On the selection of maximal Cheeger sets
Buttazzo, Giuseppe; Carlier, Guillaume; Comte, Myriam (2007), On the selection of maximal Cheeger sets, Differential and Integral Equations, 20, 9, p. 991-1004
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00140075/en/
Journal nameDifferential and Integral Equations
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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 ). 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.
Subjects / KeywordsCheeger sets; p-Laplacian approximation; concave penalization; 1-Laplacian type operators
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