Calculus of Variations and Geometric Measure Theory

M. Novaga - E. Paolini - E. Stepanov - V. M. Tortorelli

Isoperimetric planar clusters with infinitely many regions

created by paolini on 10 Oct 2022
modified on 17 Nov 2022


Submitted Paper

Inserted: 10 oct 2022
Last Updated: 17 nov 2022

Year: 2022

ArXiv: 2210.05286 PDF


An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ 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 $\sum \sqrt a_k < \infty$. We also show the existence of a bounded minimizer with the property $P(\mathbf E)=\mathcal H^1(\partial \mathbf E)$, where $\partial \mathbf E$ denotes the measure theoretic boundary of the cluster. We also provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.