#  Core shells and double bubbles
in a weighted nonlocal isoperimetric problem

Stanley Alama Lia Bronsard Xinyang Lu Chong Wang Department of Mathematics and
Statistics, McMaster University. E-mail<EMAIL_ADDRESS>of
Mathematics and Statistics, McMaster University. E-mail:
<EMAIL_ADDRESS>of Mathematical Sciences, Lakehead University.
Email<EMAIL_ADDRESS>of Mathematics, Washington and Lee
University. Email<EMAIL_ADDRESS>
###### Abstract

We consider a sharp-interface model of $ABC$ triblock copolymers, for which
the surface tension $\sigma_{ij}$ across the interface separating phase $i$
from phase $j$ may depend on the components. We study global minimizers of the
associated ternary local isoperimetric problem in ${\mathbb{R}}^{2}$, and show
how the geometry of minimizers changes with the surface tensions
$\sigma_{ij}$, varying from symmetric double-bubbles for equal surface
tensions, through asymmetric double bubbles, to core shells as the values of
$\sigma_{ij}$ become more disparate. Then we consider the effect of nonlocal
interactions in a droplet scaling regime, in which vanishingly small particles
of two phases are distributed in a sea of the third phase. We are particularly
interested in a degenerate case of $\sigma_{ij}$ in which minimizers exhibit
core shell geometry, as this phase configuration is expected on physical
grounds in nonlocal ternary systems.

##  1 Introduction

In this paper we continue our study of ternary systems, in which three
constituents or phases interact through both short range attractive and long
range repulsive forces. A prominent example of such ternary systems are the
$ABC$ triblock copolymers, linear chains of molecules consisting of three
subchains, joined covalently to each other. A subchain of type $A$ monomer is
connected to one of type $B$, which in turn is connected to another subchain
of type $C$ monomer. Because of the repulsive forces between different types
of monomers, different types of subchain tend to segregate. However, since
subchains are chemically bonded in molecules, segregation can lead to a phase
separation only at microscopic level, where $A,B$ and $C$-rich micro-domains
emerge, forming morphological phases, many of which have been observed
experimentally: see Figure 1.

Figure 1: Cross-sectional TEM image of PMMA-b-PVCa-b-PSt triblock copolymer
thin flim. The inset image shows schematic illustration of Core-Shell
Cylindrical phase (blue: Pst, red: PVCa, yellow: PMMA) [31]. Reproduced from
the Royal Society of Chemistry 2018 (Link to this Open Access Article).

A triblock copolymer can be described as a stable critical point of an energy
derived with Nakazawa and Ohta's density functional theory for triblock
copolymers [17, 22]. This is a diffuse interface model, a nonlocal version of
the vector-valued Cahn-Hilliard energy. In this paper we consider periodic
configurations in two dimensions, so we pose our problem in the flat unit
torus ${\mathbb{T}^{2}}=\left[-\frac{1}{2},\frac{1}{2}\right]^{2}$. The system
is determined via a vector-valued density function $u=(u_{1},u_{2},u_{0})\in
L^{1}({\mathbb{T}^{2}},\mathbb{R}^{3})$, in which each scalar function
$u_{i},i=0,1,2$, gives the density of one constituent of the mixture. The free
energy of the system is

$\displaystyle\mathcal{E}^{\epsilon}(u):=\frac{1}{2}\int_{{\mathbb{T}^{2}}}\left[\frac{\epsilon^{2}}{2}|\nabla
u|^{2}+W(u)\right]dx+\sum_{i,j=1}^{2}\frac{\epsilon\gamma_{ij}}{2}\int_{{\mathbb{T}^{2}}}\int_{{\mathbb{T}^{2}}}G(x-y)\;u_{i}(x)\;u_{j}(y)dxdy.$
(1.1)

$W$ is a triple-well potential which achieves minimum value $0$ at exactly
three points: $\alpha_{1}=(1,0,0),\alpha_{2}=(0,1,0)$ and
$\alpha_{0}=(0,0,1)$, and $G$ is the Laplace Green's function on
${\mathbb{T}^{2}}$, of mean zero. The functional $\mathcal{E}^{\epsilon}$ is
defined on

$\displaystyle\left\\{u=(u_{1},u_{2},u_{0})\,:\,u_{i}\in
H^{1}({\mathbb{T}^{2}};\mathbb{R}),\ i=0,1,2;\ \sum_{i=0}^{2}u_{i}=1;\
\frac{1}{|{\mathbb{T}^{2}}|}\int_{{\mathbb{T}^{2}}}u_{i}(x)dx=M_{i},i=1,2\right\\}.$
(1.2)

Using $\Gamma$-convergence [9, 21], the variational structure of
$\mathcal{E}^{\epsilon}$ is connected to minimization of an associated sharp-
interface model,

$\mathcal{E}(u):=\sum_{0\leq i<j\leq
2}\sigma_{ij}\mathcal{H}^{1}(\partial\Omega_{i}\cap\partial\Omega_{j})+\sum_{i,j=1}^{2}\frac{\gamma_{ij}}{2}\int_{{\mathbb{T}^{2}}}\int_{{\mathbb{T}^{2}}}G(x-y)\;u_{i}(x)\;u_{j}(y)dxdy,$
(1.3)

where the densities $u_{i}$ are replaced by phase domains described by
characteristic functions $u_{i}=\chi_{\Omega_{i}}$ of sets $\Omega_{i}$,
$i=0,1,2$, of finite perimeter.

The constants $\sigma_{ij}$, $i\neq j$, represent surface tension along the
interfaces separating the phase domains, and they are positive, material-
dependent constants. Their values are determined from the $\Gamma$-limit of
the vector-valued Cahn-Hilliard part of $\mathcal{E}^{\epsilon}$ [6, 26], as a
geodesic distance in a degenerate Riemannian metric induced on
${\mathbb{R}}^{2}$ by $\sqrt{W(u)}$,

$\displaystyle\sigma_{ij}=\inf\left\\{\sqrt{2}\int_{0}^{1}\sqrt{W(\zeta(t))}|\zeta^{\prime}(t)|dt:\zeta\in
C^{1}([0,1];\mathbb{R}^{3}),\zeta(0)=\alpha_{i},\zeta(1)=\alpha_{j}\right\\}.$
(1.4)

Conversely, given positive constants $\sigma_{ij}=\sigma_{ji}$, $i\neq j$, one
might want to engineer a potential $W$ for which $\sigma_{ij}$ give the
surface tensions along each interface in $\mathcal{E}(u)$. This may or may not
be possible, as the definition (1.4) imposes a necessary condition for
$\sigma_{ij}$ to arise as coefficients of the perimeter in $\mathcal{E}(u)$ in
any $\Gamma$-limit of $\mathcal{E}^{\epsilon}$, in the form of a set of
triangle inequalities,

$\sigma_{ij}\leq\sigma_{ik}+\sigma_{kj},\quad 0\leq i,j,k\leq 2.$ (1.5)

Our previous paper [1] discusses the special case
$\sigma_{01}=\sigma_{02}=\sigma_{12}=1$, which arises (for instance) when $W$
is symmetric with respect to permutations of the $u_{i}$-axes. In this paper
we consider the case of unequal surface tensions $\sigma_{ij}$, in which the
geometry of minimizers can be quite different. Although it is not motivated by
$\Gamma$-convergence, we also consider a case where the triangle inequalities
(1.5) are violated, $\sigma_{02}>\sigma_{01}+\sigma_{12}$.

When the surface tensions $\sigma_{ij}$ obey the triangle inequalities (1.5),
the variational problem (1.3) may conveniently be posed with characteristic
functions lying in BV spaces,

$\displaystyle\left\\{u=(u_{1},u_{2},u_{0}):u_{i}\in
BV({\mathbb{T}^{2}};\\{0,1\\}),i=0,1,2;\sum_{i=0}^{2}u_{i}=1;\frac{1}{|{\mathbb{T}^{2}}|}\int_{{\mathbb{T}^{2}}}u_{i}(x)dx=M_{i},i=1,2\right\\}.$
(1.6)

Here $BV({\mathbb{T}^{2}};\\{0,1\\})$ is the space of functions of bounded
variation that only take two values: $0$ and $1$. Thus, each
$u_{i}=\chi_{\Omega_{i}}$ where $\Omega_{1},\Omega_{2},\Omega_{0}$ is a
partition of ${\mathbb{T}^{2}}$. The weighted perimeter is then re-expressed
in terms of the total variations of the components $u_{i}$,

$P_{\sigma}(\Omega_{1},\Omega_{2},\Omega_{0})=\sum_{0\leq i<j\leq
2}\sigma_{ij}\mathcal{H}^{1}(\partial\Omega_{i}\cap\partial\Omega_{j})=\frac{1}{2}\sum_{i=0}^{2}\beta_{i}\int_{\mathbb{T}^{2}}|\nabla
u_{i}|,$ (1.7)

with weights

$\beta_{1}=\sigma_{01}+\sigma_{12}-\sigma_{02},\qquad\beta_{2}=\sigma_{02}+\sigma_{12}-\sigma_{01},\qquad\beta_{0}=\sigma_{01}+\sigma_{02}-\sigma_{12}.$
(1.8)

When (1.5) holds, each $\beta_{i}\geq 0$. Moreover, at most one of
$\beta_{i}=0$, since we assume each $\sigma_{ij}>0$. We note that since
$u_{0}=1-u_{1}-u_{2}$, the state of the system is completely determined by the
pair $(u_{1},u_{2})$, or indeed any pair of the triple $(u_{1},u_{2},u_{0})$.
In particular, the weighted perimeter $P_{\sigma}$ is equivalent to the
standard BV norm on the cluster $(\Omega_{1},\Omega_{2},\Omega_{0})$, and thus
is coercive and lower semicontinuous with respect to $L^{1}$ convergence in
this case. The energy may then be expressed as

$\displaystyle\mathcal{E}(u)=\frac{1}{2}\sum_{i=0}^{2}\beta_{i}\int_{\mathbb{T}^{2}}|\nabla
u_{i}|+\sum_{i,j=1}^{2}\frac{\gamma_{ij}}{2}\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}G(x-y)\;u_{i}(x)\;u_{j}(y)dxdy.$

Our goal in this paper is to characterize global minimizers of this
functional, both in the absence of the nonlocal interaction (i.e.,
$\gamma_{ij}=0$) and in an appropriate ``droplet scale'' limit, in which
minimizers form a very dilute lattice of particles of two minority phases in a
sea of the third phase [8, 2, 1].

### The local isoperimetric problem

Figure 2:  Numerical simulations: six stationary states in local ternary
systems under different values of $\sigma_{02}$. Each stationary state evolves
from random initial data. (a) A standard double bubble with $\sigma_{02}=1$.
(b)-(d) Weighted double bubbles with $\sigma_{02}=1.6,1.8,1.9$ respectively.
(e) A tangential core shell with $\sigma_{02}=2$. (f) A generalized core shell
with $\sigma_{02}=3$. For all six simulations, $\sigma_{01}=\sigma_{12}=1$,
$M_{1}=0.12$, $M_{2}=0.04$.

Figure 3: Numerical simulations. Compared to Figure 2, here we reverse the
values of $M_{1}$ and $M_{2}$, that is, $M_{1}=0.04$, $M_{2}=0.12$. For all
six simulations, $\sigma_{01}=\sigma_{12}=1$, and from (a) to (f),
$\sigma_{02}=1,1.6,1.8,1.9,2,$ and $3$, respectively.

In section 2 we consider minimizers of the local isoperimetric problem,
minimizing the perimeter among clusters in ${\mathbb{R}}^{2}$ with given areas
$M_{1},M_{2}$, in the absence of the nonlocal term. We pose the problem in all
of ${\mathbb{R}}^{2}$ both because it is a natural setting, and because it
plays an important role in the global minimization of the nonlocal energy in
the droplet regime, considered in section 4. The qualitative nature of
minimizers depends strongly on the choice of $\sigma_{ij}$, and in particular
on whether the triangle inequalities (1.5) hold strictly or not. In case of
strict triangle inequalities, isoperimetric sets with nonzero prescribed mass
$M_{1},M_{2}>0$ are double bubbles [11, 12, 13, 18, 15, 23, 24]; see (a) to
(d) in Figure 2 and Figure 3 . This is consistent with our result in our first
paper, [1], in the unweighted case $\sigma_{ij}=1$, $i\neq j$, in which the
isoperimetric sets are standard double bubbles with equal angles ${2\pi\over
3}$ at the triple junction points. In the weighted case, the angles will be
uniquely determined by the surface tensions via Young's law [33, 25, 32, 29].

The case of equality in one of (1.5) is also interesting, and yields a
completely different geometry for minimizers. If $\beta_{1}=0$, then we show
that the optimal geometry is that of a core shell,
$\mathcal{C}^{M_{1}}_{M_{2}}$, with an inner disk of area $M_{2}$ surrounded
by an annular region of area $M_{1}$; see (e,f) in Figure 2 and Figure 3.
While this is a degenerate case, physically it is highly relevant as core
shell constructions are often observed in nature. Indeed, for an ABC triblock
copolymer it seems more natural that minimizers form core shells than double
bubbles, because of the linear structure of the blocks. Since the bonding of
the chains can only occur in linear order, one might expect that between
regions of phase A and phase C there must be a phase B region, and that
adjacency of phase A and C states should be energetically unfavorable.

This is indeed a degenerate setting, and the geometry of core shell minimizers
is not uniquely determined by the perimeter alone, since the position of the
interior disk in a core shell is arbitrary, from the point of view of the
weighted perimeter alone. The addition of the nonlocal term will resolve this
degeneracy, as we will see in Proposition 4.4.

For completeness, in numerical simulations, we also include the case
$\beta_{1}<0$. In this case, the optimal geometry is also a core shell; see
(f) in Figure 2 and Figure 3.

Finally, one may also consider the case $\beta_{0}=0$, in which case
minimization prefers the separation of the cluster into disjoint single
bubbles. Again, this is a degenerate case, and after adding the nonlocal term,
the relative position of the bubbles is not determined by the perimeter alone.
Indeed, we expect that the effect of the nonlocal repulsive term will be to
push the two constituents far apart, as in the binary case.

### Nonlocal effects

It is well-known that the Green's function term in $\mathcal{E}(u)$ competes
directly with the local isoperimetric term, in the sense that it is maximized
by disks. In order to analyze its role in global minimizers we adopt the
droplet regime scaling, introduced by Choksi-Peletier [8], a critical scaling
in which both the isoperimetric and nonlocal terms in the energy act at the
same energy scale. This choice of material parameters (discussed in some
detail in section 4) represents a dilute limit as the masses of phases $A$ and
$B$ both tend to zero, but with correspondingly large interaction coefficients
on the Green's function so that a rescaled characteristic function of each's
phase domain behaves as Dirac delta measures in the limit. Introducing a small
parameter $\eta>0$, which gives the length scale of a ``droplet'' of phase $A$
or $B$, we consider the rescaled energy

$E_{\eta}(v_{\eta})=\frac{\eta}{2}\sum_{i=0}^{2}\beta_{i}\int_{\mathbb{T}^{2}}|\nabla
v_{i,\eta}|+\sum_{i,j=1}^{2}\
\frac{\Gamma_{ij}}{2|\log\eta|}\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}G_{\mathbb{T}^{2}}(x-y)v_{i,\eta}(x)v_{j,\eta}(y)dxdy,$
(1.9)

for
$v_{\eta}=(v_{1,\eta},v_{2,\eta})=\eta^{-2}\chi_{\Omega_{\eta}}\in[BV({\mathbb{T}^{2}};\\{0,\eta^{-2}\\})]^{2}$
with prescribed masses, $\int_{{\mathbb{T}^{2}}}v_{i,\eta}=M_{i}$, $i=1,2$.
Here $\Gamma_{ij}=|\log\eta|\eta^{3}\gamma_{ij}$; see Section 4 for more
details.

Figure 4:  Numerical simulations: three stationary states in nonlocal ternary
systems under different values of $\sigma_{02}$. Each stationary state evolves
from random initial data. (a) Well-organized standard double bubbles when
$\sigma_{02}=1$. (b) Well-organized weighted double bubbles when
$\sigma_{02}=1.5$. (e) Well-organized core shells when $\sigma_{02}=2$. For
all three simulations,
$\sigma_{01}=\sigma_{12}=1,M_{1}=0.10,M_{2}=0.05,\gamma_{11}=16,000,\gamma_{12}=\gamma_{21}=0,\gamma_{22}=54,000$.
With changing $\sigma_{02}$, here we can see the transition from multiple
double bubbles to multiple core shells.

