Applications of Boundary Harnack Inequities for p Harmonic Functions to Problems in Harmonic Analysis, PDE, and Function Theory

The proposer and co-author, Kaj Nyström, have recently proved a boundary Harnack inequality for the ratio of two positive p harmonic functions, vanishing on a portion of the boundary of a Lipschitz or sufficiently flat domain (in the sense of Reifenberg). This proposal is concerned with applications of boundary Harnack and related inequalities for p harmonic functions to classical problems in Harmonic Analysis (Martin boundary and boundary regular- ity of p harmonic functions), PDE (regularity of the free boundary), and Function Theory (the dimension of p harmonic measure). The P1 and co-authors have made significant applications of these inequalities to problems in the above areas, but still numerous important questions remain open. Some of these questions are discussed in this proposal. Intellectual Merit: The p Laplacian, 1 C p < oc,p ~ 2, has received little attention com- pared to its cousin, the Laplacian, primarily because it's non-linear structure makes even very basic questions, nontrivial. Thus, before the publications of the PT and co-authors, there was essentially no literature on such fundamental questions as boundary Harnack inequalities for p harmonic functions, the p Martin boundary, and p harmonic measure, when 1