# The standard lens cluster in $\mathbb{R}^{2}$
uniquely minimizes relative perimeter

Stan Alama, Lia Bronsard, and Silas Vriend

###### Abstract

In this article we consider the isoperimetric problem for partitioning the
plane into three disjoint domains, one having unit area and the remaining two
having infinite area. We show that the only solution, up to rigid motions of
the plane, is a lens cluster consisting of circular arcs containing the finite
area region, attached to a single axis, with two triple junctions where the
arcs meet at 120 degree angles. In particular, we show that such a
configuration is a local minimizer of the total perimeter functional, and on
the other hand any local minimizer of perimeter among clusters with the given
area constraints must coincide with a lens cluster having this geometry. Some
known results and conjectures on similar problems with both finite and
infinite area constraints are presented at the conclusion.

##  1 Introduction

In a previous paper [4] we have conjectured that the minimal cluster for the
planar partitioning problem with one unit area chamber and two improper
chambers (i.e. of infinite area) is the standard lens cluster depicted in
Figure 1. The boundary set of this cluster consists of two collinear rays
emanating from a pair of standard triple junctions which are joined by a pair
of symmetric circular arcs, meeting at 120 degree angles. The center
corresponding to each circular arc lies on the midpoint of the opposite
circular arc, and these conditions uniquely determine the geometry of the
lens, also known as a vesica piscis (or fish’s bladder.) In this paper we show
that this configuration is the unique solution to the partitioning problem, in
the sense that any other configuration which minimizes perimeter locally (i.e.
among compactly supported variations preserving the area of the proper
chamber) must be congruent to the unit area lens cluster.

Figure 1: The standard lens cluster in $\mathbb{R}^{2}$

All partitioning problems are extensions and generalizations of the classical
isoperimetric problem: finding the shape of given area (or Lebesgue measure,
in the general case) whose boundary has the smallest possible perimeter. More
recently attention has turned to partitioning problems in which the plane (or
space) is subdivided into several disjoint but complementary domains, each
with given measure. This gave rise to the famous Double-Bubble Conjecture, to
determine the optimal shape of a pair of planar regions with given measure
whose combined perimeter is minimized. The solution is a double-bubble
composed of arcs of circles (spheres, in higher dimensions), meeting at 120
degree angles at the junction points [15]; this was proven by a group of
undergraduate students at Williams in [7]. An analogous question may be posed
in $n$ dimensions, and other generalizations include partitions with any
number of chambers of (finite) prescribed measure. The proof that double-
bubbles, domains composed of spherical caps meeting at 120 angles, minimize
surface area in $\mathbb{R}^{3}$ with two given volumes, was given by
Hutchings, Morgan, Ritoré, and Ros [8], and generalized to any dimension by
Reichardt [16]. The case of triple-bubbles, with three prescribed areas in the
plane, was handled by Wichiramala [20], but the complexity of the method has
limited progress on multiple-bubble problems with more than three given areas.

Multiple bubble configurations also arise in certain models of block
copolymers: nonlocal variational problems for multiphase materials exhibiting
short range attractive forces but longer range repulsion between different
phases. In an appropriate limit (described by Gamma-convergence) minimizers of
a ternary model exhibit double-bubble configurations [17, 1], while a
quaternary model (with four phases) produces triple-bubble minimizers [3].

It is in connection with the triblock copolymer models that we encountered the
lens cluster problem treated in this paper. If one considers a partitioning
problem in the torus, with three phases, two of which occupy nearly all of the
total area with the third accounting for only a tiny fraction, we expect
minimizers of the nonlocal triblock copolymer energy to form a lamellar
pattern with the two majority phases, with tiny droplets of the third phase
aligned on each lamellar stripe. By blowing up the droplets in the limit of
vanishing area of the third phase we expect to recover the lens shape in the
plane, and the characterization of the vesica piscis as the unique minimizer
(up to symmetries) of this problem will be critical to the analysis of the
small-area limit of the triblock problem. (See [2].)

Isoperimetric minimizers also arise as stationary or asymptotic solutions to
geometrical flows. In work of Bellettini-Novaga [6] and Schnürer et al. [18],
the lens configuration appears as the limiting configuration of a planar
network with two triple junctions under curve-shortening flow.

### Clusters

In studying the partitioning problem we adopt the notation and vocabulary of
clusters as developed in Maggi [12]. Unlike the clusters defined in [12], our
configurations will have two chambers with infinite measure, and so we
introduce a definition which extends the framework of clusters to the more
general setting in which there may be several chambers with infinite measure.

Let $\mathbb{N}_{0}$ denote the set $\mathbb{N}\cup\\{0\\}$. For greater
generality, we set the definitions to follow in dimension $n\geq 2$. Let
$\absolutevalue{E}$ denote the Lebesgue measure of a set $E$, let
$\operatorname{\mathcal{H}}^{k}$ denote the $k$-dimensional Hausdorff measure,
and let $P(E)$ denote the perimeter of $E$.

######  Definition 1.1.

An $(N,M)$-cluster $\mathcal{X}=(\mathcal{E},\mathcal{F})$ in $\mathbb{R}^{n}$
is a pair of finite families of sets of locally finite perimeter

$\mathcal{E}=\\{\mathcal{E}(h)\\}_{h=1}^{N},\quad\mathcal{F}=\\{\mathcal{F}(i)\\}_{i=1}^{M},\quad
N,M\in\mathbb{N}_{0},$

which satisfies the following list of properties:

  1. (i)

Proper chambers: the sets $\mathcal{E}(h)$, $1\leq h\leq N$, satisfy
$0<\absolutevalue{\mathcal{E}(h)}<\infty$.

  2. (ii)

Improper chambers: the sets $\mathcal{F}(i)$, $1\leq i\leq M$, satisfy
$\absolutevalue{\mathcal{F}(i)}=\infty$.

  3. (iii)

Null overlap: the sets $\mathcal{E}(h)$ and $\mathcal{F}(i)$ have pairwise
Lebesgue null overlap, that is,

$\displaystyle\absolutevalue{\mathcal{E}(h)\cap\mathcal{E}(k)}=0,$
$\displaystyle\quad 1\leq h<k\leq N,$
$\displaystyle\absolutevalue{\mathcal{E}(h)\cap\mathcal{F}(i)}=0,$
$\displaystyle\quad 1\leq h\leq N,\;1\leq i\leq M,$
$\displaystyle\absolutevalue{\mathcal{F}(i)\cap\mathcal{F}(j)}=0,$
$\displaystyle\quad 1\leq i<j\leq M.$

  4. (iv)

Full measure: following [12] Section 30.3], we have

$\absolutevalue{\mathbb{R}^{n}\setminus\left(\bigcup_{h=1}^{N}\mathcal{E}(h)\cup\bigcup_{i=1}^{M}\mathcal{F}(i)\right)}=0.$

######  Remark 1.2.

We would like to immediately highlight a key difference between this
definition and those of [12]: the proper chambers are not assumed to have
finite perimeter a priori. Nevertheless, we will show in Proposition 3.2 that,
for the case of $(1,2)$-clusters which minimize perimeter locally, the proper
chamber must have finite total perimeter.

We remark that as a consequence of the full measure axiom, any $(N,M)$-cluster
necessarily has $M\geq 1$. Proper $N$-clusters, having $N$ disjoint chambers
$\mathcal{E}=\\{\mathcal{E}(h)\\}_{h=1}^{N}$ of finite measure and perimeter,
are defined in [12]. Given such an $N$-cluster we may always append the
exterior chamber to obtain an $(N,1)$-cluster consistent with this definition.
Conversely, any $(N,1)$-cluster whose proper chambers have finite perimeter
must be an $N$-cluster as defined in [12], together with its exterior chamber
(up to a null set). Note that [12] implicitly includes the complement of an
$N$-cluster as an improper chamber by denoting it $\mathcal{E}(0)$. Note also
that Definition 1.1 with $N=0$ extends the notion of an improper $M$-cluster
from [12].

It will often be convenient to denote the proper and improper chambers of a
cluster with a single index, and so we also write an $(N,M)$-cluster as
$\mathcal{X}=\\{\mathcal{X}(j)\\}_{j=1}^{N+M}$ by setting

$\mathcal{X}(j)=\begin{cases}\mathcal{E}(j)&\text{if }1\leq j\leq N,\\\
\mathcal{F}(j-N)&\text{if }N+1\leq j\leq N+M.\end{cases}$

######  Remark 1.3.

The chambers of an $(N,M)$-cluster are not assumed to be indecomposable (i.e.
measure-theoretically connected; see [12] Exercise 16.9). Thus, a cluster
could also have several connected components in any one or more of its
chambers.

######  Definition 1.4.

Let $\overline{\mathbb{R}}=\mathbb{R}\cup\\{\pm\infty\\}$ denote the extended
real numbers, and let $\mathcal{X}=(\mathcal{E},\mathcal{F})$ be an
$(N,M)$-cluster. The volume vector
$\mathbf{m}(\mathcal{X})\in\overline{\mathbb{R}}^{N+M}$ is defined by

$\displaystyle\mathbf{m}(\mathcal{X})$
$\displaystyle=(\absolutevalue{\mathcal{E}(1)},\dots,\absolutevalue{\mathcal{E}(N)},\absolutevalue{\mathcal{F}(1)},\dots,\absolutevalue{F(M)})$
$\displaystyle=(\absolutevalue{\mathcal{E}(1)},\dots,\absolutevalue{\mathcal{E}(N)},\underbrace{\infty,\dots,\infty}_{M\text{
times}}).$

In dimension $n=2$, we call $\mathbf{m}(\mathcal{X})$ the area vector.

So, for example, a double-bubble cluster in $\mathbb{R}^{2}$ is described as a
$(2,1)$-cluster, with two disjoint chambers $\mathcal{E}(1),\mathcal{E}(2)$ of
prescribed areas $|\mathcal{E}(1)|=\mathbf{m}_{1}=A_{1}$,
$|\mathcal{E}(2)|=\mathbf{m}_{2}=A_{2}$, and the exterior chamber
$\mathcal{F}=\mathbb{R}^{2}\setminus\overline{\mathcal{E}(1)\cup\mathcal{E}(2)}$
having infinite area. The area vector is thus
$\mathbf{m}=(A_{1},A_{2},\infty)$.

