Optimizing improved Hardy inequalities
type: Item
instance of: scholarly article
title: Optimizing improved Hardy inequalities
zbMATH Open document ID: 1030.26018
author: Stathis Filippas
author: Achilles Tertikas
published in: Journal of Functional Analysis
publication date: 2002-09-08T00:00:00Z
review text: Let \(N\geq 3\) and \(\Omega\) be a bounded domain in \({\mathbb R}^N\) such that \(0\in \Omega\). The goal of the authors is to study a general improved Hardy inequality: For all \(u\in H^1_0(\Omega)\), \[ \int_{\Omega} \left|\nabla u\right|^2\geq \left(\frac{N-2}2\right)^2\int_{\Omega} \frac{\left|u\right|^2}{\left|x\right|^2}dx+b\int_{\Omega} Vu^2dx \] for a suitable chosen potential \(V\). First, they find necessary conditions on the potential \(V\), under which the previous inequality can or cannot be further improved. They show that the best constant \(b\) is never achieved in \(H^1_0(\Omega)\). Moreover, a ``series expansion'' for the Hardy inequality is obtained. In establishing these results, they derive various sharp improved Hardy-Sobolev inequalities.
reviewed by: Grzegorz Karch
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35A25
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 1797597
zbMATH Keywords: Hardy-Sobolev inequalities
zbMATH Keywords: best constant
MaRDI profile type: Publication
OpenAlex ID: W2089589660
cites work: An improved Hardy-Sobolev inequality and its application
cites work: Existence and non-existence of the first eigenvalue of the perturbed Hardy–Sobolev operator
cites work: A unified approach to improved $L^p$ Hardy inequalities with best constants
cites work: Sobolev inequalities with remainder terms
cites work: Q4538008
cites work: Extremal functions for Hardy's inequality with weight
cites work: Q4378149
cites work: Weak eigenfunctions for the linearization of extremal elliptic problems
cites work: Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier
cites work: Q4938492
cites work: Q4264768
cites work: Hardy inequalities with optimal constants and remainder terms
cites work: Rearrangements and convexity of level sets in PDE
cites work: Q4693168
cites work: Q3997067
cites work: On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term
cites work: Schrödinger semigroups
cites work: Critical phenomena in linear elliptic problems
cites work: Domain of existence and blowup for the exponential reaction-diffusion equation
cites work: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential
DOI: 10.1006/JFAN.2001.3900
description: scientific article; zbMATH DE number 1797597