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?

  • 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 Moos
    May 29, 2021 at 10:44
  • $\begingroup$ Yes, you are correct. I am also interested at the least regularity needed for the boundary to have a mean curvature that is in $L^p$. $\endgroup$ May 30, 2021 at 2:09
  • $\begingroup$ 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 Moos
    May 30, 2021 at 9:54
  • $\begingroup$ Thank @LeoMoos! The reference you recommend is helpful. $\endgroup$ May 31, 2021 at 23:11
  • $\begingroup$ 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 Moos
    Jun 1, 2021 at 13:36


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.