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.
Categories: math.OC math.AP math.DG