A remark on Hardy type inequalities with remainder terms

Angelo Alvino; Roberta Volpicelli; Bruno Volzone

doi:10.3934/dcdss.2011.4.801

Article Contents

2011, Volume 4, Issue 4: 801-807. Doi: 10.3934/dcdss.2011.4.801

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A remark on Hardy type inequalities with remainder terms

1.
Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Complesso Monte S. Angelo, Via Cintia, 80126 Naples
2.
Dipartimento per le Tecnologie, Università degli Studi di Napoli

Received: September 30, 2009

Revised: January 31, 2010

Published: July 2011

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Abstract

In this paper we focus our attention to some Hardy type inequalities with a remainder term. In particular we find the best value of the constant $h$ for the inequalities

$\int_{\Omega}|\nabla u|^2 dx \geq c \int_{\Omega}\frac{u^2}{|x|^2} dx+ h\int_{\Omega}\frac{u^2}{|x|}dx, \forall u\in H_0^1( \Omega) $

$ \int_{\Omega}|\nabla u|^2dx\geq c\int_{\Omega} \frac{u^2}{|x|^2}dx+ h(\int_{\Omega}|\nabla u| dx)^2, \forall u\in H_0^1 (\Omega)$

where $c\geq 0$ is smaller than the optimal Hardy constant $(N-2)^2/4$.

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Mathematics Subject Classification: Primary: 35J20, 26D10; Secondary: 46E35.

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