Phase field approximation for Plateau's problem: a curve geodesic distance penalty approach

TL;DR

This paper introduces a phase field model combining Ambrosio-Torterelli energy with a geodesic distance term to approximate solutions to Plateau's problem, supported by Gamma-convergence analysis and numerical experiments.

Contribution

It proposes a novel phase field approximation for Plateau's problem that generalizes previous Steiner's problem approaches, with theoretical and numerical validation.

Findings

Gamma-convergence proven for a simplified case

Numerical schemes successfully approximate Plateau's problem solutions

Model effectively captures minimal surface configurations

Abstract

This work focuses on a phase field approximation of Plateau's problem. Inspired by Reifenberg's point of view, we introduce a model that combines the Ambrosio-Torterelli energy with a geodesic distance term, which can be considered as a generalization of the approach developed by Bonnivard, Lemenant and Santambrogio to approximate solutions to Steiner's problem. First, we present a Gamma-convergence analysis of this model in the simple case of a single curve located on the edge of a cylinder. In a numerical section, we detail the numerical optimisation schemes used to minimize this energy for numerous examples, for which good approximations of solutions to Plateau's problem are found.