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?
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1$\begingroup$ 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 $C^2$? $\endgroup$– Leo MoosMay 29, 2021 at 10:44
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$\begingroup$ Yes, you are correct. I am also interested at the least regularity needed for the boundary to have a mean curvature that is in $L^p$. $\endgroup$– Jullienne FranzMay 30, 2021 at 2:09
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$\begingroup$ 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 $C^{1,1-n/p}$. $\endgroup$– Leo MoosMay 30, 2021 at 9:54
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$\begingroup$ Thank @LeoMoos! The reference you recommend is helpful. $\endgroup$– Jullienne FranzMay 31, 2021 at 23:11
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$\begingroup$ 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 $x \mapsto \lvert x \rvert^{1+\alpha}$. $\endgroup$– Leo MoosJun 1, 2021 at 13:36
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