Isoperimetric planar clusters with infinitely many regions

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 $\partial mathbfE$ denotes the measure theoretic boundary of the cluster. We also provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.