arXiv subjects

Eve Machefert

Publications and source records attributed to Eve Machefert.

3 recordsLinked to original sources

arXiv2026

Existence and Regularity of Minimizers for a Plateau Approximation Problem

In this paper, we study the functional introduced by the author in collaboration with Bonnivard, Bretin, and Lemenant, which is designed to approximate Plateau's problem. We establish the existence of a minimizer and prove its H{\"o}lder regularity. Our results may be viewed as a generalization to higher-dimensional surfaces of the one-dimensional work of Bonnivard, Lemenant, and Millot on the approximation of the Steiner problem.

math.AP↗

arXiv2025

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

We study the regularity of quasi-minimal sets (in the sense of David and Semmes) with a boundary condition, which can be interpreted as quasi-minimizers of Plateau's problem in co-dimension one. For these Plateau-quasi-minimizers, we establish the optimal regularity, which is a characterization by bi-John domains with Ahlfors regular boundaries. This requires to investigate the Ahlfors regularity and also the uniform rectifiability of those sets, up to the boundary.

math.OC↗

arXiv2025

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

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.

math.OC↗