As in [8, 1], finite energy configurations $E_{\eta}(v_{\eta})\leq C$ have a
concentration structure (see Lemma 4.1) by which $v_{\eta}$ splits into an at
most countable collection of indecomposable clusters (in the sense of Maggi
[16, Chapter 29]) with diameter $O(\eta)$. For illustration, let's take this
structure as an ansatz: first, assume we have an at most countable collection
of finite perimeter 2-clusters in ${\mathbb{R}}^{2}$,
$\\{A^{k}\\}_{k\in{\mathbb{N}}}$, where $A^{k}=(A^{k}_{1},A^{k}_{2})$, with
$A^{k}_{i}\in{\mathbb{R}}^{2}$, $i=1,2$, and
$\sum_{k=1}^{\infty}|A_{i}^{k}|=M_{i}$, i=1,2. Then choose distinct points
$\xi^{k}\in{\mathbb{T}^{2}}$, and consider the configuration,

$\Omega_{\eta}=\bigcup_{k\in{\mathbb{N}}}(\eta
A^{k}+\xi^{k}),\quad\text{and}\quad v_{\eta}=\eta^{-2}\chi_{\Omega_{\eta}}.$

Substituting into $E_{\eta}$ yields:

$\displaystyle E_{\eta}(v_{\eta})$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{i=0}^{2}\frac{\eta}{2}\beta_{i}\int_{{\mathbb{T}^{2}}}|\nabla
v_{i,\eta}^{k}|+{\Gamma_{ij}\over
2|\log\eta|}\sum_{k,\ell=1}^{\infty}\sum_{i,j=1}^{2}\int_{{\mathbb{T}^{2}}}\int_{{\mathbb{T}^{2}}}v_{i,\eta}^{k}(x)\,G_{\mathbb{T}^{2}}(x-y)\,v_{j,\eta}^{\ell}(y)\,dx\,dy$
$\displaystyle=\sum_{k=1}^{\infty}\left[P_{\sigma}(A^{k})+\sum_{i,j=1}^{2}{\Gamma_{ij}\over
4\pi}|A^{k}_{i}|\,|A^{k}_{j}|\right]+O(|\log\eta|^{-1})$
$\displaystyle=\sum_{k=1}^{\infty}\mathcal{G}_{\sigma}(A^{k})+O(|\log\eta|^{-1}),$

with limiting energy of each component cluster $(A_{1},A_{2})$,

$\mathcal{G}_{\sigma}(A):=P_{\sigma}(A)+\sum_{i,j=1}^{2}{\Gamma_{ij}\over
4\pi}|A_{i}|\,|A_{j}|.$ (1.10)

The blow-up energy $\mathcal{G}_{\sigma}(A^{k})$ is the weighted version of
the energy of clusters studied in [1], and a ternary extension of the one
analyzed in the binary case in [8] in two dimensions. The nonlocality of
$E_{\eta}$ is expressed in the quadratic terms which depend on the mass: the
larger are $M_{1},M_{2}$, the greater the need to split the phase domains into
more and more droplets. However, the nonlocal interaction does not affect the
geometry of minimizers, which are studied in section 2 for the various choices
of weights $\sigma_{ij}$. The deformation in the shape of double-bubbles, from
the equal-angles case (with equal weights $\sigma_{ij}=1$) to cases of very
different weight values, including well-organized weighted double bubbles and
well-organized core shells, are illustrated in Figure 4.

As in the unweighted case (and the binary case,) we prove two
$\Gamma$-convergence results to describe the behavior of minimizers (or low
energy states) of $E_{\eta}$. The first limit (Theorem 4.8) describes the
splitting of masses and the fine scale geometry of the indecomposable clusters
as minimizers of $\mathcal{G}_{\sigma}$ in ${\mathbb{R}}^{2}$. Let

$\displaystyle e_{0}(m)$ $\displaystyle:=\min\left\\{\mathcal{G}_{\sigma}(A)\
|\ A=(A_{1},A_{2})\text{ 2-cluster, with $|A_{i}|=m_{i}$,
$i=1,2$}\right\\},\quad\text{and}$ (1.11) $\displaystyle\overline{e_{0}}(M)$
$\displaystyle:=\inf\left\\{\sum_{k=1}^{\infty}e_{0}(m^{k}):m^{k}=(m_{1}^{k},m_{2}^{k}),\
m_{i}^{k}\geq 0,\ \sum_{k=1}^{\infty}m_{i}^{k}=M_{i},i=1,2\right\\}.$ (1.12)

Then, sequences of minimizers $v_{\eta}$ of $E_{\eta}$ with mass
$M=(M_{1},M_{2})$ converge as measures $v_{\eta}\rightharpoonup
v_{0}=\sum_{k=1}^{\infty}m^{k}\delta_{x^{k}}$, where
$x^{k}\in{\mathbb{T}^{2}}$ and $\\{m^{k}\\}_{k\in{\mathbb{N}}}$ are determined
by $\overline{e_{0}}(M)$.

The second $\Gamma$-limit (Theorem 4.23) exploits the remainder terms, of
order $O(|\log\eta|^{-1})$ in the expansion of $E_{\eta}(v_{\eta})$ above, to
locate the centers $x^{k}$ of the droplets, and in the case of core shells, to
determine the explicit geometry of the shells. We recall from the discussion
of the local isoperimetric problem above that when $\beta_{1}=0$ the perimeter
term is degenerate, and the position of the interior disk is not determined by
minimizing $P_{\sigma}$. In Proposition 4.4 we show that the detailed geometry
of core shells is determined at the $O(|\log\eta|^{-1})$ level, depending on
the relative values of the interaction coefficients $\Gamma_{11},\Gamma_{12}$.
When the repulsion between phases is weak, that is, $\Gamma_{12}$ is small,
concentric core shells reduce the energy at second order, while for stronger
repulsion $\Gamma_{12}>\Gamma_{11}$ core shells' inner disks should be tangent
to the exterior circle.

### Core shell assemblies

Ren & Wang have constructed critical points of $\mathcal{E}(u)$ representing
dilute lattices of core shells [19] and single bubbles [20], and stationary
solutions with double-bubble lattices were constructed by Ren & Wei [23].
Numerical studies of periodic minimizers suggest that these assemblies do
appear as minimizers of $\mathcal{E}(u)$ in many parameter regimes [28], and
it is an interesting and challenging problem to verify this rigorously. At
least in the droplet regime limit described above, a first step is to study
the limiting energy $\mathcal{G}_{\sigma}$, with given total mass
$M=(M_{1},M_{2})$ and weights $\sigma$. In our first paper [1] we proved many
properties of the minimizers of $\mathcal{G}_{\sigma}$ in the case of equal
weights. We showed that the number of droplets is finite, and the size of the
constituent components is bounded in terms of the coefficients and masses. We
showed that, while a minimizer can exhibit both double-bubbles and single-
bubbles, there can only be one species of single-bubble if there is
coexistence of single- and double-bubbles.

In section 3 we take up the same questions in case $\beta_{1}=0$, and the
local isoperimetric problem favors core shell minimizers. In Lemma 3.1 we show
that there is a lower bound $m^{-}$ on the mass of any droplet constituent,
whether single-bubble or component of a core shell, which depends only on the
weights $\sigma_{ij}$, the coefficients $\Gamma_{ij}$, and the total masses
$M_{1},M_{2}$. From this we may conclude (Corollary 3.2) that minimizing
configurations of $\mathcal{G}_{\sigma}$ can have only finitely many
nontrivial indecomposable components. We note that the finiteness of
components in the binary case studied by Choksi & Peletier [8] was proven
using the concavity of the perimeter; as in our study of double-bubbles [1],
the perimeter in the ternary case is not globally concave, and so more
delicate arguments are required.

As was the case with double-bubbles, many open questions remain. Numerical
studies show regimes in which minimizers appear to have only core shells, but
we have no theorem which shows that this must be the case. On the other hand,
simulations also show that coexistence of core shells and single bubbles can
occur when one chooses values of $M_{i},\Gamma_{ij}$ which are very different
from each other, so any result in this direction would have to take the ranges
of values of the parameters into account.

### On clusters

In this paper we use the framework of clusters of finite perimeter sets in
${\mathbb{R}}^{2}$, as set out in [16, Part IV]. A 2-cluster in
${\mathbb{R}}^{2}$ is a disjoint pairing $A=(A_{1},A_{2})$ of finite perimeter
sets, $|A_{1}\cap A_{2}|=0$, each of finite Lebesgue measure. We write the
mass as $m=(m_{1},m_{2})$, $m_{i}=|A_{i}|$, $i=1,2$. It will be convenient for
us to permit one of the chambers to be empty, so the case where one of
$m_{i}=0$ is allowed. The exterior domain
$A_{0}={\mathbb{R}}^{2}\setminus\overline{(A_{1}\cup A_{2})}$ has infinite
measure, but its perimeter is included in the total weighted perimeter of the
cluster. As $\widetilde{A}_{0}={\mathbb{R}}^{2}\setminus
A_{0}=\overline{(A_{1}\cup A_{2})}$ has the same perimeter, it will often be
convenient to replace $A_{0}$ by $\widetilde{A}_{0}$ in calculating the
perimeter, that is:

$P_{\sigma}(A)=\frac{1}{2}\left[\beta_{1}P_{{\mathbb{R}}^{2}}(A_{1})+\beta_{2}P_{{\mathbb{R}}^{2}}(A_{2})+\beta_{0}P_{{\mathbb{R}}^{2}}(\overline{(A_{1}\cup
A_{2})})\right].$

The chambers of a cluster $A=(A_{1},A_{2})$ do not need to be connected, and
indeed in studying the nonlocal problem we expect that they will split into
disjoint components. The proper measure theoretic definition is that of
indecomposability: a set $E$ is indecomposable if whenever $E=E_{1}\cup E_{2}$
with $|E_{1}\cap E_{2}|=0$ and
$P_{{\mathbb{R}}^{2}}(E)=P_{{\mathbb{R}}^{2}}(E_{1})+P_{{\mathbb{R}}^{2}}(E_{2})$,
then one of $|E_{1}|,|E_{2}|=0$. (See [5].)

### Numerical methods

Figure 5:  One sample numerical simulation. The system starts from random
initial data and converges to a steady state of well-organized core shells. At
the steady state, all core shells are concentric, of equal size, and
distributed in a perfect hexagon pattern. Here $\sigma_{01}=\sigma_{12}=1$,
$\sigma_{02}=2$, $M_{1}=0.12$, $M_{2}=0.04$, $\gamma_{11}=4,000$,
$\gamma_{12}=\gamma_{21}=0,\gamma_{22}=20,000$.

To minimize the free energy (1.1), we consider the $L^{2}$ gradient flow
dynamics. Periodic boundary conditions are used here. To fulfill the mass
constraints, we adopt a modified augmented Lagrange multiplier approach. The
coupled nonlocal Allen-Cahn equations with mass constraints are first
reformulated via a linear splitting scheme and then be solved efficiently by
using the semi-implicit scheme to discretize the time variable and the
spectral method to discretize the space variables. The numerical simulations
start from random initial configurations satisfying the mass constraints.

### Acknowledgements

The authors gratefully acknowledge the support of the Natural Science and
Engineering Research Council (Canada) though the Discovery Grants program.

##  2 The local isoperimetric problem

In this section, we discuss among all the partitions of $\mathbb{R}^{2}$, the
minimizers of the local part $P_{\sigma}$; that is, minimizing

$P_{\sigma}(A_{1},A_{2},A_{0}):=\sum_{0\leq i<j\leq
2}\sigma_{ij}\mathcal{H}^{1}(\partial A_{i}\cap\partial A_{j}),$ (2.1)

among

$\\{(A_{1},A_{2},A_{0}):|A_{1}\cap
A_{2}|=0;|A_{i}|=m_{i},i=1,2;A_{0}={\mathbb{R}}^{2}\setminus\overline{(A_{1}\cup
A_{2})}\\}.$ (2.2)

As noted above, since $A_{0}$ and $\widetilde{A}_{0}=\overline{A_{1}\cup
A_{2}}$ have the same boundary, the perimeter above may be expressed in terms
of $A_{1},A_{2}$ alone, and when convenient we may replace the exterior domain
with the union $\overline{A_{1}\cup A_{2}}$. In particular, we may re-express
the total perimeter as

$\mathcal{P}_{\sigma}(A)=P_{\sigma}(A_{1},A_{2},\overline{A_{1}\cup A_{2}}).$
(2.3)

In case the $\sigma_{ij}$ satisfy the triangle inequalites (1.5) we have the
equivalent formulation of this isoperimetric problem in terms of BV
characteristic functions, given in (1.7). We are also interested in the case
of $\sigma_{02}>\sigma_{01}+\sigma_{12}$, in which the representation (1.7) is
not coercive in the BV norm, and in this case a more careful treatment of the
minimization problem is required.

The geometry of minimizers depends strongly on the surface tension values
$\sigma_{ij}$, and we consider each case separately.

###  2.1 Pattern 1: One Double bubble

When we assume that the triangle inequalities (1.5) hold with strict
inequalities in each, minimizers of (2.1) with $m_{1}\neq 0\neq m_{2}$ are
double bubbles. This has been proven (in any dimension) by Lawlor [15]. This
geometrical problem was already studied (in the context of grain boundaries)
by Mullins & Smith [25, 32]. Minimizers consist of smooth circular arcs which
meet at triple junctions. The angle formed at a triple junction must satisfy
Young's Law (also known as a Herring Condition): at each triple junction
point, the normal vectors $n_{ij}$ to the arc separating phases $i$ and $j$
must satisfy the balancing condition,

$\sum_{i\neq j}\sigma_{ij}\,n_{ij}=0.$

This is equivalent to

${\sin\theta_{1}\over\sigma_{02}}={\sin\theta_{2}\over\sigma_{01}}={\sin\theta_{0}\over\sigma_{12}};$
(2.4)

see Figure 6.

Figure 6:  At triple junction points the angles $\theta_{i}$ between the
normal vectors $n_{ki}$ and $n_{kj}$ are determined by the surface tensions
$\sigma_{ij}$ via (2.4). On the left, a symmetric double bubble, with equal
values of $\sigma_{ij}$; on the right, $\sigma_{02}>\sigma_{12}=\sigma_{01}$

.

The special case of identical weights $\sigma_{01}=\sigma_{02}=\sigma_{12}=1$,
has been studied in our previous paper [1]. In this case,

$\displaystyle P_{\sigma}(A_{1},A_{2},A_{0})=\sum_{0\leq i<j\leq
2}\mathcal{H}^{1}(\partial A_{i}\cap\partial
A_{j})=\frac{1}{2}\sum_{i=0}^{2}P_{\mathbb{R}^{2}}(A_{i}),$ (2.5)

which is a two component isoperimetric problem. The standard double bubble
satisfying the $120$ degree requirement at the two triple junction points
(where three interfaces meet) is the unique solution to this isoperimetric
problem [11, 13, 18]; see (a) in Figure 2 and Figure 3.

###  2.2 Pattern 2: One Core Shell

Core shell geometries can be expected in the degenerate case of the triangle
inequality (1.5) where

$\sigma_{02}\geq\sigma_{01}+\sigma_{12},$ (2.6)

and that the other two inequalities in (1.5) hold strictly. With this
hypothesis, the cost $\sigma_{02}$ of a transition from phase 2 to the
background zero-phase is at least as large as a ``composite interface''
passing through phase 1, so intuitively we expect a shell structure to
minimize perimeter.

######  Definition 2.1.

Given masses $m_{1},m_{2}>0$, we define the class of generalized core shells,
denoted $\mathcal{C}_{m_{2}}^{m_{1}}$, to consist of pairs $(A_{1},A_{2})$,
$|A_{i}|=m_{i}$, $i=1,2$, with an inner disk $A_{2}$ of mass $m_{2}$, and an
outer annulus $A_{1}$ of mass $m_{1}$.

That is, $A_{2}=B_{r_{2}}(p_{2})$ and $A_{1}=B_{r_{1}}(p_{1})\setminus
B_{r_{2}}(p_{2})$, with $r_{2}=\sqrt{M_{2}/\pi}$,
$r_{1}=\sqrt{(M_{1}+M_{2})/\pi}$, and centers $p_{1},p_{2}\in{\mathbb{R}}^{2}$
chosen such that $A_{2}=B_{r_{2}}(p_{2})\subset B_{r_{1}}(p_{1})$. Note that
this definition does not require the two circles (the boundary of the disk and
the outer boundary of the annulus) to be concentric; see (f) in Figure 2 and
3. In addition, this definition permits a special case in which the two
circles are tangent to each other; see (e) in Figure 2 and 3. Indeed, the
position of the inner circle $A_{2}$ does not affect the total perimeter of
the configuration, and so the local isoperimetric problem cannot distinguish
between these generalized core shells.

We expect the same geometry of minimizers in the more extreme case where
$\sigma_{02}>\sigma_{01}+\sigma_{12}$. In this situation we cannot rely on the
equivalent formulation (1.3) in BV in order to ensure existence of a minimizer
or lower semicontinuity, but we may still assert that core shell
configurations $\mathcal{C}_{m_{2}}^{m_{1}}$ must have smaller perimeter than
any other competitor. Hence, we state our geometry result for both cases at
once:

######  Theorem 2.2.

Let $\sigma_{ij}$ be given, such that

$\sigma_{02}\geq\sigma_{01}+\sigma_{12},$

and let $m_{1},m_{2}>0$ be given. Then the minimizer of $\mathcal{P}_{\sigma}$
is a core shell, ie, of the class $\mathcal{C}_{m_{2}}^{m_{1}}$.

