A phase field model for the optimization of the Willmore energy in the class of connected surfaces

TL;DR

This paper introduces a phase field model to optimize the Willmore energy of connected surfaces with prescribed area within a confined domain, proving its convergence to the sharp interface limit.

Contribution

It develops a novel phase field approach for connected surface optimization, incorporating topological constraints and providing a Gamma-convergence proof.

Findings

Model converges to sharp interface limit

Effectively enforces connectedness constraint

Applicable to confined surface optimization problems

Abstract

We consider the problem of minimizing the Willmore energy connected surfaces with prescribed surface area which are confined to a finite container. To this end, we approximate the surface by a phase field function $u$ taking values close to +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of $u$. A diffuse interface approximation for the area functional, as well as for the Willmore energy are well known. We address the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. In this article, we provide a proof of Gamma-convergence of our model to the sharp interface limit.