To define perimeter, we set the following notation for the interfaces, in
other words, the boundary arcs of the chambers of the cluster:

######  Definition 1.5.

The interfaces of an $(N,M)$-cluster $\mathcal{X}$ are the
$\operatorname{\mathcal{H}}^{n-1}$-rectifiable sets

$\mathcal{X}(j,k)=\partial^{*}\mathcal{X}(j)\cap\partial^{*}\mathcal{X}(k),\quad
1\leq j<k\leq N+M.$

The boundary set $\partial\mathcal{X}$ of an $(N,M)$-cluster $\mathcal{X}$ is
defined by

$\partial\mathcal{X}=\bigcup_{1\leq j\leq N+M}\partial\mathcal{X}(j).$

We are interested in configurations which minimize perimeter, but in case the
number of improper chambers $M\geq 2$ the total perimeter will necessarily be
infinite. Hence we must express the isoperimetric problems in terms of a
relative perimeter:

######  Definition 1.6.

The relative perimeter of $\mathcal{X}$ in $F\subset\mathbb{R}^{n}$ is defined
by

$P(\mathcal{X};F)=\sum_{1\leq j<k\leq
N+M}\operatorname{\mathcal{H}}^{n-1}(F\cap\mathcal{X}(j,k)).$

The perimeter of $\mathcal{X}$ is defined by

$P(\mathcal{X})=P(\mathcal{X};\mathbb{R}^{n})=\sum_{1\leq j<k\leq
N+1}\operatorname{\mathcal{H}}^{n-1}(\mathcal{X}(j,k)).$

Similarly, while it is natural to seek a global minimizer of perimeter among
all clusters with given mass vector $\mathbf{m}(\mathcal{E})=\mathbf{m}$, for
clusters with $M\geq 2$ improper chambers we are limited to studying local
minimizers, which minimize perimeter only with respect to compactly supported
perturbations of $\mathcal{X}$. For any set $E$ of locally finite perimeter,
let $\mu_{E}$ denote the $\mathbb{R}^{n}$-valued Radon measure (Gauss-Green
measure, see [12, Proposition 12.1]) for which

$\int_{E}\text{div}\,T=\int_{\mathbb{R}^{n}}T\cdot d\mu_{E},\quad\text{for all
$T\in C_{c}^{1}(\mathbb{R}^{n};\mathbb{R}^{n}).$}$

Then, we define:

######  Definition 1.7.

Given an $(N,M)$-cluster $\mathcal{X}$ in $\mathbb{R}^{n}$, we say that
$\mathcal{X}$ is a locally minimizing $(N,M)$-cluster in $\mathbb{R}^{n}$ if
$\operatorname{\mathrm{spt}}\mu_{\mathcal{X}(j)}=\partial X(j)$ for every
$1\leq j\leq N+M$, and, moreover,

$P(\mathcal{X};B_{r})\leq P(\mathcal{X}^{\prime};B_{r})$

for each ball $B_{r}$ of radius $r>0$ centred at the origin, and every
$(N,M)$-cluster $\mathcal{X}^{\prime}$ satisfying
$\mathbf{m}(\mathcal{X}^{\prime})=\mathbf{m}(\mathcal{X})$ and

$\mathcal{X}^{\prime}(j)\Delta\mathcal{X}(j)\subset\subset B_{r}\quad\text{for
each }1\leq j\leq N+M.$

######  Definition 1.8.

By an $(N,M)$-partitioning problem in $\mathbb{R}^{n}$ we mean a variational
problem where given a volume vector
$\mathbf{m}=(\mathbf{m}_{1},\dots,\mathbf{m}_{N+M})\in\overline{\mathbb{R}}^{N+M}$
with

$\displaystyle\begin{cases}0<\mathbf{m}_{j}<\infty&\text{if }1\leq j\leq N,\\\
\mathbf{m}_{j}=\infty&\text{if }N+1\leq j\leq N+M,\end{cases}$

we seek an $(N,M)$-cluster $\mathcal{X}$ which is a local minimizer in the
sense of Definition 1.7.

The resolution of the double-bubble conjecture by [7, 8] thus proves that for
any choices of $A_{1},A_{2}>0$, the $(2,1)$-partitioning problem in
$\mathbb{R}^{n}$ with volume vector $\mathbf{m}=(A_{1},A_{2},\infty)$ is a
double-bubble, consisting of spherical interfaces meeting at 120 degree angles
at the junctions. Such a configuration is both a global and local minimizer of
the perimeter.

### The Main Result

######  Definition 1.9.

The standard lens cluster in $\mathbb{R}^{2}$ is the $(1,2)$-cluster
$\mathcal{X}_{\infty}=(\mathcal{E},\mathcal{F})$ whose chambers are given by

$\displaystyle\mathcal{E}(1)=\\{(x,y):\absolutevalue{x}\leq\tfrac{\sqrt{3}}{2}R,\;\absolutevalue{y}\leq
u(x)\\},$ $\displaystyle\mathcal{F}(1)=\\{y\geq
0\\}\setminus\operatorname{\mathrm{Int}}(\mathcal{E}(1)),\quad\mathcal{F}(2)=\\{y\leq
0\\}\setminus\operatorname{\mathrm{Int}}(\mathcal{E}(1)),$

where $u:[-\frac{\sqrt{3}}{2}R,\frac{\sqrt{3}}{2}R]\to\mathbb{R}$ is
determined by

$\displaystyle u(x)$ $\displaystyle=\sqrt{R^{2}-x^{2}}-\frac{1}{2}R,$

and the radius of curvature $R$ is determined by

$R^{-1}=\sqrt{\frac{2\pi}{3}-\frac{\sqrt{3}}{2}}.$ (1)

An elementary computation shows that $\absolutevalue{\mathcal{E}(1)}=1$.

Note that the two interfaces of $\mathcal{E}(1)$ are circular arcs of the same
radius $R$, and the center of each arc is located on the midpoint
$(0,\pm{\tfrac{R}{2}})$ of the opposite arc.

In this article, we prove the

######  Theorem 1.10 (Main Result).

The standard lens cluster is (up to translation and rotational symmetry) the
unique locally minimizing cluster for the $(1,2)$-partitioning problem in
$\mathbb{R}^{2}$, with area vector $\mathbf{m}=(1,\infty,\infty)$.

The proof of Theorem 1.10 will be in two parts. First, we show that the
standard lens cluster is a local minimizer, in the sense of Definition 1.7.
For this, we use a strategy suggested to us by Frank Morgan, thinking of the
lens cluster as a limit of double-bubbles as the area of one chamber diverges
to infinity. This limit is described in Section 2. Next, we show that any
local minimizer must reduce to the standard lens cluster, by a sequence of
constructions which would reduce the local perimeter were the configuration to
have a different geometry. The proof of Theorem 1.10 is contained in Section
3. Some conjectures on the geometry of minimizing $(N,M)$-clusters for larger
values of $M,N$ are included in Section 4.

##  2 A Convergence Result

In this section we show that the standard lens cluster may be obtained as a
local limit of double bubbles with a very large chamber. We first define a
weak notion of convergence for clusters.

######  Definition 2.1.

Let $N,M,N^{\prime},M^{\prime}$ be such that $N+M=N^{\prime}+M^{\prime}$. The
cluster distance in $F\subset\mathbb{R}^{n}$ between an $(N,M)$-cluster
$\mathcal{X}$ and an $(N^{\prime},M^{\prime})$-cluster $\mathcal{X}^{\prime}$
in $\mathbb{R}^{n}$ is defined as

$d_{F}(\mathcal{X},\mathcal{X}^{\prime})=\sum_{j=1}^{N+M}\absolutevalue{F\cap(\mathcal{X}(j)\Delta\mathcal{X}^{\prime}(j))}.$

We set
$d(\mathcal{X},\mathcal{X}^{\prime})=d_{\mathbb{R}^{n}}(\mathcal{X},\mathcal{X}^{\prime})$.
Notice that in general we could have
$d(\mathcal{X},\mathcal{X}^{\prime})=\infty$.

######  Definition 2.2.

Fix $N,M$ and let $\mathcal{X}$ be an $(N^{\prime},M^{\prime})$-cluster such
that $N^{\prime}+M^{\prime}=N+M$.

  * •

We say that a sequence of $(N,M)$-clusters
$\\{\mathcal{X}_{k}\\}_{k\in\mathbb{N}}$ in $\mathbb{R}^{n}$ locally converges
to $\mathcal{X}$, and write
$\mathcal{X}_{k}\xrightarrow{\mathrm{loc}}\mathcal{X}$, if for every compact
set $K\subset\mathbb{R}^{n}$ we have $d_{K}(\mathcal{X}_{k},\mathcal{X})\to 0$
as $k\to\infty$.

  * •

We say that $\\{\mathcal{X}_{k}\\}_{k\in\mathbb{N}}$ converges to
$\mathcal{X}$, and write $\mathcal{X}_{k}\to\mathcal{X}$, if
$d(\mathcal{X}_{k},\mathcal{X})\to 0$ as $k\to\infty$.

######  Remark 2.3.

For clusters with $M\geq 2$, we can in general only talk about local
convergence. Note that cluster type need not be preserved under cluster
convergence, and cluster distance is indexing-dependent.

The key idea in the existence part of the proof of the main result is that the
standard lens cluster is the limit in local cluster convergence of a suitably
chosen family of standard double bubbles (with their exteriors). We make this
precise below.

######  Lemma 2.4.

If $\mathcal{X}$ is an $(N,M)$-cluster in $\mathbb{R}^{n}$, then for every
$F\subset\mathbb{R}^{n}$ we have

$P(\mathcal{X};F)=\frac{1}{2}\sum_{1\leq j\leq N+M}P(\mathcal{X}(j);F).$

In particular, if $U$ is open in $\mathbb{R}^{n}$ and
$\mathcal{X}_{k}\xrightarrow{\mathrm{loc}}\mathcal{X}$, then

$P(\mathcal{X};U)\leq\liminf_{k\to\infty}P(\mathcal{X}_{k};U).$

###### Proof.

This follows from a straightforward adaptation of [12] Proposition 29.4. ∎

######  Lemma 2.5.

For $A>0$, consider the minimizing double bubble
$\mathcal{D}_{A}=\\{\mathcal{D}_{A}(1),\mathcal{D}_{A}(2),\mathcal{D}_{A}(3)\\}$,
$\mathcal{D}_{A}(3)=\mathbb{R}^{2}\setminus\overline{\mathcal{D}_{A}(1)\cup\mathcal{D}_{A}(2)}$,
with area vector $(1,A,\infty)$, oriented so that the circle supporting the
interface $\mathcal{D}_{A}(2,3)$ is tangent to the $x$-axis at the origin and
centered on a point of the positive $y$-axis; see Figure 4. Then as
$A\to\infty$,

$\mathcal{D}_{A}\xrightarrow{\mathrm{loc}}\mathcal{X}_{\infty}.$

Furthermore, for any bounded open set $U$,

$P(\mathcal{X}_{\infty};U)\leq\liminf_{A\to\infty}P(\mathcal{D}_{A};U).$

###### Proof.

Fix $r>\tfrac{\sqrt{3}}{2}R$. By monotonicity of $d_{F}$ with respect to $F$
and boundedness of each compact set, it suffices to check that for each open
ball $B_{r^{\prime}}$ centered at $0$ with $r^{\prime}\geq r$, we have

$d_{B_{r^{\prime}}}(\mathcal{D}_{A},\mathcal{X}_{\infty})\xrightarrow[A\to\infty]{}0.$

Let $r^{\prime}\geq r$ be arbitrary. We aim to derive estimates on
$d_{B_{r^{\prime}}}(\mathcal{D}_{A},\mathcal{X}_{\infty})$ via geometrical
information about $\mathcal{D}_{A}$ (see [9]). A double bubble is bounded by
three circular arcs $C_{0},C_{1},C_{2}$ of radii $r_{1},r_{2}$ and $r_{0}$,
where $r_{0}$ is the radius of the common boundary of the two lobes of a
double bubble with area vector $(m_{1},m_{2},\infty)$. Denote by
$\theta_{1},\theta_{2}$, and $\theta_{0}$ the angles associated with the three
arcs, as in Figure 2. The double bubble $\mathcal{D}_{A}$ corresponds to the
case $m_{1}=1,m_{2}=A$. We have fixed translation invariance by selecting
$\mathcal{D}_{A}$ so that the circle $C_{2}$ supporting the interface
$\mathcal{D}_{A}(2,3)$ is tangent to the $x$-axis at the origin. Note this
implies that in the limit $A\to\infty$ the large circle $C_{2}$ will approach
the $x$-axis. By axial symmetry we conclude that the centers
$p_{0},p_{1},p_{2}$ of the three circles each lies on the $y$-axis. We denote
by $q$ either one of the triple junction points (which are placed
symmetrically about the $y$-axis) where the three circular arcs meet.

