For domains of class $C^{1,1}$, I know that the mean curvature of its boundary belongs to $L^{\infty}(\partial \Omega)$. Is $C^{1,1}$ regularity the least regularity needed for the mean curvature to be in $L^{\infty}(\partial \Omega)$? How about the conditions on the boundary of a domain to have bounded mean curvatures? Which references discusses this kind of topic?

Lectures on Geometric Measure Theory? I believe this applies here, and gives the regularity as $C^{1,1-n/p}$. $\endgroup$