Mathematical Sciences: Theory of Capacities

工作将做调查几何分析 域的性质，因为它们涉及部分解 微分方程 工作集中在边界上 因为它是在边界的性质， 解决办法往往是模糊的。 在这些概念中 在测量边界规则性的过程中， 最重要的是能力，或者更准确地说， 一套的能力。 虽然有许多类型的 一个人可以定义的能力，理论的来源几乎肯定在于 用静电学和测量导体 支持一项指控 这本身就是一个关于解决方案的问题 一个偏微分方程。 能力理论中的几个问题将是 在这项工作中处理。 第一个问题是关于 退化椭圆方程研究中的加权容量 方程 人们希望检查边界正则点 或者解的可移除奇点。 的能力 (set函数）是由微分算子构建的。 某些基本问题还没有完全得到回答。 它们包括加权容量如何取决于权重，至少 就null set而言。 一个人也会想成为 能够将规则点分类为等价类 根据简并的类型。 在更大的范围内， 容量的等价性和正则性概念 非线性方程才刚刚出现。 这项研究将有应用的理论，部分 微分方程和势能理论。

Work will be done investigating geometric-analytic properties of domains as they relate to solutions of partial differential equations. The work concentrates on the boundaries of domains for it is at the boundary where the nature of solutions tends to be obscure. Among the concepts which have evolved in the measurement of boundary regularity, one of the most important is that of capacity, or more precisely, the capacity of a set. Although there are a number of types of capacity one may define, the source of theory almost surely rests with electrostatics and measurement of how well a conductor can support a charge. This in itself is a question about solutions of a partial differential equations. Several questions in the theory of capacities will be addressed in this work. The first is concerned with nature of weighted capacities arising in the study of degenerate elliptic equations. One wishes to chacterize the boundary regular points or the removable singularities for solutions. The capacities (set functions) are built out of the differential operator. Certain fundamental questions as yet are incompletely answered. They include how weighted capacity depends on the weight at least as far as null sets are concerned. One would also want to be able to classify regular points into equivalence classes according to types of degeneracy. On a larger scale, results on equivalence of capacities and concepts of regularity for nonlinear equations are only just appearing. This research will have application to the theory of partial differential equations and potential theory.