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Overdetermined problems and relative Cheeger sets in unbounded domains

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