# Least regularity of boundary to have Lipschitz or bounded mean curvature?

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?

• Am I correct in assuming that you are interested in a weak definition of mean curvature - via the first variation formula for example - seeing as it is only classically defined when $\partial \Omega$ is $C^2$? May 29, 2021 at 10:44
• Yes, you are correct. I am also interested at the least regularity needed for the boundary to have a mean curvature that is in $L^p$. May 30, 2021 at 2:09
• Have you read the chapter on Allard regularity in Leon Simon's Lectures on Geometric Measure Theory? I believe this applies here, and gives the regularity as $C^{1,1-n/p}$. May 30, 2021 at 9:54
• Thank @LeoMoos! The reference you recommend is helpful. May 31, 2021 at 23:11
• I'm glad you found it helpful - if it answers your question you could post it below. The analysis should be a bit simpler in your case because $\partial \Omega$ is the boundary of Caccioppoli set. Note also you should recover this exponent by looking at graphs of functions like $x \mapsto \lvert x \rvert^{1+\alpha}$. Jun 1, 2021 at 13:36