Figure 2: An asymmetric double bubble in $\mathbb{R}^{2}$ [1]

The equations relating areas, angles, and radii are

$\displaystyle
r_{1}^{2}(\theta_{1}-\cos\theta_{1}\sin\theta_{1})+r_{0}^{2}(\theta_{0}-\cos\theta_{0}\sin\theta_{0})$
$\displaystyle=$ $\displaystyle 1,$ (2) $\displaystyle
r_{2}^{2}(\theta_{2}-\cos\theta_{2}\sin\theta_{2})-r_{0}^{2}(\theta_{0}-\cos\theta_{0}\sin\theta_{0})$
$\displaystyle=$ $\displaystyle A,$ (3) $\displaystyle r_{1}\sin\theta_{1}$
$\displaystyle=$ $\displaystyle r_{0}\sin\theta_{0},$ (4) $\displaystyle
r_{2}\sin\theta_{2}$ $\displaystyle=$ $\displaystyle r_{0}\sin\theta_{0},$ (5)
$\displaystyle(r_{1})^{-1}-(r_{2})^{-1}$ $\displaystyle=$
$\displaystyle(r_{0})^{-1},$ (6)
$\displaystyle\cos\theta_{1}+\cos\theta_{2}+\cos\theta_{0}$ $\displaystyle=$
$\displaystyle 0.$ (7)

As $A\to\infty$, $r_{2}\to\infty$ and $\theta_{2}\to\pi$. Hence from equations
(6) and (4), we see that

$|r_{1}-r_{0}|\to 0$ (8)

and

$|\theta_{1}-\theta_{0}|\to 0.$ (9)

Since $\mathcal{D}_{A}$ is a minimizing double bubble, the tangents to the
arcs meet at angles of $2\pi/3$. Now form the triangle $\triangle
p_{0}p_{2}q$. The edges $\overline{p_{0}q}$ and $\overline{p_{2}q}$ are radii
of the circular arcs, and thus are normal lines to the transition curves at
$q$. They must therefore meet at angle $2\pi/3$, and so adding the interior
angles of the triangle $\triangle p_{0}p_{2}q$ we obtain
$\theta_{0}=\theta_{2}-{\frac{2\pi}{3}}$. As a consequence, we conclude that
$\theta_{0}$ is convergent, $\theta_{0}\to{\frac{\pi}{3}}$ as $A\to\infty$.
From (9) we also have $\theta_{1}\to{\frac{\pi}{3}}$.

To pass to the limit in the radii, we note that $|\mathcal{D}_{A}(1)|=1$, and
so $0<r_{1}<1/\sqrt{\pi}$ holds for all $A$. By (8) both $r_{0},r_{1}$ are
bounded, so there exists a subsequence along which both converge, to the same
limiting value. Using equation (2) we conclude that
$r_{1}^{-1}\to\sqrt{2\pi/3-\sqrt{3}/2}=:R^{-1}$ which yields the radius of the
circles in the standard lens cluster $\mathcal{X}$. From (8) we also have
$r_{0}^{-1}\to R^{-1}$.

To show convergence of the centers of the circles $C_{0},C_{1}$ we first note
that the large circle $C_{2}$ converges locally uniformly to the $x$-axis in
the sense of parametrized curves. As the centers of $C_{0},C_{1}$ must lie on
the $y$-axis, and the radii $r_{0},r_{1}$ converge, the triple point $q$
remains bounded, and since it also lies on $C_{2}$, passing to a further
subsequence if necessary, it converges to a point on the $x$-axis (see Figure
3). The center $p_{0}$ (resp. $p_{1}$) must lie on the intersection of the
$y$-axis with the circle of radius $r_{0}$ (resp. $r_{1}$) centered at $q$,
which consists of two points. In the limit, $r_{0},r_{1}\to R$ and the centers
converge to the two points lying on the intersection of the $y$-axis and the
circle of radius $R$ centered at the limiting triple point $q$. The same is
true of the circle of radius $R$ centered at the opposing triple point. The
centers $p_{0},p_{1}$ lie at the intersection of these two circles, and are
symmetrically placed with respect to the line connecting the two triple
points. Now consider the isosceles triangle $\triangle p_{0}p_{1}q$. Again,
the radius vectors of the circles are normal vectors to the curves. As the
tangents meet at angle $2\pi/3$, the angle $\angle p_{0}qp_{1}$ at the triple
point is $\pi/3$, and therefore $\triangle p_{0}p_{1}q$ is equilateral, of
length $R$. In particular, the segment joining $p_{0}$ and $p_{1}$ has length
$R$ and we conclude that $p_{0}=(0,-R/2)$, $p_{1}=(0,R/2)$ and
$q=\frac{\sqrt{3}}{2}R$.

Figure 3: Convergence to the standard double-bubble

Finally, the boundary arcs being circular, they represent parametrized
piecewise $C^{\infty}$ arcs. We have already noted that the large circle
$C_{2}$ converges locally uniformly to the $x$-axis. Since the radii and
centers of $C_{0},C_{1}$ converge, each circle converges in the sense of
parametrized curves to a limiting circle, and so each of the arcs in the
double bubble $\mathcal{D}_{A}$ converges, locally uniformly in the sense of
parametrized curves, to a circular arc. The local uniform convergence then
implies the convergence of the associated clusters in the distance defined by
Definition 2.1.

Since $r^{\prime}\geq r>\tfrac{\sqrt{3}}{2}R$, the lens is inside
$B_{r^{\prime}}$ and the conclusion follows. ∎

##  3 Proof of Theorem 1.10

We now prove that the standard lens cluster is the unique locally minimizing
$(1,2)$-cluster, up to rigid motions of the plane, with area vector
$\mathbf{m}=(1,\infty,\infty)$.

### Minimality of $\mathcal{X}_{\infty}$

We first show existence of a local minimizer: in particular, that
$\mathcal{X}_{\infty}$ as defined in Definition 1.9 is a local minimizer of
the perimeter functional. The strategy is to argue by contradiction and assume
that there is a compactly supported perturbation of the lens cluster which
decreases perimeter. If so, we construct a competitor for the double-bubble
problem with a large chamber, which would have less perimeter than a
minimizing double bubble. The argument relies on the following corollary of
the double bubble theorem:

######  Theorem 3.1 ([7]).

Fix $A>0$, and consider the family of planar partitioning problems,

$\inf\\{P(\mathcal{E}):\mathbf{m}(\mathcal{E})=(1,A^{\prime},\infty),\
A^{\prime}\geq A\\},$

where $\mathcal{E}$ ranges over all planar $(2,1)$-clusters with area vector
$(1,A^{\prime},\infty)$. Then, up to translation and rotation, the unique
minimizer of this problem is the standard double bubble $\mathcal{D}_{A}$ with
area vector $(1,A,\infty)$.

From now on, we will refer to the standard double bubble
$\mathcal{X}_{\infty}$ as $\mathcal{X}$. Assume toward a contradiction that
$\mathcal{X}$ is not a locally minimizing $(1,2)$-cluster for the area vector
$(1,\infty,\infty)$. Negating Definition 1.7, it follows that there exists an
open ball $B_{r}$ of radius $r>0$ centered at $0$ and a $(1,2)$-cluster
$\mathcal{X}^{\prime}$ such that the area vectors of
$\mathcal{X}=\mathcal{X}_{\infty}$ and $\mathcal{X}^{\prime}$ agree,

$\mathbf{m}(\mathcal{X}^{\prime})=\mathbf{m}(\mathcal{X}),$

the modifications are compactly contained in $B_{r}$,

$\mathcal{X}(j)\Delta\mathcal{X}^{\prime}(j)\subset\subset B_{r}\text{ for
each }1\leq j\leq 3,$ (10)

and $\mathcal{X}^{\prime}$ improves on $\mathcal{X}$ within $B_{r}$,

$P(\mathcal{X}^{\prime};B_{r})<P(\mathcal{X};B_{r}).$

By (10), it follows that $\mathcal{X}(1),\mathcal{X}^{\prime}(1)\subset\subset
B_{r}$. Note that we make no a priori assumptions about the regularity of the
interfaces of $\mathcal{X}^{\prime}$ within $B_{r}$. Denote by $K$ a compact
set such that

$\mathcal{X}(j)\Delta\mathcal{X}^{\prime}(j)\subset K\subset
B_{r}\quad\text{for each }1\leq j\leq 3.$

By enlarging $K$ if necessary, we can assume that $K$ is a closed ball
centered at $0$ containing $\mathcal{X}(1)$ and $\mathcal{X}^{\prime}(1)$ in
its interior.

Since $\mathbf{m}(\mathcal{X}^{\prime})=\mathbf{m}(\mathcal{X})$, we have that
$\absolutevalue{\mathcal{X}^{\prime}(1)}=1$. One of
$\mathcal{X}^{\prime}(2)\cap B_{r}$ or $\mathcal{X}^{\prime}(3)\cap B_{r}$
must have the same or strictly less relative area within $B_{r}$ compared to
$\mathcal{X}(2)\cap B_{r}$ or $\mathcal{X}(3)\cap B_{r}$, respectively. As
$\mathcal{X}$ is symmetric about the $x$-axis, without loss of generality we
can assume that the upper region $\mathcal{X}^{\prime}(2)\cap B_{r}$ has area
at least that of $\mathcal{X}(2)\cap B_{r}$. Since $\mathcal{X}$ and
$\mathcal{X}^{\prime}$ only differ in $K$, we can assume,

$\absolutevalue{\mathcal{X}^{\prime}(2)\cap
K}\geq\absolutevalue{\mathcal{X}(2)\cap K}.$ (11)

Set

$\varepsilon=P(\mathcal{X};B_{r})-P(\mathcal{X}^{\prime};B_{r})>0.$ (12)

Consider the one-parameter family of $(2,1)$-clusters $\mathcal{D}_{A}$
associated with the minimizing double-bubbles defined in Lemma 2.5, with area
vectors $(1,A,\infty)$; see Figure 4 for a diagram. By Lemma 2.5, we know that
$\mathcal{D}_{A}$ locally converges to $\mathcal{X}$ in the sense of cluster
distance as $A\to\infty$, i.e.,

$\mathcal{D}_{A}\xrightarrow{\mathrm{loc}}\mathcal{X},$

and furthermore that

$P(\mathcal{X};U)\leq\liminf_{A\to\infty}P(\mathcal{D}_{A};U)$

for any bounded open set $U$. Note that for all $A$ sufficiently large,
$\partial\mathcal{D}_{A}(2)\cap B_{r}$ consists of piecewise smooth circular
arcs which are graphs over the $x$-axis. By construction of $\mathcal{D}_{A}$,
we fix translation invariance so that the large circle $C_{2}$ is tangent to
the axis at the origin. Note in fact that the graphs forming
$\partial\mathcal{D}_{A}(2)\cap B_{r}$ are positive functions.

Figure 4: The $(2,1)$-cluster $\mathcal{D}_{A}$ (in blue) locally approximates
the standard lens cluster $\mathcal{X}$ (in black) for large $A$. The lens
$\mathcal{X}(1)$ is contained in nested sets $K$ (in yellow), $K^{\prime}$ (in
green), and $B_{r}$ (in grey). The region $\mathcal{R}_{A}$ to be added to
$\mathcal{D}_{A}(2)$ is shaded (in pink)

Let $K^{\prime}$ be a closed ball concentric with $K$ and $B_{r}$ such that
$K\subset K^{\prime}\subset B_{r}$. Define the open annulus
$\mathcal{A}=\operatorname{\mathrm{Int}}(K^{\prime})\setminus K$ and the set
$\mathcal{R}_{A}=\mathcal{A}\cap(\mathcal{X}(2)\setminus\mathcal{D}_{A}(2))$.
We consider two cases. First suppose that for $A$ sufficiently large, the area
of $\mathcal{D}_{A}(2)\cap K$ is smaller than the area of $\mathcal{X}(2)\cap
K$, that is, for some $A_{0}>0$ we have

$\absolutevalue{\mathcal{D}_{A}(2)\cap K}\leq\absolutevalue{\mathcal{X}(2)\cap
K},\ \text{for all }A\geq A_{0}.$ (13)

In this case, we use the variation $\mathcal{X}^{\prime}$ to define a
competitor $\mathcal{D}^{\prime}_{A}$ to the double bubble $\mathcal{D}_{A}$
in the following way: $\mathcal{D}^{\prime}_{A}$ coincides with
$\mathcal{X}^{\prime}$ in $K$, and agrees with the double-bubble
$\mathcal{D}_{A}$ outside of $K^{\prime}$. In the annular region
$\mathcal{A}=K^{\prime}\setminus K$ we connect the two by taking
$\mathcal{D}^{\prime}_{A}(2)=\mathcal{D}_{A}(2)\cup\mathcal{R}_{A}$ while
$\mathcal{D}^{\prime}_{A}(3)=\mathcal{D}_{A}(3)\setminus\mathcal{R}_{A}$ (see
Figure 4). (Note that the area of the chamber $\mathcal{D}_{A}^{\prime}(2)$
may not be exactly equal to $A$, but we will see below that it is at least as
large.) In constructing this competitor, we have modified the perimeter in a
predictable way, namely we have added the small arc
$\partial\mathcal{R}_{A}\cap\partial K^{\prime}$ and replaced the arc
$\mathcal{D}_{A}(2)\cap\mathcal{A}$ with the segments of the $x$-axis
contained in the annular region $\mathcal{A}$. By choosing $A$ sufficiently
large, we can make this smaller than $\frac{\varepsilon}{2}$ with
$\varepsilon$ as defined by (12) and obtain:

