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The planar Plateau's problem via capillarity

Kennedy Obinna Idu

Abstract

The Plateau's problem seeks to determine a surface of minimal area which spans a given boundary. It is widely studied for its varied mathematical formulations, applications and relevance to physical models such as soap films. We revisit the problem and study a soap film model in the spirit of capillarity formulations in two dimensions. Our approach introduces a nonlocal geometric potential in the variational length minimization scheme. This incorporates effects of thickness of soap films and provides insight into addressing the so-called collapsing phenomenon and other observable physical phenomena and properties.

The planar Plateau's problem via capillarity

Abstract

The Plateau's problem seeks to determine a surface of minimal area which spans a given boundary. It is widely studied for its varied mathematical formulations, applications and relevance to physical models such as soap films. We revisit the problem and study a soap film model in the spirit of capillarity formulations in two dimensions. Our approach introduces a nonlocal geometric potential in the variational length minimization scheme. This incorporates effects of thickness of soap films and provides insight into addressing the so-called collapsing phenomenon and other observable physical phenomena and properties.

Paper Structure

This paper contains 11 sections, 25 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Variational formulation of problem
  3. Review of the Maggi et al. model
  4. Structure of paper
  5. Preliminaries
  6. Fractional Sobolev spaces
  7. Existence and Non-collapsing of Minimizers
  8. Proof of \ref{['thm:Existence']}: Existence of minimizers
  9. Finite-energy curves and proof of \ref{['Thm: CharFiniteEnergy']}
  10. Convergence to the capillarity model
  11. Plateau problem as capillarity limit

Key Result

Theorem 1.1

There exists a solution $\gamma\in \mathcal{E}$ to the minimization problem (Main).

Figures (2)

  • Figure 1: Perturbation of curve near fixed points
  • Figure 2: The Quadrifolium traversed in the order $1$-$4$-$3$-$2$ is an example of a curve in $\bar{\mathcal{E}}$ which is not in $\bar{\mathcal{E}}_R$.

Theorems & Definitions (60)

  • Theorem 1.1: Existence of minimizers
  • Theorem 1.2: Non-collapsing for finite-energy curves
  • Theorem 1.3: Partial $\Gamma-$convergence as $\delta\rightarrow 0$
  • Corollary 1.4: Convergence of minimizers
  • Theorem 1.5: $\Gamma-$convergence to Plateau problem
  • Proposition 1.6
  • Theorem 1.7
  • Definition 2.1: Plane curve
  • Remark 2.2
  • Remark 2.3
  • ...and 50 more