In the case of equality in (1.5),

$\sigma_{02}=\sigma_{01}+\sigma_{12},$ (2.7)

the proof is straightforward. So we start with the proof of this case.

###### Proof of Theorem 2.2, case of equality in (2.6).

Applying (2.7) to (2.1), for any admissible cluster $A_{1},A_{2},A_{0}$,

$\displaystyle P_{\sigma}(A_{1},A_{2},A_{0})$ $\displaystyle=$
$\displaystyle\sigma_{01}\left[\mathcal{H}^{1}(\partial A_{0}\cap\partial
A_{1})+\mathcal{H}^{1}(\partial A_{0}\cap\partial
A_{2})\right]+\sigma_{12}\left[\mathcal{H}^{1}(\partial A_{1}\cap\partial
A_{2})+\mathcal{H}^{1}(\partial A_{0}\cap\partial A_{2})\right]$
$\displaystyle=$ $\displaystyle\sigma_{01}P_{{\mathbb{R}}^{2}}(A_{1}\cup
A_{2})+\sigma_{12}P_{{\mathbb{R}}^{2}}(A_{2}).$

Since the area of $A_{1}\cup A_{2}$ is fixed, that is, $|A_{1}\cup
A_{2}|=m_{1}+m_{2}$, by the isoperimetric inequality,
$P_{{\mathbb{R}}^{2}}(A_{1}\cup A_{2})$ is minimized when $\partial(A_{1}\cup
A_{2})$ is a circle. Similarly, the area of $A_{2}$ is fixed,
$P_{{\mathbb{R}}^{2}}(A_{2})$ is minimized when $\partial A_{2}$ is a circle.
Since $A_{2}\subset A_{1}\cup A_{2}$, we obtain a core shell, of the form
$\mathcal{C}_{m_{2}}^{m_{1}}$. That is, given any admissible cluster
$(A_{1},A_{2},A_{0})$ with masses $m_{1},m_{2}$, its weighted total perimeter
is bounded below by that of a core shell in $\mathcal{C}_{m_{2}}^{m_{1}}$; a
posteriori a minimizer exists, which is a core shell. Moreover, the inequality
will be strict in case either $A_{1}\cup A_{2}$ or $A_{2}$ are not disks, and
so the class of minimizers is exactly $\mathcal{C}_{m_{2}}^{m_{1}}$. ∎

When the inequality (2.6) is strict the situation is more delicate, as the
weighted perimeter functional is no longer lower semicontinuous. We need some
preliminary results.

######  Lemma 2.3.

Given $\sigma_{ij}$, with $\sigma_{02}>\sigma_{01}+\sigma_{12}$, then we can
construct explicit sequences $u_{i,n}=\mathbf{1}_{A_{i,n}}$, $i=0,1,2$,
converging to some $u_{i}=\mathbf{1}_{A_{i}}$ in the strong $L^{1}$ topology,
such that

$\lim_{n\to+\infty}P_{\sigma}(A_{1,n},A_{2,n},A_{0,n})<P_{\sigma}(A_{1},A_{2},A_{0}).$

###### Proof.

Consider a sequence of configurations like in Figure 7, and we denote by
$T_{n}$ (resp. $B_{n}$) the thin layer (resp. small ball) of type I material
wrapping around the lobe of type II constituent (resp. hollowed out from the
lobe of type I material). Then by construction, we can choose $|B_{n}|\to 0$
as $n\to+\infty$, and the thickness of $T_{n}$ also goes to zero. For any $n$,
the perimeter of $A_{2,n}$ is completely insulated from $A_{0,n}$, and the
contribution of the perimeter between type II and $T_{n}$ is

$+\sigma_{12}\mathcal{H}^{1}(\partial A_{2,n}\cap\partial T_{n}),$

while that between $T_{n}$ and $A_{0,n}$ is

$+\sigma_{01}\mathcal{H}^{1}(\partial A_{0,n}\cap\partial T_{n}).$

As the thickness of $T_{n}$ goes to zero, both

$\mathcal{H}^{1}(\partial A_{0,n}\cap\partial
T_{n}),\qquad\mathcal{H}^{1}(\partial A_{2,n}\cap\partial T_{n})$

converge to $\mathcal{H}^{1}(\partial A_{2}\cap\partial A_{0})$. Thus

$\lim_{n\to+\infty}\sigma_{12}\mathcal{H}^{1}(\partial A_{2,n}\cap\partial
T_{n})+\sigma_{01}\mathcal{H}^{1}(\partial A_{0,n}\cap\partial
T_{n})=(\sigma_{01}+\sigma_{12})\mathcal{H}^{1}(\partial A_{2}\cap\partial
A_{0}).$ (2.8)

But in the limit case, the boundary of the lobe of type II constituent will
have no layer of type I constituent insulating it anymore, so it will
contribute

$+\sigma_{02}\mathcal{H}^{1}(\partial A_{2}\cap\partial A_{0})$

to the perimeter term. This is greater than the sum (2.8). As the other terms
are continuous when passing to the limit $n\to+\infty$, i.e.

$\displaystyle\lim_{n\to+\infty}$
$\displaystyle\sigma_{1j}\mathcal{H}^{1}(\partial(A_{1,n}\setminus
B_{n})\cap\partial A_{j,n})=\sigma_{1j}\mathcal{H}^{1}(\partial
A_{1}\cap\partial A_{j}),\qquad j=0,2$ $\displaystyle\lim_{n\to+\infty}$
$\displaystyle\sigma_{01}\mathcal{H}^{1}(\partial(A_{1,n}\setminus
B_{n})\cap\partial B_{n})=0,$

we infer

$\lim_{n\to+\infty}P_{\sigma}(A_{1,n},A_{2,n},A_{0,n})<P_{\sigma}(A_{1},A_{2},A_{0}),$

as desired. ∎

######  Corollary 2.4.

As a consequence of Lemma 2.3, the full nonlocal energy $\mathcal{E}(u)$ is
also not lower semicontinuous with respect to the strong $L^{1}$ topology.

###### Proof.

It suffices to notice that the interaction term is continuous with respect to
the convergence from Lemma 2.3. ∎

As a consequence of Lemma 2.3, the existence of optimal configurations is
significantly more challenging when $\sigma_{02}>\sigma_{01}+\sigma_{12}$, as
it prevents us from using most classical arguments relying on minimizing
sequences. Thus, we have to rely on an ad hoc construction.

We require a geometrical lemma which proves an inner-cone condition for
minimizers of $\mathcal{P}_{\sigma}$. We recall that, since we are working
with sets $A_{1},A_{2},A_{0}$ of finite perimeter, both $\partial
A_{i}\cap\partial A_{0}$, $i=1,2$, are $\mathcal{H}^{1}$-rectifiable and hence
Lipschitz regular. The unit tangent vectors thus exist $\mathcal{H}^{1}$-a.e,
but there may be corners on any of the components of the interfaces, and the
following lemma restricts the sharpness of these angles. Define the angle
$\alpha_{0}$ as the the solution in $(0,\frac{\pi}{2})$ of

$1-\sin\alpha-\sqrt{\frac{\pi}{2}\sin(2\alpha)}=0.$

######  Lemma 2.5.

Let $m_{1},m_{2}>0$, and $(A_{1},A_{2},A_{0})$ in (2.2) be given. If there
exists a point $x_{0}\in\Sigma:=\partial A_{i}\cap\partial A_{j}$ such that
the interior (to $A_{i}$) angle between the right and left tangent lines to
$\Sigma$ at $x_{0}$ has amplitude $\alpha<\alpha_{0}$, then there exists a
perturbation of $(A_{1},A_{2},A_{0})$ with the same respective masses but
lower (weighted) perimeter.

###### Proof.

Assume there exists a point $x_{0}\in\Sigma:=\partial A_{i}\cap\partial A_{j}$
such that the angle between the right and left tangent lines to $\Sigma$ at
$x_{0}$ has amplitude $2\alpha>0$. Then we perturb the entire configuration in
the following way:

  1. 1.

first, we choose points $p_{\varepsilon},q_{\varepsilon}\in\Sigma$ such that
the path distance on $\Sigma$ between $p_{\varepsilon},q_{\varepsilon}$ and
$x_{0}$ is $\varepsilon$. Due to the Lipschitz regularity, $\Sigma$ is
approximated in first order, near $x_{0}$, by its tangent line, therefore we
have also

$\delta_{\varepsilon}:=|x_{0}-p_{\varepsilon}|=|x_{0}-q_{\varepsilon}|=\varepsilon+o(\varepsilon).$

  2. 2.

Second, we connect $p_{\varepsilon}$ and $q_{\varepsilon}$ with a straight
line segment $\llbracket p_{\varepsilon},q_{\varepsilon}\rrbracket$, and
define

$\widetilde{\Sigma}_{\varepsilon}:=\Sigma\setminus\\{\text{portion of
}\Sigma\text{ between }p_{\varepsilon}\text{ and
}q_{\varepsilon}\\}\cup\llbracket p_{\varepsilon},q_{\varepsilon}\rrbracket.$

Geometrically, it is clear that

$|p_{\varepsilon}-q_{\varepsilon}|=2\delta_{\varepsilon}\sin\alpha+o(\varepsilon),$

thus

$\mathcal{H}^{1}(\widetilde{\Sigma}_{\varepsilon})=\mathcal{H}^{1}(\Sigma)-2\delta_{\varepsilon}(1-\sin\alpha)+o(\varepsilon).$

  3. 3.

The construction in the previous step, however, alters the total masses of
each type of constituent. Indeed, it is clear that the mass of type $i$
constituent inside the region delimited by

$\\{\text{portion of }\Sigma\text{ between }p_{\varepsilon}\text{ and
}q_{\varepsilon}\\}\qquad\text{and}\qquad\llbracket
p_{\varepsilon},q_{\varepsilon}\rrbracket,$

whose area is

$A_{\varepsilon}:=\frac{1}{2}\varepsilon^{2}\sin(2\alpha)+o(\varepsilon^{2}),$

has been simply removed, and been replaced with type $j$ constituent. To
balance this issue, we remove a ball of area $A_{\varepsilon}$ from type $j$
constituent, and replace it with type $i$ constituent. Our final competitor is
the set obtained in this way.

By construction, such a competitor from the previous three steps will have the
exact same total masses for each type of constituent as the original
configuration. The perimeter between types $i$ and $j$ constituents has been
decreased by $2\delta_{\varepsilon}(1-\sin\alpha)+o(\varepsilon)$ due to the
construction in Step 2, and increased by $2\sqrt{\pi A_{\varepsilon}}$ in Step
3. Since

$2\delta_{\varepsilon}(1-\sin\alpha)-2\sqrt{\pi
A_{\varepsilon}}=2\varepsilon[1-\sin\alpha-\sqrt{\frac{\pi}{2}\sin(2\alpha)}]+o(\varepsilon),$

which, since we assumed $\alpha<\alpha_{0}$, where $\alpha_{0}$ is the
solution of

$1-\sin\alpha-\sqrt{\frac{\pi}{2}\sin(2\alpha)}=0,$

becomes positive for all sufficiently small $\varepsilon$, we get that such a
construction produces a competitor with less total (weighted) perimeter. The
proof is thus complete. ∎

With this geometrical lemma we can then show that in the case of strict
inequality in (2.6) that there should be no interfaces between phases 2 and 0.

######  Lemma 2.6.

Let $\sigma_{ij}$ be given, such that

$\sigma_{02}\geq\sigma_{01}+\sigma_{12}.$

Let $m_{1},m_{2}>0$ be given, and let $(A_{1},A_{2},A_{0})$ in (2.2) be a
minimizer of (2.1). Then $\mathcal{H}^{1}(\partial A_{0}\cap\partial
A_{2})=0$, i.e. $A_{2}$ is separated from the background $A_{0}$ by $A_{1}$.

###### Proof.

Case I: $\sigma_{02}>\sigma_{01}+\sigma_{12}$. Assume the opposite, i.e.
$\mathcal{H}^{1}(\partial A_{0}\cap\partial A_{2})>0$. We construct a
competitor with lower perimeter in the following way:

  1. 1.

first, remove a ball $B_{\varepsilon}$ of area $\varepsilon$ from $A_{1}$, and
fill it with background constituent $A_{0}$. This step creates a new perimeter
$\partial B_{\varepsilon}$, between $A_{1}$ and the background $A_{0}$. Thus
the perimeter is increased by

$+2\sigma_{01}\sqrt{\pi\varepsilon}.$

Figure 7: Construction of a competitor with lower perimeter

  2. 2.

Next, we claim that we can add a thin layer around $\partial A_{2}\cap\partial
A_{0}$ of thickness $\varepsilon/\mathcal{H}^{1}(\partial A_{2}\cap\partial
A_{0})$, so the total mass of type I constituent is preserved. This adds a new
perimeter, again between $A_{1}$ and the background $A_{0}$, of length

$+\sigma_{01}\mathcal{H}^{1}(\partial A_{2}\cap\partial A_{0})+f(\varepsilon)$

for some function $f(\varepsilon)\to 0$ as $\varepsilon\to 0$. However, this
step also completely erases the former boundary $\partial A_{2}\cap\partial
A_{0}$, and transforms it into a boundary between $A_{1}$ and $A_{2}$: thus
the perimeter decreases by a term

$(-\sigma_{02}+\sigma_{12})\mathcal{H}^{1}(\partial A_{2}\cap\partial A_{0}).$

Assuming that step 2 can be achieved, we complete the proof of Lemma 2.6 and
verify the details of the construction of the modified domain below. The above
construction induces a change in the perimeter of

$+2\sigma_{01}\sqrt{\pi\varepsilon}+f(\varepsilon)-(\sigma_{02}-\sigma_{01}-\sigma_{12})\mathcal{H}^{1}(\partial
A_{2}\cap\partial A_{0})<0,$

which contradicts the minimality of $(A_{1},A_{2},A_{0})$.

It remains to verify the details of the construction in step 2 above,
constructing the insulating layer as in Figure 7.

Step 1. Choosing the points. Take a curve $\Gamma$ on $\partial
A_{2}\cap\partial A_{0}$, and let $\gamma:[0,1]\longrightarrow\Gamma$ be a
constant speed parameterization. Furthermore, impose that the angle between
the tangents $\gamma^{\prime}(0)$ and $\gamma^{\prime}(s)$, $s\in[0,1]$, never
exceeds $2\pi$. Without loss of generality, assume the overall net turning
from $\gamma^{\prime}(0)$ to $\gamma^{\prime}(1)$ is in the counterclockwise
sense, and we denote such turning by

$A:=\angle\gamma^{\prime}(0)\gamma^{\prime}(1)\in[0,2\pi].$

Let

$\Gamma_{n}:=\bigcup_{i=1}^{n}\llbracket t_{i-1,n},t_{i,n}\rrbracket,\qquad
t_{i,n}:=\gamma(s_{i,n}),\quad s_{i,n}:=\frac{i}{n},$

be the piecewise linear curve through all the $t_{i,n}$. Note that this
construction ensures $\mathcal{H}^{1}(\Gamma_{n})\to\mathcal{H}^{1}(\Gamma)$.
Now, for each $t_{i,n}$, $i=1,\cdots,n-1$, denote by $\nu_{i,n}^{\pm}$ the
left/right exterior (i.e. pointing towards the background) unit normal to
$\Gamma_{n}$ at $t_{i,n}$, and let

$t_{i,n}^{\varepsilon,\pm}:=t_{i,n}+\varepsilon\nu_{i,n}^{\pm},\qquad
i=1,\cdots,n-1.$

For $i=0,n$, denote by $\nu_{i,n}$ the exterior unit normal to $\Gamma_{n}$.

Step 2. Constructing the layer. Connect $t_{0,n}$ to $t_{0,n}^{\varepsilon}$,
and $t_{n,n}$ to $t_{n,n}^{\varepsilon}$, and then $t_{i-1,n}^{\varepsilon,+}$
to $t_{i,n}^{\varepsilon,-}$, $i=1,\cdots,n$, with line segments (with the
convention $t_{0,n}^{\varepsilon,+}=t_{0,n}^{\varepsilon}$,
$t_{n,n}^{\varepsilon,-}=t_{n,n}^{\varepsilon}$).

Figure 8: A schematic representation of the construction.

The resulting set

$\llbracket t_{0,n}^{\varepsilon},t_{0,n}\rrbracket\cup\llbracket
t_{n,n}^{\varepsilon},t_{n,n}\rrbracket\cup\bigcup_{i=1}^{n}\llbracket
t_{i-1,n}^{\varepsilon,+},t_{i,n}^{\varepsilon,-}\rrbracket$

however, might have two issues:

  1. 1.

first, if the angle in $t_{i,n}$ is convex, then the line segments $\llbracket
t_{i-1,n}^{\varepsilon,+},t_{i,n}^{\varepsilon,-}\rrbracket$ and $\llbracket
t_{i,n}^{\varepsilon,+},t_{i+1,n}^{\varepsilon,-}\rrbracket$ do not intersect.

  2. 2.

