An existence result for a quadrature surface free boundary problem

Abstract

The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.

Shape stability of a quadrature surface problem in infinite Riemannian manifolds

“…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 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…”

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

Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem

“…For more details about the methods used for solving this problem see the (Gustafsson and Shahgholian, Introduction, 1996). Using the maximum principle together with the compatibility condition of the Neumann problem, the authors gave sufficient condition of existence for problem Q S ( f, k) (Barkatou and al., 2005).…”