Harmonic approximation and its applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 0991.31007
publication date: 2002-09-01T00:00:00Z
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31-02
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 31B20
Mathematics Subject Classification ID: 31B25
zbMATH DE Number: 1679783
zbMATH Keywords: harmonic approximation
zbMATH Keywords: survey
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: accessibility
zbMATH Keywords: fine topology
zbMATH Keywords: finely harmonic functions
zbMATH Keywords: fine cluster sets
zbMATH Keywords: non-tangential cluster sets
author: Stephen J. Gardiner
MaRDI profile type: Publication
title: Harmonic approximation and its applications
review text: This article is a survey originally delivered by the author as a series of four lectures in July 2000. Almost all of the work discussed is due to the author, and an account of results obtained before 1995 is given in his monograph [Harmonic approximation, London Math. Soc. Lecture Note Ser. 221, Cambridge Univ. Press, Cambridge (1995; Zbl 0826.31002)]. The topics treated are briefly indicated below. NEWLINENEWLINENEWLINELecture 1. Let \(\Omega\) be an open subset of \(\mathbb{R}^n\) \((n\geq 2)\) and let \(E\) be a relatively closed subset of \(\Omega\). For a subset \(A\) of \(\mathbb{R}^n\), let \({\mathcal H}(A)\) denote the space of functions that are hamonic on some open set containing \(A\). Under which conditions on \(\Omega\) and \(E\) is it possible to approximate functions in \(C(E)\cup{\mathcal H}(E^0)\) by functions in \({\mathcal H}(\Omega)\), the approximation being either uniform on \(E\) or (if \(E\) is non-compact) tangential in some sense? Precise answers are given; the conditions involve both the Euclidean topology and the fine topology. NEWLINENEWLINENEWLINELecture 2. Harmonic approximation is used to answer an old question about the Dirichlet problem: which open sets \(\Omega\) have the property that for each \(f\in C(\partial \Omega)\) there exists \(h_f\in {\mathcal H}(\Omega)\) such that \(h_f(x)\to f(y)\) as \(x\to y\) for every regular \(y\in\partial \Omega\) and \(h_f\) is bounded near every irregular \(y\in \partial \Omega\)? Here \(\partial \Omega\) is the Euclidean boundary of \(\Omega\), so (if \(\Omega\) is unbounded) functions in \(C(\partial \Omega)\) can be wild near \(\infty\). NEWLINENEWLINENEWLINELecture 3. The author returns to the theory of harmonic approximation. Here the question is: given a particular relatively closed subset \(E\) of \(\Omega\), which functions in \(C(E)\cap{\mathcal H}(E^0)\) can be approximated uniformly on \(E\) by functions in \({\mathcal H}(\Omega)\)? A precise answer is given involving the notions of…
reviewed by: D. H. Armitage
description: scientific article; zbMATH DE number 1679783