This proposal is concerned with the dimension of a measure associated with a positive solution to the p Laplace equation, often called a p harmonic function, which vanishes continuously on the boundary of a domain and related problems. If p is two the p Laplace equation reduces to Laplace's equation and the Green's function for Laplace's equation, with pole at a fixed point in the domain, gives harmonic measure. This measure has had numerous applications in Potential Theory and applied areas throughout the twentieth century. In the mid 1980's mathematicians began to study the Hausdorff dimension of harmonic measure. In an important paper, Makarov proved that harmonic measure in simply connected planar domains always has dimension one. Jones and Wolf showed that the dimension of this measure is always less than or equal to one in any planar domain. The proposer has obtained endpoint type analogues of Makarov's work for p harmonic measures, when p is between one and infinity. In this proposal he discusses possible techniques for studying p harmonic measures in arbitrary planar domains and also in certain domains in higher dimensional Euclidean space.
Laplace's equation was used widely throughout the nineteenth and twentieth centuries to explain physical processes. Its nonlinear cousin, the p Laplacian, has only recently found applications in mathematical modeling (glacier formation, image processing) perhaps because this PDE is difficult to work with. The PI's work with coauthors gives the p Laplacian strong visibility in an area previously reserved for the Laplacian.The PI and coauthors have developed a p harmonic toolbox which enabled them to make significant progress on problems previously considered solvable only for Laplace's equation. The PI believes that the techniques, results, and problems discussed in his proposal will play an important role in popularizing the p Laplacian and also have applications in mathematical modeling. These problems - techniques involve a nice mixture of harmonic analysis, complex function theory, and partial differential equations, so should be attractive to researchers and graduate students in a wide area of mathematical and applied analysis.
