# Uniform ball property and existence of optimal shapes for a wide class of geometric functionals

### Jérémy Dalphin

Université de Lorraine, Vandœuvre-Lès-Nancy, France

## Abstract

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 $B⊂R_{n}$, $n⩾2$, we consider a specific class $O_{ε}(B)$ of open sets $Ω⊂B$ satisfying a uniform $ε$-ball condition. First, we recall that this geometrical property $Ω∈O_{ε}(B)$ can be equivalently characterized in terms of $C_{1,1}$-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 $C_{1,1}$-regular minimizer among $Ω∈O_{ε}(B)$ 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:

where $n$, $H$, $K$ respectively denote the unit outward normal vector, the scalar mean curvature and the Gaussian curvature. We only assume continuity of $j_{0},j_{1},j_{2}$ with respect to the set of variables and convexity of $j_{1},j_{2}$ with respect to the last variable, but no growth condition on $j_{1},j_{2}$ 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. 211–260

DOI 10.4171/IFB/401