The weighted Hardy constant

TL;DR

This paper investigates weighted Hardy inequalities near domain boundaries, establishing conditions for their validity, calculating optimal constants, and applying results to self-adjointness of elliptic operators.

Contribution

It provides new criteria for the validity of weighted Hardy inequalities in various domain types and determines explicit optimal constants, with applications to elliptic operator self-adjointness.

Findings

Hardy inequality validity linked to Davies' weak Hardy inequality.

Explicit optimal constants for smooth, convex, and uniform domains.

Application of inequalities to criteria for self-adjointness of elliptic operators.

Abstract

Let $\Omega$ be a domain in $R^d$ and $d_\Gamma$ the Euclidean distance to the boundary $\Gamma$. We investigate whether the weighted Hardy inequality \[ \|d_\Gamma^{\delta/2-1}\varphi\|_2\leq a_\delta\,\|d_\Gamma^{\delta/2}\,(\nabla\varphi)\|_2 \] is valid, with $\delta\geq 0$ and $a_\delta>0$, for all $\varphi\in C_c^1(\Gamma_r)$ and all small $r>0$ where $\Gamma_r=\{x\in\Omega: d_\Gamma(x)<r\}$. First we prove that if $\delta\in[0,2\rangle$ then the inequality is equivalent to the weighted version of Davies' weak Hardy inequality on $\Omega$ with equality of the corresponding optimal constants. Secondly, we establish that if $\Omega$ is a uniform domain with Ahlfors regular boundary then the inequality is satisfied for all $\delta\geq 0$, and all small $r$, with the exception of the value $\delta=2-(d-d_H)$ where $d_H$ is the Hausdorff dimension of $\Gamma$. Moreover, the optimal constant $a_\delta(\Gamma)$ satisfies $a_\delta(\Gamma)\geq 2/|(d-d_H)+\delta-2|$. Thirdly, if $\Omega$ is a $C^{1,1}$-domain or a convex domain $a_\delta(\Gamma)=2/|\delta-1|$ for all $\delta\geq0$ with $\delta\neq1$. The same conclusion is correct if $\Omega$ is the complement of a convex domain and $\delta>1$ but if $\delta\in[0,1\rangle$ then $a_\delta(\Gamma)$ can be strictly larger than $2/|\delta-1|$. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators.