Isoperimetric planar clusters with infinitely many regions

2022/10/11 by Novaga, Matteo, Paolini, Emanuele, Stepanov, Eugene +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.05286

Abstract

An infinite cluster \mathbf E in \mathbb R^d is a sequence of disjoint measurable sets E_k⊂ \mathbb R^d, k∈ \mathbb N, called regions of the cluster. Given the volumes a_k≥ 0 of the regions E_k, a natural question is the existence of a cluster \mathbf E which has finite and minimal perimeter 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 of the areas a_k with ∑ √ a_k < ∞. We also show the existence of a bounded minimizer with the property P(\mathbf E)=\mathcal H¹(∂ \mathbf E), where ∂ mathbf E denotes the measure theoretic boundary of the cluster. We also provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.

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