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Local representability as Dirichlet solutions

机译：作为Dirichlet解决方案的局部可表示性

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摘要

A bounded Euclidean domain R is said to be a Dirichlet domain if every quasibounded harmonic function on R is represented as a generalized Dirichlet solution on R. As a localized version of this, R is said to be locally a Dirichlet domain at a boundary point y ∈ partial deriv R if there is a regular domain U containing y such that every quasibounded harmonic function on U ∩ R with vanishing boundary values on R ∩ partial deriv U is represented as a generalized Dirichlet solution on U ∩ R. The main purpose of this paper is to show that the following three statements are equivalent by pairs: R is a Dirichlet domain; R is locally a Dirichlet domain at every boundary point y ∈ partial deriv R; R is locally a Dirichlet domain at every boundary point y ∈ partial deriv R except for points in a boundary set of harmonic measure zero. As an application it is shown that if every boundary point of R is graphic except for points in a boundary set of harmonic measure zero, then R is a Dirichlet domain, where a boundary point y ∈ partial deriv R is said to be graphic if there are neighborhood V of y and an orthogonal (or polar) coordinate x = (x',x~d) (or x = rξ) such that V ∩ R is represented as one side of a graph of a continuous function x~d = φ(x') (or r = φ(ξ)).
机译：如果R上的每个拟有界谐波函数都表示为R上的广义Dirichlet解，则有界欧几里得域R被称为Dirichlet域。作为其局部化形式，R被认为是在边界点y上的局部Dirichlet域。 ∈偏导数R，如果存在一个包含y的规则域U，使得U∩R上的每个拟有界谐波函数在R∩偏导数U上的边界值都消失了，则表示为U∩R上的广义Dirichlet解。本文将证明以下三个语句成对等效：R是Dirichlet域； R在每个边界点y∈偏导R处局部为Dirichlet域； R在每个边界点y∈偏导数R处局部是Dirichlet域，除了谐波量度为零的边界集中的点。作为应用表明，如果R的每个边界点都是图形的，除了谐波量为零的边界集中的点，则R是Dirichlet域，如果存在，则边界点y∈偏导数R就是图形的是y的邻域V和正交坐标（或极坐标）x =（x'，x〜d）（或x =rξ），使得V∩R表示为连续函数x〜d =的图的一侧φ（x'）（或r =φ（ξ））。

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