A geometric characterization of a sharp Hardy inequality

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}.

The results were improved to $C^2$ domains