Mathematical Sciences: Potential Theory in n-space

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Mathematical research concerned with potential theory and the boundary properties of harmonic functions will be done. Much of the best work done in potential theory occurs in the two- dimensional setting where the connections with complex analysis of holomorphic functions can be exploited. In higher dimensions, "real variable" techniques have to be developed in conjunction with advances in partial differential equations. A typical gap in the lore is the lack of information on the roots of the gradient along the boundary. In two dimensions, this set has zero measure. Work will be done to clarify the situation in higher dimensions. A second thrust of this project will focus on the Oksendal problem for higher dimensions (and related issues). If one accepts the idea of a density (measure) on the boundary of a domain which produces harmonic extensions of continuous boundary functions, it has been shown that this measure is concentrated on a set of dimension 1 (precisely) in the two-dimensional case. The natural conjecture is that n-dimensional domains support harmonic measure on sets of dimension n-1, regardless of how complicated the boundary may be. The preceding effort requires careful analysis of the Green's function of a domain. Several questions have been raised concerning the geometry of the level sets of the Greens's function which will be addressed in this research. At the base of this work is the fundamental question of how Green's lines deform under continuous deformation of a domain. //