On Hardy-Rellich type inequalities in \(\mathbb R^N\)
type: Item
instance of: scholarly article
title: On Hardy-Rellich type inequalities in \(\mathbb R^N\)
zbMATH Open document ID: 1175.26034
published in: Applied Mathematics Letters
publication date: 2009-06-10T00:00:00Z
review text: The following result is proved. Theorem. For all \(a,b\in\mathbb R\) and \(u\in C_0^{\infty}(\mathbb R^N\setminus\{0\})\) one has \[ \biggl[(n-2-\gamma)\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{\gamma}}\,dx+ 2\gamma\int_{\mathbb R^N}\frac{(x\cdot\nabla u)^2}{|x|^{\gamma+2}}\,dx\biggr]^2\leq 4\biggl(\int_{\mathbb R^N}\frac{|\Delta u|^2}{|x|^{2b}}\,dx\biggr)\biggl(\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{2a}}\,dx\biggr) \] where \(\gamma:=a+b+1\). In addition, if \(\gamma\leq n-2\), then \[ \biggl|\frac{N+a+b-1}{2}\biggr|\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{a+b+1}}\,dx\leq \biggl(\int_{\mathbb R^N}\frac{|\Delta u|^2}{|x|^{2b}}\,dx\biggr)^{1/2} \biggl(\int_{\mathbb R^N}\frac{|\nabla u|^2}{|x|^{2a}} \,dx\biggr)^{1/2}, \] where the constant \(|\frac{N+a+b-1}{2}|\) is sharp.
reviewed by: Szymon Wąsowicz
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5563917
zbMATH Keywords: Hardy inequality
zbMATH Keywords: Hardy-Rellich inequalities
author: David G. Costa
MaRDI profile type: Publication
full work available at URL: j.aml.2008.02.018
OpenAlex ID: W2061638214
cites work: Q4378149
cites work: Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities
cites work: Best constants in the Hardy-Rellich inequalities and related improvements
DOI: 10.1016/J.AML.2008.02.018
description: scientific article; zbMATH DE number 5563917