Overdetermined problems and relative Cheeger sets in unbounded domains

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 overd...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Danilo Gregorin Afonso, Iacopetti, Alessandro, Pacella, Filomena
Format: Article
Language:English
Subjects:
Boundary value problems
Cylinders
Domains
Online Access:Get full text
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Summary: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.
ISSN:2331-8422