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It is a well known result that if $\Omega\subset \Bbb{R}^N$ is an open set, with regular boundary (smooth, or Lipschitz) then the problem $ \min_{E \subset \Omega |E|=c} Per(E)$ has a solution, where $c \in (0,|\Omega|)$ and $Per(E)$ is the perimeter of $E$ with respect to $\Omega$. It is also known that if $\Omega$ contains an open ball of volume $c$ then the minimizer is a ball.

Now, sometimes the geometry of $\Omega$ doesn't allow for a ball of radius $c$ to fit in it (for $c$ sufficiently large, well chosen). I was wondering it you could provide me a reference where the following reasonable facts are proved:

  • if $c$ is chosen such that $c<|\Omega|$ and a ball of volume $c$ doesn't fit in $\Omega$ then the minimizer $E$ must touch the boundary of $\Omega$ on a set of positive $\mathcal{H}^{N-1}$ measure, i.e. positive perimeter

  • the portions of the minimizer $E$ which are in the interior of $\Omega$ must have piecewise constant mean curvature.

Thank you.

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    I think that the "well known result" is not true as stated. Let $\Omega$ be a ball $B$ with a long thin tube attached to it. You can bound the volume $|B|/2$ within the tube by using very little (relative) perimeter.2012-08-05

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