$\displaystyle P(\mathcal{D}_{A}^{\prime};B_{r})$ $\displaystyle\leq
P(\mathcal{X}^{\prime};B_{r})+\frac{\varepsilon}{2}$ $\displaystyle\leq
P(\mathcal{X};B_{r})-\varepsilon+\frac{\varepsilon}{2}$
$\displaystyle<\liminf_{A\to\infty}P(\mathcal{D}_{A};B_{r}).$

This means that we can find some area $A\geq A_{0}$ for which
$P(\mathcal{D}_{A}^{\prime};B_{r})<P(\mathcal{D}_{A};B_{r})$. On the other
hand, we have also modified the area of this competitor in a predictable way,
and using (11) and (13) we find:

$\displaystyle\absolutevalue{\mathcal{D}_{A}^{\prime}(2)}$
$\displaystyle=\absolutevalue{\mathcal{D}_{A}^{\prime}(2)\cap
K}+\absolutevalue{\mathcal{D}_{A}^{\prime}(2)\cap\mathcal{A}}+\absolutevalue{\mathcal{D}_{A}^{\prime}(2)\setminus{K^{\prime}}}$
$\displaystyle\geq\absolutevalue{\mathcal{X}^{\prime}(2)\cap
K}+\absolutevalue{\mathcal{D}_{A}(2)\cap\mathcal{A}}+\absolutevalue{\mathcal{D}_{A}(2)\setminus{K^{\prime}}}$
$\displaystyle\geq\absolutevalue{\mathcal{X}(2)\cap
K}+\absolutevalue{\mathcal{D}_{A}(2)\setminus K}$
$\displaystyle\geq\absolutevalue{\mathcal{D}_{A}(2)\cap
K}+\absolutevalue{\mathcal{D}_{A}(2)\setminus K}$
$\displaystyle=\absolutevalue{\mathcal{D}_{A}(2)}=A.$

This contradicts the minimality of the double-bubble $\mathcal{D}_{A}$ as per
Theorem 3.1 and thus (13) does not hold.

In case (13) fails, we can find a sequence $A_{n}\to\infty$ for which:

$\absolutevalue{\mathcal{D}_{A_{n}}(2)\cap
K}>\absolutevalue{\mathcal{X}(2)\cap K}.$ (14)

Consider the vertical translates $\mathcal{D}_{A_{n}}+(0,t)$ of
$\mathcal{D}_{A_{n}}$. As $\partial\mathcal{D}_{A_{n}}(2)\cap B_{r}$ consists
of graphs of positive functions, the area
$\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t))\cap B_{r}}$ is continuous and
strictly monotone decreasing to zero as $t\to\infty$. Thus we can define
translates $t_{n}>0$ for which:

$\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\cap
K}=\absolutevalue{\mathcal{X}(2)\cap K},\ \text{for all }n.$ (15)

Next we claim that $t_{n}\to 0$. Indeed, suppose not, so that extracting a
subsequence if necessary, we may assume that $t_{n}\to t>0$. Observe that

$\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}/2)\cap
K}>\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n})\cap
K}=\absolutevalue{\mathcal{X}(2)\cap K},\ \text{for all }n.$

Therefore by passing to the limit we have $\mathcal{D}_{A_{n}}(2)\cap
K\to\mathcal{X}(2)\cap K$ locally and hence we conclude that
$\absolutevalue{(\mathcal{X}(2)+(0,t/2)\cap
K}\geq\absolutevalue{\mathcal{X}(2)\cap K}$ which contradicts the strict
monotonicity of the area under vertical translation. Hence $t_{n}\to 0$ and
therefore we may also conclude from Lemma 2.5 that
$\mathcal{D}_{A_{n}}+(0,t_{n})\to\mathcal{X}$ locally.

Now construct the competitor as before but replacing $\mathcal{D}_{A}$ with
$\mathcal{D}_{A_{n}}+(0,t_{n})$, namely $\mathcal{D}_{A_{n}}^{\prime\prime}$
agrees with $\mathcal{X}^{\prime}$ in $K$, coincides with
$\mathcal{D}_{A_{n}}+(0,t_{n})$ outside of $K^{\prime}$, and with
$\mathcal{D}^{\prime\prime}_{A_{n}}(2)={\mathcal{{R}}}_{A_{n}}\cup(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))$
and
$\mathcal{D}^{\prime\prime}_{A_{n}}(3)=(\mathcal{D}_{A_{n}}(3)+(0,t_{n}))\setminus\mathcal{R}_{A_{n}}$
in $\mathcal{A}$. As before the modification of the perimeter only occurs in
the annular region and by taking the area $A_{n}$ sufficiently large, the
modification can be made smaller than $\frac{\varepsilon}{2}$ where
$\varepsilon$ is defined by (12). So the perimeter of the competitor now
satisfies:

$\displaystyle P(\mathcal{D}_{A_{n}}^{\prime\prime};B_{r})$ $\displaystyle\leq
P(\mathcal{X};B_{r})-\frac{\varepsilon}{2}$ $\displaystyle<\liminf
P(\mathcal{D}_{A_{n}}+(0,t_{n});B_{r}).$

Further using (11) and (15) the area of the competitor satisfies:

$\displaystyle\absolutevalue{\mathcal{D}_{A_{n}}^{\prime\prime}(2)}$
$\displaystyle=\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\setminus
K^{\prime}}+\absolutevalue{{\mathcal{{R}}}_{A_{n}}\cup(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\cap\mathcal{A}}+\absolutevalue{\mathcal{X}^{\prime}(2)\cap
K}$ $\displaystyle>\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\setminus
K^{\prime}}+\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\cap\mathcal{A}}+\absolutevalue{\mathcal{X}(2)\cap
K}$ $\displaystyle=\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\setminus
K^{\prime}}+\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\cap\mathcal{A}}+\absolutevalue{(\mathcal{D}_{A_{n}}(2)+(0,t_{n}))\cap
K}$ $\displaystyle\geq\absolutevalue{\mathcal{D}_{A_{n}}(2)+(0,t_{n})}.$

This once again contradicts the minimality of the double bubble with area
$(1,A_{n})$ for $A_{n}$ sufficiently large. Therefore the lens cluster
$\mathcal{X}$ is a minimizer in the mixed planar partitioning problem with
area vector $(1,\infty,\infty)$.

### Uniqueness of the minimizer

Let $\mathcal{M}=(\mathcal{M}(1),\mathcal{M}(2),\mathcal{M}(3))$ be a locally
minimizing $(1,2)$-cluster for the mixed planar partitioning problem with area
vector $(1,\infty,\infty)$. We will show that (up to symmetries of the
problem) necessarily $\mathcal{M}=\mathcal{X}$, the standard lens cluster. We
recall the notation
$\mathcal{M}(j,k)=\partial^{*}\mathcal{M}(j)\cap\partial^{*}\mathcal{M}(k)$
for the interfaces between the chambers.

First, we recall Remark 1.2, and prove that a locally minimizing $(N,M)=(1,2)$
cluster $\mathcal{M}$ (which a priori is defined as having locally finite
perimeter in each chamber) must in fact have a proper chamber $\mathcal{M}(1)$
with finite total perimeter in $\mathbb{R}^{2}$.

######  Proposition 3.2.

Suppose $\mathcal{M}=(\mathcal{M}(1),\mathcal{M}(2),\mathcal{M}(3))$ is a
cluster in $\mathbb{R}^{2}$ satisfying:

  1. (i)

$|\mathcal{M}(1)|=1$;

  2. (ii)

$|\mathcal{M}(2)|=\infty=|\mathcal{M}(3)|$:

  3. (iii)

Each $\mathcal{M}(j)$ has locally finite perimeter;

  4. (iv)

$\mathcal{M}$ is locally minimizing.

Then $\mathrm{Per}_{\mathbb{R}^{2}}(\mathcal{M}(1))<\infty$.

###### Proof.

Assume for a contradiction that
$\mathrm{Per}_{\mathbb{R}^{2}}(\mathcal{M}(1))=\infty$. Then we may choose
$R>0$ for which

$\mathrm{Per}_{B_{R}}(\mathcal{M}(1))\geq 3\sqrt{\pi}.$ (16)

As

$1=|\mathcal{M}(1)|=\int_{0}^{\infty}\int_{\partial
B_{r}}\mathbf{1}_{\mathcal{M}(1)}\,ds_{r}\,dr,$

by Fubini’s Theorem there exists $R_{0}\in[R,R+1]$ so that

$\int_{\partial B_{R_{0}}}\mathbf{1}_{\mathcal{M}(1)}(x)\,ds_{R_{0}}\leq
1\quad\text{ie.,}\quad\operatorname{\mathcal{H}}^{1}\left(\mathcal{M}(1)\cap\partial
B_{R_{0}}\right)\leq 1,$ (17)

and thus $\mathcal{M}(1)$ intersects $\partial B_{R_{0}}$ on arcs with total
arclength at most one. Define

$\varepsilon:=\int_{B_{R_{0}}}\mathbf{1}_{\mathcal{M}(1)}(x)\,dx\leq 1.$

Inside $B_{R_{0}}$, $\partial\mathcal{M}(1)$ must intersect either
$\partial\mathcal{M}(2)$ or $\partial\mathcal{M}(3)$ (or both). Suppose it’s
true for $\partial\mathcal{M}(2)$, so
$\operatorname{\mathcal{H}}^{1}(\partial\mathcal{M}(1)\cap\partial\mathcal{M}(2)\cap
B_{R_{0}})>0$.

Now, construct a competitor $\tilde{\mathcal{M}}$ as follows, by modification
of $\mathcal{M}$ inside $\overline{B_{R_{0}}}$. Choose
$r=\sqrt{\varepsilon/\pi}$ so that the ball $|B_{r}(0)|=\varepsilon$. Then, we
define

$\displaystyle\tilde{\mathcal{M}}(1):=\left[\mathcal{M}(1)\setminus
B_{R_{0}}\right]\cup B_{r}(0),$
$\displaystyle\tilde{\mathcal{M}}(2):=\left(\mathcal{M}(2)\cup[\mathcal{M}(1)\cap
B_{R_{0}}]\right)\setminus B_{r}(0),$
$\displaystyle\tilde{\mathcal{M}}(3):=\mathcal{M}(3)\setminus B_{r}(0).$

Outside of $B_{R_{0}}$, nothing changes. The value of $r$ is chosen so that
the area constraint $|\tilde{\mathcal{M}}(1)|=1$ is satisfied. Inside
$B_{R_{0}}$, perimeter is decreased by gluing $\mathcal{M}(1)\cap B_{R_{0}}$
onto $\mathcal{M}(2)$, as some interfaces will disappear. On the other hand,
new interfaces may be created along $\partial B_{R_{0}}$, but by (17) the
added perimeter is bounded by one. We may thus calculate the change in
perimeter of the proper chamber: by (16), (17), and the choice of $r$,

$\displaystyle\mathrm{Per}_{B_{R_{0}}}\left(\mathcal{M}(1)\right)-\mathrm{Per}_{B_{R_{0}}}\left(\tilde{\mathcal{M}}(1)\right)$
$\displaystyle\geq
3\sqrt{\pi}-\mathrm{Per}_{B_{R_{0}}}(B_{r})-\operatorname{\mathcal{H}}^{1}\left(\mathcal{M}(1)\cap\partial
B_{R_{0}}\right)$ $\displaystyle\geq
3\sqrt{\pi}-2\sqrt{\pi}-1=\sqrt{\pi}-1>0.$

We conclude that the perimeter
$P(\tilde{\mathcal{M}};B_{R_{0}})<P(\mathcal{M};B_{R_{0}})$, which contradicts
the local minimality of $\mathcal{M}$. Hence
$\mathrm{Per}_{\mathbb{R}^{2}}(\mathcal{M}(1))<\infty$. ∎

We now prove the uniqueness statement via a series of claims.

