Existence and Regularity of Minimizers for a Plateau Approximation Problem

TL;DR

This paper proves the existence and regularity of minimizers for a functional approximating Plateau's problem, extending previous work to higher dimensions and providing mathematical guarantees for the solutions.

Contribution

It introduces a new functional for approximating Plateau's problem and establishes the existence and Hölder regularity of its minimizers, generalizing prior one-dimensional results.

Findings

Existence of minimizers for the functional.

Hölder regularity of the minimizers.

Extension of previous one-dimensional results to higher dimensions.

Abstract

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.