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

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.