Second, if the angle in $t_{i,n}$ is concave, then the line segments
$\llbracket t_{i-1,n}^{\varepsilon,+},t_{i,n}^{\varepsilon,-}\rrbracket$ and
$\llbracket t_{i,n}^{\varepsilon,+},t_{i+1,n}^{\varepsilon,-}\rrbracket$ do
cross each other.

To overcome it, we do the following:

  1. 1.

if the angle in $t_{i,n}$ is convex, then we connect the line segments
$,t_{i,n}^{\varepsilon,-}$ and $t_{i,n}^{\varepsilon,+}$ with an arc of circle
$\omega_{i,n}$ centered in $t_{i,n}$.

  2. 2.

If the angle in $t_{i,n}$ is concave, then define
$p_{i,n}^{\varepsilon}:=\llbracket
t_{i-1,n}^{\varepsilon,+},t_{i,n}^{\varepsilon,-}\rrbracket\cap\llbracket
t_{i,n}^{\varepsilon,+},t_{i+1,n}^{\varepsilon,-}\rrbracket$. Then, replace
these two segments with $\llbracket
t_{i-1,n}^{\varepsilon,+},p_{i,n}^{\varepsilon}\rrbracket\cup\llbracket
p_{i,n}^{\varepsilon},t_{i+1,n}^{\varepsilon,-}\rrbracket$

Let $\Gamma_{n}^{\varepsilon}$ be the resulting curve. By taking the limit
$n\to+\infty$, we obtain a curve $\Gamma^{\varepsilon}$ that plays the role of
the outer boundary of the ``insulating layer'' from Figure 7.

Step 3. Estimating the length of $\Gamma_{n}^{\varepsilon}$. We want to
estimate from above the difference
$\mathcal{H}^{1}(\Gamma^{\varepsilon})-\mathcal{H}^{1}(\Gamma)$, by first
estimating
$\mathcal{H}^{1}(\Gamma_{n}^{\varepsilon})-\mathcal{H}^{1}(\Gamma_{n})$, and
then pass to the limit $n\to+\infty$.

Around a convex angle of amplitude $2\alpha$, there is

$2\varepsilon\beta,\qquad\beta:=\frac{\pi}{2}-\alpha,$

extra length. See Figure 9.

Figure 9: Geometry around a convex angle

Around a concave angle of amplitude $2\alpha$, there is

$2\varepsilon\tan\beta,\qquad\beta:=\frac{\pi}{2}-\alpha,$

less length. See Figure 10.

Figure 10: Geometry around a concave angle

Recalling that we have the extra pieces $\llbracket
t_{0,n}^{\varepsilon},t_{0,n}\rrbracket$ and $\llbracket
t_{n,n}^{\varepsilon},t_{n,n}\rrbracket$, which might add $2\varepsilon$ in
length, we have

$\mathcal{H}^{1}(\Gamma_{n}^{\varepsilon})-\mathcal{H}^{1}(\Gamma_{n})\leq
2\varepsilon\Big{[}1+\sum_{i=1}^{N_{+}}\beta_{i}^{+}-\sum_{i=1}^{N_{-}}\tan\beta_{i}^{-}\Big{]},\qquad\beta_{i}^{\pm}:=\frac{\pi}{2}-\alpha_{i}^{\pm},$

where the $\alpha_{i}^{+}$ (resp. $\alpha_{i}^{-}$) are the convex (resp.
concave) angles.

Note that the tangent turns in opposite directions around a convex angle
compared to a concave one, thus the total turning of the tangent is

$\displaystyle A^{+}-A^{-}=A\in[0,2\pi],\qquad
A^{+}:=\sum_{i=1}^{N_{+}}\beta_{i}^{+},\quad
A^{-}:=\sum_{i=1}^{N_{-}}\beta_{i}^{-}.$ (2.9)

Thus we need to bound

$\sum_{i=1}^{N_{+}}\beta_{i}^{+}-\sum_{i=1}^{N_{-}}\tan\beta_{i}^{-}$

from above, subject to (2.9). Using the convexity of $\tan$, and the fact that
$\tan\theta\geq\theta$,

$\displaystyle\sum_{i=1}^{N_{+}}\beta_{i}^{+}-\sum_{i=1}^{N_{-}}\tan\beta_{i}^{-}$
$\displaystyle\leq A^{+}-N^{-}\tan\frac{A^{-}}{N^{-}}\leq A^{+}-A^{-}=A.$

Thus, combining all the above estimates,

$\mathcal{H}^{1}(\Gamma_{n}^{\varepsilon})-\mathcal{H}^{1}(\Gamma_{n})\leq
2(A+1)\varepsilon.$

The right hand side is now independent of $n$, which allows to take the
$\liminf$ for $n\to+\infty$: our construction ensured
$\mathcal{H}^{1}(\Gamma_{n})\to\mathcal{H}^{1}(\Gamma)$, while
$\mathcal{H}^{1}(\Gamma^{\varepsilon})\leq\liminf_{n\to+\infty}\mathcal{H}^{1}(\Gamma_{n}^{\varepsilon})$,
hence

$\mathcal{H}^{1}(\Gamma^{\varepsilon})-\mathcal{H}^{1}(\Gamma)\leq
2(A+1)\varepsilon.$

∎

We may now complete the proof of Theorem 2.2, in the case of strict inequality
in (2.6).

###### Proof of Theorem 2.2, remaining case.

With the conclusion of Lemma 2.6, the conclusion of Theorem 2.2 follows as in
the case of equality. Indeed, since $\mathcal{H}^{1}(\partial
A_{0}\cap\partial A_{2})=0$, we again obtain the following identity for the
total weighted perimeter,

$\displaystyle P_{\sigma}(A_{1},A_{2},A_{0})$ $\displaystyle=$
$\displaystyle\sigma_{01}\mathcal{H}^{1}(\partial A_{0}\cap\partial
A_{1})+\sigma_{12}\mathcal{H}^{1}(\partial A_{1}\cap\partial A_{2})$
$\displaystyle=$ $\displaystyle\sigma_{01}P_{{\mathbb{R}}^{2}}(A_{1}\cup
A_{2})+\sigma_{12}P_{{\mathbb{R}}^{2}}(A_{2}).$

Again, $A_{2}\subset A_{1}\cup A_{2}$, and each is optimized by choosing a
disk of the appropriate area, and thus Theorem 2.2 is proven in both cases.

∎

It is interesting to think of the core shell as a limit case of the weighted
double bubbles as we increase the surface tension $\sigma_{02}$ to a point
where equality is attained in (1.5). The numerical experiments in Figure 2 and
Figure 3 illustrate this process. Beginning from the symmetric situation (all
$\sigma_{ij}$ equal), we increase $\sigma_{02}$; so to reduce (2.1),
$\mathcal{H}^{1}(\partial A_{0}\cap\partial A_{2})$ will decrease and we
observe a weighted double bubble will be the solution; see (b) to (d) in
Figure 2 and Figure 3. If we continue increasing $\sigma_{02}$, when
$\sigma_{02}=\sigma_{01}+\sigma_{12}$, a generalized core-shell with two
circles tangential to each other is the minimizer; see (e) in Figure 2 and
Figure 3. As the location of the core shell is not determined by the geometry
problem, we will see that it is the second order Gamma convergence and the
interaction term $\Gamma_{12}$ that will determine the location.

###  2.3 Pattern 3: Two single bubbles

When $\sigma_{12}\geq\sigma_{01}+\sigma_{02}$, and strict inequality holds in
the other two of (1.5), we expect to have single bubble configurations. This
case is similar to the previous one, except now it is the interface between
$A_{1}$ and $A_{2}$ which is effectively penalized, and minimizers should
prefer to insert a layer of $A_{0}$ between these two components. The
consequence is that the optimal geometry separates the two minority phases
into disjoint balls.

######  Lemma 2.7.

Let $\sigma_{ij}$ be given, such that

$\sigma_{12}\geq\sigma_{01}+\sigma_{02}.$

Let $m_{1},m_{2}>0$ be given, and let $(A_{1},A_{2},A_{0})$ in (2.2) be a
minimizer of (2.1). Then $\mathcal{H}^{1}(\partial A_{1}\cap\partial
A_{2})=0$, and the minimizer consists of disjoint balls $A_{1}$, $A_{2}$.

###### Proof.

We can follow the same steps as in the proof of Lemma 2.6 in case
$\sigma_{12}=\sigma_{01}+\sigma_{02}$, but the situation is simpler when
$\sigma_{12}>\sigma_{01}+\sigma_{02}$. Suppose for a contradiction that there
exists a minimizer of (2.1) with $\mathcal{H}^{1}(\partial A_{1}\cap\partial
A_{2})>0$. Let $A_{2}^{\prime}$ be a translation of $A_{2}$, chosen such that
$\overline{A_{1}}\cap\overline{A_{2}^{\prime}}=\emptyset$, and
$A_{0}^{\prime}={\mathbb{R}}^{2}\setminus\overline{(A_{1}\cup
A_{2}^{\prime})}$. Then, $|A_{2}^{\prime}|=|A_{2}|$, and
$\mathcal{H}^{1}(\partial A_{1}\cap\partial A_{2}^{\prime})=0$, while the
other components of the boundary have the same perimeter as before. Thus,

$P_{\sigma}(A_{1},A_{2}^{\prime},A_{0}^{\prime})<P_{\sigma}(A_{1},A_{2},A_{0}),$

as $\sigma_{01}+\sigma_{02}<\sigma_{12}$, which contradicts the minimality of
$(A_{1},A_{2},A_{0})$.

Now that we have established $\mathcal{H}^{1}(\partial A_{1}\cap\partial
A_{2})=0$, the total perimeter splits into the weighted sum of the perimeters
of $A_{1}$ and $A_{2}$, and each is minimized independently, resulting in a
disjoint union of two balls of the given masses. ∎

As in the core shell case, the degeneracy of the weighted perimeter is felt
through the nonuniqueness of minimizing configurations; the relative positions
of the two bounded components of a minimizing cluster is arbitrary.

##  3 The Geometry of Core Shell Configurations

In this section we study the combined effect of the local (weighted)
isoperimetric energy and the nonlocal interaction energy in the formation of
core shell assemblies obtained by minimization of $E_{\eta}$ (see (1.9)) in
the droplet regime limit. Following our analysis of the isoperimetric problem
in the previous section, this entails making the choice

$\sigma_{02}=\sigma_{01}+\sigma_{12}$ (3.1)

in (1.5), in order that core shells are energetically preferred in
$P_{\sigma}$. As described in the Introduction, in Theorem 4.8 we will prove a
$\Gamma$-convergence result for $E_{\eta}$ as $\eta\to 0$ to the limiting
energy defined in (1.11), (1.12), for any $\sigma_{ij}$ satisfying the
triangle inequalities (1.5), including the case of equality above.

Figure 11:  Another numerical simulation when
$\sigma_{02}=\sigma_{01}+\sigma_{12}$. These core shells are well-organized:
concentric, of equal size, and distributed in a hexagon pattern. Here
$\sigma_{02}=2$, $\sigma_{01}=\sigma_{12}=1$, $M_{1}=0.12$, $M_{2}=0.06$,
$\gamma_{11}=20,000$, $\gamma_{12}=\gamma_{21}=0$, $\gamma_{22}=100,000$.

By Theorem 2.2, with the choice (3.1) we have an explicit form for the local
part of the energy,

$\displaystyle\inf\left\\{\sum_{0\leq i<j\leq
2}\sigma_{ij}\mathcal{H}^{1}(\partial A_{i}\cap\partial
A_{j}):|A_{i}|=m_{i},i=1,2;A_{0}=\mathbb{R}^{2}\setminus\overline{(A_{1}\cup
A_{2})}\right\\}$ (3.2) $\displaystyle=$ $\displaystyle
2\sigma_{01}\sqrt{\pi(m_{1}+m_{2})}+2\sigma_{12}\sqrt{\pi m_{2}}$

which represents the total perimeter of a generalized core-shell when
$m_{1},m_{2}>0$. Then, substituting in (1.10) and (1.11),

$\displaystyle e_{0}(m)$ $\displaystyle=$ $\displaystyle
2\sigma_{01}\sqrt{\pi(m_{1}+m_{2})}+2\sigma_{12}\sqrt{\pi
m_{2}}+\sum_{i,j=1}^{2}\frac{\Gamma_{ij}m_{i}m_{j}}{4\pi}.$ (3.3)

In the case of $m_{1}>0,m_{2}=0$,

$\displaystyle e_{0}(m)=e_{0}(m_{1},0)=2\sigma_{01}\sqrt{\pi
m_{1}}+\frac{\Gamma_{11}m_{1}^{2}}{4\pi}.$ (3.4)

In the case of $m_{1}=0,m_{2}>0$,

$\displaystyle e_{0}(m)=e_{0}(0,m_{2})=2\sigma_{01}\sqrt{\pi
m_{2}}+2\sigma_{12}\sqrt{\pi
m_{2}}+\frac{\Gamma_{22}m_{2}^{2}}{4\pi}=2\sigma_{02}\sqrt{\pi
m_{2}}+\frac{\Gamma_{22}m_{2}^{2}}{4\pi}.$ (3.5)

In the above two cases, a core-shell is degenerated to a single bubble.

As in the previous studies of the droplet scaling for binary [8] and ternary
systems [1], the limiting minimization problem (1.12) is very subtle since the
division of the total masses $M=(M_{1},M_{2})$ is determined by minimization
itself. In particular, although core shells are favored when both constituents
$m_{1},m_{2}>0$ are nontrivial, minimizers may exhibit a mixed state of core
shells and single bubbles. In addition, it is reasonable to expect that
minimizers of $\overline{e_{0}}(M)$ will only have finitely many connected
components, and that each constituent bubble should have a minimum size; these
facts are known for the binary case, and in some parameter regimes, for
unweighted ternary systems [8, 1]. Such results are the goal of this section.

We recall (1.12)

$\displaystyle\overline{e_{0}}(M)=\inf\left\\{\sum_{k=1}^{\infty}e_{0}(m^{k}):m^{k}=(m_{1}^{k},m_{2}^{k}),\
m_{i}^{k}\geq 0,\sum_{k=1}^{\infty}m_{i}^{k}=M_{i},i=1,2\right\\},$

where $M=(M_{1},M_{2})$.

First we show that having too small a mass is not energetically advantageous.

######  Lemma 3.1.

There exists a lower bound $m^{-}$ depending on

$M_{i},\;\sigma_{01},\text{ and }\Gamma_{ij},\qquad i,j=1,2,$

such that no single bubble or core shell in a minimizing configuration can
have total mass less than

$m^{-}:=\frac{32\pi^{3}\sigma_{01}^{2}}{(1+\sqrt{2})^{2}(\Gamma_{11}+2\Gamma_{12}+\Gamma_{22})^{2}(M_{1}+M_{2})^{2}}.$

###### Proof.

Consider an arbitrary minimizing configuration. If there is only one bubble
(single bubble or core shell) then the thesis is trivial. Pick two bubbles
(single bubble or core shell) whose masses are $m_{i}^{k}$, $i,k=1,2$, where
the index $i$ denotes the constituent type, and $m_{1}^{k}$ (resp.
$m_{2}^{k}$) is the ``outer'' (resp. ``inner'') shell. We allow for
$m_{i}^{k}=0$ corresponding to the case of a single bubble. Their energy
contribution is thus

$\displaystyle\sum_{k=1}^{2}2\sqrt{\pi}\Big{[}\sigma_{01}\sqrt{m_{1}^{k}+m_{2}^{k}}+\sigma_{12}\sqrt{m_{2}^{k}}\Big{]}+\sum_{i,j,k=1}^{2}\frac{\Gamma_{ij}}{4\pi}m_{i}^{k}m_{j}^{k}.$
(3.6)

Combining them into a bubble, whose inner disk (resp. outer shell) has mass
$m_{2}^{1}+m_{2}^{2}$ (resp. $m_{1}^{1}+m_{1}^{2}$), then the energy
contribution becomes

$\displaystyle
2\sqrt{\pi}\Bigg{[}\sigma_{01}\sqrt{\sum_{k=1}^{2}(m_{1}^{k}+m_{2}^{k})}+\sigma_{12}\sqrt{\sum_{k=1}^{2}m_{2}^{k}}\Bigg{]}+\sum_{i,j=1}^{2}\frac{\Gamma_{ij}}{4\pi}\Big{(}\sum_{k=1}^{2}m_{i}^{k}\Big{)}\Big{(}\sum_{k=1}^{2}m_{j}^{k}\Big{)}.$
(3.7)

Denote $m^{k}:=m_{1}^{k}+m_{2}^{k}$, $k=1,2$. By subtracting (3.6) from (3.7)
we get

$\displaystyle 2\sqrt{\pi}\Bigg{[}\sigma_{01}$
$\displaystyle\sqrt{\sum_{k=1}^{2}(m_{1}^{k}+m_{2}^{k})}+\sigma_{12}\sqrt{\sum_{k=1}^{2}m_{2}^{k}}\Bigg{]}+\sum_{i,j=1}^{2}\frac{\Gamma_{ij}}{4\pi}\Big{(}\sum_{k=1}^{2}m_{i}^{k}\Big{)}\Big{(}\sum_{k=1}^{2}m_{j}^{k}\Big{)}$
$\displaystyle-\sum_{k=1}^{2}2\sqrt{\pi}\Big{[}\sigma_{01}\sqrt{m_{1}^{k}+m_{2}^{k}}+\sigma_{12}\sqrt{m_{2}^{k}}\Big{]}-\sum_{i,j,k=1}^{2}\frac{\Gamma_{ij}}{4\pi}m_{i}^{k}m_{j}^{k}$
$\displaystyle=2\sqrt{\pi}\sigma_{01}[\sqrt{m^{1}+m^{2}}-\sqrt{m^{1}}-\sqrt{m^{2}}]+2\sqrt{\pi}\sigma_{12}[\sqrt{m_{2}^{1}+m_{2}^{2}}-\sqrt{m_{2}^{1}}-\sqrt{m_{2}^{2}}]$
$\displaystyle\quad+\frac{1}{2\pi}\Big{[}\Gamma_{11}m_{1}^{1}m_{1}^{2}+\Gamma_{12}(m_{1}^{1}m_{2}^{2}+m_{1}^{2}m_{2}^{1})+\Gamma_{22}m_{2}^{1}m_{2}^{2}\Big{]}.$

By the optimality of our initial configuration, we need the above term to be
nonnegative, i.e.

