Isoperimetric planar clusters with infinitely many regions

In this paper we study infinite isoperimetric clusters. An infinite cluster $ {\bf{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 $ {\bf{E}} $ with given volumes $ a_k\ge 0 $ of the regions $ E_k $, having finite perimeter $ P({\bf{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 < \infty $. We also show the existence of a bounded minimizer with the property $ P({\bf{E}}) = \mathcal H^1({\tilde\partial} {\bf{E}}) $, where $ {\tilde\partial} {\bf{E}} $ denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.