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

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The p Laplacian for most values of p is a nonlinear divergence form degenerate elliptic PDE. Solutions to this PDE (called p harmonic functions) are relatively nice in that they are invariant under rotations, translations, dilations, and also remain solutions under multiplication by constants. Still the nonlinearity of this PDE makes even basic questions difficult to answer. Recently the principal investigator and coauthor Nystrom have proved a boundary Harnack inequality (including Holder continuity) for the ratio of two positive p harmonic functions vanishing on a portion of a Lipschitz domain. This project proposal is concerned with applications of this boundary Harnack inequality to problems concerning the p Martin boundary, the dimension of p harmonic measure, and to certain two phase free boundary problems. Most physical models involve linear PDE (in their principal part). Laplace`s equation is one of the best known linear PDE and is often used in mathematical models, as well as to describe physical phenomena. This proposal is concerned with extending some classical results for Laplace's equation to its cousin the nonlinear p Laplace equation. Recent technology developed by the proposer and coauthors make this extension now possible. It is hoped that our work will eventually lead to greater use of the p Laplace equation in mathematical modeling and in general in the sciences.