Dimension of p Harmonic Measure and Related Topics

This proposal is concerned with the dimension of a measure associated with a positive p harmonic function, 1 < p < �‡, vanishing continuously on the boundary of a domain and related problems. For fixed p the PI has endpoint type estimates for this measure in a simply connected domain in two dimensions and with coauthors has also obtained estimates for the dimension of this measure in higher dimensions in a sufficiently flat Reifenberg domain when p . n. Still numerous important questions remain open. Some of these questions are discussed in this proposal. Intellectual Merit: The p Laplacian, 1 < p < �‡, p = 2, has received little attention compared to the Laplacian, primarily because its nonlinear structure makes even very basic questions nontrivial. The PI and coauthors have developed a p harmonic toolbox which enabled them to make significant progress on problems previously considered only for Laplace�fs equation. These include the above dimension problem, boundary Harnack inequalities, and regularity - free boundary regularity for p harmonic functions. The PI hopes that the techniques, results, and problems discussed in this proposal will play an important role in popularizing the p Laplacian. Broad Impact: The PI believes that this proposal has a nice mixture of problems in harmonic analysis, complex function theory, and partial differential equations, so should be attractive to researchers and graduate students in these areas. The proposer currently has a PhD student, Murat Akman, who is studying jointly with the proposer some of the problems listed in this proposal. In general the PI is committed to working with graduate students through one on one lectures and joint written collaborations. From his efforts he hopes to produce classical analysts that are capable of doing quality research at a high level. C-