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∈∂R if there is a regular domain U containing y such that every quasibounded harmonic function on U∩R with vanishing boundary values on $\overline{R}$∩∂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∈∂R; R is locally a Dirichlet domain at every boundary point y∈&\part;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∈∂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)).