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An existence theorem for sliding minimal sets

Guy David, Camille Labourie

TL;DR

This work proves an existence theorem for the sliding boundary variant of the Plateau problem for $2$-dimensional sets in $\mathbb{R}^n$, under the assumption that the boundary $\Gamma$ is a finite union of $C^{1+\alpha}$ curves with good access to the complement of its convex hull. The authors construct a minimizer by analyzing a minimizing sequence, passing to a weak limit of the associated Hausdorff measures to obtain a sliding minimal limit $E_\infty$, and then building intrinsic Lipschitz retractions onto $E_\infty$ to recover a minimizer in the original class ${\cal E}(E_0,\Gamma)$. The approach hinges on boundary regularity for sliding almost minimal sets (via Dvv) and stability results for limits of such sets (via La), together with a detailed blow-up analysis and a gluing procedure for retractions. The framework also accommodates variants with different functionals and boundary geometries, and yields local biLipschitz parameterizations of certain almost minimal configurations, highlighting the interplay between geometric measure theory, regularity, and variational methods in sliding boundary problems.

Abstract

We prove an existence theorem for the sliding boundary variant of the Plateau problem for $2$-dimensional sets in $\mathbb{R}^n$. The simplest case of sufficient condition is when $n=3$ and the boundary $Γ$ is a finite disjoint union of smooth closed curves contained in the boundary of a convex body, but the main point of our sufficient condition is to prevent the limits in measure of a minimizing sequence to have singularities of type $\mathbb{Y}$ along $Γ$.

An existence theorem for sliding minimal sets

TL;DR

This work proves an existence theorem for the sliding boundary variant of the Plateau problem for -dimensional sets in , under the assumption that the boundary is a finite union of curves with good access to the complement of its convex hull. The authors construct a minimizer by analyzing a minimizing sequence, passing to a weak limit of the associated Hausdorff measures to obtain a sliding minimal limit , and then building intrinsic Lipschitz retractions onto to recover a minimizer in the original class . The approach hinges on boundary regularity for sliding almost minimal sets (via Dvv) and stability results for limits of such sets (via La), together with a detailed blow-up analysis and a gluing procedure for retractions. The framework also accommodates variants with different functionals and boundary geometries, and yields local biLipschitz parameterizations of certain almost minimal configurations, highlighting the interplay between geometric measure theory, regularity, and variational methods in sliding boundary problems.

Abstract

We prove an existence theorem for the sliding boundary variant of the Plateau problem for -dimensional sets in . The simplest case of sufficient condition is when and the boundary is a finite disjoint union of smooth closed curves contained in the boundary of a convex body, but the main point of our sufficient condition is to prevent the limits in measure of a minimizing sequence to have singularities of type along .

Paper Structure

This paper contains 12 sections, 14 theorems, 323 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Almost minimal sets and minimizing sequences
  3. Good access
  4. Blow-up limits of Einfty at a point
  5. A description of Einfty in annuli centered at x0 = 0
  6. Intrinsic projections on small Lipschitz graphs
  7. Projections along the spheres Sr
  8. Contraction of Einfty to the origin
  9. A retraction onto Einfty defined near Einfty inter B(0,r0)
  10. We glue the retractions
  11. Variants of the main theorem
  12. Local biLipschitz parameterizations of some almost minimal sets

Key Result

Theorem 1.1

Let a compact set $\Gamma \subset \mathbb{R}^n$ be the union of a finite family of disjoint closed curves of class $C^{1+\alpha}$, with $\alpha > 0$. Assume in addition that we will explain what this means in Section S3. Let $E_0$ be any compact subset of $\mathbb{R}^n$, and define ${\cal E} = {\cal E}(E_0,\Gamma)$ as above. Then there exists a solution for the sliding Plateau problem associated

Figures (3)

  • Figure 1: The set $E_\infty$ in Case 2
  • Figure 2: Case 2, the intersection with $S_r = \partial B(0,r)$
  • Figure 3: Case 2, typical behavior of $\sigma_1 : S_r \cap E \to S_{tr} \cap E$

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Definition 3.1: Good access to the complement
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2: Covering of $X \cap \partial B(0,1)$
  • proof
  • Lemma 4.3
  • ...and 10 more