On the selection of maximal Cheeger sets

Given a bounded open subset Ω of Rd and two positive weight functions f and g, the Cheeger sets of Ω are the subdomains C of finite perimeter of Ω that maximize the ratio ∫ C f(x) dx / ∫ ∂∗C g(x) dH 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 Ω is convex and f ≡ 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 f(u − εΦ(u) ) dx for a strictly convex function Φ, gives, as ε → 0, the desired maximal set.