Shape Derivative of the First Eigenvalue of the 1-laplacian

where Ḣ 1 (Ω) is the closure of C ∞ 0 (Ω) in the Sobolev space H 1 1 (Ω) of functions in L(Ω) with one derivative in L. The purpose of this paper is the study of the dependence of λ1,Ω under regular perturbations by diffeomorphisms of Ω, i.e. we want to compute the first variation, the so-called shape derivative, of the functional Ω → λ1,Ω. General results about the stability of λ1,Ω under perturbations of Ω have been obtained in [17]. In particular the authors of [17] found the shape derivative of Ω → λ1,Ω in the case of regular perturbations by diffeomorphisms close to homotheties. We want to extend this result to the case of a general perturbation by diffeomorphisms. Let us recall some known facts about λ1,Ω (see e.g. [17, 20]). A natural space to study λ1,Ω is the space BV (Ω) of functions of bounded variations (see, for instance, [2, 11, 14, 26]). By standard properties of BV (Ω), we can also define λ1,Ω by λ1,Ω = inf 8