Fix two open smooth bounded domains $\Omega_-$ and $\Omega_+$ with $\overline{\Omega}_-\subset \Omega_+$ in a complete Riemannian manifold ($\mathbb{R}^n$ is already interesting to me).

I was wondering if there is any literature on the following question: For $V\in (0, |\Omega_+|-|\Omega_-|)$ let $$ P_{min}(V)=\min\{ |\partial^* E|: \Omega_-\subset E\subset \Omega_+, |E\backslash \Omega_-|=V\} $$ where here $E$ is a set of finite perimeter. That is we are trying to find the region containing $\Omega_-$ and contained in $\Omega_+$ with that encloses a fixed amount of volume (namely $V+|\Omega_-|$) and has least area.

Standard compactness arguments for sets of finite perimeter should give a minimizer $E_{min}(V)$ with $|\partial^* E_{min}(V)|=P_{min}(V)$ that has constant mean curvature (in a weak sense) in $\Omega_+\backslash \overline{\Omega}_-$. I would also expect something like smoothness for the (reduced) boundary of this minimizer away from a codimension 7 singular set in this set. I would also expect something like free boundary behavior at $\partial \Omega_-\cup \partial \Omega_+$.

I'm particularly interested in what happens as $V\to 0$. I would expect that (perhaps under some additional geometric conditions on $\Omega_-$) that the minimizers should like like little ``bumps" on $\Omega_-$ when $V$ is below some critical value.

This seems like a fairly natural problem to study, but haven't been able to find anything on it from a cursory search of the literature.


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