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 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.
Highlights
In mathematical engineering, many practical applications are modelled by minimization processes
A first natural question arises from this setting: is our problem well posed i.e. does such a design exist? To answer this question, it is necessary to study the existence of minimizers to the following kind of shape optimization problems: inf J(Ω), Ω∈A
Where J : Ω → J(Ω) is a real-valued functional defined over a set A of admissible shapes Ω ⊆ R and c : (Rn), that may include some additional constraints
Summary
Many practical applications are modelled by minimization processes. We are interested in the existence of solutions to such shape optimization problems when the functional is defined on the boundary of the domain and depends on the first- and second-order geometric properties of the associated (hyper-)surface:. If Σ ⊂ B is a non-empty compact C1,1-hypersurface of Rn, there exists ε > 0 such that its inner domain Ω ∈ Oε(B) Equipped with this class of admissible shapes, the main contribution of this article is to extend the existence results of [10] for functionals of the form (1.6). We recall that the C1,1-regularity is the minimum possible regularity to get existence in the case where the functionals depend on the principal curvatures of the domains, which are only L∞ for C1,1-domains To our knowledge, such a study has not been carried out in its generality and these new results may have some potential applications in applied mathematics.
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