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

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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.