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What is the minimal regularity required for a set $\Omega \subset \Bbb{R}^3 (\Bbb{R}^N)$ such that the mean curvature $H$ is a continuous function $H: \partial \Omega \to \Bbb{R}$?

I know that $C^2$ is enough, but can we have less regularity than that (for example $C^{1,1}$) and the curvature to still be continuous?

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You need second derivatives even to define $H$. Unless you are using a generalization of some sort? – user53153 Jan 14 '13 at 15:36
up vote 2 down vote accepted

I would say that $C^{1,1}$ is not enough to guarantee continuity of the mean curvature. In $\mathbb{R}^2$, think of a stadium-like domain (the convex hull of two tangent balls having the same radius R): then $H=1/R$ on the curved parts, but $H=0$ on the straight lines.

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