$\displaystyle 0$ $\displaystyle\leq
2\sqrt{\pi}\sigma_{01}[\sqrt{m^{1}+m^{2}}-\sqrt{m^{1}}-\sqrt{m^{2}}]+2\sqrt{\pi}\sigma_{12}[\sqrt{m_{2}^{1}+m_{2}^{2}}-\sqrt{m_{2}^{1}}-\sqrt{m_{2}^{2}}]$
$\displaystyle\quad+\frac{1}{2\pi}\Big{[}\Gamma_{11}m_{1}^{1}m_{1}^{2}+\Gamma_{12}(m_{1}^{1}m_{2}^{2}+m_{1}^{2}m_{2}^{1})+\Gamma_{22}m_{2}^{1}m_{2}^{2}\Big{]}.$
(3.8)

Thus,

$0\leq
2\sqrt{\pi}\sigma_{01}[\sqrt{m^{1}+m^{2}}-\sqrt{m^{1}}-\sqrt{m^{2}}]+\frac{1}{2\pi}\Big{[}\Gamma_{11}m_{1}^{1}m_{1}^{2}+\Gamma_{12}(m_{1}^{1}m_{2}^{2}+m_{1}^{2}m_{2}^{1})+\Gamma_{22}m_{2}^{1}m_{2}^{2}\Big{]}$

Assume without loss of generality that $m^{1}\geq m^{2}$. Note that

$\displaystyle\sqrt{m^{1}}+\sqrt{m^{2}}-\sqrt{m^{1}+m^{2}}$
$\displaystyle=\frac{2\sqrt{m^{1}m^{2}}}{\sqrt{m^{1}}+\sqrt{m^{2}}+\sqrt{m^{1}+m^{2}}}\geq\frac{2}{2+\sqrt{2}}\sqrt{m^{2}},$

and

$\displaystyle\Gamma_{11}m_{1}^{1}m_{1}^{2}+\Gamma_{12}(m_{1}^{1}m_{2}^{2}+m_{1}^{2}m_{2}^{1})+\Gamma_{22}m_{2}^{1}m_{2}^{2}$
$\displaystyle\leq\Gamma_{11}m^{1}m^{2}+\Gamma_{12}(m^{1}m^{2}+m^{2}m^{1})+\Gamma_{22}m^{1}m^{2}$
$\displaystyle=(\Gamma_{11}+2\Gamma_{12}+\Gamma_{22})m^{1}m^{2}$
$\displaystyle\leq(\Gamma_{11}+2\Gamma_{12}+\Gamma_{22})(M_{1}+M_{2})m^{2}$

Thus (3.8) becomes

$\displaystyle 0$
$\displaystyle\leq-\frac{4\sqrt{\pi}\sigma_{01}}{2+\sqrt{2}}\sqrt{m^{2}}+\frac{1}{2\pi}(\Gamma_{11}+2\Gamma_{12}+\Gamma_{22})(M_{1}+M_{2})m^{2}$
$\displaystyle\Longrightarrow
m^{2}\geq\frac{32\pi^{3}\sigma_{01}^{2}}{(1+\sqrt{2})^{2}(\Gamma_{11}+2\Gamma_{12}+\Gamma_{22})^{2}(M_{1}+M_{2})^{2}},$

concluding the proof since $m^{1}\geq m^{2}$. ∎

######  Corollary 3.2.

(Finiteness) For any $M=(M_{1},M_{2}),\ M_{1},M_{2}>0$, a minimizing
configuration for $\overline{e_{0}}(M)$ has finitely many nontrivial
components. That is, there exist $N<\infty$ and pairs $m^{1},\dots,m^{N}$,
with $m^{k}=(m_{1}^{k},m_{2}^{k})\neq(0,0)$, for which
$\overline{e_{0}}(M)=\sum_{k=1}^{N}e_{0}(m^{k})$.

###### Proof.

Lemma 3.1 gives that each single bubble / core shell must have total mass at
least $m^{-}$. Since the total combined mass of types I and II constituents is
$M_{1}+M_{2}$, we have

$\\#\text{core shells}+\\#\text{single
bubbles}\leq\frac{M_{1}+M_{2}}{m^{-}}=:N.$

∎

######  Lemma 3.3.

There exist computable lower bounds $m^{-}_{i}$, depending on

$M_{i},\sigma_{01},\sigma_{12},\text{ and }\Gamma_{ij},\qquad i,j=1,2,$

such that any bubble of type $i$ constituent, be it a single bubble, or lobe
in a core shell, in a minimizing configuration must have mass at least
$m^{-}_{i}$.

###### Proof.

If type $i$ constituent, $i=1,2$, is entirely in one bubble, then the lower
bound on $m_{i}^{-}$ is automatically true. The proof is slightly different
between the cases where there are at least two core shells, and where there is
only a single core shell with the rest being single bubbles.

Case 1: at least two core shells. Denote by $m_{i}^{k}$, $k=1,2$ the masses of
their lobes. By Lemma 3.1 we have $m_{1}^{k}+m_{2}^{k}\geq m^{-}$, $k=1,2$,
and assume that $m_{2}^{2}$ is the smallest (for the other scenarios, the
proof is similar). We need to show that $m_{2}^{2}$ cannot be both too small.
The energy contribution of these two bubbles is

$\displaystyle\sum_{k=1}^{2}2\sqrt{\pi}\Big{[}\sigma_{01}\sqrt{m_{1}^{k}+m_{2}^{k}}+\sigma_{12}\sqrt{m_{2}^{k}}\Big{]}+\sum_{i,j,k=1}^{2}\frac{\Gamma_{ij}}{4\pi}m_{i}^{k}m_{j}^{k}.$
(3.9)

Now we do the following construction: move the entire bubble of type II
constituent, with mass $m_{2}^{2}$, from the second bubble to the first, Then
take the equivalent amount $m_{2}^{2}$ of type I constituent from the first
bubble and move it to the second. Therefore, this construction a core shell
with inner disk (resp. outer shell) of mass $m_{2}^{1}+m_{2}^{2}$ (resp.
$m_{1}^{1}-m_{2}^{2}$), and a single bubble of type I constituent of mass
$m_{1}^{2}+m_{2}^{2}$. Their energy contribution is thus

$\displaystyle
2\sqrt{\pi}\Big{(}\sigma_{01}\sqrt{m_{1}^{1}+m_{2}^{1}}+\sigma_{12}\sqrt{m_{2}^{1}+m_{2}^{2}}\Big{)}$
$\displaystyle+\frac{\Gamma_{11}(m_{1}^{1}-m_{2}^{2})^{2}+2\Gamma_{12}(m_{1}^{1}-m_{2}^{2})(m_{2}^{1}+m_{2}^{2})+\Gamma_{22}(m_{2}^{1}+m_{2}^{2})^{2}}{4\pi}$
$\displaystyle+2\sqrt{\pi}\sigma_{01}\sqrt{m_{1}^{2}+m_{2}^{2}}+\frac{\Gamma_{11}(m_{1}^{2}+m_{2}^{2})^{2}}{4\pi}.$
(3.10)

Subtracting (3.10) from (3.9) gives

$\displaystyle 2\sqrt{\pi}\sigma_{12}$
$\displaystyle\Big{[}\sqrt{m_{2}^{1}}+\sqrt{m_{2}^{2}}-\sqrt{m_{2}^{1}+m_{2}^{2}}\Big{]}$
$\displaystyle-\frac{1}{2\pi}\bigg{[}\Gamma_{11}({m_{1}^{2}-m_{1}^{1}+m_{2}^{2}})+\Gamma_{12}(m_{1}^{1}-m_{2}^{1}-m_{1}^{2}-m_{2}^{2})+\Gamma_{22}m_{2}^{1}\bigg{]}m_{2}^{2},$

and due to the optimality of our initial configuration, such a difference must
be negative. Thus we need

$\displaystyle 0$ $\displaystyle\geq
2\sqrt{\pi}\sigma_{12}\Big{[}\sqrt{m_{2}^{1}}+\sqrt{m_{2}^{2}}-\sqrt{m_{2}^{1}+m_{2}^{2}}\Big{]}-\frac{1}{2\pi}\bigg{[}\Gamma_{11}(m_{1}^{2}-m_{1}^{1}+m_{2}^{2})+\Gamma_{12}(m_{1}^{1}-m_{2}^{1}-m_{1}^{2}-m_{2}^{2})+\Gamma_{22}m_{2}^{1}\bigg{]}m_{2}^{2}$
$\displaystyle\geq
2\sqrt{\pi}\sigma_{12}\frac{2\sqrt{m_{2}^{1}m_{2}^{2}}}{\sqrt{m_{2}^{1}}+\sqrt{m_{2}^{2}}+\sqrt{m_{2}^{1}+m_{2}^{2}}}-\frac{1}{2\pi}\Big{[}\Gamma_{11}(m_{1}^{2}+m_{2}^{2})+\Gamma_{12}m_{1}^{1}+\Gamma_{22}m_{2}^{1}\Big{]}m_{2}^{2},$

and using

$m_{2}^{1}\geq m_{2}^{2}$

the previous line gives

$\displaystyle 0$ $\displaystyle\geq
2\sqrt{\pi}\sigma_{12}\frac{2\sqrt{m_{2}^{1}m_{2}^{2}}}{\sqrt{m_{2}^{1}}+\sqrt{m_{2}^{2}}+\sqrt{m_{2}^{1}+m_{2}^{2}}}-\frac{1}{2\pi}\Big{[}\Gamma_{11}(m_{1}^{2}+m_{2}^{2})+\Gamma_{12}m_{1}^{1}+\Gamma_{22}m_{2}^{1}\Big{]}m_{2}^{2}$
$\displaystyle\geq\frac{4\sqrt{\pi}\sigma_{12}}{(2+\sqrt{2})}\sqrt{m_{2}^{2}}-\frac{1}{2\pi}\Big{[}\Gamma_{11}(M_{1}+M_{2})+\Gamma_{12}M_{1}+\Gamma_{22}M_{2}\Big{]}m_{2}^{2}.$

Therefore, for such inequality to hold, we need

$\frac{1}{2\pi}\Big{[}\Gamma_{11}(M_{1}+M_{2})+\Gamma_{12}M_{1}+\Gamma_{22}M_{2}\Big{]}\sqrt{m_{2}^{2}}\geq\frac{4\sqrt{\pi}\sigma_{12}}{(2+\sqrt{2})},$

and the proof of this case is complete.

Case 2: only one core shell. Using the same arguments from [8], we may
conclude that all single bubbles of the same type constituent have the same
mass. Several cases are possible.

  1. 1.

If there are two single bubbles of type $i$ constituent, both with mass
$m^{k}$, then combining them into one single bubble changes the energy by

$\displaystyle 2\sqrt{\pi}\sigma_{0i}$
$\displaystyle[\sqrt{2m^{k}}-2\sqrt{m^{k}}]-\frac{\Gamma_{ii}}{2\pi}[2|m^{k}|^{2}-|2m^{k}|^{2}]=-2\sqrt{\pi}(2-\sqrt{2})\sigma_{0i}\sqrt{m^{k}}+\frac{\Gamma_{ii}}{\pi}|m^{k}|^{2}.$

Such change cannot be negative, as it would contradict the optimality of the
initial configuration, hence a necessary condition is

$2\pi\sqrt{\pi}(2-\sqrt{2})\sigma_{0i}\Gamma_{ii}^{-1}\leq|m^{k}|^{3/2},$

thus prohibiting $m^{k}$ from being too small. Then, by noting that the Euler-
Lagrange equation contain terms that diverge when a mass gets too small, we
obtain a computable lower on the masses.

  2. 2.

If there is only one other single bubble, then the entire configuration is
made of a core shell and a single bubble.

We first show that, in the core-shell, the type I constituent is forming the
outer annulus contacting the background, while the type II constituent is
always concentrated in a ball. To this aim, we need to compare the energies of
the following configurations:

    1. (a)

a single bubble (of mass $m_{3}$, and whatever constituent type), plus a core-
shell where the type I constituent (of mass $m_{1}$) is forming the outer
annulus and the type II constituent (of mass $m_{2}$) is concentrated in a
ball.

    2. (b)

The same single bubble (of mass $m_{3}$, and whatever constituent type), plus
a core-shell where the type II constituent (of mass $m_{2}$) is forming the
outer annulus and the type I constituent (of mass $m_{1}$) is concentrated in
a ball.

The energy contribution of the single bubble is the same in both cases, as
well as the interaction terms. Thus only the (weighted) perimeters of the
core-shells are different: in Case (a), this is equal to

$\displaystyle
2\sqrt{\pi}(\sigma_{12}\sqrt{m_{2}}+\sigma_{01}\sqrt{m_{1}+m_{2}}),$ (3.11)

while in Case (b), this is equal to

$2\sqrt{\pi}(\sigma_{12}\sqrt{m_{1}}+\sigma_{02}\sqrt{m_{1}+m_{2}}),$

which, since $\sigma_{02}=\sigma_{01}+\sigma_{12}$, is equal to

$\displaystyle
2\sqrt{\pi}(\sigma_{12}\sqrt{m_{1}}+\sigma_{01}\sqrt{m_{1}+m_{2}}+\sigma_{12}\sqrt{m_{1}+m_{2}}).$
(3.12)

It is clear that (3.12) is always larger than (3.11), hence Case (b) is
energetically unfavorable.

Thus we are left with two cases to consider:

    1. (a)

a single bubble (of type I constituent and mass $M_{1}-\eta$), plus a core-
shell where the type I constituent (of mass $\eta$) is forming the outer
annulus and the type II constituent (of mass $M_{2}$) is concentrated in a
ball. The energy is thus

$\displaystyle f_{1}(\eta)$
$\displaystyle=2\sqrt{\pi}[\sigma_{12}\sqrt{M_{2}}+\sigma_{01}\sqrt{M_{2}+\eta}+\sigma_{01}\sqrt{M_{1}-\eta}]$
$\displaystyle\qquad+\frac{1}{4\pi}\Big{[}\Gamma_{11}((M_{1}-\eta)^{2}+\eta^{2})+2\Gamma_{12}M_{2}\eta+\Gamma_{22}M_{2}^{2}\Big{]},$

and

$\displaystyle f_{1}^{\prime}(\eta)$
$\displaystyle=\sigma_{01}\sqrt{\pi}\Big{(}\frac{1}{\sqrt{M_{2}+\eta}}-\frac{1}{\sqrt{M_{1}-\eta}}\Big{)}+\frac{1}{2\pi}\Big{[}\Gamma_{11}(2\eta-
M_{1})+\Gamma_{12}M_{2}\Big{]}.$

Since $f_{1}^{\prime}(\eta)\to-\infty$ as $\eta\nearrow M_{1}$, we conclude
that the single ball cannot be too small. In order to prevent critical
configurations arising for very small (yet positive) $\eta$, we notice that
the (algebraic) equation

$\displaystyle 0=f_{1}^{\prime}(\eta)$
$\displaystyle=\sigma_{01}\sqrt{\pi}\Big{(}\frac{1}{\sqrt{M_{2}+\eta}}-\frac{1}{\sqrt{M_{1}-\eta}}\Big{)}+\frac{1}{2\pi}\Big{[}\Gamma_{11}(2\eta-
M_{1})+\Gamma_{12}M_{2}\Big{]}$

can be transformed in some 8th order polynomials by repeatedly taking the
squares:

$\displaystyle 0=f_{1}^{\prime}(\eta)$
$\displaystyle=\sigma_{01}\sqrt{\pi}\Big{(}\frac{1}{\sqrt{M_{2}+\eta}}-\frac{1}{\sqrt{M_{1}-\eta}}\Big{)}+\frac{1}{2\pi}\Big{[}\Gamma_{11}(2\eta-
M_{1})+\Gamma_{12}M_{2}\Big{]}$
$\displaystyle\Longrightarrow\frac{1}{\sqrt{M_{1}-\eta}}-\frac{1}{\sqrt{M_{2}+\eta}}=\frac{1}{2\pi\sigma_{01}\sqrt{\pi}}\Big{[}\Gamma_{11}(2\eta-
M_{1})+\Gamma_{12}M_{2}\Big{]}$
$\displaystyle\Longrightarrow\sqrt{M_{2}+\eta}-\sqrt{M_{1}-\eta}=\frac{\sqrt{(M_{1}-\eta)(M_{2}+\eta)}}{2\pi\sigma_{01}\sqrt{\pi}}\Big{[}\Gamma_{11}(2\eta-
M_{1})+\Gamma_{12}M_{2}\Big{]}$ $\displaystyle\Longrightarrow
M_{2}+M_{1}-2\sqrt{(M_{1}-\eta)(M_{2}+\eta)}=\frac{(M_{1}-\eta)(M_{2}+\eta)}{4\pi^{3}\sigma_{01}^{2}}\Big{[}\Gamma_{11}(2\eta-
M_{1})+\Gamma_{12}M_{2}\Big{]}^{2}$ $\displaystyle\Longrightarrow
4(M_{1}-\eta)(M_{2}+\eta)=\bigg{[}\frac{(M_{1}-\eta)(M_{2}+\eta)}{4\pi^{3}\sigma_{01}^{2}}\Big{[}\Gamma_{11}(2\eta-
M_{1})+\Gamma_{12}M_{2}\Big{]}^{2}-(M_{2}+M_{1})\bigg{]}^{2}$ (3.13)

As such, any solution of $f_{1}^{\prime}(\eta)=0$ is also root of (3.13), an
8th order polynomial, where the coefficients depend only on $M_{i}$,
$\Gamma_{ij}$, $\sigma_{ij}$. Therefore, the smallest (positive) root
$\eta_{*}$ depends only on $M_{i}$, $\Gamma_{ij}$, $\sigma_{ij}$, and no other
critical configuration with $0<\eta<\eta_{*}$ can exist.

