Fine properties of Cheeger sets and the Prescribed Mean Curvature problem in weakly regular domains

The two main problems we study are the \emph{Cheeger problem} and the \emph{Prescribed Mean Curvature problem}. The former consists in finding the subsets $E$ of a given ambient set $\Omega$ that realize the Cheeger constant, i.e. such that \[ \frac{P(E)}{|E|} = \inf \left \{ \frac{P(A)}{|A|} \right\} = h_1(\Om)\,, \] where the infimum is sought amongst all subsets of $\Om$ with positive volume; the latter is the non-linear partial differential equation given by \[ \div(Tu) = \div\left( \frac{\grad u}{\sqrt{1+|\grad u|^2}} \right) = H\,, \] which consists in finding functions $u$ whose graph has mean curvature $H$. At a first sight these two problems do not seem to be related but in the special case of a positive, constant prescribed mean curvature $H$ on $\Omega$, a necessary and sufficient condition to existence of solutions and uniqueness up to translations is that $H$ equals the Cheeger constant of $\Omega$ and $\Omega$ is a minimal Cheeger set. On one hand, we study a generalization of the Cheeger problem considering volumes with positive, non-vanishing $L^\infty$ weights and perimeters weighted through a function $g(x,\nu_\Omega (x))$ depending both on the point $x \in \de \Omega$ and the outer normal to $\Om$ at $x$. Then, we prove that any connected minimizer admits a Poincar\'e trace inequality, as well as the standard Sobolev embeddings. On the other hand, in the case of the standard Cheeger problem in dimension $2$ we show that, for simply connected sets $\Om$ that satisfy a ``no-bottleneck'' condition, the maximal Cheeger set $E$ equals the union of all balls contained in $\Om$ whose radius is $r=h_1^{-1}(\Om)$. Moreover, the inner Cheeger formula $|[\Om]^r|=\pi r^2$ holds, where $[\Om]^r$ denotes the set of points of $\Om$ at distance greater or equal than $r$ from $\de \Omega$. This result generalizes a property so far proved only for convex sets and planar strips. Concerning the Prescribed Mean Curvature problem, we show existence and uniqueness of solutions of the PMC equation only assuming that $\Omega$ is a \emph{weakly regular} open set, i.e., when $\Omega$ satisfies a Poincar\'e trace inequality and its perimeter agrees with the $(n-1)$-dimensional Hausdorff measure of the topological boundary. Under these assumptions, we show that uniqueness up to vertical translations is equivalent to several other properties. Namely, that the domain is maximal, i.e. no solutions for the same prescribed datum $H$ can exist in any set $\widetilde \Omega$ strictly containing $\Om$; that $\Om$ is critical, i.e. among all its subsets, it is the only one for which the inequality $|\int_A H| \le P(A)$ becomes an equality; that there exists a solution which solves the capillarity problem in a tube of cross-section $\Om$ with vertical contact angle, i.e. that it satisfies a tangential boundary condition in an integral sense or in a ``weak trace'' sense. Moreover, whenever the perimeter of $\Om$ agrees with the inner Minkowski content of $\Om$, this tangential ``weak trace'' condition assumes the stronger form $Tu(x) \to \nu_\Om (z)$ in a measure-theoretic sense, as $x\in \Omega$ approaches a point $z$ in the ``super-reduced boundary''. Finally, when the prescribed datum $H$ is positive and non-vanishing, we observe again the link between the Cheeger problem and the Prescribed Mean Curvature problem, as being critical corresponds to $\Omega$ being a minimal Cheeger set with Cheeger constant $1$, for the Cheeger problem with the standard perimeter and volume weighted through $H$.