Shape derivative of the Cheeger constant

Abstract

ABSTRACT. This paper deals with the existence of the shape derivative of the Cheeger constant h 1 (Ω) of a bounded domain Ω. We prove that if Ω admits a unique Cheeger set, then the shape derivative of h 1 (Ω) exists, and we provide an explicit formula. A counterexample shows that the shape derivative may not exist without the uniqueness assumption. INTRODUCTIONLet Ω ⊂ R n be a bounded domain. The Cheeger constant of Ω is defined asHere P(E; R n ) is the distributional perimeter of E measured with respect to R n , while |E| is the n−dimensional Lebesgue measure of E. A set C ⊂ Ω for which the infimum is attained is called a Cheeger set.The problem of finding a Cheeger set for a given domain Ω has extensively received attention in the last decades, starting from the original work of Jeff Cheeger [5]. For an introductory survey on the Cheeger problem we refer to [18]; here we recall that for every bounded domain Ω with Lipschitz boundary there exists at least one Cheeger set. Uniqueness does not hold in general, but it is guaranteed if we assume Ω to be convex; in this case the Cheeger set turns out to be convex and of class C 1,1 (see [1]). The Cheeger constant can be obtained as the limit for p → 1 of the first eigenvalue λ p (Ω) of the p−Laplacian under Dirichlet boundary conditions (see [12]), and corresponds to the first eigenvalue of the 1−Laplacian (see [14]).Shape analysis roughly consists in studying the regularity and the optimisation of a functional J : Ω ∈ A → J(Ω) ∈ R defined over some class A of subsets Ω ⊂ R n . Due to its physical relevance, a particularly important class of functionals are the ones defined in terms of the eigenvalues of some operator. A lot of works have been dedicated for instance to the study of the dependence of the eigenvalues of the Laplacian as functions of the domain under various boundary conditions. We refer for example to the monograph [11] for an introduction to the field of shape analysis.In order to optimize J over A it is important to determine how sensitive is J under perturbation of a given set Ω. Given a smooth vector field V ∈ C ∞ c (R n ; R n ), define F t : R n → R n as F t (x) = (Id + tV )(x). We then perturb Ω in the direction V by considering the sets Ω t = F t (Ω). The shape derivative of J in the direction V at Ω is then defined as J(Ω,V ) ′ := lim t→0 J(Ω t ) − J(Ω) t .

Reverse Cheeger inequality for planar convex sets

“…If Ω is a convex planar set, C its Cheeger set, and V ∈ C 1 (R n ; R n ) a diffeomorphism, then the following shape derivative formula holds true (see [21]):…”

“…exists due to the fact that both λ 1 (Ω) and h 1 (Ω) are shape differentiable (see [21]). We observe that Ω is a critical point of the functional J if and only if…”

A Blaschke–Lebesgue theorem for the Cheeger constant

“…In order to derive optimality conditions, a classical idea is to impose that the first order shape derivative of h at a critical Reuleaux polygon vanishes for every small deformation which preserves the constraints of B 1 N . For a generic convex set Ω, denoting by C Ω its (unique, see [1]) Cheeger set, the first order shape derivative of h at Ω in direction V ∈ C 1 (R 2 ; R 2 ) reads (see [21])…”