    2. (b)

A single bubble (of type II constituent and mass $M_{2}-\zeta$), plus a core-
shell where the type I constituent (of mass $M_{1}$) is forming the outer
annulus and the type II constituent (of mass $\zeta$) is concentrated in a
ball. The energy is thus

$\displaystyle f_{2}(\eta)$
$\displaystyle=2\sqrt{\pi}[\sigma_{12}\sqrt{\eta}+\sigma_{01}\sqrt{M_{1}+\eta}+\sigma_{02}\sqrt{M_{2}-\eta}]$
$\displaystyle\qquad+\frac{1}{4\pi}\Big{[}\Gamma_{22}((M_{2}-\eta)^{2}+\eta^{2})+2\Gamma_{12}M_{1}\eta+\Gamma_{11}M_{1}^{2}\Big{]},$

and

$\displaystyle f_{2}^{\prime}(\eta)$
$\displaystyle=\sqrt{\pi}\Big{(}\frac{\sigma_{01}}{\sqrt{M_{1}+\eta}}+\frac{\sigma_{12}}{\sqrt{\eta}}-\frac{\sigma_{02}}{\sqrt{M_{2}-\eta}}\Big{)}+\frac{1}{2\pi}\Big{[}\Gamma_{22}(2\eta-
M_{2})+\Gamma_{12}M_{1}\Big{]}.$

Since

$\lim_{\eta\to 0}f_{2}^{\prime}(\eta)=+\infty,\qquad\lim_{\eta\to
M_{2}}f_{2}^{\prime}(\eta)=-\infty,$

this means that there exists $\eta_{0}>0$, depending only on $M_{i}$,
$\sigma_{ij}$, $\Gamma_{ij}$, such that there cannot be any stable
configurations (and thus no optimal ones) if $\eta<\eta_{0}$ or
$\eta>M_{2}-\eta_{0}$. Consequently, this means that neither the single
bubble, nor any lobes of the core-shell, can have too small a mass.

Combining all the above cases concludes the proof. ∎

######  Lemma 3.4.

For any $M=(M_{1},M_{2}),M_{1},M_{2}>0$, when
$\sigma_{02}=\sigma_{01}+\sigma_{12}$ and $\Gamma_{12}=0$, a minimizing
configuration for $\overline{e_{0}}(M)$ can not have two different type of
single bubbles.

###### Proof.

Assume in a minimizing configuration for $\overline{e_{0}}(M)$, there are two
single bubbles $(m_{1}^{k},0)$ and $(0,m_{2}^{l})$. Then replacing these two
single bubbles by a core shell $(m_{1}^{k},m_{2}^{l})$ will reduce the energy
$\sum_{k=1}^{\infty}e_{0}(m^{k})$ since

$\displaystyle
e_{0}(m_{1}^{k},0)+e_{0}(0,m_{2}^{l})-e_{0}(m_{1}^{k},m_{2}^{l})$
$\displaystyle=$ $\displaystyle 2\sigma_{01}\sqrt{\pi
m_{1}^{k}}+\frac{\Gamma_{11}(m_{1}^{k})^{2}}{4\pi}+2\sigma_{02}\sqrt{\pi
m_{2}^{l}}+\frac{\Gamma_{22}(m_{2}^{l})^{2}}{4\pi}$
$\displaystyle-\left[2\sigma_{01}\sqrt{\pi(m_{1}^{k}+m_{2}^{l})}+2\sigma_{12}\sqrt{\pi
m_{2}^{l}}+\frac{\Gamma_{11}(m_{1}^{k})^{2}}{4\pi}+\frac{\Gamma_{22}(m_{2}^{l})^{2}}{4\pi}\right]$
$\displaystyle=$ $\displaystyle
2\sigma_{01}\sqrt{\pi}\left[\sqrt{m_{1}^{k}}+\sqrt{m_{2}^{l}}-\sqrt{(m_{1}^{k}+m_{2}^{l})}\right]>0.$

This contradicts with the minimality. ∎

Existence of Core Shells: Lemma 3.4 implies that for any
$M=(M_{1},M_{2}),M_{1},M_{2}>0$, when $\sigma_{02}=\sigma_{01}+\sigma_{12}$
and $\Gamma_{12}=0$, any minimizing configuration for $\overline{e_{0}}(M)$
must contain at least one generalized core shell.

######  Remark 3.1.

In a binary system [8] have shown that all bubbles are of equal size, by using
the concavity of $e_{0}(m)$ with respect to the single mass parameter $m$. In
simulations it appears that the same phenomenon should hold for double bubbles
and core shells, that all components of a minimizer of the same type are
congruent. However, unlike the binary case, in ternary systems (3.3) a direct
calculation of the Hessian shows it to be indefinite, and so the observed
congruence of forms remains an open question.

##  4 The nonlocal problem in the droplet regime

To see the effects of the nonlocal interaction we consider the Droplet
rescaling, as in Choksi-Peletier ([8]; see also [1, 3].)

###  4.1 The Droplet Regime

We take a periodic domain $\mathbb{T}^{2}$. In the remainder of the paper, we
assume that $\sigma_{ij}$ satisfies (1.5), and so the perimeter term may be
expressed in terms of total variation norms. Let $u_{i}=\chi_{\Omega_{i}}$,
$i=1,2$. As remarked upon earlier, while we must keep track of the perimeter
of the exterior domain $\Omega_{0}=(\overline{\Omega_{1}\cup\Omega_{2}})^{c}$,
the perimeter term is the same for its complement,
$\widetilde{\Omega}_{0}=\overline{\Omega_{1}\cup\Omega_{2}}.$ Thus, it will be
convenient to define

$u_{0}=\chi_{\Omega_{1}\cup\Omega_{2}},$

as it yields the same total variation,

$\int_{\mathbb{T}^{2}}|\nabla
u_{0}|=\int_{\mathbb{T}^{2}}|\nabla(1-u_{1}-u_{2})|=\int_{\mathbb{T}^{2}}|\nabla\chi_{\Omega_{0}}|.$

Thus, the local part of the functional may be expressed in terms of the two
``small'' sets $\Omega_{1},\Omega_{2}$ and the constants $\beta_{i}\geq 0$,
$i=0,1,2$ as:

$\displaystyle P_{\sigma}(\Omega_{1},\Omega_{2})=\sum_{0\leq i<j\leq
2}\sigma_{ij}\mathcal{H}^{1}(\partial\Omega_{i}\cap\partial\Omega_{j})=\frac{1}{2}\sum_{i=0}^{2}\beta_{i}\int_{\mathbb{T}^{2}}|\nabla
u_{i}|,$

with $\beta_{i}$ determined by (1.8), and so the energy becomes:

$\displaystyle\mathcal{E}(u)=\frac{1}{2}\sum_{i=0}^{2}\beta_{i}\int_{\mathbb{T}^{2}}|\nabla
u_{i}|+\sum_{i,j=1}^{2}\frac{\gamma_{ij}}{2}\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}G(x-y)\;u_{i}(x)\;u_{j}(y)dxdy.$

######  Remark 4.1.

We note throughout that as long as the triangle inequalities (1.5) hold (even
with $\beta_{i}=0$ in the Core Shell or Single Bubble cases), the weighted
perimeter $P_{\sigma}(\Omega_{1},\Omega_{2})$ is equivalent to the standard
norm for the cluster in $BV({\mathbb{T}^{2}};\\{0,1\\})$. That is, there
exists a constant $C=C(\sigma_{ij})$ with

${1\over
C}P_{\sigma}(\Omega_{1},\Omega_{2})\leq\sum_{i=0}^{2}\int_{{\mathbb{T}^{2}}}|\nabla\chi_{\Omega_{i}}|\leq
CP_{\sigma}(\Omega_{1},\Omega_{2}).$

In particular, the compactness and convergence proofs done for the unweighted
case $\beta_{i}=1$, $\forall i=0,1,2$, in [1] may be carried through to the
weighted case with straightforward modifications.

As in [8, 1], we introduce a new parameter $\eta$ which is to represent the
characteristic length scale of the droplet components. Thus, areas scale as
$\eta^{2}$, and so we choose mass constraints on $u=(u_{1},u_{2})$,

$\int_{\mathbb{T}^{2}}u_{i}=\eta^{2}M_{i}$

for some fixed $M_{i}$, $i=1,2$. We then rescale $u_{i}$ as

$\displaystyle v_{i,\eta}=\frac{u_{i}}{\eta^{2}},\quad
i=0,1,2,\quad\text{with}\quad{\int}_{\mathbb{T}^{2}}v_{i,\eta}=M_{i},\quad
i=1,2,$ (4.1)

and thus the appropriate space for droplet configurations is:

$X_{\eta}:=\left\\{v_{\eta}=(v_{1,\eta},v_{2,\eta}):\ \eta^{2}v_{i,\eta}\in
BV({\mathbb{T}^{2}};\\{0,1\\}),i=1,2;\ v_{1,\eta}v_{2,\eta}=0\ a.e.,\
\int_{\mathbb{T}^{2}}v_{i,\eta}=M_{i},\;i=1,2\right\\},$ (4.2)

We recall that we represent the exterior domain equivalently as
$v_{0,\eta}=v_{1,\eta}+v_{2,\eta}$.

In terms of $v_{\eta}\in X_{\eta}$, the energy takes the form,

$\displaystyle\mathcal{E}(u)$ $\displaystyle=$
$\displaystyle\eta\left(\frac{\eta}{2}\sum_{i=0}^{2}\beta_{i}\int_{\mathbb{T}^{2}}|\nabla
v_{i,\eta}|+\sum_{i,j=1}^{2}\frac{\gamma_{ij}\eta^{3}}{2}\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}v_{i,\eta}(x)v_{j,\eta}(y)dxdy\right)$
(4.3)

Finally, we scale the interaction matrix $\gamma=[\gamma_{ij}]$ in such a way
that both terms in (4.3) contribute at the same order in $\eta$. This is
accomplished by choosing

$\displaystyle\gamma_{ij}=\frac{1}{|\log\eta|\eta^{3}}\Gamma_{ij},$

with fixed constants $\Gamma_{ij}\geq 0$. Dividing by $\eta$, we thus obtain
an $O(1)$ valued energy for $v_{\eta}\in X_{\eta}$, defined by:

$\displaystyle
E_{\eta}(v_{\eta}):=\frac{1}{\eta}\mathcal{E}(u)=\frac{\eta}{2}\sum_{i=0}^{2}\beta_{i}\int_{\mathbb{T}^{2}}|\nabla
v_{i,\eta}|+\sum_{i,j=1}^{2}\
\frac{\Gamma_{ij}}{2|\log\eta|}\int_{\mathbb{T}^{2}}\int_{\mathbb{T}^{2}}G_{\mathbb{T}^{2}}(x-y)v_{i,\eta}(x)v_{j,\eta}(y)dxdy,$
(4.4)

and $E_{\eta}(v_{\eta})=+\infty$ otherwise. This is the droplet scaling of
[8], which we adopt in the remainder of the paper.

###  4.2 Expanding the energy in droplets

The advantage of the droplet regime for nonlocal isoperimetric problems is
that it separates length scales in the energy, so that both the local
isoperimetric effects and the nonlocal repulsive interaction are both observed
but at different scales in $\eta$. As a result of this balancing of the
strengths of the competing terms, any finte energy configuration $v_{\eta}$
decomposes into an at most countable number of well-separated droplets, each
of diameter $O(\eta)$.

In the following we assume that $(v_{\eta})_{\eta>0}$ are a family in
$X_{\eta}$ with bounded energy, $\exists C>0$ with $E_{\eta}(v_{\eta})\leq C$,
for all $\eta>0$. Such configurations may be decomposed into indecomposable
clusters (that is, connected in a measure-theoretic sense) of diameter of
$O(\eta)$, for which the BV norm separates exactly. As in [1] we may prove:

######  Lemma 4.1.

Assume (1.5) holds, and let $v_{\eta}=\eta^{-2}\chi_{\Omega_{\eta}}\in
X_{\eta}$ with $\eta\int_{{\mathbb{T}^{2}}}|\nabla v_{\eta}|\leq C$. Then,
there exists an at most countable collection
$v_{\eta}^{k}=\eta^{-2}\chi_{\Omega_{\eta}^{k}}\in X_{\eta}$, with clusters
$\Omega_{\eta}^{k}=(\Omega_{1,\eta}^{k},\Omega_{2,\eta}^{k})$, such that

  1. (a)

$\Omega_{i,\eta}^{k}\cap\Omega_{j,\eta}^{\ell}=\emptyset$, for $k\neq\ell$ and
$i,j=1,2$.

  2. (b)

$v_{\eta}=\sum_{k=1}^{\infty}v_{\eta}^{k}$ in $X_{\eta}$; in particular,

$\int_{{\mathbb{T}^{2}}}|\nabla
v_{i,\eta}|=\sum_{k=1}^{\infty}\int_{{\mathbb{T}^{2}}}|\nabla
v_{i,\eta}^{k}|,\quad i=0,1,2.$

  3. (c)

There exists $C>0$ with $\operatorname{diam}(\Omega_{\eta}^{k})\leq C\eta$ for
all $k\in{\mathbb{N}}$.

###### Proof.

Lemma 4.1 was proven in the case of equal weights in [1], and the same proof
may be employed here. However, it is also a consequence of the more general
Decomposition Theorem [5, Theorem 1], for finite perimeter sets in
${\mathbb{R}}^{n}$; we sketch the argument here for completeness. Indeed,
applying the Decomposition Theorem in [5] to
$\Omega_{0,\eta}=\bar{\Omega}_{1,\eta}\cup\bar{\Omega}_{2,\eta}$ for each
$\eta>0$ we obtain an at most countable disjoint collection of finite
perimeter sets,

$\Omega_{0,\eta}=\bigcup_{k}\Omega_{0,\eta}^{k},$

for which each $\Omega_{0,\eta}^{k}$ is indecomposable. In two dimensions, the
diameter of each indecomposable component is controlled by its perimeter, and
so (c) holds for each $\Omega^{k}_{0,\eta}$. Using the
$\\{\Omega^{k}_{0,\eta}\\}$ to separate $\Omega_{i,\eta}$ into essentially
disjoint components
$\Omega_{i,\eta}^{k}=\Omega_{i,\eta}\cap\Omega^{k}_{0,\eta}\,$, $i=1,2$, the
decomposition of the perimeters (b) then follows. ∎

In order to prove Gamma-convergence results and convergence of minimizers we
need to expand the energy in terms of the indecomposable clusters
$\Omega_{\eta}^{k}=(\Omega_{1,\eta}^{k},\Omega_{2,\eta}^{k})$. Since the
diameter of the support $\Omega_{\eta}^{k}$ is small compared to
${\mathbb{T}^{2}}$, we may think of each $v_{i,\eta}^{k}$ as a function on
${\mathbb{R}}^{2}$. This enables us to blow up each component at scale $\eta$
to determine the fine structure of the configuration $v_{\eta}$. For each
fixed $k\in{\mathbb{N}}$ choose any point
$\xi_{\eta}^{k}\in\Omega_{0,\eta}^{k}\subset{\mathbb{T}^{2}}$. Then for
$i=1,2,$ we define a pair $z_{i}^{k}(x):\mathbb{R}^{2}\rightarrow\\{0,1\\}$ ,
$i=1,2$, by

$\displaystyle z_{i}^{k}(x):=\eta^{2}v_{i,\eta}^{k}(\xi_{\eta}^{k}+\eta
x)=\chi_{A_{i,\eta}^{k}}(x),\quad i=1,2,$ (4.5)

where
$A_{i,\eta}^{k}=\eta^{-1}\left(\Omega_{i,\eta}^{k}-\xi_{\eta}^{k}\right)\subset{\mathbb{R}}^{2}$,
$i=1,2$, with $A^{k}_{\eta}=(A^{k}_{1,\eta},A^{k}_{2,\eta})$ a finite
perimeter 2-cluster in ${\mathbb{R}}^{2}$. Then,

$\displaystyle\int_{\mathbb{R}^{2}}z_{i}^{k}=\int_{{\mathbb{T}^{2}}}v_{i,\eta}^{k},\text{
and }\int_{\mathbb{R}^{2}}|\nabla
z_{i}^{k}|=\eta\int_{{\mathbb{T}^{2}}}|\nabla v_{i,\eta}^{k}|.$ (4.6)

As above, we set
$A^{k}_{0,\eta}=\bar{A}^{k}_{1,\eta}\cup\bar{A}^{k}_{2,\eta}\,$, which has the
same perimeter as the exterior domain.

