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arXiv · 2210.05286

Isoperimetric planar clusters with infinitely many regions

Matteo Novaga·Emanuele Paolini·Eugene Stepanov·Vincenzo Maria Tortorelli
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Abstract

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.

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Matteo Novaga, Emanuele Paolini, Eugene Stepanov, Vincenzo Maria Tortorelli. 2022-10-11. Isoperimetric planar clusters with infinitely many regions. https://arxiv.org/abs/2210.05286

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