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A stability inequality for the planar lens partition

Marco Bonacini, Riccardo Cristoferi, Ihsan Topaloglu

Abstract

Recently it has been shown that the unique locally perimeter minimizing partitioning of the plane into three regions, where one region has finite area and the other two have infinite measure, is given by the so-called standard lens partition. Here we prove a sharp stability inequality for the standard lens; hence strengthening the local minimality of the lens partition in a quantitative form. As an application of this stability result we consider a nonlocal perturbation of an isoperimetric problem.

A stability inequality for the planar lens partition

Abstract

Recently it has been shown that the unique locally perimeter minimizing partitioning of the plane into three regions, where one region has finite area and the other two have infinite measure, is given by the so-called standard lens partition. Here we prove a sharp stability inequality for the standard lens; hence strengthening the local minimality of the lens partition in a quantitative form. As an application of this stability result we consider a nonlocal perturbation of an isoperimetric problem.
Paper Structure (10 sections, 10 theorems, 122 equations, 5 figures)

This paper contains 10 sections, 10 theorems, 122 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions and main result
  3. Locally isoperimetric partitions: definitions
  4. Main result: stability of the planar standard lens
  5. Selection principle and improved convergence
  6. Selection principle
  7. Improved convergence
  8. Stability of the lens among smooth perturbations
  9. An application: small-mass minimizers for an isoperimetric problem with nonlocal perturbation
  10. A "volume-fixing variation" lemma

Key Result

Theorem 2.3

Let $d=2$ and let $\mathcal{E}_0$ be a locally isoperimetric partition in $\mathbb R^2$. Then $\partial\mathcal{E}_0$ is connected and there exist a finite family of points $\{p_i\}_{i\in I}$ (vertices) and a finite family $\{\gamma_j\}_{j\in J}$ of closed curves with boundary such that where $\mathrm{int}(\gamma)$ and $\mathrm{bd}(\gamma)$ denote the interior and the boundary points of the curve

Figures (5)

  • Figure 1: Some planar locally isoperimetric partitions: the standard double bubble, the standard lens, the peanut, and the Reuleaux triangle. All the highlighted angles are 120 degree angles.
  • Figure 2: Two-dimensional construction showing that the constant $\kappa$ in \ref{['eq:quantitative']} should depend on the diameter $R$ of the perturbation.
  • Figure 3: The standard lens partition $\mathcal{L}_{m}$ as in Definition \ref{['def:lens']}.
  • Figure 4: The set $O_R$ constructed in Definition \ref{['def:eye']}, depending whether $\mathcal{E}_0$ has two infinite regions (left) or three infinite regions (right).
  • Figure 5: A translated smooth perturbation of the standard lens partition $\mathcal{L}_{m}$.

Theorems & Definitions (30)

  • Definition 2.1: Partition
  • Definition 2.2: Locally isoperimetric partition
  • Theorem 2.3: Structure of planar locally isoperimetric partitions
  • Definition 2.4: Standard lens partition
  • Theorem 2.5: Stability of the standard lens
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • ...and 20 more