For a 2-cluster $A=(A_{1},A_{2})$, $A_{i}\subset{\mathbb{R}}^{2}$, $i=1,2$ of
finite perimeter, and $A_{0}=A_{1}\cup A_{2}$, we define

$\mathcal{G}_{\sigma}(A):=P_{\sigma}(A)+\sum_{i,j=1}^{2}\frac{\Gamma_{ij}}{4\pi}|A_{i}|\,|A_{j}|,$

and for $m=(m_{1},m_{2})$,

$\displaystyle e_{0}(m)$ $\displaystyle:=\min\left\\{\mathcal{G}_{\sigma}(A)\
|\ A=(A_{1},A_{2})\text{ 2-cluster, with $|A_{i}|=m_{i}$, $i=1,2$}\right\\}$

By the results of Section 2, the minimum in $e_{0}(m)$ is attained for all
$m=(m_{1},m_{2})$ nonzero, with the geometry of minimizers determined by the
choice of weights $\sigma_{ij}$, a double bubble, core shell, or single bubble
(which in particular occurs if either $m_{i}=0$, $i=1$ or $2$.) We expect that
the effect of the nonlocal interaction will be to enforce splitting of the
mass into several droplets, but that energy minimization will determine the
distribution of the droplet masses. To this end, we define

$\displaystyle\overline{e_{0}}(M):=\inf\left\\{\sum_{k=1}^{\infty}e_{0}(m^{k}):m^{k}=(m_{1}^{k},m_{2}^{k}),\
m_{i}^{k}\geq 0,\ \sum_{k=1}^{\infty}m_{i}^{k}=M_{i},i=1,2\right\\},$ (4.7)

where $M=(M_{1},M_{2})$.

Expanding $v_{\eta}$ into component clusters, and separating the on-diagonal
terms in the double sum,

$\displaystyle E_{\eta}(v_{\eta})$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{i=0}^{2}\frac{\eta}{2}\beta_{i}\int_{{\mathbb{T}^{2}}}|\nabla
v_{i,\eta}^{k}|+{\Gamma_{ij}\over
2|\log\eta|}\sum_{k,\ell=1}^{\infty}\sum_{i,j=1}^{2}\int_{{\mathbb{T}^{2}}}\int_{{\mathbb{T}^{2}}}v_{i,\eta}^{k}(x)\,G_{\mathbb{T}^{2}}(x-y)\,v_{j,\eta}^{\ell}(y)\,dx\,dy$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{i=0}^{2}\frac{\beta_{i}}{2}\int_{\mathbb{R}^{2}}|\nabla
z_{i}^{k}|+{\Gamma_{ij}\over
2|\log\eta|}\sum_{k,\ell=1}^{\infty}\sum_{i,j=1}^{2}\int_{A_{i}^{k}}\int_{A_{j}^{\ell}}G_{\mathbb{T}^{2}}(\xi^{k}_{\eta}+\eta\tilde{x}-\xi^{\ell}_{\eta}-\eta\tilde{y})\,d\tilde{x}\,d\tilde{y}$
$\displaystyle=\sum_{k=1}^{\infty}\left[P_{\sigma}(A^{k}_{\eta})+\sum_{i,j=1}^{2}{\Gamma_{ij}\over
4\pi}|A^{k}_{i,\eta}|\,|A^{k}_{j,\eta}|\right]+|\log\eta|^{-1}\Phi_{\eta}$
$\displaystyle=\sum_{k=1}^{\infty}\mathcal{G}_{\sigma}(A^{k}_{\eta})+|\log\eta|^{-1}\Phi_{\eta},$
(4.8)

where the remainder terms include a first piece depending on only the geometry
of the indecomposable clusters, and a second piece containing off-diagonal
terms which encodes the interactions between them,

$\displaystyle\Phi_{\eta}$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{i,j=1,2}{\Gamma_{ij}\over
2}\int_{A^{k}_{i,\eta}}\int_{A^{k}_{j,\eta}}\left({1\over
2\pi}\log{1\over|x-y|}+R_{{\mathbb{T}^{2}}}(\eta(x-y))\right)dx\,dy$
$\displaystyle\qquad+\sum_{k\neq\ell=1}^{\infty}\sum_{i,j=1,2}{\Gamma_{ij}\over
2}\int_{A_{i,\eta}^{k}}\int_{A_{j,\eta}^{\ell}}G_{\mathbb{T}^{2}}(\xi_{\eta}^{k}+\eta\tilde{x}-\xi_{\eta}^{\ell}-\eta\tilde{y})\,d\tilde{x}\,d\tilde{y}$
$\displaystyle=:\Phi^{1}(\\{A^{k}_{\eta}\\}_{k\in{\mathbb{N}}})+\Phi^{2}_{\eta}(\\{A^{k}_{\eta}\\}_{k\in{\mathbb{N}}},\\{\xi^{k}_{\eta}\\}_{k\in{\mathbb{N}}})$
(4.9)

###  4.3 The first order limit

The decomposition (4.8) induced by Lemma 4.1 suggests a first order Gamma
limit for the functional $E_{\eta}$, in which the mass splits via
$\overline{e}_{0}(M)$, encoding the geometric information from the weighted
isoperimetric energy $\mathcal{G}_{\sigma}$.

We define a class of measures with countable support on ${\mathbb{T}^{2}}$,

$Y:=\left\\{v_{0}=\sum_{k=1}^{\infty}(m_{1}^{k},m_{2}^{k})\,\delta_{x^{k}}\ |\
m_{i}^{k}\geq 0,\ x^{k}\in{\mathbb{T}^{2}}\ \text{distinct points}\right\\},$

and a functional on $Y$,

$\displaystyle
E_{0}(v_{0}):=\left\\{\begin{array}[]{rcl}\sum_{k=1}^{\infty}\overline{e_{0}}(m^{k}),&&\text{
if }v_{0}\in Y,\\\ \infty,&&\text{ otherwise. }\end{array}\right.$ (4.12)

As in [1], we have a first Gamma-convergence result:

######  Theorem 4.2 (First $\Gamma$-Limit).

We have

$\displaystyle E_{\eta}\overset{\Gamma}{\longrightarrow}E_{0}\ \text{ as }\
\eta\rightarrow 0.$

That is,

  1. (a)

Let $v_{\eta}\in X_{\eta}$ be a sequence with
$\sup_{\eta>0}E_{\eta}(v_{\eta})<\infty$. Then there exists a subsequence
$\eta\to 0$ and $v_{0}\in Y$ such that $v_{\eta}\rightharpoonup v_{0}$ (in the
weak topology of the space of measures), and

$\displaystyle\underset{\eta\rightarrow 0}{\lim\inf}E_{\eta}(v_{\eta})\geq
E_{0}(v_{0}).$

  2. (b)

Let $v_{0}\in Y$ with $E_{0}(v_{0})<\infty$. Then there exists a sequence
$v_{\eta}\rightharpoonup v_{0}$ weakly as measures, such that

$\displaystyle\underset{\eta\rightarrow 0}{\lim\sup}E_{\eta}(v_{\eta})\leq
E_{0}(v_{0}).$

As the proof is very similar to that of [1, Theorem 3.2], we provide only a
sketch here. For the lower semicontinuity result (a), we take any sequence
$(v_{\eta})_{\eta>0}$ with uniformly bounded energy $E_{\eta}(v_{\eta})\leq
C$, and apply [1, Lemma 3.5]. Note that the proof of Lemma 3.5 of [1], and of
all the convergence results in that paper, were given for the case of equal
weights $\beta_{i}=1$, $i=0,1,2$, but the proofs are identical for the
weighted case, as long as the triangle inequalities (1.5) hold. It states
that, up to a subsequence $\eta\to 0$ (not relabelled,) there exist at most
countably many 2-clusters $\\{A^{k}\\}$ in ${{\mathbb{R}}^{2}}$ and
corresponding points $\\{x_{\eta}^{k}\\}$ in ${\mathbb{T}^{2}}$, such that the
domain $\Omega_{\eta}=\bigcup_{k\in{\mathbb{N}}}\Omega_{\eta}^{k}$ (with
indecomposable clusters $\Omega_{\eta}^{k}$,) satisfies:

$\left|A^{k}\
\triangle\,\left(\eta^{-1}\left[\Omega_{\eta}^{k}-x_{\eta}^{k}\right]\right)\right|=\left|A^{k}\
\triangle\,A^{k}_{\eta}\right|\xrightarrow{\eta\rightarrow 0}0,\qquad\forall
k.$ (4.13)

Moreover, the masses of the clusters are preserved in the limit,

$M_{i}=\lim_{\eta\to
0}\sum_{k=1}^{\infty}\eta^{-2}|\Omega_{i,\eta}^{k}|=\sum_{k=1}^{\infty}|A_{i}^{k}|,\quad
i=1,2,$ (4.14)

and the total energy is bounded below by $\overline{e_{0}}(M)$.

$\liminf_{\eta\to
0}E_{\eta}(v_{\eta})\geq\sum_{k=1}^{\infty}\mathcal{G}(A^{k})\geq\overline{e_{0}}(M).$
(4.15)

At this point, note that (4.15) is the lower bound of the first Gamma
convergence result, Theorem 4.2. The remainder of the Gamma convergence then
follows the proof of Theorem 3.2 in [1].

The geometrical structure can be made more precise for minimizers: the
limiting clusters after blow-up must minimize the weighted perimeter
functional $\mathcal{G}_{\sigma}$ with the given mass distribution imposed by
$\overline{e_{0}}(M)$. Indeed, it is enough to assume that
$E_{\eta}(v_{\eta})\to\overline{e_{0}}(M)$ to obtain this conclusion:

######  Proposition 4.3.

Let $v_{\eta}^{\ast}=\eta^{-2}\chi_{\Omega_{\eta}}\in X_{\eta}$ such that
$\displaystyle\lim_{\eta\to 0}E_{\eta}=\overline{e_{0}}(M)$. Then, there
exists a subsequence $\eta\to 0$ (still denoted by $\eta$) such that:

  1. (i)

there exist connected clusters $(A^{k})_{k\in{\mathbb{N}}}$ in
${{\mathbb{R}}^{2}}$ and points $x_{\eta}^{k}\in{\mathbb{T}^{2}}$,
$k\in{\mathbb{N}}$, for which

$\eta^{-2}\left|\Omega_{\eta}\ \triangle\ \bigcup_{k=1}^{\infty}\left(\eta
A^{k}+x_{\eta}^{k}\right)\right|\xrightarrow{\eta\rightarrow 0}0,$ (4.16)

and

$M_{i}=\sum_{k=1}^{\infty}|A_{i}^{k}|;$

  2. (ii)

each $A^{k}$ is a minimizer of $\mathcal{G}_{\sigma}$:

$\mathcal{G}_{\sigma}(A^{k})=e_{0}(m^{k}),\qquad
m^{k}=(m_{1}^{k},m_{2}^{k})=|A^{k}|;$ (4.17)

Moreover,

$\overline{e_{0}}(M)=\lim_{\eta\to
0}E_{\eta}(v_{\eta})=\sum_{k=1}^{\infty}\mathcal{G}_{\sigma}(A^{k})=\sum_{k=1}^{\infty}e_{0}(m^{k}).$
(4.18)

  3. (iii)

If $k\neq\ell$, $d(x^{k}_{\eta},x^{\ell}_{\eta})\gg\eta$; more precisely,

$\lim_{\eta\to 0}{|\log
d(x^{k}_{\eta},x^{\ell}_{\eta})|\over|\log\eta|}=0,\quad\forall k\neq\ell.$

######  Remark 4.2.

In Corollary 3.2 we prove that in the core shell case $\beta_{1}=0$,
minimizers of $\overline{e_{0}}(M)$ may only have finitely many nontrivial
clusters, $A^{1},\dots,A^{K}$, for some $K\in{\mathbb{N}}$. This was also the
case in the unweighted case, and in the cases of single bubbles
($\beta_{0}=0$) as proven in [1, Theorem 1.1].

The proof of Proposition 4.3 follows exactly those of Lemmas 3.5 and 3.6 in
[1] (recalling from Remark 4.1 that the weighted perimeter controls the
relevant BV norms in the convergence steps.)

###  4.4 Second order limit

To obtain the Second Gamma Limit, Theorem 4.5, and the more detailed
description of limiting configurations, we must analyze the remainder term in
(4.9). As the analysis in the case of strict triangle inequalities (1.5)
(leading to minimizers with double bubbles) or in case $\beta_{0}=0$ (yielding
only single bubbles) is essentially identical to the unweighted case of our
previous paper [1], we will concentrate on the case $\beta_{1}=0$, in which
core shells are favored.

We subtract the first $\Gamma$-limit from $E_{\eta}$, which (from the
decomposition (4.8), (4.9)) we expect to have scale $|\log\eta|^{-1}$. For
$v_{\eta}\in X_{\eta}$, let

$\displaystyle
F_{\eta}(v_{\eta}):=|\log\eta|\left[E_{\eta}(v_{\eta})-\overline{e_{0}}(M)\right].$
(4.19)

For the second $\Gamma$-limit we consider $v_{\eta}\in X_{\eta}$ for which
$F_{\eta}(v_{\eta})$ is bounded. For these $v_{\eta}$, Proposition 4.3
applies, and we may thus assume that (along subsequences) the detailed
concentration structure described by (4.13), (4.14), (4.15) is observed. Given
Lemma 2.6, each limiting cluster $A^{k}=(A^{k}_{1},A^{k}_{2})$ is either a
core shell $\mathcal{C}^{m^{k}_{1}}_{m^{k}_{2}}$ (if $m_{1}^{k},m_{2}^{k}>0$,)
or a single bubble (in case one of $m_{i}^{k}=0$.) We recall that minimization
of $\mathcal{G}_{\sigma}$ does not determine the centers of core shells; this
will be addressed below.

Next we define the second order limit function, starting with its domain,
which is suggested from the first order limit Theorem 4.2 and the structure
described in Proposition 4.3. For $K\in\mathbb{N}$, $m_{1}^{k}\geq 0$,
$m_{2}^{k}\geq 0$ and $m_{1}^{k}+m_{2}^{k}>0$, the sequence
$K\otimes(m_{1}^{k},m_{2}^{k})$ is defined by

$\displaystyle(K\otimes(m_{1}^{k},m_{2}^{k}))^{k}:=\left\\{\begin{aligned}
&(m_{1}^{k},m_{2}^{k}),&1\leq k\leq K,\\\ &(0,0),&K+1\leq
k<\infty.\end{aligned}\right.$

Let ${\cal{M}}_{M}$ be the set of optimal sequences made of all clusters for
the problem (4.7):

$\displaystyle{\cal{M}}_{M}:=\Bigg{\\{}K\otimes(m_{1}^{k},m_{2}^{k}):K\otimes(m_{1}^{k},m_{2}^{k})\text{
minimizes \eqref{mine0} for }M_{i},i=1,2,$ $\displaystyle\text{ and }\
\overline{e_{0}}(m^{k})=e_{0}(m^{k}),\ m^{k}=(m_{1}^{k},m_{2}^{k})\Bigg{\\}}.$

Let $Y_{M}$ denote the space of all measures
$v_{0}=\sum_{k=1}^{K}m^{k}\delta_{x^{k}}$ with $\\{x^{1},\dots,x^{K}\\}$
distinct points in ${\mathbb{T}^{2}}$ and $K\otimes m^{k}\in\mathcal{M}_{M}$.
That is, $v_{0}\in Y_{M}$ represents the limit (in the sense of distributions)
of a sequence of energy minimizers $v_{\eta}$ of mass $M$, which obey the
conclusions of Proposition 4.3. We recall that the finiteness of these
components is a consequence of Corollary 3.2.

