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On the selection of maximal Cheeger sets

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Date
2007-03
Link to item file
http://hal.archives-ouvertes.fr/hal-00140075/en/
Dewey
Analyse
Sujet
Cheeger sets; p-Laplacian approximation; concave penalization; 1-Laplacian type operators
Journal issue
Differential and Integral Equations
Volume
20
Number
9
Publication date
09-2007
Article pages
991-1004
Publisher
Khayyam Publishing
URI
https://basepub.dauphine.fr/handle/123456789/3452
Collections
  • CEREMADE : Publications
Metadata
Show full item record
Author
Buttazzo, Giuseppe
Carlier, Guillaume
Comte, Myriam
Type
Article accepté pour publication ou publié
Abstract (EN)
Given a bounded open subset Ω of ℝd and two positive weight functions ƒ et g, the Cheeger sets of Ω are the subdomains C of finite perimeter of Ω that maximize the ratio ∫cƒ(x)dx/∫∂*c g(x)dΗd-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 Ω is convex and ƒ ≡ g ≡ 1 as recently proved in [4]). However, there always exists a unique maximal (in the sense of inclusion) 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 → 1, a selection criterion for determining the maximal Cheeger set. On the contrary, a different perturbation scheme, based on the constrained maximization of ∫Ω ƒ(u-εΦ(u))dx for a strictly convex function Φ, gives, as ε→0, the desired maximal set.

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