A Riemannian Point of View for a Quadrature Surface Free Boundary Problem

Shape stability of a quadrature surface problem in infinite Riemannian manifolds

“…It has been intensively studied at least during the last forty years, see for example [39], [19] and the references therein for more details. Among these works, some authors have established an intimate link between the existence of quadrature surfaces and the solution of free boundary problems governed by overdetermined partial differential equations, see for instance [25], [37], [38], [17] and references therein. The quadrature surface problem (2) can be tackled by a shape optimization approach when µ is regular enough, for instance by taking it in L 2 (Ω), supp(µ) ⊂ Ω. Fore more details see for instance [9] and [17].…”

“…In this work, we focus on two-dimensional shapes as subsets. And considering [9], [17], we think that it is possible to write our work in high dimensions and even if Ω is an open set with boundary of a compact N −dimensional Riemannian manifolds noted M. One of our main question is the following: Is it possible to express the Hessian of a shape functional to get sufficient conditions so that the critical domain of the functional J assumes its minimum? To answer this question, we study the positiveness of the quadratic form of the functional J which is related to the quadrature surface that is nothing but the following free boundary problem…”

“…Among these works, some authors have established an intimate link between the existence of quadrature surfaces and the solution of free boundary problems governed by overdetermined partial differential equations, see for instance [25], [37], [38], [17] and references therein. The quadrature surface problem (2) can be tackled by a shape optimization approach when µ is regular enough, for instance by taking it in L 2 (Ω), supp(µ) ⊂ Ω. Fore more details see for instance [9] and [17]. Before proceeding further, let us remind that in optimisation or in the study of minimal action, one of the essential questions is the characterization of an optimum if it exists.…”

“…In [9], [17], there are all details on existence results of quadrature surface by using shape optimization tools.…”