Local representability as Dirichlet solutions

Abstract

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 $y\in \partial R$ if there is a regular domain $U$ containing $y$ such that every quasibounded harmonic function on $U\cap R$ with vanishing boundary values on $\overline{R}\cap \partial U$ is represented as a generalized Dirichlet solution on $U\cap R$. 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 $y\in \partial R$; $R$ is locally a Dirichlet domain at every boundary point $y\in \partial R$ 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 $y\in \partial R$ is said to be graphic if there are neighborhood $V$ of $y$ and an orthogonal (or polar) coordinate $x=(x\text{'},x^d)$ (or $x=r\xi$) such that $V\cap R$ is represented as one side of a graph of a continuous function $x^d=\varphi (x\text{'})$ (or $r=\varphi \left(\xi \right)$).