Overdetermined problems and relative Cheeger sets in unbounded domains

Abstract

In this paper, we study a partially overdetermined mixed boundary value problem for domains $\Omega$ contained in an unbounded set $\mathcal{C}$. We introduce the notion of the Cheeger set relative to $\mathcal{C}$ and show that if a domain $\Omega \subset \mathcal{C}$ 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 $\Gamma$ inside $\mathcal{C}$. In the case when $\mathcal{C}$ is a cylinder, we obtain further results whenever the relative boundary of $\Omega$ or the surface $\Gamma$ is a graph on the base of the cylinder.