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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the best constant for Hardy's inequality in $\mathbb R^n$


Authors: Moshe Marcus, Victor J. Mizel and Yehuda Pinchover
Journal: Trans. Amer. Math. Soc. 350 (1998), 3237-3255
MSC (1991): Primary 49R05, 35J70
MathSciNet review: 1458330
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Abstract: Let $\Omega $ be a domain in $\mathbb R^n$ and $p\in (1,\infty)$. We consider the (generalized) Hardy inequality $\int _\Omega |\nabla u|^p\geq K\int _\Omega |u/\delta |^p$, where $\delta (x)=\operatorname{dist}\,(x,\partial \Omega )$. The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant $\mu _p(\Omega )=\inf _{\stackrel{\circ}{W}_{1,p}(\Omega )}\left (\int _\Omega |\nabla u|^p\,/\,\int _\Omega |u/\delta |^p \right )$ and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth $n$-dimensional domains, $\mu _p(\Omega )\leq c_p$, where $c_p=(1-{1\over p})^p$ is the one-dimensional Hardy constant. Moreover it is shown that $\mu _p(\Omega )=c_p$ for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for $p=2$, it is proved that $\mu _2(\Omega )<c_2=1/4$ if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but $\mu _p=c_p$ for convex domains.


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Additional Information

Moshe Marcus
Affiliation: Department of Mathematics, Technion, Haifa, Israel
Email: marcusm@tx.technion.ac.il

Victor J. Mizel
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: vm09+@andrew.cmu.edu

Yehuda Pinchover
Affiliation: Department of Mathematics, Technion, Haifa, Israel
Email: pincho@tx.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02122-9
PII: S 0002-9947(98)02122-9
Keywords: Rayleigh quotient, concentration effect, essential spectrum.
Received by editor(s): September 5, 1996
Article copyright: © Copyright 1998 American Mathematical Society