Harmonic Analysis and Partial Differential Equations

University Of Chicago, Chicago IL

Investigators

Abstract

In this proposal,the PI will concentrate on the study of "locally flat" sets, the geometry of the measures that they support,and the connections with harmonic measure.Also,the PI will study boundary value problems under minimal smoothness conditions,and the existence and regularity of solutions to two-phase free boun dar problems.The PI will also study non-linear evolution problems,focusing on Liouville theorems for parabolic regularizations of conservation laws,and on the connection between oscillatory integrals,restriction theorems for the Fourier transform,and the well-posedness of solutions to non-linear dispersive equations and systems. The main guiding principle in the proposal is the development of various aspec ts of harmonic analysis,potential theory and geometric measure theory,with a view to applications to a number of challenging problems in linear and non-lin ear partial differential equations,and the implementation of these applications. It is expected that this research will lead to the development of new graduate courses,to Phd dissertations,and to the sposoring of a number of postdoc's reas erch.