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.

中文翻译：

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均匀球性质和存在多种几何泛函的最佳形状

在本文中，我们对涉及曲面几何（法线、曲率）的形状优化问题感兴趣。我们考虑 $\mathbb{R}^{n}$ 中的一类超曲面 s 满足均匀球条件，并且我们证明了 $C^{1,1}$-一般几何泛函和缺点的正则极小值的存在涉及表面一阶和二阶性质的训练，例如 $\mathbb{R}^{3}$ 形式的问题： $$ \inf \int_{\partial \Omega} j_0 [ \mathbf{x },\mathbf{n}(\mathbf{x}) ] dA (\mathbf{x}) + \int_{\partial \Omega} j_1 [ \mathbf{x},\mathbf{n}(\mathbf{x} }),H(\mathbf{x}) ] dA (\mathbf{x}) + \int_{\partial \Omega} j_2 [\mathbf{x},\mathbf{n}(\mathbf{x}), K(\mathbf{x})] dA (\mathbf{x}), $$ 其中 $\mathbf{n}$、$H$ 和 $K$ 分别表示法线、标量平均曲率和高斯曲率。