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

2025/06/27 by Matthieu Bonnivard, Élie Bretin, Bonnivard, Matthieu +5 · 1 citation
Computer Science · Materials Science · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Electromagnetic Scattering and Analysis #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2506.22273

openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Optimal regularity up to the boundary for Plateau-quasi-minimizers
• A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation
• Plateau borders in soap films and Gauss' capillarity theory
• A penalized Allen-Cahn equation for the mean curvature flow of thin structures
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• Variational approximation of size-mass energies for k-dimensional currents
• On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers
• A direct approach to Plateau's problem
• Approximation of length minimization problems among compact connected sets
• A Modica-Mortola approximation for the Steiner Problem
• Existence and Soap Film Regularity of Solutions to Plateau's Problem
• Soap Film Solutions to Plateau's Problem
• Normal and Integral Currents

Cited by

• Optimal regularity up to the boundary for Plateau-quasi-minimizers

Related

• Plateau's Problem: What's Next
• Numerical analysis for the Plateau problem by the method of fundamental solutions
• On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers
• Keeping it together: a phase field version of path-connectedness and its implementation
• Soap Film Solutions to Plateau's Problem