Isoperimetric planar clusters with infinitely many regions

1.
Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy
2.
Scuola Normale Superiore, Piazza dei Cavalieri 6, Pisa, Italy
3.
St. Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, St. Petersburg, Russia
4.
Faculty of Mathematics, Higher School of Economics, Moscow, Russia

Received: 12 November 2021 Revised: 10 May 2022 Accepted: 08 July 2022 Published: 21 April 2023

In this paper we study infinite isoperimetric clusters. An infinite cluster $ {\bf{E}} $ in $ \mathbb R^d $ is a sequence of disjoint measurable sets $ E_k\subset \mathbb R^d $, called regions of the cluster, $ k = 1, 2, 3, \dots $ A natural question is the existence of a cluster $ {\bf{E}} $ with given volumes $ a_k\ge 0 $ of the regions $ E_k $, having finite perimeter $ P({\bf{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 $ \sum \sqrt a_k < \infty $. We also show the existence of a bounded minimizer with the property $ P({\bf{E}}) = \mathcal H^1({\tilde\partial} {\bf{E}}) $, where $ {\tilde\partial} {\bf{E}} $ denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.
- isoperimetric clusters,
- isoperimetric sets,
- regularity
Citation: Matteo Novaga, Emanuele Paolini, Eugene Stepanov, Vincenzo Maria Tortorelli. Isoperimetric planar clusters with infinitely many regions[J]. Networks and Heterogeneous Media, 2023, 18(3): 1226-1235. doi: 10.3934/nhm.2023053

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Abstract

In this paper we study infinite isoperimetric clusters. An infinite cluster $ {\bf{E}} $ in $ \mathbb R^d $ is a sequence of disjoint measurable sets $ E_k\subset \mathbb R^d $, called regions of the cluster, $ k = 1, 2, 3, \dots $ A natural question is the existence of a cluster $ {\bf{E}} $ with given volumes $ a_k\ge 0 $ of the regions $ E_k $, having finite perimeter $ P({\bf{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 $ \sum \sqrt a_k < \infty $. We also show the existence of a bounded minimizer with the property $ P({\bf{E}}) = \mathcal H^1({\tilde\partial} {\bf{E}}) $, where $ {\tilde\partial} {\bf{E}} $ denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.

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