Hardy-Sobolev inequalities in unbounded domains and heat kernel estimates
type: Item
instance of: scholarly article
title: Hardy-Sobolev inequalities in unbounded domains and heat kernel estimates
zbMATH Open document ID: 1273.26026
author: Konstantinos T. Gkikas
published in: Journal of Functional Analysis
publication date: 2013-03-01T00:00:00Z
review text: The author studies Hardy-Sobolev inequalities in unbounded domains. Specifically, new geometric assumptions on the external domains \(\Omega \subset \mathbb{R}^n\), \(n\geq 3\), are introduced and this assumption was used to derive a Hardy inequality with an optimal constant involving the distance on the boundary. Furthermore, it is shown that under some additional assumptions on \(\Omega\), this Hardy inequality can be improved by adding a Sobolev-type term with critical exponent. Several results on Hardy and Hardy-Sobolev-type inequalities in exterior domains are obtained. Finally, the Harnack inequality up to the boundary for the positive solutions of the problem \(u_{t} =\triangle u+\frac1{4}\frac{u}{\operatorname{dist}^{2}(x,\delta,\Omega)}\) and the heat kernel estimates for small times are derived, proved and discussed.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 6140428
zbMATH Keywords: Hardy inequalities
zbMATH Keywords: Hardy-Sobolev inequalities
zbMATH Keywords: distance function
zbMATH Keywords: critical exponent
zbMATH Keywords: exterior domain
zbMATH Keywords: unbounded domain
zbMATH Keywords: Harnack inequality
zbMATH Keywords: heat kernel estimates
MaRDI profile type: Publication
full work available at URL: j.jfa.2012.11.007
OpenAlex ID: W2046780459
DOI: 10.1016/J.JFA.2012.11.007
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description: scientific article; zbMATH DE number 6140428