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

Abstract

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: $$ \inf \int_{\partial \Omega} j_0 [ \mathbf{x},\mathbf{n}(\mathbf{x}) ] dA (\mathbf{x}) + \int_{\partial \Omega} j_1 [ \mathbf{x},\mathbf{n}(\mathbf{x}),H(\mathbf{x}) ] dA (\mathbf{x}) + \int_{\partial \Omega} j_2 [\mathbf{x},\mathbf{n}(\mathbf{x}),K(\mathbf{x})] dA (\mathbf{x}), $$ 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.