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On the best constant for Hardy's inequality in

Authors:	Moshe Marcus Victor J Mizel Yehuda Pinchover

Institution:	Department of Mathematics, Technion, Haifa, Israel ; Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 ; Department of Mathematics, Technion, Haifa, Israel

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 -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 , 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.

Keywords:	Rayleigh quotient concentration effect essential spectrum

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