Besov spaces and when the underlying structures are minimally smooth.