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

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in ${\mathbb{R}}^n$ satisfying a uniform ball condition and we prove the exist ence of a $C^{1,1}$-regular minimizer for general geometric functionals and cons traints involving the first- and second-order properties of surfaces, such as in ${\mathbb{R}}^3$ problems of the form: $$\inf {\int }_{\mathrm{\partial }\mathrm{\Omega }}{j}_0[\mathbf{x},\mathbf{n}(\mathbf{x})]dA(\mathbf{x})+{\int }_{\mathrm{\partial }\mathrm{\Omega }}{j}_1[\mathbf{x},\mathbf{n}(\mathbf{x}),H(\mathbf{x})]dA(\mathbf{x})+{\int }_{\mathrm{\partial }\mathrm{\Omega }}{j}_2[\mathbf{x},\mathbf{n}(\mathbf{x}),K(\mathbf{x})]dA(\mathbf{x}),$$ where $\mathbf{n}$, $H$, and $K$ respectively denotes the normal, the scalar mea n curvature and the Gaussian curvature. We gives some various applications in th e modelling of red blood cells such as the Canham-Helfrich energy and the Willmo re functional.