Mathematical Sciences: Potential Theory in n-space

与势能理论有关的数学研究， 调和函数的边界性质。 多 在势能理论中最好的工作发生在这两个方面- 维度设置与复杂分析的连接 全纯函数可以被利用。 高等 “真实的变量”技术必须在 与偏微分方程的进展相结合。 一 知识中的典型差距是缺乏有关根源的信息 沿边界沿着的梯度。 在二维空间中， 零度量。 将开展工作， 更高的维度 该项目的第二个重点将集中在奥克森达尔 更高维度的问题（以及相关问题）。 如果一个 接受一个密度（测量）的概念， 产生连续边界调和扩张的域 事实证明，这项措施主要集中在 在二维情况下，一组维度为1（精确地）。 自然的猜想是，n维域支持 n-1维集合上的调和测度，无论如何 边界可能很复杂。 前面的工作需要仔细分析 域的绿色函数。 有人提出了几个问题 关于绿党的水平集的几何形状 这将在本研究中得到解决。 在基地 这项工作的根本问题是如何绿色的线 在域的连续变形下变形。 //

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. //