ISOPERIMETRIC PLANAR CLUSTERS WITH INFINITELY MANY REGIONS MATTEO NOVAGA EMANUELE PAOLINI EUGENE STEPANOV

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ISOPERIMETRIC PLANAR CLUSTERS WITH INFINITELY

MANY REGIONS

MATTEO NOVAGA, EMANUELE PAOLINI, EUGENE STEPANOV,

AND VINCENZO MARIA TORTORELLI

Abstract. An inﬁnite cluster Ein Rdis a sequence of disjoint measurable

sets Ek⊂Rd,k∈N, called regions of the cluster. Given the volumes ak≥0

of the regions Ek, a natural question is the existence of a cluster Ewhich has

ﬁnite and minimal perimeter P(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 akwith P√ak<∞. We also show the existence of a

bounded minimizer with the property P(E) = H1(∂E), where ∂Edenotes the

measure theoretic boundary of the cluster. We also provide several examples

of inﬁnite isoperimetric clusters for anisotropic and fractional perimeters.

Contents

1. Introduction 1

2. Notation and preliminaries 3

2.1. Perimeters and boundaries 3

2.2. Auxiliary results 4

3. Main result 5

4. Some examples 7

References 9

1. Introduction

A ﬁnite cluster Eis a sequence E= (E1,...Ek, . . . , EN)of measurable sets, such

that |Ek∩Ej|= 0 for k6=j, where |·|denotes the Lebesgue measure (usually called

volume). The sets Ejare called regions of the cluster Eand E0:= Rd\S∞

k=1 Ek

is called external region. We denote the sequence of volumes of the regions of the

cluster Eas

(1) m(E) := (|E1|,|E2|,...,|EN|)

Date: October 12, 2022.

2010 Mathematics Subject Classiﬁcation. Primary 53C65. Secondary 49Q15, 60H05.

Key words and phrases. Isoperimetric clusters, isoperimetric sets, regularity.

The ﬁrst and second authors are members of the INDAM/GNAMPA and were supported by

the PRIN Project 2019/24. The work of the third author has been partially ﬁnanced by the RFBR

grant #20-01-00630 A .

arXiv:2210.05286v1 [math.AP] 11 Oct 2022

2 NOVAGA, PAOLINI, STEPANOV, AND TORTORELLI

Figure 1. The Apollonian gasket, on the left-hand side, is a clus-

ter with minimal fractional perimeter. On the right-hand side a

similar construction with squares: this is a minimal cluster with

respect to the perimeter induced by the Manhattan distance.

and we call perimeter of the cluster the quantity

(2) P(E) := 1

2"P(E0) +

k=1

P(Ek)#,

where Pis the Caccioppoli perimeter. A cluster Eis called minimal, or isoperimet-

ric, if

P(E) = min {P(F): m(F) = m(E)}.

In this paper we consider inﬁnite clusters, i.e., inﬁnite sequences E= (Ek)k∈Nof

essentially pairwise disjoint regions: |Ej∩Ei|= 0 for i6=j(this can be interpreted

as a model for a soap foam). Note that a ﬁnite cluster with Nregions, can also

be considered a particular case of an inﬁnite cluster for example by posing Ek=∅

for k > N . Clusters with inﬁnitely many regions of equal areas were considered

in [12], where it has been shown that the honeycomb cluster is the unique minimizer

with respect to compact perturbations. Inﬁnite clusters have been considered also

in [17, 13, 3], where the variational curvature is prescribed, and in [23], where

existence of generalized minimizers for both ﬁnite and inﬁnite isoperimetric clusters

has been proven in the general setting of homogeneous metric measure spaces.

An interesting example of inﬁnite cluster, detailed in Example 4.1 (see Figure 1)

is the Apollonian packing of a circle (see [15]). In fact this cluster is composed

by isoperimetric regions and hence should trivially have minimal perimeter among

clusters with regions of the same areas. Actually, it turns out that this cluster

has inﬁnite perimeter and hence all clusters with same prescribed areas have in-

ﬁnite perimeter too. Note that very few explicit examples of minimal clusters

are known [10, 26, 14, 25, 19]. Nevertheless, quite curiously, Apollonian packings

give nontrivial examples of inﬁnite isoperimetric clusters for fractional perimeters

[4, 7, 6], as shown in Example 4.1. An even simpler example of an inﬁnite isoperi-

metric planar cluster is given in Example 4.2 (see Figure 1 again) where the Cacciop-

poli perimeter is replaced by an anisotropic perimeter functional [16, 21, 22, 5, 8].

Our main result, Theorem 3.1, states that if d= 2 (planar case), given any

sequence of positive numbers a= (a1, a2, . . . , ak, . . . )such that P∞

k=0 √ak<+∞,

there exists a minimal cluster Ein R2with m(E) = a. The assumption on ais

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ISOPERIMETRICPLANARCLUSTERSWITHINFINITELYMANYREGIONSMATTEONOVAGA,EMANUELEPAOLINI,EUGENESTEPANOV,ANDVINCENZOMARIATORTORELLIAbstract.AninniteclusterEinRdisasequenceofdisjointmeasurablesetsEkRd,k2N,calledregionsofthecluster.Giventhevolumesak0oftheregionsEk,anaturalquestionistheexistenceofaclusterEwh...

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ISOPERIMETRIC PLANAR CLUSTERS WITH INFINITELY MANY REGIONS MATTEO NOVAGA EMANUELE PAOLINI EUGENE STEPANOV

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