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The lens cluster and triod cluster uniquely minimize the anisotropic perimeter in $\mathbb{R}^2$

Paula Benitez

Abstract

(N, M)-clusters are partitions of $\mathbb{R}^d$ into N+M regions, where N chambers have prescribed finite measure and M chambers have infinite measure. Locally minimizing clusters are the configurations which minimize the perimeter among all competitors with compact support satisfying the same measure constraints. The characterization of these partitions has been widely studied for the standard (isotropic) perimeter. In the present paper, we investigate the corresponding problem for anisotropic perimeters, considering a general anisotropy. More specifically, we focus on (1,2)-clusters and (1,3)-clusters in $\mathbb{R}^2$. Our main results provide a geometric characterization of these local minimizers: for regular (smooth, symmetric, and uniformly convex) anisotropies, we prove that a cluster is a local minimizer if and only if, up to translations, it is a standard anisotropic lens cluster in the (1,2)-cluster case, or a standard anisotropic triod cluster in the (1,3)-cluster case. In addition, using an approximation argument, we extend the minimizing property of these configurations to general anisotropies.

The lens cluster and triod cluster uniquely minimize the anisotropic perimeter in $\mathbb{R}^2$

Abstract

(N, M)-clusters are partitions of into N+M regions, where N chambers have prescribed finite measure and M chambers have infinite measure. Locally minimizing clusters are the configurations which minimize the perimeter among all competitors with compact support satisfying the same measure constraints. The characterization of these partitions has been widely studied for the standard (isotropic) perimeter. In the present paper, we investigate the corresponding problem for anisotropic perimeters, considering a general anisotropy. More specifically, we focus on (1,2)-clusters and (1,3)-clusters in . Our main results provide a geometric characterization of these local minimizers: for regular (smooth, symmetric, and uniformly convex) anisotropies, we prove that a cluster is a local minimizer if and only if, up to translations, it is a standard anisotropic lens cluster in the (1,2)-cluster case, or a standard anisotropic triod cluster in the (1,3)-cluster case. In addition, using an approximation argument, we extend the minimizing property of these configurations to general anisotropies.

Paper Structure

This paper contains 12 sections, 17 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Partitions and anisotropic perimeter
  4. Wulff shape and initial results
  5. Regularity theory and local estimates
  6. The minimization problem in $B_R$
  7. The $(1,2)$-cluster
  8. The $(1,3)$-cluster
  9. Main Results
  10. The $(1,2)$-cluster
  11. The $(1,3)$-cluster
  12. Acknowledgments

Key Result

Theorem 2.8

Let $E \subseteq \mathbb{R}^2$ be a set of locally finite perimeter, and let $x \in \mathbb{R}^2$ be fixed. Then, for a.e. $r > 0$, one has that $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Local minimizing clusters for the plane. Columns from left to right represent configurations with one, two, and three infinite chambers, respectively. The rows show an increasing number of finite chambers.
  • Figure 2: Construction of the Wulff anisotropic lens.
  • Figure 3: Construction of the anisotropic Reuleaux triangle.
  • Figure 4: Configuration of the minimal $(1,2)$-cluster inside $B_R$ for a given $R$.
  • Figure 5: Example of a possible minimizer with a non lens shape, given by the anisotropic density $\phi(\nu)= |\nu_1|+|\nu_2|.$
  • ...and 1 more figures

Theorems & Definitions (63)

  • Definition 2.1: Partition
  • Definition 2.3: Locally minimizing partition / locally isoperimetric partition
  • Definition 2.4: Eventually flat partitions
  • Definition 2.5: Anisotropic perimeter of a set
  • Theorem 2.8: Vol'pert
  • Definition 2.9: Wulff shape
  • Lemma 2.10
  • proof
  • Lemma 2.11
  • proof
  • ...and 53 more