Isoperimetric planar clusters with infinitely many regions

In this paper we study infinite isoperimetric clusters. An infinite cluster $\mathbf{E}$ in ${\mathbb{R}}^d$ is a sequence of disjoint measurable sets $E_k\subset {\mathbb{R}}^d$, called regions of the cluster, $k=1,2,3,\dots$ A natural question is the existence of a cluster $\mathbf{E}$ with given volumes $a_k\ge 0$ of the regions $E_k$, having finite perimeter $P(\mathbf{E})$, which is minimal among all the clusters with regions having the same volumes. We prove that such a cluster exists in the planar case $d=2$, for any choice of the areas $a_k$ with $\sum {\sqrt{a}}_k<\mathrm{\infty }$. We also show the existence of a bounded minimizer with the property $P(\mathbf{E})={\mathcal{H}}^1(\tilde{\mathrm{\partial }}\mathbf{E})$, where $\tilde{\mathrm{\partial }}\mathbf{E}$ denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.