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

Abstract

In this article, we study shape optimization problems involving the geometry of surfaces (normal vector, principal curvatures). Given $\varepsilon > 0$ and a fixed non-empty large bounded open hold-all $B \subset \mathbb{R}^{n}$, $n \geqslant 2$, we consider a specific class $\mathcal{O}{\varepsilon}(B)$ of open sets $\Omega \subset B$ satisfying a uniform $\varepsilon$-ball condition. First, we recall that this geometrical property $\Omega \in \mathcal{O}{\varepsilon}(B)$ can be equivalently characterized in terms of $C^{1,1}$-regularity of the boundary $\partial \Omega \neq \emptyset$, 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 $\Omega \in \mathcal{O}\_{\varepsilon}(B)$ for a general range of geometric functionals and constraints defined on the boundary $\partial \Omega$, involving the first- and second-order properties of surfaces, such as problems of the form: $$ \inf\_{\Omega \in \mathcal{O}{\varepsilon}(B)} \int{\partial \Omega} \left( \begin{matrix} \ \ \end{matrix} j\_{0} \left\[ \mathbf{x},\mathbf{n}\left(\mathbf{x}\right) \right] + j\_{1} \left\[ \mathbf{x},\mathbf{n}\left(\mathbf{x}\right),H\left( \mathbf{x} \right)\right] + j\_{2}\left\[\mathbf{x},\mathbf{n}\left(\mathbf{x}\right),K\left(\mathbf{x}\right)\right] \begin{matrix} \ \ \end{matrix} \right) dA \left( \mathbf{x}\right), $$ where $\mathbf{n}$, $H$, $K$ 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.