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.
Intellectual Merit:and Broad Impact Laplace's equation was used widely throughout the nineteenth and twentieth centuries to explain such physical processes as gravity, electricity, and steady state heat flow. Its nonlinear cousin, the p Laplacian, has only recently found applications in mathematical modeling (glacier formation, image processing). Still (in my opinion) this equation has yet to be fully utilized, primarily because its nonlinear structure makes the p Laplacian very difficult to work with, i.e, the sum or difference of two solutions need not be a solution to the p Laplace equation. As a consequence many basic questions remain unanswered for this partial differential equation. During this grant coauthors and myself made a careful study of certain solutions to the p Laplace equation, referred to as p harmonic functions. We developed a p harmonic toolbox which enabled us to investigate and solve problems previously reserved for Laplace's equation. In this sense our work is groundbreaking and should lead to a popularizing of the p Laplace equation both in theoretical and practical applications. Summary of Results During this grant coauthors and myself published papers on boundary Harnack inequalities and the p Martin boundary boundary problem in Lipschitz and vanishing Reifenberg flat domains. We also published papers on (a) Regularity of p harmonic functions near the boundary of Lipschitz and Reifenberg flat domains, (b) Free boundary regularity for p harmonic functions in certain two phase free boundary problems (c) The dimension of p harmonic measure. Boundary Harnack inequalities for Laplace's equation were originally studied by Dahlberg, Wu, and Ancona more or less simultaneously in the mid 70' s and were later generalized to more general domains by Kenig and Jerison. Regularity below the continuous threshold was also first considered by Dahlberg and later generalized by Kenig and Jerison in several papers, during the early 80's. More recent work on this topic, (since around 2000) is due to Kenig and Toro. Regularity in two phase free boundary problems was first considered for Laplace's equation by Caffarelli, Alt, and Friedman in the early 80's and also later by Caffarelli himself in several papers around the mid 80's. My work with coauthors on the dimension of a measure associated with a positive p harmonic function, vanishing on a portion of the boundary in certain domains (called p harmonic measure), was particularly satisfying to me, as my background is in complex function theory. As a young man I was very much interested in the Bieberbach Conjecture for univalent functions (solved by de Brange in the early 80's) and the Dimension Problem for harmonic measure (solved by Makarov in the mid 80's in simply connected planar domains and soon after by Jones - Wolff in essentially all planar domains). My work on p harmonic measure began in the early 90' s when I generalized a theorem of Bourgain, concerning the dimension of harmonic measure in Euclidean n space to the p harmonic setting. During this grant, after many years of trying, coauthors and myself were able to obtain for p harmonic measure, a complete analogue of Makarov's theorem in simply connected domains and also the natural generalization of Jones - Wolff to p harmonic measure in Euclidean n space ! Finally the techniques used in solving the above problems involve a nice mixture of harmonic analysis, complex function theory, and partial differential equations, so should be attractive to researchers and graduate students in these areas. Hopefully our efforts will eventually lead to the long sought after ` line in the mathematical textbooks of future generations. '
