Hardy-type inequalities
type: Item
instance of: scholarly article
title: Hardy-type inequalities
zbMATH Open document ID: 1083.26012
DOI: 10.4171/JEMS/12
author: Juan Dávila
author: L. Dupaigne
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2004-09-24T00:00:00Z
review text: Though the well-known Hardy-Sobolev inequality \[ K^2\int_\Omega u^2/| x|^2\,dx\leq\int_\Omega|\nabla u|^2\,dx, \quad \Omega\subset\mathbb R^n,\;n\geq3,\;u\in C^\infty_c(\Omega), \] holds with an optimal constant, the equality is never achieved. Therefore the inequality has been improved in various ways. Recall, for instance, the inequality \[ \gamma \biggl(\int_\Omega| u|^p\biggr)^{2/p}+ K^2\int_\Omega u^2/| x|^2 \leq \int_\Omega |\nabla u|^2, \quad \gamma>0, \;\Omega \text{ bounded, }1\leq p<2n/(n-2), \] proved by \textit{H. Brézis} and \textit{J. L. Vázquez} [Rev. Mat. Univ. Complutense Madr. 10, No. 2, 443--469 (1997; Zbl 0894.35038)]. The authors consider the operators of the form \(L=-\Delta-\mu/d(x)^2\), where \(\mu\in\mathbb R^n\), \(d(x)=\text{dist}(x,\Sigma)\), \(x\in\Omega\), \(\Omega\subset\mathbb R^n\) is an open bounded set, \(\Sigma\) is a compact smooth manifold without boundary of codimension \(k\neq2\), and in this connection they are interested in the question whether an inequality analogous to the above one holds. They prove that there exist constants \(C>0\), \(\gamma>0\) such that any \(u\in C^\infty_c(\Omega\setminus\Sigma)\) satisfies \[ \gamma\biggl(\int_\Omega| u|^p\biggr)^{2/p}+ H^2\int_\Omega u^2/d^2 \leq \int_\Omega |\nabla u|^2+ C\int_\Omega u^2, \] where \(d(x)=\text{dist}(x,\Sigma)\), \(H=(k-2)/2\), \(1\leq p<p_k\) and \(p_k\) is given by \(\frac1{p_k}=\frac12-\frac2{k(n-k+2)}\) for \(k>2\), \(\frac1{p_1}=\frac12-\frac1{n+1}\) for \(k=1\). Moreover, there exist \(\beta>0\) and a neighbourhood \(\Omega_\beta=\{x\in\Omega: d(x,\Sigma)<\beta\}\) of \(\Sigma\) such that for any \(u\in C^\infty_c(\Omega_\beta\subset\Sigma)\) the inequality \[ H^2\int_\Omega u^2/d^2\leq\int_\Omega|\nabla u|^2 \] holds. The authors admit that the values of \(p_k\) are probably not optimal and they expect that these values might be replaced with \(2n/(n-2)\). Finally, they extend some of their earlier results [Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 1, …
reviewed by: Jiří Rákosník
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 2102879
zbMATH Keywords: Hardy-Sobolev inequality
zbMATH Keywords: additional terms
zbMATH Keywords: distance function
zbMATH Keywords: manifold without boundary
zbMATH Keywords: linear boundary value problem
zbMATH Keywords: pointwise estimate
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2102879