Isoperimetric planar clusters with infinitely many regions

TL;DR

This paper proves the existence of minimal perimeter infinite clusters in the plane with given area conditions, and explores examples involving anisotropic and fractional perimeters.

Contribution

It establishes the existence of minimal perimeter infinite clusters in the plane under specific volume conditions and provides examples with complex perimeter notions.

Findings

Existence of minimal perimeter infinite clusters in 2D for certain volume sequences.

Construction of bounded minimizers with boundary measure equality.

Examples of infinite clusters with anisotropic and fractional perimeters.

Abstract

An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a natural question is the existence of a cluster $\mathbf E$ which has finite and minimal perimeter $P(\mathbf E)$ among all clusters with regions having such volumes. We prove that such a cluster exists in the planar case $d=2$, for any choice of the areas $a_k$ with $\sum \sqrt a_k < \infty$. We also show the existence of a bounded minimizer with the property $P(\mathbf E)=\mathcal H^1(\partial \mathbf E)$, where $\partial mathbf E$ denotes the measure theoretic boundary of the cluster. We also provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.