A geometric characterization of a sharp Hardy inequality

TL;DR

This paper characterizes when the distance function is superharmonic in terms of weak mean convexity of the boundary and establishes sharp Hardy inequalities on such domains, highlighting the geometric conditions needed.

Contribution

It provides a geometric characterization of weakly mean convex domains via superharmonicity of the distance function and proves sharp Hardy inequalities in this setting.

Findings

Distance function is superharmonic iff boundary is weakly mean convex.

Sharp Hardy inequalities hold on weakly mean convex $C^{2}$ domains.

Weakly mean convexity condition is necessary and cannot be weakened.

Abstract

In this paper, we prove that the distance function of an open connected set in $\mathbb R^{n+1}$ with a $C^{2}$ boundary is superharmonic in the distribution sense if and only if the boundary is {\em weakly mean convex}. We then prove that Hardy inequalities with a sharp constant hold on {weakly mean convex} $C^{2}$ domains. Moreover, we show that the {weakly mean convexity} condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis-Marcus \cite{BM}.