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

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 e > 0 and a large hold-all B ⊂ ℝn , n ≥ 2, we consider the class of admissible shapes Ω ⊂ B satisfying an e -ball condition. The main contribution of this paper is to prove the existence of a minimizer in this class for problems of the form . 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.