We now describe the terms appearing in the second $\Gamma$-limit, which arise
from passage to the limit in $\Phi_{\eta}$, defined in (4.9). First, the
``self-interaction'' terms making up
$\Phi^{1}(\\{A^{k}_{\eta}\\}_{k\in{\mathbb{N}}})$. For $m=(m_{1},m_{2})$,
$m_{1},m_{2}\geq 0$ and not both zero, define

$f_{0}(m)=\inf\biggl{\\{}\sum_{i,j=1,2}\frac{\Gamma_{ij}}{2}\left[\frac{1}{2\pi}\int_{A_{i}}\int_{A_{j}}\log{1\over|x-y|}dx\,dy+m_{i}\,m_{j}\,R_{{\mathbb{T}^{2}}}(0)\right]\
:\ \\\ \ \ A=(A_{1},A_{2})\ \text{minimizes $\mathcal{G}_{\sigma}$ with
$|A_{\ell}|=m_{\ell}$, $\ell=1,2$}\biggr{\\}}$ (4.20)

The minimization here is only pertinent in case $m_{1},m_{2}>0$, that is,
where $A=\mathcal{C}^{m_{1}}_{m_{2}}$ is a core shell; for single bubbles the
minimizers of $\mathcal{G}_{\sigma}$ with one of $m_{i}=0$ are unique up to
rigid motion, and the set above is a singleton. It is in this term that energy
minimization of the nonlocal energy resolves the degeneracy in the local
isoperimetric problem, in favor of centered core shells:

######  Proposition 4.4.

Let $\beta_{1}=0$, and $m=(m_{1},m_{2})$ with $m_{1},m_{2}>0$.

  1. (a)

If $\Gamma_{11}>\Gamma_{12}$, then the minimum in $f_{0}(m)$ is attained by a
concentric core shell $A=\mathcal{C}^{m_{1}}_{m_{2}}$.

  2. (b)

If $\Gamma_{11}<\Gamma_{12}$, then the minimum in $f_{0}(m)$ is attained by a
core shell $A=\mathcal{C}^{m_{1}}_{m_{2}}$ whose inner boundary circle is
tangent to the outside circle.

###### Proof.

Given $m_{1},m_{2}>0$, from Lemma 2.6 we know that all minimizers of
$\mathcal{G}_{\sigma}$ with $m_{1},m_{2}>0$ are core shells, that is they are
composed of two (round) balls $B_{1}=B_{r_{1}}(0)$, $B_{2}=B_{r_{2}}(p)$, with
$r_{1}>r_{2}$, and the outer domain $A_{1}=B_{1}\setminus B_{2}$, with inner
disk $A_{2}=B_{2}$. The distance between $p$ (the center of $B_{2}$) and $0$
must be small enough that $A_{2}\subset A_{1}$. We must choose $p$ to minimize
the quantity

$\sum_{i,j=1}^{2}\frac{\Gamma_{ij}}{2}\left[I_{ij}+m_{i}m_{j}R_{{\mathbb{T}^{2}}}(0)\right],\quad\text{where}\quad
I_{ij}=\frac{1}{2\pi}\int_{A_{i}}\int_{A_{j}}\log{1\over|x-y|}dx\,dy.$

Notice that $I_{22}$ is independent of the location of $p$, as are of course
the terms $m_{i}m_{j}R_{{\mathbb{T}^{2}}}(0)$.

For any $x\in\mathbb{R}^{2}\setminus B_{2}$, by the Mean Value Theorem for
harmonic functions,

$\displaystyle\int_{B_{2}}\log{1\over|x-y|}dy=|B_{2}|\log{1\over|x-p|}.$

Thus,

$\displaystyle I_{12}$ $\displaystyle=\frac{m_{2}}{2\pi}\int_{B_{1}\setminus
B_{2}}\log{1\over|x-p|}dx=\frac{m_{2}}{2\pi}\left[\int_{B_{1}}\log{1\over|x-p|}dx-\int_{B_{2}}\log{1\over|x-p|}dx\right]$
$\displaystyle=\frac{m_{2}}{2\pi}\int_{B_{1}}\log{1\over|x-p|}dx-I_{22}.$

We claim that $I_{12}=I_{12}(p)$ is maximized when $p=0$ and minimized when
the circles are tangent. By symmetry we may assume $p=(t,0)$, with $t\geq 0$
lying in an interval $t\in[0,\alpha]$ for which $A_{2}\subset A_{1}$.
Differentiating,

${d\over dt}I_{12}(t,0)={1\over
2\pi}\int_{B_{1}}{x_{1}-t\over|x-(t,0)|^{2}}dx.$ (4.21)

Let $\omega_{t}=\\{x\in B_{{\color[rgb]{1,0,0}1}}\ |\ x_{1}\geq t\\}$, and
$\tilde{\omega}_{t}$ its reflection in the line $\\{x_{1}=t\\}$. As the
integrand in (4.21) is odd around the line $\\{x_{1}=t\\}$, and
$B_{1}\setminus(\omega_{t}\cup\tilde{\omega}_{t})\subset\\{x_{1}<t\\}$, we
have $\forall t\in(0,\alpha]$,

${d\over dt}I_{12}(t,0)={1\over
2\pi}\int_{B_{1}\setminus(\omega_{t}\cup\tilde{\omega}_{t})}{x_{1}-t\over|x-(t,0)|^{2}}dx<0.$

Thus, $I_{12}(t,0)$ is even and strictly monotone decreasing in
$t\in[0,\alpha]$, so it attains its maximum value at $t=0$ and minimum at the
extreme value $t=\alpha$, where the circles are tangent. This concludes the
proof of the claim.

To complete the argument, we note that,

$\displaystyle I_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{A_{1}}\int_{A_{1}}\log{1\over|x-y|}dx\,dy=\frac{1}{2\pi}\int_{A_{1}}\int_{B_{1}}\log{1\over|x-y|}dx\,dy-
I_{12}$ $\displaystyle=$ $\displaystyle-
I_{12}+\frac{1}{2\pi}\left[\int_{B_{1}}\int_{B_{1}}\log{1\over|x-y|}dx\,dy-\int_{B_{2}}\int_{B_{1}}\log{1\over|x-y|}dx\,dy\right]$
$\displaystyle=$
$\displaystyle-2I_{12}+\frac{1}{2\pi}\int_{B_{1}}\int_{B_{1}}\log{1\over|x-y|}dx\,dy-
I_{22}.$

As $I_{22}$ and
$\frac{1}{2\pi}\int_{B_{1}}\int_{B_{1}}\log{1\over|x-y|}dx\,dy$ are
independent of the location of $p$, we have

$\displaystyle\sum_{i,j=1}^{2}\frac{\Gamma_{ij}}{2}I_{ij}$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{11}}{2}(-2I_{12}(p))+\Gamma_{12}I_{12}(p)+\text{
terms independent of }p$ (4.22) $\displaystyle=$
$\displaystyle-(\Gamma_{11}-\Gamma_{12})I_{12}(p)+\text{ terms independent of
}p.$

The stated conclusions then follow, depending on the sign of the coefficient
$(\Gamma_{11}-\Gamma_{12})$. ∎

We note that in the case $\Gamma_{11}<\Gamma_{12}$ it may be difficult to
observe tangential core shells, since that choice of strong repulsion between
the two phases may result in core shells being split into single bubbles.

With this satisfactory resolution of the question of the specific geometry of
optimal core shells, we may now define the second $\Gamma$-limit functional,
by including the formal limits of the two remainder terms from (4.9). For
$v_{0}\in Y_{M}$, we define

$F_{0}(v_{0})=\sum_{k=1}^{K}f_{0}(m^{k})+\sum_{k,\ell=1\atop
k\neq\ell}^{K}\sum_{i,j=1}^{2}\frac{\Gamma_{ij}}{2}m_{i}^{k}m_{j}^{\ell}G_{\mathbb{T}^{2}}(x_{i}^{k}-x_{j}^{\ell}),$
(4.23)

and $F_{0}(v_{0})=+\infty$ otherwise. The first term in (4.23) is thus already
completely determined by the first $\Gamma$-limit and the fine structure of
core shells as determined by Proposition 4.4, and only the spatial
distribution of droplets is at play in $F_{0}$.

######  Theorem 4.5 (Second $\Gamma$-limit).

We have

$\displaystyle F_{\eta}\overset{\Gamma}{\longrightarrow}F_{0}\ \text{ as }\
\eta\rightarrow 0.$

That is,

  1. (a)

Let $v_{\eta}\in X_{\eta}$ be a sequence with
$\sup_{\eta>0}F_{\eta}(v_{\eta})<\infty$. Then there exists a subsequence
$\eta\to 0$ and $v_{0}\in Y_{M}$ such that $v_{\eta}\rightharpoonup v_{0}$ (in
the weak topology of the space of measures), and

$\displaystyle\underset{\eta\rightarrow 0}{\lim\inf}F_{\eta}(v_{\eta})\geq
F_{0}(v_{0}).$

  2. (b)

Let $v_{0}\in Y_{M}$ with $F_{0}(v_{0})<\infty$. Then there exists a sequence
$v_{\eta}\rightharpoonup v_{0}$ weakly as measures, such that

$\displaystyle\underset{\eta\rightarrow 0}{\lim\sup}F_{\eta}(v_{\eta})\leq
F_{0}(v_{0}).$

The tools required for the proof of the second limit are essentially included
in our paper [1], in the unweighted case. We provide a slightly more detailed
sketch here for the case $\beta_{1}=0$, as the paper [1] concentrated on the
first $\Gamma$-limit and the structure of minimizers and did not give details
for the second $\Gamma$-limit.

###### Sketch of the proof of Theorem 4.5.

We begin with compactness and the lower limit. Take a sequence $\eta_{n}\to 0$
and assume $v_{\eta_{n}}\in X_{\eta}$ is a sequence with bounded
$F_{\eta_{n}}(v_{\eta_{n}})$. (Recall $F_{\eta}$ is defined in (4.19).) For
convenience, we abuse notation and denote this sequence (and all forthcoming
subsequences) simply as $v_{\eta}$, and $\eta\to 0$ along this sequence (and
eventually, certain subsequences.) By the boundedness of $F_{\eta}(v_{\eta})$
we immediately conclude that

$\lim_{\eta\to 0}E_{\eta}(v_{\eta})=\overline{e_{0}}(M),$

and hence Lemma 4.1 and Proposition 4.3 (Lemma 3.5 in [1]) apply to the
sequence $v_{\eta}$. As a consequence,
$v_{\eta}=\eta^{-2}\chi_{\Omega_{\eta}}$ splits into at most countably many
indecomposable clusters, $\Omega_{\eta}=\bigcup_{k}\Omega_{\eta}^{k}$, whose
energy $E_{\eta}(v_{\eta})$ breaks down via (4.8). Moreover, by Corollary 3.2
in the core shell case $\beta_{1}=0$ minimizers of $\overline{e_{0}}(M)$ can
only have a finite number $K\in{\mathbb{N}}$ of nontrivial components. Hence,

$\displaystyle E_{\eta}(v_{\eta})-\overline{e_{0}}(M)$
$\displaystyle=\sum_{k=1}^{K}\mathcal{G}_{\sigma}(A^{k}_{\eta})-\overline{e_{0}}(M)+|\log\eta|^{-1}\Phi_{\eta}(A_{\eta}^{{\color[rgb]{1,0,0}k}},\xi^{k}_{\eta})$
$\displaystyle\geq\sum_{k=1}^{K}e_{0}(|A^{k}_{\eta}|)-\overline{e_{0}}(M)+|\log\eta|^{-1}\Phi_{\eta}(A^{k}_{\eta},\xi^{k}_{\eta})$
$\displaystyle\geq|\log\eta|^{-1}\Phi_{\eta}(A^{k}_{\eta},\xi^{k}_{\eta}),$

since $\overline{e_{0}}(M)\leq\sum_{k=1}^{K}e_{0}(m^{k})$ whenever
$\sum_{k=1}^{K}m^{k}=M$. That is,
$\Phi_{\eta}(A^{k}_{\eta},\xi^{k}_{\eta})\leq F_{\eta}(v_{\eta})\leq C$ are
uniformly bounded as $\eta\to 0$.

Next, we pass to the limit in
$\Phi^{1}(\\{A^{k}_{\eta}\\}_{k\in{\mathbb{N}}})$ in (4.9). By (4.16), for
each $k$, the clusters $|A^{k}_{\eta}\triangle A^{k}|\to 0$. In addition, by
Lemma 4.1 (c), the diameters diam$\,{A^{k}_{\eta}}\leq C$ are uniformly
bounded in $k,\eta$. As $\log{1\over|x-y|}$ is locally integrable, we may then
pass to the limit using dominated convergence,

$\lim_{\eta\to
0}\Phi^{1}(\\{A^{k}_{\eta}\\}_{k\in{\mathbb{N}}})=\sum_{i,j=1,2}\frac{\Gamma_{ij}}{2}\left[\frac{1}{2\pi}\int_{A_{i}}\int_{A_{j}}\log{1\over|x-y|}dx\,dy+m_{i}\,m_{j}\,R_{{\mathbb{T}^{2}}}(0)\right]\geq\sum_{k=1}^{K}f_{0}(m^{k}).$
(4.24)

Since $\Phi^{1}$ converges, we conclude that the second interaction term
$\Phi^{2}_{\eta}(\\{A^{k}_{\eta}\\}_{k\in{\mathbb{N}}},\\{\xi^{k}_{\eta}\\}_{k\in{\mathbb{N}}})$
is also bounded above. To pass to the limit in this term, we first note that
if for some pair $k\neq\ell$,
$d(\xi^{k}_{\eta},\xi^{\ell}_{\eta})\geq\delta_{k,\ell}>0$ for all $\eta>0$,
then by a similar argument as above we may pass to the limit in this $k,\ell$
term and obtain the corresponding term in $F_{0}$ (see (4.23).) We claim that
this must be the case for all pairs $k\neq\ell$. Indeed, we assume for some
$k\neq\ell$ (and a subsequence $\eta\to 0$ that
$r_{\eta}^{k,\ell}:=d(\xi^{k}_{\eta},\xi^{\ell}_{\eta})\to 0$, then recall
that by Proposition 4.3 (iii) we must nevertheless have for $i,j=1,2$,

$\displaystyle\int_{A_{i,\eta}^{k}}\int_{A_{j,\eta}^{\ell}}G_{\mathbb{T}^{2}}(\xi_{\eta}^{k}+\eta\tilde{x}-\xi_{\eta}^{\ell}-\eta\tilde{y})\,d\tilde{x}\,d\tilde{y}$
$\displaystyle=\int_{A_{i,\eta}^{k}}\int_{A_{j,\eta}^{\ell}}\left[{1\over
2\pi}\log{1\over|\xi^{k}_{\eta}-\xi^{\ell}_{\eta}|}+R_{\mathbb{T}^{2}}(\xi_{\eta}^{k}+\eta\tilde{x}-\xi_{\eta}^{\ell}-\eta\tilde{y})\right]d\tilde{x}\,d\tilde{y}$
$\displaystyle=|A_{i,\eta}^{k}||A_{j,\eta}^{\ell}|\left({1\over 2\pi}|\log
r^{k,\ell}_{\eta}|+R_{\mathbb{T}^{2}}(0)+o(1)\right).$

Since $|A^{k}|>0$ for $k=1,\dots,K$, there exists $i,j\in\\{1,2\\}$ for which
$|A^{k}_{i}||A^{\ell}_{j}|>0$, and that term will be unbounded above. This
contradicts the upper bound on
$\Phi^{2}_{\eta}(\\{A^{k}_{\eta}\\}_{k\in{\mathbb{N}}},\\{\xi^{k}_{\eta}\\}_{k\in{\mathbb{N}}})$,
and hence we conclude that the points $\\{\xi^{k}_{\eta}\\}_{k=1,\dots,K}$
must remain distinct as $\eta\to 0$. Passing to a further subsequence if
necessary, we may assume $\xi^{k}_{\eta}\to x^{k}\in{\mathbb{T}^{2}}$, with
distinct $x^{k}\neq x^{\ell}$, $k\neq\ell$, and

$\liminf_{\eta\to 0}F_{\eta}(v_{\eta})\geq\liminf_{\eta\to
0}\Phi_{\eta}(A^{k}_{\eta},\xi^{k}_{\eta})\geq F_{0}(v_{0}),$

where $v_{0}=\sum_{k=1}^{K}m^{k}\delta_{x^{k}}\in Y_{M}$. Note that (repeating
an argument from above) by (4.16) and the uniform boundedness of the diameters
of $A^{k}_{\eta}$, we may also conclude that $v_{\eta}\rightharpoonup v_{0}$
in the weak topology on the space of measures on ${\mathbb{T}^{2}}$. This
concludes the proof of the compactness and lower limit of the second
$\Gamma$-limit.

The construction of a recovery sequence for part 2 is straightforward: given
points $x^{k}$ and masses $m^{k}=(m^{k}_{1},m^{k}_{2})$, it suffices to
assemble a superposition of $\eta$-rescaled minimizers of $e_{0}(m^{k})$ at
each $x^{k}$. When the minimizing cluster $A^{k}$ is a core shell, the central
disk is positioned centered when $\Gamma_{12}$ is small according to
Proposition 4.4.∎

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