Isoperimetric planar clusters with infinitely many regions

In this paper we study infinite isoperimetric clusters. An infinite cluster E in Rᵈ is a sequence of disjoint measurable sets Eₖ⊂ Rᵈ, called regions of the cluster, k = 1, 2, 3, A natural question is the existence of a cluster E with given volumes aₖ≥ 0 of the regions Eₖ, having finite perimeter P(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 aₖ with ∑ √ aₖ < ∞. We also show the existence of a bounded minimizer with the property P(E) = H¹(∂ E), where ∂ E denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.