  1. 1.

Each interface $\mathcal{M}(j,k)$ is an analytic arc of constant curvature,
and so either a straight line or an arc of a circle. This follows from Almgren
[5] (see also p. 391 or Remark 30.4 in [12], and Taylor [19] for the three-
dimensional case.) Note that Almgren and Maggi only consider the case of a
single improper chamber, but the regularity theory holds locally along the
interfaces and so the proofs are identical.

Since the chambers of $\mathcal{M}$ have piecewise smooth boundary, we may
assume that each is an open set, and speak of its connected components (as
opposed to indecomposable measurable sets, the proper notion of connectivity
for sets of locally finite perimeter.)

  2. 2.

The transitions $\mathcal{M}(2,3)$, between phase domains $\mathcal{M}(2)$ and
$\mathcal{M}(3)$ are straight line segments, rays, or full lines. As there is
no mass constraint for $\mathcal{M}(2),\mathcal{M}(3)$, there is no Lagrange
multiplier in the Euler-Lagrange equation, and so minimizers must have zero
curvature at points of $\mathcal{M}(2,3)$.

  3. 3.

The transitions $\mathcal{M}(1,2),\mathcal{M}(1,3)$ between the finite area
component $\mathcal{M}(1)$ and the infinite area components
$\mathcal{M}(2),\mathcal{M}(3)$ are circular arcs, each of the same constant
curvature independent of the connected component of $\mathcal{M}(1)$ or the
neighboring phase $\mathcal{M}(2),\mathcal{M}(3)$. This follows from the
regularity theory, and the calculation of the Euler-Lagrange equation in the
area constrained case, as in [12, Theorem 17.20]. The choice of Lagrange
multiplier should be the same at any regular point in
$\partial\mathcal{M}(1)$, and so is independent of the connected component or
of the choice of interface $\mathcal{M}(1,3),\mathcal{M}(1,2)$.

  4. 4.

Singular points of $\mathcal{M}(j,k)$ occur when analytic arcs meet; these can
only occur at triple junction points, that is, points at which all three
phases meet, and the tangents to the arcs form 120 degree angle junctions.
Again, this follows from the regularity theory, and the classification of
planar cone clusters [12, Theorem 30.7]. As an immediate consequence, we
observe that the lines forming the transitions $\mathcal{M}(2,3)$ cannot
cross, and can only terminate on components of the finite area domain
$\mathcal{M}(1)$.

  5. 5.

The infinite area components $\mathcal{M}(2),\mathcal{M}(3)$ need not be
connected, but they cannot have bounded connected components. Assume that
(say) $\mathcal{M}(2)$ has a connected component $\Omega$ which is bounded.
Suppose $\partial\Omega$ has nontrivial intersection with
$\partial\mathcal{M}(3)$. In that case, we create a new cluster,
$\tilde{\mathcal{M}}=(\tilde{\mathcal{M}}(1),\tilde{\mathcal{M}}(2),\tilde{\mathcal{M}}(3))$
which is identical to $\mathcal{M}$ except that $\Omega$ is assigned to
$\tilde{\mathcal{M}}(3)$, that is,
$\tilde{\mathcal{M}}(3)=\mathcal{M}(3)\cup\Omega$,
$\tilde{\mathcal{M}}(2)=\mathcal{M}(2)\setminus\overline{\Omega}$, and
$\tilde{\mathcal{M}}(1)=\mathcal{M}(1)$. Then, $\tilde{\mathcal{M}}$ is an
admissible cluster which agrees with $\mathcal{M}$ outside of a compact set
(containing $\Omega$), but has smaller total perimeter, as a component of
$\mathcal{M}(2,3)$ is erased by the modification. This contradicts local
minimality of $\mathcal{M}$, and so we obtain the desired conclusion. If
instead the boundary $\partial\Omega$ is disjoint from
$\partial\mathcal{M}(3)$, then the component $\Omega$ must be contained inside
a connected component $\tilde{\Omega}$ of $\mathcal{M}(1)$. If there were such
an island, by translating it inside $\tilde{\Omega}$ until it contacts the
outer boundary of $\tilde{\Omega}$ we create a competing configuration with
the same area of $\mathcal{M}(1)$, which agrees with $\mathcal{M}$ outside of
a compact set, and with the same perimeter. However, the new configuration has
a singular point where the phases are tangent, violating the 120 degree
condition at junctions.

  6. 6.

Each connected component of $\mathcal{M}(1)$ has exactly two triple junction
points on its boundary.  That is, for any connected component $\Omega$ of
$\mathcal{M}(1)$, $\partial\Omega$ consists of two disjoint arcs of circles
(of the same curvature, for each arc and each connected component), one from
$\mathcal{M}(1,2)$ and one from $\mathcal{M}(1,3)$.

The proof of this claim is subtle, and will be done in several steps. First,
any connected component $\Omega$ of $\mathcal{M}(1)$ must be bounded. This
follows from our definition of improper cluster, in particular that
$\mathcal{M}(1)$ is a set of (globally) finite perimeter. In the plane, the
perimeter controls the diameter of a connected component, which must therefore
by finite.

Next, any connected component $\Omega$ of $\mathcal{M}(1)$ must contact both
of $\mathcal{M}(2),\mathcal{M}(3)$ on its boundary. First, note that by the
previous paragraph $\Omega$ cannot be multiply connected, with an island of
either $\mathcal{M}(2)$ or $\mathcal{M}(3)$ inside. Next suppose that
$\mathcal{M}$ is a local minimizer with a component $\Omega$ whose boundary is
disjoint from (say) $\mathcal{M}(3)$. By translating $\Omega$ inside
$\mathcal{M}(2)$ until it is tangent to a regular point of $\mathcal{E}(2,3)$
we create an admissible variation of $\mathcal{M}$ with the same area
constraint and the same perimeter, but with a singular point which violates
the 120 degree angle condition at junctions between the three phases. Hence it
is impossible to have a connected component of $\mathcal{M}(1)$ without triple
junctions. As the outer boundary of $\Omega$ is a closed curve, for each
transition of $\mathcal{M}(2)$ to $\mathcal{M}(3)$ on the curve there must be
a second transition with the opposite orientation, and so at least a pair of
junctions on $\partial\Omega$.

Now suppose there are $2n$ transitions on $\partial\Omega$; we will show that
necessarily $n=1$. Consider a large circle $C_{R}$ of radius $R$ which
contains the component $\Omega$ inside. Since each connected component of
$\mathcal{M}(2)$, $\mathcal{M}(3)$ is unbounded, by choosing $R$ sufficiently
large $C_{R}$ must intersect both unbounded sets, and so there must be at
least two transitions between the unbounded phases on $C_{R}$. We claim that
the number of arcs in $C_{R}\cap\mathcal{M}(2)$ and $C_{R}\cap\mathcal{M}(3)$
is also at least $n$ in each case. In case $n=1$ there is nothing to prove, so
assume $n\geq 2$. Take two connected arcs in
$\partial\Omega\cap\mathcal{M}(1,2)$, and assume (for a contradiction) that
they determine the same connected component $\Sigma$ of
$\mathcal{M}(2)\cap\overline{B_{R}}\setminus\Omega$. For any points $x,y$, one
in each of the distinct connected arcs of
$\partial\Omega\cup\mathcal{M}(1,2)$, we may then connect them with a simple
path inside $\Sigma$. Then, that path unioned with the arc along
$\partial\Omega$ connecting $x,y$, defines a bounded domain which must enclose
a (connected) component of $\mathcal{M}(3)$. But this contradicts our Step 5,
and so distinct connected arcs on $\mathcal{M}(2)\cap\partial\Omega$ define
distinct connected (and unbounded) components of
$\mathcal{M}(2)\cap\overline{B_{R}}\setminus\Omega$. Thus the number of
connected component arcs in $C_{R}\cap\mathcal{M}(2)$ must be at least $n$
(and similarly for $C_{R}\cap\mathcal{M}(3)$.)

Finally, to arrive at a contradiction we suppose $n\geq 2$ and make a
construction suggested to us by F. Morgan. Fix a circle $C_{r}$ which
circumscribes $\partial\Omega$. By the previous paragraph, there are at least
$n$ components of each of $\mathcal{M}(2)$, $\mathcal{M}(3)$ cut by $C_{R}$.
The endpoints of these arcs lie on the boundaries $\partial\mathcal{M}(2)$,
$\partial\mathcal{M}(3)$, and each may be connected to distinct triple
junction points on $\partial\Omega$ by distinct curves forming part of the
respective boundaries. The total length of these connecting curves inside
$B_{R}\setminus\Omega$ is thus at least $2n(R-r)$. Now, consider the following
admissible modification of $\mathcal{M}$: along $C_{R}$, one of
$\mathcal{M}(2)\cap C_{R}$ and $\mathcal{M}(3)\cap C_{R}$ can occupy at most
half, $L_{R}\leq\pi R$, of the circumference. Suppose it is $\mathcal{M}(3)$.
We then create a new configuration by excising the region $\mathcal{M}(3)\cap
B_{R}$ and reassigning this set to $\mathcal{M}(2)$. In doing so, we have
introduced a new transition between $\mathcal{M}(3)$ and $\mathcal{M}(2)$
along $C_{R}$, which adds $L_{R}$ to the total perimeter, but effectively
removed the transitions between $\mathcal{M}(3)$ and $\mathcal{M}(2)$ in the
region $B_{R}\setminus\Omega$, saving at least $2n(R-r)$ in perimeter. (See
Figure 5 for an example.) The new configuration $\tilde{\mathcal{M}}$ agrees
with $\mathcal{M}$ outside of $C_{R}$, and has perimeter,

$P(\tilde{\mathcal{M}};B_{R})-P(\mathcal{M};B_{R})\leq\pi R+2\pi
r-2n(R-r)=(\pi-2n)R+(2\pi+2n)r<0,$

choosing $R$ sufficiently large, when $n\geq 2$. This again contradicts the
local minimality of $\mathcal{M}$, and so we must have $n=1$, and each
connected component $\Omega$ of $\mathcal{M}(1)$ has exactly one pair of
triple junctions on $\partial\Omega$.

Figure 5:  $\Omega$ meets $2n$ alternating unbounded components of
$\mathcal{M}(2)$ and $\mathcal{M}(3)$, and is enclosed by circles $C_{R}$ and
$C_{r}$. Reassigning the regions in $\mathcal{M}(3)\cap B_{R}$ to
$\mathcal{M}(2)$ introduces transitions along $C_{R}$ but eliminates at least
$2n(R-r)$ of perimeter inside $B_{R}$.

  7. 7.

Each connected component of $\mathcal{M}(1)$ is isomorphic to a single lens
shape (each with the same area). From the previous step, we know that a
component $\Omega$ of $\mathcal{M}$ is simply connected and has exactly one
pair of triple junctions, $q_{1},q_{2}$. Each arc in $\partial\Omega$ is a
circle of the same radius $s$; call $p_{1},p_{2}$ the centers of these
circles, and draw the rhombus connecting $p_{1},q_{1},p_{2},q_{2}$ (each side
of length $s$). As the triple points meet with tangents of 120 degrees, the
radii (which are normal to the arcs at the triple points) also meet at an
angle of 120 degrees. Hence the opposite angles to the rhombus at
$p_{1},p_{2}$ each measure 60 degrees, and the rhombus decomposes as two
equilateral triangles, via the diagonal of length $s$ connecting
$q_{1},q_{2}$. The circular arcs then describe a lens, with area uniquely
determined by the radius $s$, and so each connected component of
$\mathcal{M}(1)$ is congruent.

  8. 8.

