A geometric characterization of a sharp Hardy inequality

Details A geometric characterization of a sharp Hardy inequality A geometric characterization of a sharp Hardy inequality

2011-03-28

arxiv.org/abs/1103.5429v3

Mathematics

Analysis of PDEs

The results were improved to $C^2$ domains

Scientific paper

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}.

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