Overdetermined problems and relative Cheeger sets in unbounded domains

In this paper we study a partially overdetermined mixed boundary value problem for domains $\mathrm{\Omega }$ contained in an unbounded set $\mathcal{C}$. We introduce the notion of Cheeger set relative to $\mathcal{C}$ and show that if a domain $\mathrm{\Omega }\subset \mathcal{C}$ admits a solution of the overdetermined problem, then it coincides with its relative Cheeger set. We also study the related problem of characterizing constant mean curvature surfaces $\mathrm{\Gamma }$ inside $\mathcal{C}$. In the case when $\mathcal{C}$ is a cylinder we obtain further results whenever the relative boundary of $\mathrm{\Omega }$ or the surface $\mathrm{\Gamma }$ is a graph on the base of the cylinder.