There are a finite number of connected components of $\mathcal{M}(1)$, and it
is a bounded set. This follows directly from the finite area constraint and
the previous step.

  9. 9.

Outside of a sufficiently large disk $B_{R}$, $\mathcal{M}(2)$ and
$\mathcal{M}(3)$ coincide with complementary half-planes, and $\mathcal{M}(1)$
is symmetric with respect to reflection in the separating plane. Fix any one
of the connected components $\Omega$ of $\mathcal{M}(1)$. By Step 7 we may
choose axes for which $\Omega$ is centered at the origin, and whose triple
points lie on the $x$-axis. By the 120 degree angle condition, a part of the
transition set $\mathcal{M}(2,3)$ must lie along the $x$-axis outside of
$\Omega$. Either these are rays extending out along the $x$-axis to infinity,
or else they terminate at some other connected component of $\mathcal{M}(1)$.
In the latter case, those other components of $\mathcal{M}(1)$ are again
congruent to a standard lens shape, and by the 120 degree angle condition
those additional lenses must connect to $\mathcal{M}(2,3)$ as segments or rays
along the $x$-axis. Since there are only finitely many connected components of
$\mathcal{M}(1)$ this process must terminate, and there is a first and last
component of $\mathcal{M}(1)$ lying along the $x$-axis, with a component of
$\mathcal{E}(2,3)$ lying on opposing rays along the axis. We claim that this
exhausts all connected components of $\mathcal{M}(1)$, and so the whole set
$\mathcal{M}(1)$ lies along the $x$-axis, and is symmetric with respect to
reflection in the axis. Indeed, if there were another connected component
$\tilde{\Omega}$ of $\mathcal{M}(1)$ which did not lie on the axis, we would
repeat the above procedure to reveal a distinct second line in the plane on
which $\tilde{\Omega}$ is centered. Other components could lie on this second
line, but being finite in number there will be a first and last one along the
line, and another component of $\mathcal{M}(2,3)$ would exist as a pair of
collinear rays, distinct from the $x$-axis. However, this contradicts Step 5,
which states that on any sufficiently large circle there can be only a single
pair of transition points between $\mathcal{M}(2)$ and $\mathcal{M}(3)$. In
conclusion, there can be no other rays of $\mathcal{M}(2,3)$ other than along
the $x$-axis, and thus outside of a large disk $\mathcal{M}(2)$ and
$\mathcal{M}(3)$ coincide with the upper and lower half-planes, and all
connected components of $\mathcal{M}(1)$ must be symmetrically placed along
the $x$-axis.

  10. 10.

$\mathcal{M}(1)$ consists of a single unit lens, centered on the $x$-axis. If
there were several connected components in $\mathcal{M}(1)$, they would be
aligned on the $x$-axis, and so each could be translated horizontally without
changing the total area or perimeter, until they were in contact at their
junction points. This modified configuration coincides with $\mathcal{M}$
outside of a ball, and it therefore also a local minimizer. However, the
contact at the junction points would then create singularities which
contradict the 120 degree angle condition imposed by regularity theory, and so
having multiple components in $\mathcal{M}$ is not possible. By Step 7, the
single component is a unit lens.

This proves uniqueness. ∎

##  4 Some conjectures

Here we collect some known results and conjectures concerning minimizing
planar $(N,M)$-clusters; see Table 1 for a summary.

$\begin{array}[]{c|cccc}\leavevmode\hbox to21.87pt{\vbox
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.2pt}{1.2pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{N}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{ {}}{}{
{}}{}}{}{{}}{}{{ {}}{}{ {}}{}}
{}{}{}\pgfsys@moveto{-11.56668pt}{9.2333pt}\pgfsys@lineto{9.9pt}{-9.2333pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&0&1&2&3\\\ \hline\cr
1&\mathbb{R}^{2}&\text{Disc}&\text{Std.\ $2$-bubble}&\text{Std.\
$3$-bubble}\\\ 2&\text{Two half-
planes}^{{\dagger}}&\text{Lens}&\text{Peanut}^{*}&\\\ 3&\text{Steiner
partition}^{{\dagger}}&\text{Tailor's chalk}^{*}&&\end{array}$ Table 1: Known
and conjectured minimizing $(N,M)$-clusters in $\mathbb{R}^{2}$, for $N,M\leq
3$. († indicates minimality among cone-like improper $M$-clusters; ∗ indicates
conjectured minimal clusters.)

The case $N=0$ of improper clusters is discussed in [12, Proposition 30.9]:
among cone-like clusters, a pair of complementary half-planes is minimizing
for $(0,2)$-clusters, and the Steiner partition (three rays meeting at 120
degree angles) is the unique $(0,3)$-minimizer. The case $M=1$ (a single
improper chamber) reduces to the $N$-bubble problem. Results in the plane for
$N=2$ are by Foisy et al. [7], and by Wichiramala [20] for $N=3$. Simpler
alternative proofs using calibrations were more recently given by Lawlor in
[10] and [11], for double and triple bubbles respectively. Resolution of the
case of $N\leq\min\\{5,n+1\\}$ bubbles in $\mathbb{R}^{n}$ was proven by
Milman and Neeman [13], who have additional results on quintuple-bubbles in
higher dimensions [14].

With the exception of the $(1,2)$ lens cluster, the cases with $M\geq 2$
remain open. We conjecture that the minimizer among the $(2,2)$-clusters with
area vector $(1,1,\infty,\infty)$ is the peanut shape illustrated in Figure 6;
and among $(1,3)$-clusters with area vector $(1,\infty,\infty,\infty)$, we
believe the unique minimizer should be the “tailor’s chalk” configuration, as
in Figure 7.

Figure 6: The peanut cluster in $\mathbb{R}^{2}$ Figure 7: The tailor’s chalk
cluster in $\mathbb{R}^{2}$

##  5 Acknowledgements

We would like to thank Frank Morgan, who suggested the idea for the proof of
the main result, and from whom we enjoyed insightful correspondence. Thank you
to Heather Lowe, whose expertise as a sewist led her to suggest the name of
the tailor’s chalk cluster. SA and LB were supported by NSERC (Canada)
Discovery Grants. SV was supported by an NSERC Canada Graduate Scholarship
(Master’s) and an Ontario Graduate Scholarship.

## References

  * [1] Alama, S., Bronsard, L., Lu, X., and Wang, C. Periodic minimizers of a ternary non-local isoperimetric problem.  Indiana Univ. Math. J. 70 (2021), 2557–2601.
  * [2] Alama, S., Bronsard, L., Lu, X., and Wang, C. Decorated phases in a model of tri-block co-polymers.  In preparation.
  * [3] Alama, S., Bronsard, L., Lu, X., and Wang, C. On a quaternary nonlocal isoperimetric problem.  Quarterly of Applied Math. (2023).  To appear.
  * [4] Alama, S., Bronsard, L., and Vriend, S. On a free-endpoint isoperimetric problem in $\mathbb{R}^{2}$.  arXiv preprint arXiv:2304.10531 (2023).
  * [5] Almgren, Jr., F. J. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints.  Mem. Amer. Math. Soc. 4, 165 (1976), viii+199.
  * [6] Bellettini, G., and Novaga, M. Curvature evolution of nonconvex lens-shaped domains.  Journal für die reine und angewandte Mathematik 2011, 656 (2011).
  * [7] Foisy, J., Garcia, M. A., Brock, J. F., Hodges, N., and Zimba, J. The standard double soap bubble in $\mathbb{R}^{2}$ uniquely minimizes perimeter.  Pacific Journal of Mathematics 159, 1 (1993), 47 – 59.
  * [8] Hutchings, M., Morgan, F., Ritore, M., and Ros, A. Proof of the double bubble conjecture.  Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 45–49.
  * [9] Isenberg, C. The science of soap films and soap bubbles. Tieto Ltd., Clevedon,,, 1978.  With a foreword by George Porter.
  * [10] Lawlor, G. R. Double bubbles for immiscible fluids in $\mathbb{R}^{n}$.  J. Geom. Anal. 24, 1 (2014), 190–204.
  * [11] Lawlor, G. R. Perimeter-minimizing triple bubbles in the plane and the 2-sphere.  Analysis and Geometry in Metric Spaces 7, 1 (2019), 45–61.
  * [12] Maggi, F. Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory.  No. 135 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2012.
  * [13] Milman, E., and Neeman, J. The structure of isoperimetric bubbles on $\mathbb{R}^{n}$ and $\mathbb{S}^{n}$, 2022.
  * [14] Milman, E., and Neeman, J. Plateau bubbles and the quintuple bubble theorem on $\mathbb{S}^{n}$, 2023\.
  * [15] Morgan, F. Soap bubbles in $\mathbb{R}^{2}$ and in surfaces.  Pacific Journal of Mathematics 165, 2 (1994), 347–361.
  * [16] Reichardt, B. W. Proof of the double bubble conjecture in $\mathbb{R}^{n}$.  J. Geom. Anal. 18, 1 (2008), 172–191.
  * [17] Ren, X., and Wei, J. Asymmetric and symmetric double bubbles in a ternary inhibitory system.  SIAM J. Math. Anal. 46, 4 (2014), 2798–2852.
  * [18] Schnürer, O., Azouani, A., Georgi, M., Hell, J., Jangle, N., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., and Smith, B. Evolution of convex lens-shaped networks under the curve shortening flow.  Transactions of the American Mathematical Society 363, 5 (2011), 2265–2294.
  * [19] Taylor, J. E. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces.  Ann. of Math. (2) 103, 3 (1976), 489–539.
  * [20] Wichiramala, W. Proof of the planar triple bubble conjecture.  Journal für die reine und angewandte Mathematik 2004, 567 (2004).


# Broadband Brillouin scattering study of ferroelectric instability of barium
sodium niobate

S. Kojima e-mail<EMAIL_ADDRESS>
(Received July 18, 2022)

###### Abstract

The barium sodium niobate (BNN) with tungsten-bronze structure is one of well-
known optical crystals for electro-optic and nonlinear optic applications.
This paper reviews the ferroelectric instability of BNN crystals. BNN is a
uniaxial ferroelectric with a spontaneous polarization along the tetragonal
$c$-axis. There is no report on the observation of an optical soft mode
responsible for a ferroelectric phase transition. In the vicinity of the Curie
temperature, $T_{\text{C}}=560$℃, an intense central peak (CP) related to the
polarization fluctuations along the $c$-axis was observed by the broadband
Brillouin scattering experiment. The relaxation time determined by the CP
width shows the critical slowing down towards $T_{\text{C}}$. This fact
indicates that the ferroelectric instability of BNN is an order-disorder type.

Key words: Brillouin scattering, ferroelectric, order-disorder, central peak,
barium sodium niobate

###### Abstract

Íîáàò áàðþ-íàòðþ (BNN) ç ñòðóêòóðîþ âîëüôðàìîâî¿ áðîíçè îäíèì ç äîáðå âäîìèõ
îïòèíèõ êðèñòàëâ, ÿê âèêîðèñòîâóþòüñÿ äëÿ åëåêòðîîïòèíèõ äîñëäæåíü òà ó
íåëéíéíé îïòèö. Ó äàíé ðîáîò ðîçãëÿäàòüñÿ ñåãíåòîåëåêòðèíà íåñòéêñòü â
êðèñòàëàõ BNN. BNN îäíîâñíèì ñåãíåòîåëåêòðèêîì, â ÿêîìó ñïîíòàííà ïîëÿðèçàöÿ
íàïðÿìëåíà âçäîâæ òåòðàãîíàëüíî¿ îñ $c$. Ó ëòåðàòóð íåìà çãàäîê ïðî
ñïîñòåðåæåííÿ îïòèíî¿ ì’ÿêî¿ ìîäè, âäïîâäàëüíî¿ çà ñåãíåòîåëåêòðèíèé ôàçîâèé
ïåðåõä ó öüîìó êðèñòàë. Â îêîë òåìïåðàòóðè Êþð $T_{\text{C}}=560$℃ â ñïåêòðàõ
øèðîêîñìóãîâîãî ðîçñþâàííÿ Áðëëþåíà ñïîñòåðãàòüñÿ íòåíñèâíèé öåíòðàëüíèé ïê,
ïîâ’ÿçàíèé ç ôëóêòóàöÿìè ïîëÿðèçàö¿ óäîâæ îñ $c$. àñ ðåëàêñàö¿, ÿêèé
âèçíààòüñÿ øèðèíîþ öåíòðàëüíîãî ïêà, âèÿâëÿ êðèòèíå ñïîâëüíåííÿ ïðè íàáëèæåíí
äî $T_{\text{C}}$. Öåé ôàêò ñâäèòü ïðî òå, ùî ñåãíåòîåëåêòðèíà íåñòéêñòü ó
BNN-ñïîëóêàõ òèïó ‘‘ëàä-áåçëàä’’.

