Année 2024-2025

A phase field approximation for the Reifenberg Plateau's problem

par Eve Machefert

Europe/Paris
3L15 (Laboratoire de Mathématiques d'Orsay)

3L15

Laboratoire de Mathématiques d'Orsay

Description

The goal of this work is to use a phase field method to approximate the notorious Plateau problem, which consists of finding a surface of minimum area that spans a given curve. To this aim, we want to generalise to Plateau's problem, with a Reifenberg formulation, the functional introduced by M. Bonnivard, A. Lemenant and F. Santambrogio for Steiner's problem (search for the shortest path connecting a given set of points). The novelty of this approach is to deal with the topological constraint by penalising some geodesic distance, which must be defined. In particular, I will explain a Gamma-convergence type result for this generalised functional. Then, following an approach proposed by M. Bonnivard, E. Bretin and A. Lemenant in the case of Steiner's problem, I will present numerical simulations for Plateau's problem that illustrate the applicability of the method in practice.