A subscription is required to access this article.
In this article, we study shape optimization problems involving the geometry of surfaces (normal vector, principal curvatures). Given and a fixed non-empty large bounded open hold-all , , we consider a specific class of open sets satisfying a uniform -ball condition. First, we recall that this geometrical property can be equivalently characterized in terms of -regularity of the boundary , 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 -regular minimizer among for a general range of geometric functionals and constraints defined on the boundary , involving the first- and second-order properties of surfaces, such as problems of the form:
where , , respectively denote the unit outward normal vector, the scalar mean curvature and the Gaussian curvature. We only assume continuity of with respect to the set of variables and convexity of with respect to the last variable, but no growth condition on are imposed here regarding the last variable. Finally, we give various applications in the modelling of red blood cells such as the Canham-Helfrich energy and the Willmore functional.
Cite this article
Jérémy Dalphin, Uniform ball property and existence of optimal shapes for a wide class of geometric functionals. Interfaces Free Bound. 20 (2018), no. 2, pp. 211–260DOI 10.4171/IFB/401