Ключов слова: ðîçñþâàííÿ Áðëëþåíà, ñåãíåòîåëåêòðèê, ëàä-áåçëàä, öåíòðàëüíèé
ïê, íîáàò áàðþ-íàòðþ

##  1 Introduction

Ferroelectricity is defined by the existence of switchable spontaneous
polarization by an external electric field. Ferroelectric phenomenon was
identified for the first time in 1920 by Valasek on the study of Rochelle salt
[1]. The microscopic origin of ferroelectricity has two typical cases, namely,
the displacive type and order-disorder type. In the displacive type, an
infrared active soft optic mode exists in a paraelectric phase, and the
freezing of a soft mode displacement induces a spontaneous polarization [2,
3]. The softening of a soft mode frequency has been observed by far-infrared
spectroscopy, Raman scattering, and neutron inelastic scattering. On the other
hand, in the order-disorder type, the relaxation time of the polarization
fluctuations of polar molecules along a ferroelectric axis diverges at the
Curie temperature, and the aligned polar molecules induce a spontaneous
polarization. Rochelle salt belongs to this type. The critical slowing down
towards a Curie temperature has been observed by dielectric spectroscopy. In
NaNO2, Hatta observed the divergence of the relaxation time of the flipping
motion of each NO2 ion toward a Curie temperature due to the thermodynamical
slowing down process of the correlated fluctuation of polarization [4]. The
critical slowing-down of the polarization relaxation process was also observed
in triglycine sulfate [5] and Ca2Sr(C2H5CO${}_{2})_{6}$ above the Curie
temperature [6]. Another method to observe the critical slowing down is the
low-frequency inelastic light scattering. The polarization fluctuations along
a ferroelectric axis are observed as a central peak (CP). In the vicinity of a
Curie temperature, the divergence of CP intensity and the narrowing of CP
width are observed for an order-disorder phase transition. In a
K(Ta0.68Nb${}_{0.32})$O3 crystal, the relaxation time determined by the CP
width clearly shows a critical slowing down towards the Curie temperature,
$T_{\text{C}}=258$ K, indicating an order-disorder feature of the
ferroelectric phase transition [7, 8]. Up to the present, the critical slowing
down has been studied by the observation of a CP in ferroelectric phase
transitions of 12 mol% KF substituted BaTiO3 [9], LiTaO3 [10], KNN [11],
MAPbCl3 [12], BaTi2O5 [13], and K2MgWO2(PO${}_{4})_{2}$ [14].

For the ferroelectric phase transitions of relaxor ferroelectrics, the
diffusive nature was observed in the critical slowing down [15]. In
0.70Pb(Zn1/3Nb${}_{2/3})$O3–0.30PbTiO3 (PZN–7PT), the slowing down was
suppressed below the intermediate temperature $T^{*}$ and the typical critical
slowing down was not observed near $T_{\text{C}}$. The local transition from
dynamic to static PNRs at $T^{*}$ stops the farther slowing down. To describe
such a suppressed slowing down by random fields, the empirical equation of the
stretched slowing down was used in the vicinity of $T_{\text{C}}$ as given by
the following equation,

$\frac{1}{\tau_{\text{CP}}}=\frac{1}{\tau_{0}}+\frac{1}{\tau_{1}}\left(\frac{T-T_{\text{C}}}{T_{\text{C}}}\right)^{\beta},\quad(T>T_{\text{C}}),$
(1.1)

where $\beta$ is the stretched index. In the case of $\beta=1.0$, the equation
(1.1) gives a critical slowing down of normal ferroelectrics without random
fields. In the case of $\beta>1.0$, the slowing down of relaxation time is
suppressed and/or stretched by an increase of the strength of random fields.
In PZN–7PT, it is found that the value of $\beta=3.0$ gives a good
reproduction of slowing down [16].

In some ferroelectrics, the mechanism of the ferroelectricity is not simple.
The coexistence of both mechanisms or the crossover from displacive to order-
disorder nature has been reported. In LiNbO3, two-stage process involving a
displacive transition in the Nb–O cages and an order-disorder transition in
the Li–O planes was reported at the Curie temperature [17]. The unified model
theory describing both the “order-disorder” and “displacive” ferroelectric
phase transitions was proposed by introducing the model pseudospin-phonon
Hamiltonian [18]. For such phase transitions, the study of the lowest
frequency soft optic modes by Raman scattering or infrared spectroscopy is
also necessary.

##  2 Ferroelectrics with tungsten-bronze structure

Ferroelectricity has been observed in various kinds of organic and inorganic
materials. Regarding inorganic ferroelectrics, the oxygen octahedra
ferroelectrics are the most popular. One of this family is the ferroelectrics
with tungsten-bronze structure. It is technologically important in the field
of telecommunications due to its superior electro-optical, photorefractive,
and nonlinear optical properties such as second harmonic generation, and its
resistance to optical damage is high. The ferroelectricity was reported at
first in lead metaniobate, PbNb2O6 with $T_{\text{C}}=570$℃ [19]. Its
piezoelectric constant is the same order of magnitude as that of barium
titanate, BaTiO3 with $T_{\text{C}}=120$℃. The useful nonlinear coefficients
and low optical damage were reported in barium sodium niobate, Ba2NaNb5O15
(BNN), with $T_{\text{C}}=560$℃ [20]. Nowadays, a lot of tungsten-bronze type
ferroelectrics are known [2].

Figure 1: (Colour online) Projection of tungsten bronze structure on the
$c$-plane. Tetragonal and orthorhombic unit cells are shown by dotted and
solid lines, respectively.

Figure 1 shows the projection of tungsten bronze structure on the $c$-plane.
In BNN, the A1 and A2 sites are fully filled by Na and Ba, respectively, and
there is no vacancy. The B1 and B2 sites are fully occupied by Nb, while the C
sites are vacant. BNN is called a filled tungsten-bronze structure with no
charge disorder at A1 and A2 sites. BNN undergoes successive phase transitions
at 560, 300, and $-163$℃ [21]. The higher one is associated with the
ferroelectric Curie temperature at 560℃, and its crystal symmetry changes from
prototypic $4/mmm$ to ferroelectric tetragonal $4mm$ with a spontaneous
polarization $P_{3}$ along the $c$-axis. The symmetry changes from tetragonal
to incommensurate (IC) orthorhombic $mmm$ systems at 300℃, the $a$ and $b$
axes are rotated for 45∘ along the $c$-axis as shown in figure 1. The
modulation direction of the IC wave vectors is along $a$ and $b$ axes of the
orthorhombic coordinate [22]. The lowest temperature phase transition at
$-163$℃ is the reentrant ferroelastic phase transition into the tetragonal
$4mm$ phase. The pressure induced reentrant ferroelastic phase transition was
also observed at 2.2 GPa and at room temperature [23, 24]. Recently, a new
type of the IC phase transition was proposed by Ishibashi [25]. This new type
of phase transition is referred to as type III, and it is characterized by the
parabolic splitting of the doubly degenerate modes at the Brillouin zone
boundary. The related macroscopic change in the IC phase transition was
studied by Brillouin scattering [26]. Regarding a ferroelectric instability,
the accurate measurement of low-frequency polaritons was performed on the
optical phonon branch of $\mathrm{A}_{1}(z)$ symmetry. However, any evidence
of displacive nature was not found down to 18 cm-1 [27].

Since the $T_{\text{C}}$ of Ba2NaTa5O15 (BNT) is $-233$℃, the high tunability
of the $T_{\text{C}}$ of the temperature width of about 800℃ was reported for
Ba2NaNb5(1-x)Ta5xO15 (BNNT), and this is technologically important [28, 29].
The ferroelectric phase transition of the BNNT single crystals with $x=0.57$
at $T_{\text{C}}=115$℃ was studied by broadband dielectric spectroscopy up to
4 GHz. The order-disorder nature of the proper ferroelectric phase transition
was observed, and its origin is attributed to the anharmonic motion of the Nb
(Ta) atoms in a double well potential of oxygen octahedra [30].

Up to the present, no soft optic mode was observed in the tungsten-bronze
ferroelectrics by vibrational spectroscopy. Recent theoretical studies
reported the local pseudo-Jahn–Teller effect (PJTE) in transition metal B ion
center of ABO3 perovskite crystals. The vibronic coupling between the ground
and excited electronic states of the local BO6 center results in dipolar
distortions, leading to an eight-well adiabatic potential energy surface [31].
Such a situation may also occur in tungsten bronze ferroelectrics and the
order-disorder nature of ferroelectricity may exist. Therefore, the order-
disorder nature of a ferroelectric phase transition of BNN has been examined
by the broadband Brillouin scattering spectroscopy, which is a powerful tool
to observe a critical slowing down in the vicinity of an order-disorder type
phase transition temperature. In this paper, we review the broadband Brillouin
scattering study on the ferroelectric instability of a BNN crystal [32].

##  3 Broadband Brillouin scattering and ferroelectric instability

Vibrational spectroscopy i.e., infrared spectroscopy and Raman scattering
observes the vibrational modes of atoms, molecules, and crystal lattice. In
the vibrational study of inorganic ferroelectric crystals, it is possible to
observe not only the internal modes of octahedra or tetrahedra but also the
external modes such as a soft optic mode. The spectral resolution of
vibrational spectroscopy is usually 1 cm-1${}=30$ GHz or more, and it is
sufficient to detect the change of mode frequency related to a phase
transition. The resolution of 1 cm-1 is sufficient to measure the temperature
dependence of a ferroelectric soft optic mode. However, in the study of a
ferroelectric phase transition of order-disorder type, the resolution of 1
cm-1 is not sufficient to measure the critical slowing down of the relaxation
time towards the Curie temperature [33].

Polarization fluctuations related to a ferroelectric instability are detected
as a broad CP in an inelastic scattering spectrum. The colorless and
transparent BNN crystal studied was grown by Czochralski method in Tamagawa
factory, NEC, Japan. The (100) plate with the size of $2.65\times 2.25\times
0.68$ mm with two optically polished surfaces was used for Brillouin
scattering measurements. Brillouin scattering spectra were measured at the
backward scattering geometry using a $3+3$ tandem multi-pass Fabry–Perot
interferometer and a conventional photon counting system. A single frequency
green YAG laser ($\lambda=532$ nm) with power of 100 mW was used as an
exciting source. The light spot size at a sample surface was about 10 $\mu$m
using the optical microscope (BX-60) [33]. The temperature of a sample was
controlled by the heating stage of a T1500 (high T Linkam) from room
temperature up to 750℃. All the Brillouin scattering spectra were measured in
the condition that the free spectral range (FSR) and the scan range are 300
and 600 GHz, respectively. An intense polarized CP of BNN was observed in the
vicinity of the Curie temperature, $T_{\text{C}}=560$℃, in the broadband
Brillouin scattering spectra as shown in figure 2 [34]. In the polarized VV
spectrum observed at $a(cc)\bar{a}$, backward scattering geometry shows an
intense broad CP with $\mathrm{A}_{1}(z)$ symmetry, while in the depolarized
VH spectrum observed at $a(cb)\bar{a}$, backward scattering geometry does not
show an intense CP with $\mathrm{B}_{2}$ symmetry. Therefore, the polarization
fluctuations along a ferroelectric $c$-axis are the origin of an intense broad
CP.

