Overdetermined problems and relative Cheeger sets in unbounded domains

TL;DR

This paper explores the relationship between overdetermined boundary problems and relative Cheeger sets in unbounded domains, establishing conditions under which solutions coincide with Cheeger sets and analyzing related geometric problems.

Contribution

It introduces the concept of relative Cheeger sets in unbounded domains and characterizes solutions of overdetermined problems as these sets, extending geometric analysis in unbounded contexts.

Findings

Solutions of overdetermined problems coincide with their relative Cheeger sets.

Characterization of constant mean curvature surfaces within unbounded domains.

Additional results for cylindrical domains with boundary or surfaces as graphs.

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 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.