The Dirichlet problem for harmonic functions on compact sets
type: Item
instance of: scholarly article
title: The Dirichlet problem for harmonic functions on compact sets
zbMATH Open document ID: 1243.31007
author: Tony L. Perkins
published in: Pacific Journal of Mathematics
publication date: 2012-05-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.5575
full work available at URL: p11.xhtml
review text: This article presents an account of the Dirichlet problem with respect to a compact set \(K\) in \(\mathbb{R}^{n}\). The fine topology of classical potential theory plays an essential role, as does the notion of a finely harmonic function on a finely open set. Let \(\partial _{f}K\) denote the fine boundary of \(K\), and \(fH^{c}(K)\) be the collection of finely continuous functions on \(K\) that are finely harmonic on the fine interior of \(K\) and continuous and bounded on \(\partial _{f}K\). Theorem 6.5 says that, for every bounded continuous function \(\phi \) on \(\partial _{f}K\), there is a unique function \(h_{\phi }\in fH^{c}(K)\) that equals \(\phi \) on \(\partial _{f}K\). Further, for each \(x\in K\), the value of \(h_{\phi }(x)\) is given by the integral of \(\phi \) against a certain probability measure on \(\partial _{f}K\).
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31C40
zbMATH DE Number: 6029188
zbMATH Keywords: potential theory
zbMATH Keywords: fine topology
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
OpenAlex ID: W1981173115
arXiv ID: 1004.5575
DOI: 10.2140/PJM.2011.254.211
description: scientific article; zbMATH DE number 6029188