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 C 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 Z 11ak < co. We also show the existence of a bounded minimizer with the property P(E) = Tl1( partial differential & SIM;E), where partial differential & SIM;E denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.