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

TL;DR

This paper proves the existence of optimal shapes with regular boundaries for a broad class of geometric surface functionals involving curvature, under a uniform ball condition, with applications to modeling biological membranes.

Contribution

It establishes the existence of smooth minimizers for complex geometric functionals involving curvature terms in shape optimization problems.

Findings

Existence of $C^{1,1}$-regular minimizers for geometric functionals.

Applicability to biological membrane models like Canham-Helfrich energy.

General framework for shape optimization with curvature constraints.

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.