Mathematical Sciences: Theory of Capacities

我们将研究与偏微分方程解有关的域的几何解析性质。这项工作集中在领域的边界上，因为它是在边界上，解决方案的性质往往是模糊的。在边界正则性测量中发展起来的概念中，最重要的概念之一是容量，或者更准确地说，是一个集合的容量。尽管人们可以定义多种类型的容量，但理论的来源几乎肯定是基于静电学和导体承载电荷能力的测量。这本身就是一个关于偏微分方程解的问题。能力理论中的几个问题将在这项工作中得到解决。第一部分是关于退化椭圆方程研究中出现的加权容量的性质。人们希望描述解的边界正则点或可移动奇点。容量（集合函数）是由微分算子构建的。某些基本问题还没有得到完全的回答。它们包括至少就空集而言，加权容量如何依赖于权重。人们还希望能够根据退化的类型将正则点划分为等价类。在更大的尺度上，关于非线性方程的容量等价和正则性概念的结果才刚刚出现。本研究将应用于偏微分方程理论和势理论。

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.