Valentina FranceschiLJLL, Sorbonne Université, Université de Paris, Inria, CNRS,

Paris, France

Aldo PratelliUniversità di Pisa,

Pisa, Italy

Giorgio StefaniScuola Normale Superiore,

Pisa, Italy

Abstract – We illustrate some results about minimal bubble clusters in the plane with double density. This amounts to findthe best configuration of m ∈ N regions in the plane enclosing given volumes, in order to minimize their total perimeter, whereperimeter and volume are defined by suitable densities. We focus on a particular structure of such densities, which is inspired by asub-Riemannian model, called the Grushin plane.

After an overview concerning existence of minimizers, we focus on their Steiner regularity, i.e., the fact that their boundaries aremade of regular curves meeting at 120o. We will show that this holds in a wide generality.

Although our initial motivation came from the study of the particular sub-Riemannian framework of the Grushin plane, our approachworks in wide generality and is new even in the classical Euclidean case.

The minimal partition problemIn R2 consider a volume measure V and a perimeter measure P .

Given v1, . . . , vM > 0, consider the class of M-clusters

C(v1, . . . , vM ) ={E = {E1, . . . , EM︸ ︷︷ ︸

pairwise disjoint

} : V(Ei) = vi, i = 1, . . . ,M},

v1

v2 v3

The minimal partition problem is

inf {PP (E) : E ∈ C(v1, . . . vM )} , (1)

where PP (E) =1

2

M∑

i=1

P (Ei) + P

M⋃

i=1

Ei

.

Solutions are called M-minimal clusters.

Euclidean case

P = PEucl = De Giorgi Perimeter;

V = | · | = Lebesgue measure

• [1] : ∃ + regularity of solutions proved in Rn, n ≥ 2:E minimal ⇒ ∂E smooth outside singular set Σ, Hn−1(Σ) = 0.

• [9] : Structure of singularities in the plane:E min. ⇒ ∂E = ∪ smooth CMC curves meeting in 120◦ threes,(see [13] for n = 3).

f=f(0,v)120120 120

x

y

(M = 1) Isoperimetric problem : for n ≥ 2 and all volumes solu-tions are balls.

(M = 2) For n ≥ 2 and all volumes solutions are standard doublebubbles : [2, 7, 12] .(Images from [2])

(M = 3) For n = 2 and all volumes, solutions are standard triplebubbles : [14]. (Image from [14])

(M = 4) n = 2, same volumes [10]

(“M = ∞”) n = 2, Honeycomb conjecture [6]. (Image from [6])

Minimal partition problem in the planewith double densityWe study the minimal partition problem (1), where perimeter andvolume have l.s.c. densities h, f : R2 → (0,∞):

Ph(E) =

∫

∂∗Eh(x) dH1(x), |E|f =

∫

Ef (x) dx.

(M = 1): isoperimetric problem with density. [Brock, Cañete, Cianchi, DePhilippis, Fusco, Maggi, Miranda, Morgan, Pratelli, Rosales, Saracco, ... ]

→ including the case of anisotropic densities h = h(x, ν)

NB: Existence and regularity are in general not expected.

GOAL: Identify a class of densities for which existence holds andstudy regularity properties in view of the 120◦ rule.

The Grushin plane is a 2D sub-Riemannian manifold: α ≥ 0

Grushin plane

Ph(E) = Pα(E) =

∫

∂∗E

√ν21 + x

2α1 ν

22︸ ︷︷ ︸

h(x,ν)

dH1(x), |E|f = | · |︸ ︷︷ ︸f≡1

Via a change of variables we get:

Transformed Grushin plane

Ph(E) = PEucl(E)︸ ︷︷ ︸h≡1

,

|E|fα =∫

E|(α + 1)x1|−

αα+1︸ ︷︷ ︸

fα(x)

dx. no translation invariance

Translation invariance

NB: The Grushin perimeter Pα is not translation-invariant.

An important feature of the Grushin plane is that, differently tohigher dimensional sub-Riemannian examples, isoperimetric sets canbe characterized, see [8].

Theorem 1 (M = 1 - characterization) Let v > 0. Up tovertical translations, ∃! sol. to inf{Pα(E) : |E| = v} obtained viaanisotropic dilations from Eα = {(x, y) ∈ R2 : |y| ≤ ϕα(|x|), |x| ≤1}, ϕα explicit:

-1.0 -0.5 0.5 1.0

-0.5

0.5

Existence In [3] we obtain the following result.

Theorem 2 (F., P., S., - ∃ in the Grushin plane)M-minimal clusters exist for Pα, | · |, for any M ∈ N.

Remark 1 For general densities h, f : ∃ holds if

∃Φ : Ph(Φ(E)) ≤ Ph(E), |Φ(E)|f = |E|f

.

a b

E

b’a’ a b

EΦ(E)

Φ

Regularity and 120◦ rule. In [4] we obtain the following result.

Theorem 3 (F., P., S., Steiner regularity) Let E be aM-minimal cluster for densities h, f . Assume that h is con-tinuous, and that for η > 0, 0 < β ≤ 1, ηβ > 1 we have:i) growth condition, i.e., |BEucl(x, r)|f ≲ rη, r 0, α ≥ 0.Then solutions to problems (DBV), (DBH) exist. Moreover:(DBV) E ⊂ R2 sol. to (DBV) ⇒ up to vertical translations

E = {x ∈ R2 : |x2| ≤ f (|x1|), |x1| ≤ r},

f = fv,α ∈ C([0, r]) ∩ C∞(]0, r[), r ∈]0,+∞[.f=f(0,v)120

120 120

x

yf=f( ,v)

x

y

120 in transformed coordinates

α = 0 α = 1

(DBH) E ⊂ R2 sol. to (DBH) ⇒ up to vertical translations

E = δ1k

({x ∈ R2 :

(x1, |x2|− ϕα

(√32

))∈ Eα

}),

ϕα : [0, 1] → [0,+∞[ isoperimetric profile and k = k(v,α).E

120120120

x

y

E

???

x

y

120 in transformed coordinates

α = 0 α = 1

Comparison between vertical and horizontal

(DBV) (DBH).

EV

x

y

EH

x

y

P1(EV ) > P1(EH) !

Numerical simulations: (via Brakke Surface evolverhttp://facstaff.susqu.edu/brakke/evolver/html/)

References

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problems with constraints. Mem. Amer. Math. Soc., 4(165):viii+199, 1976.

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preprint.

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clusters. preprint.

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Sub-Riemannian soap bubbles