## On the best constant for Hardy’s inequality in $\mathbb {R}^n$

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- by Moshe Marcus, Victor J. Mizel and Yehuda Pinchover PDF
- 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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## 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
- MR Author ID: 139695
- Email: pincho@tx.technion.ac.il
- Received by editor(s): September 5, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 3237-3255 - MSC (1991): Primary 49R05, 35J70
- DOI: https://doi.org/10.1090/S0002-9947-98-02122-9
- MathSciNet review: 1458330