Liouville's theorem and the restricted mean value property in the plane
type: Item
instance of: scholarly article
title: Liouville's theorem and the restricted mean value property in the plane
zbMATH Open document ID: 0919.31002
DOI: 10.1016/S0021-7824(01)80004-0
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1999-07-28T00:00:00Z
review text: Let \(U\) be a domain in \(\mathbb{R}^2\) such that the complement to \(U\) is polar and let \(r\) be a real function on \(U\) such that \(0<r\leq\| \cdot\|+M\) where \(M\) is a nonnegative constant. A nonnegative numerical function \(f\) on \(U\) is called \(r\)-supermedian if, for every \(x\in U\), the average of \(f\) on the disk of centre \(x\) and radius \(r(x)\) is smaller than or equal to \(f(x)\). The current paper provides a short proof of the fact that every lower semicontinuous \(r\)-supermedian function is constant.
Mathematics Subject Classification ID: 31A99
zbMATH DE Number: 1248319
zbMATH Keywords: Liouville's theorem
zbMATH Keywords: restricted mean value
zbMATH Keywords: median function
zbMATH Keywords: supermedian function
author: Wolfhard Hansen
reviewed by: Ilya S. Molchanov
MaRDI profile type: Publication
cites work: A converse to the mean value theorem for harmonic functions
cites work: Mean values and harmonic functions
cites work: Liouville's theorem and the restricted mean value property
cites work: Restricted mean value property on \(\mathbb{R}^ d, d\leq 2\)
description: scientific article; zbMATH DE number 1248319