Abstract
A bounded Euclidean domain is said to be a Dirichlet domain if every quasibounded harmonic function on is represented as a generalized Dirichlet solution on . As a localized version of this, is said to be locally a Dirichlet domain at a boundary point if there is a regular domain containing such that every quasibounded harmonic function on with vanishing boundary values on is represented as a generalized Dirichlet solution on . The main purpose of this paper is to show that the following three statements are equivalent by pairs: is a Dirichlet domain; is locally a Dirichlet domain at every boundary point ; is locally a Dirichlet domain at every boundary point except for points in a boundary set of harmonic measure zero. As an application it is shown that if every boundary point of is graphic except for points in a boundary set of harmonic measure zero, then is a Dirichlet domain, where a boundary point is said to be graphic if there are neighborhood of and an orthogonal (or polar) coordinate (or ) such that is represented as one side of a graph of a continuous function (or ).
Citation
Mitsuru NAKAI. "Local representability as Dirichlet solutions." J. Math. Soc. Japan 59 (2) 449 - 468, April, 2007. https://doi.org/10.2969/jmsj/05920449
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