Problems of Existence, Uniqueness and Dimension in Harmonic Analysis, Function Theory and Partial Differential Equations

Project Summary This proposal discusses two kinds of problems. The first kind involves existence and uniqueness in some overdetermined free boundary problems of p Laplacian type under minimal boundary assumptions. The proposer, coauthors, and his PhD student, have obtained existence and uniqueness in certain of these problems. The proposer would like to investigate the sharpness of these results in higher dimensions, as well as lower dimensional analogues, and also consider parabolic generalizations. Problems of the second kind are concerned with the dimension of a measure associated with a positive p harmonic function, vanishing on the boundary of a certain domain. The proposer and his PhD student have obtained analogues of well known theorems for harmonic measure in a quasicirde. The PI would like to investigate extensions of these theorems to simply connected domains and planar domains in general. Related questions involve a rate theorem for p harmonic functions and estimates of the dimension as a function of p. Intellectual Merit: The problems in the proposal are simply stated and the PI believes, quite natural and fundamental questions to ask. However there solutions are often rather complicated and appear related to some fundamental questions in harmonic analysis and function theory. For example, the Riesz transforms problem, characterizations of uniform rectifiability, the dimension and absolute continuity of elliptic measures, are all questions which arise naturally in our existence, uniqueness, dimension, problems. So the proposer's research could also lead to solutions or new insights into these related questions. Broad Impact: The PI notes that the p Laplacian is often used in mathematical modeling and free boundary - inverse type problems occur naturally in nature. Also existence and uniqueness are important questions in applications and form one of the core areas in partial differential equations. In addition to possible applications, the problems in this proposal are a good mix of harmonic analysis, function theory, and partial differential equations, so should be attractive to researchers and graduate students who are interested in classical analysis. The PI is committed to working with graduate students, through one on one lectures, and joint written collaborations. Through his efforts he hopes to produce classical analysts that are capable of doing quality research at a high level. A-l