# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## On the best constant for Hardy’s inequality in $\mathbb {R}^n$HTML articles powered by AMS MathViewer

by Moshe Marcus, Victor J. Mizel and Yehuda Pinchover
Trans. Amer. Math. Soc. 350 (1998), 3237-3255 Request permission

## 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 $\mu _p(\Omega )=\inf _{W_{1, p}(\Omega )}\left (\int _\Omega |\nabla u|^p / \int _\Omega |u/\delta |^p \right )$ 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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• 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
• MR Author ID: 139695
• Email: pincho@tx.technion.ac.il
• Received by editor(s): September 5, 1996