Volume 26, 2020
|Number of page(s)||30|
|Published online||10 December 2020|
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,
Paris Cedex 5, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 6 May 2020
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 uΩ 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 Oε(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 infΩ∈Oε(B) ∫ ∂Ωj[uΩ(x),∇uΩ(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.
Mathematics Subject Classification: 49Q10 / 49J20 / 53A05
Key words: Shape optimization / uniform ball condition / elliptic partial differential equations / geometric functionals / existence theory / boundary shape identification problems
© The authors. Published by EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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