Geodesic
Uniform ball property and existence of optimal shapes for a wide class of geometric functionals
Jérémy Dalphin1
1Université de Lorraine, Vandœuvre-Lès-Nancy, France
Interfaces and free boundaries, Tome 20 (2018) no. 2, pp. 211-260

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In this article, we study shape optimization problems involving the geometry of surfaces (normal vector, principal curvatures). Given ε>0 and a fixed non-empty large bounded open hold-all B⊂Rn, n⩾2, we consider a specific class Oε​(B) of open sets Ω⊂B satisfying a uniform ε-ball condition. First, we recall that this geometrical property Ω∈Oε​(B) can be equivalently characterized in terms of C1,1-regularity of the boundary ∂Ω=∅, and thus also in terms of positive reach and oriented distance function. Then, the main contribution of this paper is to prove the existence of a C1,1-regular minimizer among Ω∈Oε​(B) for a general range of geometric functionals and constraints defined on the boundary ∂Ω, involving the first- and second-order properties of surfaces, such as problems of the form:
DOI : 10.4171/ifb/401
Classification : 49-XX, 53-XX
Mots-clés : Shape optimization, uniform ball condition, geometric functionals, Helfrich, Willmore, curvature depending energies

Jérémy Dalphin  1

1 Université de Lorraine, Vandœuvre-Lès-Nancy, France 
Jérémy Dalphin. Uniform ball property and existence of optimal shapes for a wide class of geometric functionals. Interfaces and free boundaries, Tome 20 (2018) no. 2, pp. 211-260. doi: 10.4171/ifb/401
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