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{ö}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.

💡 Research Summary

The paper investigates a variational functional that was introduced in a previous joint work of the author with Bonnivard, Bretin, and Lemenant as a tool to approximate Plateau’s problem. The functional, denoted (E_\varepsilon(u,\ell)), couples a phase‑field energy in a three‑dimensional domain (C) with a surface term defined on the image (S_\ell) of a Lipschitz homotopy (\ell) that connects two prescribed boundary curves (\gamma_0,\gamma_1). The admissible homotopies belong to a class (\mathrm{Hom}\Lambda(\gamma_0,\gamma_1)) consisting of Lipschitz maps with a uniform Lipschitz constant bounded by (\Lambda) and whose images are (\Lambda)-upper Ahlfors regular, i.e. (\mathcal H^2(S\ell\cap B(x,r))\le \Lambda\pi r^2) for all (x) and (r).

The functional reads \