Volume mean densities for the heat equation
type: Item
instance of: scholarly article
title: Volume mean densities for the heat equation
zbMATH Open document ID: 1307.35014
author: Wolfhard Hansen
author: Ivan Netuka
published in: Potential Analysis
publication date: 2014-11-13T00:00:00Z
review text: Let \(D\) be a bounded domain in \(\mathbb R^d\times (-\infty ,0)\), \(d\geq 1\), such that \(z_0=(0,0) \in \overline D\). If \(0<a<1/e\) and \(D=\{ (x,-t)\in \mathbb R^{d+1}; |x|<\sqrt{2dt \log(1/t)}, 0<t<a\} \), then there exists a Borel measurable function \(w\) on \(D\) such that \(\inf w(D)>0\) and \(\int_D w(z)v(z)\;dz\leq v(z_0)\) for every supertemperature \(v\) on a neighbourhood of \(\overline D\). On the other hand, if there exist \(t_0,r\in (0,\infty )\) such that \(\{ (x,t)\in \mathbb R^{d+1}; 0<t<t_0, |x|<(d+1)d^{-1}\sqrt{2dt\log (r/t)}\} \subset D\), then there is no Borel measurable function \(w\) on \(D\) with these properties.
reviewed by: Dagmar Medková
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 31D05
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 6369345
zbMATH Keywords: heat ball
zbMATH Keywords: fulks measure
zbMATH Keywords: mean value density
zbMATH Keywords: representing measure
MaRDI profile type: Publication
full work available at URL: s11118-014-9411-z
OpenAlex ID: W1999673952
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DOI: 10.1007/S11118-014-9411-Z
description: scientific article; zbMATH DE number 6369345