Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and Regularity of Minimizers for a Plateau Approximation Problem

Eve Machefert

TL;DR

The paper tackles Plateau-type problems by introducing a decoupled phase-field energy $E_ps(u,ll)$ that blends a Ginzburg–Landau–type term with a geodesic-distance penalty on Lipschitz homotopies $ll$ spanning given curves. The analysis hinges on the fixed-$ll$ problem, establishing existence and Hölder regularity for the minimizer $u$ via a Golab-type lower semicontinuity result and elliptic regularity, and on the fixed-$u$ problem, proving existence of minimizers over $ll$ using Ascoli–Arzelà and a Lipschitz-surface Golab-type lemma. These two strands are then merged to obtain existence of a minimizer in the full joint problem in $(u,ll)$, with careful handling of lower semicontinuity of the surface term through measure convergence of $ll$-induced surfaces. The results generalize one-dimensional Steiner-approximation ideas to higher dimensions and provide a rigorous variational framework and regularity tools for Plateau-type phase-field approximations, with implications for Γ-convergence analyses and numerical implementations.

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{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.

Existence and Regularity of Minimizers for a Plateau Approximation Problem

TL;DR

The paper tackles Plateau-type problems by introducing a decoupled phase-field energy that blends a Ginzburg–Landau–type term with a geodesic-distance penalty on Lipschitz homotopies spanning given curves. The analysis hinges on the fixed- problem, establishing existence and Hölder regularity for the minimizer via a Golab-type lower semicontinuity result and elliptic regularity, and on the fixed- problem, proving existence of minimizers over using Ascoli–Arzelà and a Lipschitz-surface Golab-type lemma. These two strands are then merged to obtain existence of a minimizer in the full joint problem in , with careful handling of lower semicontinuity of the surface term through measure convergence of -induced surfaces. The results generalize one-dimensional Steiner-approximation ideas to higher dimensions and provide a rigorous variational framework and regularity tools for Plateau-type phase-field approximations, with implications for Γ-convergence analyses and numerical implementations.

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{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.
Paper Structure (6 sections, 14 theorems, 104 equations, 1 figure)

This paper contains 6 sections, 14 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existence and regularity of a minimizer u for l in Hom lambda fixed
  3. Existence and uniqueness with fixed $\ell$
  4. Hölder estimates for $u$ when $S_\ell$ is fixed and upper Ahlfors regular
  5. Existence of a minimizer l in Hom lambda for u fixed
  6. Existence of a minimizer in the variable (u,l) for l in Hom lambda

Key Result

Theorem 1.1

There exists a minimizing couple $(u,\ell)$ of the following problem in both variables Moreover, the minimizer $u$ is $\alpha$-Hölder continuous for all $0<\alpha<1$ with the estimate:

Figures (1)

  • Figure 1: Argument by contradiction to prove that every half-line emanating from $x_0$ intersects the closed curve $\Tilde{\Gamma}$.

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.1
  • Theorem 1.1
  • Proposition 1.1
  • Proposition 1.2
  • Lemma 1.1
  • Proposition 2.1
  • Lemma 2.1
  • ...and 19 more