Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems

Institut des Sciences du Calcul et des Données (ISCD), Sorbonne Université, 4 place de Jussieu, boîte courrier 380, 75252 Paris Cedex 5, France.

^* Corresponding author: jeremy.dalphin@mines-nancy.org

Received: 11 September 2018
Accepted: 6 May 2020

Abstract

In this article, we are interested in shape optimization problems where the functional is defined on the boundary of the domain, involving the geometry of the associated hypersurface (normal vector n, scalar mean curvature H) and the boundary values of the solution u_Ω related to the Laplacian posed on the inner domain Ω enclosed by the shape. For this purpose, given ε > 0 and a large hold-all B ⊂ ℝⁿ, n ≥ 2, we consider the class O_ε(B) of admissible shapes Ω ⊂ B satisfying an ε-ball condition. The main contribution of this paper is to prove the existence of a minimizer in this class for problems of the form inf_Ω∈Oε(B) ∫ _∂Ωj[u_Ω(x),∇u_Ω(x),x,n(x),H(x)]dA(x). We assume the continuity of j in the set of variables, convexity in the last variable, and quadratic growth for the first two variables. Then, we give various applications such as existence results for the configuration of fluid membranes or vesicles, the optimization of wing profiles, and the inverse obstacle problem.