Figure 2: (Colour online) Broadband VV and VH Brillouin scattering spectra of
a BNN crystal at 600℃ observed by $a(cc)\bar{a}$ and $a(cb)\bar{a}$ back
scattering geometry, respectively.

##  4 Critical slowing down on a ferroelectric phase transition of barium
sodium niobate

For the detailed analysis of the width of a CP, the temperature dependence of
broadband Brillouin scattering spectra of a BNN crystal was measured at the
backward scattering geometry with the free spectral range of 300 GHz as shown
in figure 3.

Figure 3: (Colour online) Broadband VV Brillouin scattering spectra of a BNN
crystal in a paraelectric phase.

Under the assumption of a single Debye relaxation process, the relaxation time
$\tau_{\text{CP}}$ was determined by the relation $\piup\times\text{(CP
width)}=\tau_{\text{CP}}^{-1}$. The relaxation process related to the order-
disorder nature of a ferroelectric phase transition has been observed as a CP
with a zero frequency shift in an inelastic scattering spectrum. In the order-
disorder phase transition, the relaxation time $\tau$ of the fluctuations of
the order parameters increases toward the phase transition point and was
called the critical slowing down. The relaxation time determined from the CP
width shows a critical slowing down in the vicinity of $T_{\text{C}}=560$℃ as
shown in figure 4 [34]. The temperature dependence of the relaxation time is
given by the following equation of the case of $\beta=1.0$ in equation (1.1)
for a first order phase transition:

$\frac{1}{\tau_{\text{CP}}}=\frac{1}{\tau_{0}}+\frac{1}{\tau_{1}}\left(\frac{T-T_{1}}{T_{1}}\right),\quad(T>T_{\text{C}}>T_{1}).$
(4.1)

For example, in the ferroelectric phase transition at $T_{\text{C}}=500$ K of
the relaxor ferroelectric 0.70Pb(Sc1/2Nb${}_{1/2})$O3–0.30PbTiO3 with the
perovskite structure, the values of the fitting parameters are $\tau_{0}=14$
ps and $\tau_{1}=0.47$ ps, and $T_{1}=500$ K [35]. The temperature dependences
of $T/I_{\text{CP}}$ of a BNN crystal are shown in figure 5. In BNN, the
fitting parameters of $1/\tau$ are $\tau_{0}=1.29$ ps, $\tau_{1}=0.73$ ps, and
$T_{1}=555$℃. The intensity of a CP $I_{\text{CP}}$ obeys the following
equation in a paraelectric phase [36]:

$\frac{T}{I_{\text{CP}}}\propto\left[\int_{0}^{\infty}{\frac{\chi^{\prime\prime}\left(\omega\right)}{\omega}\,\mathrm{d}\omega}\right]^{-1}\propto\frac{1}{\chi^{\prime}(0)}=\frac{T-T_{1}}{C},\quad(T>T_{\text{C}}>T_{1}).$
(4.2)

Here, for the first order phase transition, $T_{\text{C}}>$T1, because the
ferroelectric phase transition of BNN is the first order. In the ferroelectric
phase transition at $T_{\text{C}}=500$ K of the
0.70Pb(Sc1/2Nb${}_{1/2})$O3–0.30PbTiO3, the Curie–Weiss law also holds for
$I_{\text{CP}}/T$ above $T_{\text{C}}$ [35].

Figure 4: (Colour online) Temperature dependence of the inverse relaxation
time. The dotted line is the fitted line by the equation (4.1) above
$T_{\text{C}}=560$℃.

Figure 5: (Colour online) Temperature dependence of temperature divided by the
intensity of a central peak. The dotted line is the fitted line by the
equation (4.2) above $T_{\text{C}}=560$℃.

The experimental results of the critical slowing down of relaxation time and
the Curie–Weiss behavior of the CP intensity indicate the order-disorder
nature of a ferroelectric phase transition of BNN. In the study of the order-
disorder phase transition, Brillouin scattering is a powerful tool to detect
the critical slowing down [37].

##  5 Conclusions

For the study of the lattice instability of ferroelectrics, vibrational
spectroscopy is a powerful tool to discuss not only displacive but also order-
disorder nature. This paper reviews the experimental studies on the
ferroelectric instability of a ferroelectric phase transition of barium sodium
niobate (BNN) crystals with tungsten-bronze structure. BNN is one of well-
known optical crystals for electro-optic and nonlinear optic applications. It
shows a uniaxial ferroelectricity with a spontaneous polarization along the
tetragonal $c$-axis. In the vicinity of the Curie temperature,
$T_{\text{C}}=560$℃, an intense central peak (CP) was observed by the
broadband Brillouin scattering experiment. The CP has a strong polarization
dependence, which originates from the polarization fluctuations along the
ferroelectric $c$-axis. The CP intensity shows a maximum at $T_{\text{C}}$.
The relaxation time determined by the CP width shows a critical slowing down
towards $T_{\text{C}}$. The temperature dependence of the CP intensity shows
the Curie–Weiss behavior. These experimental results are the evidence of the
order-disorder nature of the ferroelectric instability of BNN.

## Acknowledgements

Author thanks to Prof. J. Grigas, Prof. J. Banys, Prof. M. Maczka for the
collaboration and S. Ohta, Y. Christy, K. Matsumoto, K. Suzuki, and M.
Aftebuzzamann for the discussion and experiments.

## Funding

This research was funded in part by JSPS KAKENHI, Grant No. JP17K05030.

## References

  * [1] Valasek J., Phys. Rev., 1921, 17, No. 4, 475–481, doi:10.1103/PhysRev.17.475.
  * [2] Xu Y., Ferroelectric Materials and Their Applications, North-Holland, Amsterdam, 1991.
  * [3] Blinc R., Advanced Ferroelectricity, Oxford University Press, New York, 2011.
  * [4] Yamada Y., Fujii Y., Hatta I., J. Phys. Soc. Jpn., 1968, 24, No. 5, 1053–1058, doi:10.1143/JPSJ.24.1053.
  * [5] Hill R. M., Ichiki S. K., Phys. Rev., 1962, 128, No. 3, 1140, doi:10.1103/PhysRev.128.1140.
  * [6] Nakamura E., Hosoya M., J. Phys. Soc. Jpn., 1967, 23, No. 4, 844–847, doi:10.1143/JPSJ.23.844.
  * [7] Ohta R., Zushi J., Ariizumi T., Kojima S., Appl. Phys. Lett., 2011, 98, No. 9, 092909, doi:10.1063/1.3560345.
  * [8] Rahaman M. M., Imai T., Miyazu J., Kobayashi J., Tsukada S., Helal M. A., Kojima S., J. Appl. Phys., 2014, 116, No. 7, 074110, doi:10.1063/1.4893363.
  * [9] Tsukada S., Hiraki Y., Akishige Y., Kojima S., Phys. Rev. B, 2009, 80, No. 1, 012102,
doi:10.1103/PhysRevB.80.012102.

  * [10] Hushur A., Gvasaliya S., Roessli B., Lushnikov S., Kojima S., Phys. Rev. B, 2007, 76, No. 11, 064104,
doi:10.1103/PhysRevB.76.064104.

  * [11] Kojima S., Tsukada S., Noguchi Y., Ferroelectrics, 2022, 586, No. 1, 2–9, doi:10.1080/00150193.2021.2014256.
  * [12] Hushur A., Shigematsu H., Akishige Y., Kojima S., Appl. Phys. Lett., 2005, 86, No. 11, 112903,
doi:10.1063/1.1880442.

  * [13] Lee J. W., Naqvi F. H., Ko J. H., Kim T. H., Ahn C. W., Materials, 2022, 15, No. 10, 3692,
doi:10.3390/ma15103692.

  * [14] Maczka M., Hanuza J., Majchrowski A., Kojima S., Appl. Phys. Lett., 2007, 90, No. 12, 122903,
doi:10.1063/1.2713873.

  * [15] Kojima S., Tsukada S., Ferroelectrics, 2010, 405, No. 1, 32–38, doi:10.1080/00150193.2010.482885.
  * [16] Tsukada S., Kojima S., Phys. Rev. B, 2008, 78, No. 14, 144106, doi:10.1103/PhysRevB.78.144106.
  * [17] Phillpot S. R., Gopalan V., Appl. Phys. Lett., 2004, 84, No. 11, 1916, doi:10.1063/1.1669063.
  * [18] Stamenković S., Plakida N. M., Aksienov V. L., Siklós T., Phys. Rev. B, 1976, 14, No. 11, 5080,
doi:10.1103/PhysRevB.14.5080.

  * [19] Goodman G., J. Am. Ceram. Soc., 1953, 36, No. 11, 368–372, doi:10.1111/j.1151-2916.1953.tb12820.x.
  * [20] Geusic J. E., Levinstein H. J., Rubin J. J., Singh S., Van Uitert L. G., Appl. Phys. Lett., 1967, 11, No. 9, 269,
doi:10.1063/1.1755129.

  * [21] Schneck J., Primot J., Von der Mühil R., Ravez J., Solid State Commun., 1977, 21, No. 1, 57–60,
doi:10.1016/0038-1098(77)91478-8.

  * [22] Schneck J., Denoyer F., Phys. Rev. B, 1981, 23, No. 1, 383, doi:10.1103/PhysRevB.23.383.
  * [23] Kojima S., Asaumi K., Nakamura T., Minomura S., J. Phys. Soc. Jpn., 1978, 45, 1433–1434,
doi:10.1143/JPSJ.45.1433.

  * [24] Kojima S., Nakamura T., Asaumi K., Takashige M., Minomura S., Solid State Commun., 1979, 29, No. 11, 779–783, doi:10.1016/0038-1098(79)90160-1.
  * [25] Ishibashi Y., Iwata M., Ferroelectrics, 2014, 461, No. 1, 1–9, doi:10.1080/00150193.2014.889521.
  * [26] Christy Y., Matsumoto K., Kojima S., Jpn. J. Appl. Phys., 2015, 54, 07HB02, doi:10.7567/JJAP.54.07HB02.
  * [27] Kojima S., Nakamura T., Jpn. J. Appl. Phys., 1980, 19, L609, doi:10.1143/JJAP.19.L609.
  * [28] Takashige M., Kojima S., Hamazaki S., Tsukioka F. S., Jpn. J. Appl. Phys., 1993, 32, 4384,
doi:10.1143/JJAP.32.4384.

  * [29] Kojima S., Hamazaki S., Tsukioka M., Takashige M., J. Phys. Soc. Jpn., 1993, 62, No. 4, 1097–1100,
doi:10.1143/JPSJ.62.1097.

  * [30] Mizaras R., Takashige M., Banys J., Kojima S., Grigas J., Hamazaki S., Brilingas A., J. Phys. Soc. Jpn., 1997, 66, No. 9, 2881–2885, doi:10.1143/JPSJ.66.2881.
  * [31] Bersuker I. B., Polinger V., Condens. Matter, 2020, 5, No. 4, 68, doi:10.3390/condmat5040068.
  * [32] Kojima S., Aftabuzzaman M., Dec J., Kleemann W., In: Proceedings of the Conference “IV Lithuanian–Ukrainian–Polish meeting on physics of ferroelectricity”, Palanga, Lithuania, 2016.
  * [33] Kojima S., Jpn. J. Appl. Phys., 2010, 49, 07HA01, doi:10.1143/JJAP.49.07HA01.
  * [34] Ota S., Matsumoto K., Suzuki K., Kojima S., IOP Conf. Ser.: Mater. Sci. Eng., 2014, 54, 012018,
doi:10.1088/1757-899X/54/1/012018.

  * [35] Kojima S., Tsukada S., Hidaka Y., Bokov A. A., Ye Z. G., J. Appl. Phys., 2011, 109, No. 8, 084114,
doi:10.1063/1.3581025.

  * [36] Hays W., Loudon R., Scattering of Light by Crystals, Dover Publishing, New York, 1978.
  * [37] Kojima S., Materials, 2022, 15, No. 10, 3518, doi:10.3390/ma15103518.

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