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arXiv · 2203.08940

Overdetermined problems and relative Cheeger sets in unbounded domains

Danilo Gregorin Afonso·Alessandro Iacopetti·Filomena Pacella
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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.

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Danilo Gregorin Afonso, Alessandro Iacopetti, Filomena Pacella. 2022-03-16. Overdetermined problems and relative Cheeger sets in unbounded domains. https://arxiv.org/abs/2203.08940

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