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′,
xd) (or
x=
rξ) such that
V∩
R is represented as one side of a graph of a continuous function
xd=φ(
x′) (or
r=φ(ξ)).
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