# Isoperimetric profile with obstacle

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.