Isoperimetric planar clusters with infinitely many regions

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