Mathematical Sciences: Potential Theory in n-space

对势理论和调和函数的边界性质进行数学研究。势理论中许多最好的工作发生在二维环境中，在那里可以利用与全纯函数的复分析的联系。在更高的维度，“实变量”技术必须与偏微分方程的进展结合起来发展。该算法的一个典型缺陷是缺乏沿边界的梯度根的信息。在二维空间中，这个集合的测度为零。工作将被完成以澄清更高维度的情况。这个项目的第二个重点将集中在更高维度的Oksendal问题（以及相关问题）上。如果一个人接受在产生连续边界函数的调和扩展的区域边界上的密度（测度）的思想，那么已经证明，在二维情况下，这个测度集中在维数为1的集合上（精确地）。自然的猜想是n维域在n-1维的集合上支持调和测度，不管边界有多复杂。前面的工作需要仔细分析一个域的格林函数。本文对格林函数的水平集的几何性质提出了若干问题。这项工作的基础是格林线如何在一个域的连续变形下变形的基本问题。//

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. //