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

TL;DR

This paper characterizes the local minimizers of anisotropic perimeter in two-dimensional clusters, proving that lens and triod configurations are unique minimizers under certain conditions.

Contribution

It provides a geometric characterization of local minimizers for anisotropic perimeters, identifying lens and triod clusters as unique solutions in specific cases.

Findings

Lens and triod clusters are the unique local minimizers for anisotropic perimeter in their respective cases.

The characterization holds for regular, symmetric, and uniformly convex anisotropies.

The results extend to general anisotropies via approximation arguments.

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.