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 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 ∈ ∂ 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 = r ξ ) such that V ∩ R is represented as one side of a graph of a continuous function x d = ϕ ( x ' ) (or r = ϕ ( ξ ) ).