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?