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An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjointmeasurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions ofthe cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a naturalquestion is the existence of a cluster $\mathbf E$ which has finite and minimalperimeter $P(\mathbf E)$ among all clusters with regions having such volumes.We prove that such a cluster exists in the planar case $d=2$, for any choice ofthe areas $a_k$ with $\sum \sqrt a_k < \infty$. We also show the existence of abounded minimizer with the property $P(\mathbf E)=\mathcal H^1(\partial \mathbfE)$, where $\partial mathbf E$ denotes the measure theoretic boundary of thecluster. We also provide several examples of infinite isoperimetric clustersfor anisotropic and fractional perimeters.

Isoperimetric planar clusters with infinitely many regions