@article{Novaga2023, 
author = {Matteo Novaga and Emanuele Paolini and Eugene Stepanov and Vincenzo Maria Tortorelli},
title = {Isoperimetric planar clusters with infinitely many regions},
year = {2023},
journal = {Networks and Heterogeneous Media},
volume = {18},
number = {3},
pages = {1226-1235},
keywords = {isoperimetric clusters, isoperimetric sets, regularity},
url = {https://www.sciopen.com/article/10.3934/nhm.2023053},
doi = {10.3934/nhm.2023053},
abstract = {In this paper we study infinite isoperimetric clusters. An infinite cluster              E       in              R        d   is a sequence of disjoint measurable sets        E    k    ⊂            R        d  , called regions of the cluster,    k  =  1  ,  2  ,  3  ,  … A natural question is the existence of a cluster              E       with given volumes        a    k    ≥  0 of the regions        E    k  , 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        a    k   with    ∑            a        k    <  ∞. We also show the existence of a bounded minimizer with the property    P  (            E        )  =            H        1    (                    ∂        ~                        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.}
}