A rearrangement based proof for the existence of extremal functions for the Sobolev-Poincaré inequality on \(B^n\)
type: Item
instance of: scholarly article
title: A rearrangement based proof for the existence of extremal functions for the Sobolev-Poincaré inequality on \(B^n\)
zbMATH Open document ID: 1185.26029
author: Mark Leckband
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-12-22T00:00:00Z
review text: Let \(B^n\) be the open ball in \({\mathbb R}^n\), \(n\geq 2\). Assume that \(p\) is a real number such that \(1<p\leq (n+1)/2\) and \(q=np/(n-p)\). Let \(W^{1,p}({\mathbb R}^n)\) be the completion of \(C^\infty (B^n)\) in the norm \(\int_{B^n}(|u|^p+|\nabla u|^p)\,dx<\infty\). Let \(\bar u\) denote the mean-value of \(u\) over \(B^n\). The main result of this paper asserts that, under these assumptions, there is an extremal function for \[ \lambda (n,p):=\inf\frac{\|\nabla u\|_p}{\| u-\bar u\|_q}>0. \] The proof is based on the following facts: (i) the concentrated compactness principle asserts that a maximizing sequence must either converge to an extremal or to a delta function at a point on the boundary; (ii) a maximizing sequence must either converge to an extremal or to a delta function at \(-1\) or \(1\).
reviewed by: Teodora-Liliana Rădulescu
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46N20
zbMATH DE Number: 5651224
zbMATH Keywords: rearrangement
zbMATH Keywords: Sobolev-Poincaré inequality
zbMATH Keywords: extremals
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.09.050
OpenAlex ID: W2074495189
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DOI: 10.1016/J.JMAA.2009.09.050
description: scientific article; zbMATH DE number 5651224