Differential and Integral Equations

On the selection of maximal Cheeger sets

Giuseppe Buttazzo, Guillaume Carlier, and Myriam Comte

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Abstract

Given a bounded open subset $\Omega$ of $\mathbb R^d$ and two positive weight functions $f$ and $g$, the Cheeger sets of $\Omega$ are the subdomains $C$ of finite perimeter of ${{\overline{\Omega}}}$ that maximize the ratio $$ \int_C f(x)\,dx\,\big/\int_{\partial^* C}g(x)\,d{{\mathcal H}}^{d-1} . $$ Existence of Cheeger sets is a well-known fact. Uniqueness is a more delicate issue and is not true in general (although it holds when $\Omega$ is convex and $f\equiv g\equiv 1$ as recently proved in [4]). However, there always exists a unique maximal Cheeger set, and this paper addresses the issue of how to determine this maximal set. We show that in general the approximation by the $p$-Laplacian does not provide, as $p\to 1$, a selection criterion for determining the maximal Cheeger set. On the contrary, a different perturbation scheme, based on the constrained maximization of $\int_\Omega f(u-{\varepsilon}\Phi(u)\big)\,dx$ for a strictly convex function $\Phi$, gives, as ${\varepsilon}\to0$, the desired maximal set.

Article information

Source
Differential Integral Equations Volume 20, Number 9 (2007), 991-1004.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039307

Mathematical Reviews number (MathSciNet)
MR2349376

Zentralblatt MATH identifier
1212.49019

Subjects
Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 35J20: Variational methods for second-order elliptic equations 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90] 49R50

Citation

Buttazzo, Giuseppe; Carlier, Guillaume; Comte, Myriam. On the selection of maximal Cheeger sets. Differential Integral Equations 20 (2007), no. 9, 991--1004. https://projecteuclid.org/euclid.die/1356039307.


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