The Γ-limit of the two-dimensional
Ohta-Kawasaki energy. I. Droplet density.
Dorian Goldman, Cyrill B. Muratov and Sylvia Serfaty
May 29, 2013
Abstract
This is the first in a series of two papers in which we derive a Γ-expansion fora two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also
known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In
that model, two phases appear, which interact via a nonlocal Coulomb type energy.
We focus on the regime where one of the phases has very small volume fraction, thus
creating small “droplets” of the minority phase in a “sea” of the majority phase. In
this paper we show that an appropriate setting for Γ-convergence in the consideredparameter regime is via weak convergence of the suitably normalized charge density inthe sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki
energy functional Γ-converges to a quadratic energy functional of the limit charge
density generated by the screened Coulomb kernel. A consequence of our results isthat minimizers (or almost minimizers) of the energy have droplets which are almost
all asymptotically round, have the same radius and are uniformly distributed in thedomain. The proof relies mainly on the analysis of the sharp interface version of
the energy, with the connection to the original diffuse interface model obtained via
matching upper and lower bounds for the energy. We thus also obtain an asymptotic
characterization of the energy minimizers in the diffuse interface model.

1

Introduction

In the studies of energy-driven pattern formation, one often encounters variational problems
with competing terms operating on different spatial scales [25,26,32,39,50,53,55]. Despite
the fundamental importance of these problems to a multitude of physical systems, theirdetailed mathematical studies are fairly recent (see e.g. [7–11,19,29,48]). To a great extent
this fact is related to the emerging multiscale structure of the energy minimizing patterns
and the associated difficulty of their description [8,10,16,30,35]. In particular, the popular
approach of Γ-convergence [4] is rendered difficult due to the emergence of more than two
well-separated spatial scales in suitable asymptotic limits (see e.g. [8–11, 16, 30, 35, 49]).
These issues can be readily seen in the case of the Ohta-Kawasaki model, a canonicalmathematical model in the studies of energy-driven pattern forming systems. This model,
1originally proposed in [42] to describe different morphologies observed in diblock copolymer
melts (see e.g. [3]) is defined (up to a choice of scales) by the energy functional

Z Z
Z  2
ε
1
2
(u(x) − ū)G0 (x, y)(u(y) − ū) dx dy,(1.1)
E[u] =|∇u| + W (u) dx +
2
2 Ω Ω
Ω
where Ω is the domain occupied by the material, u : Ω → R is the scalar order parameter,
W (u) is a symmetric double-well potential with minima at u = ±1, such as the usual
Ginzburg-Landau potential W (u) = 14 (1 − u2 )2 , ε > 0 is a parameter characterizing interfacial thickness, ū ∈ (−1, 1) is the background charge density, and G0 is the Neumann
Green’s function of the Laplacian, i.e., G0 solves
Z
1−∆G0 (x, y) = δ(x − y) −G0 (x, y) dx = 0,
(1.2)
,
|Ω|
Ω
where ∆ is the Laplacian in x and δ(x) is the Dirac delta-function, with Neumann boundary
conditions. Note that u is also assumed to satisfy the “charge neutrality” conditionZ1
u dx = ū.
(1.3)
|Ω| Ω
Let us point out that in addition to a number of polymer systems [15, 41, 52], this modelis also applicable to many other physical systems due to the fundamental nature of the
Coulombic non-local term in (1.1) [6,17,22,32,37,40]. Because of this Coulomb interaction,
we also like to think of u as a density of “charge”.
The Ohta-Kawasaki functional admits the following “sharp-interface” version:
Z
Z Z
ε
1
E[u] =|∇u| dx +(u(x) − ū)G(x, y)(u(y) − ū) dx dy,
(1.4)
2 Ω
2 Ω Ω
where now u : Ω → {−1, +1} and G(x, y) is the screened Green’s function of the Laplacian,i.e., it solves the Neumann problem for the equation (distinguish from (1.2))−∆G + κ2 G = δ(x − y),

(1.5)

p
where κ := 1/ W 00 (1) > 0. Note also that in contrast to the diffuse interface energy in(1.1), for the sharp interface energy in (1.4) the charge neutrality constraint in (1.3) is nolonger imposed. This is due to the fact that in a minimizer of the diffuse interface energy,
the charge of the minority phase is expected to partially redistribute into the majority
phase to ensure screening of the induced non-local field (see a more detailed discussion in
the following section).
The two terms in the energy (1.4) are competing: the second term favors u to beconstant and equal to its average ū, but since u is valued in {+1, −1} this means in effectthat it is advantageous for u to oscillate rapidly between the two phases u = +1 and
2

0

0

0

−0.5

−0.5

−0.5

−1
−1

0

−1
−1

1

1

1

0.5

0.5

0.5

0

0

0

−0.5

−0.5

−0.5

0

−1
−1

1

47

Fig. 2.

a)

0

1

−1
−1

Droplet Phases in Non-local Ginzburg-Landau Models

−1
−1

1

0

−1
−1

1

0

1

0

1

The droplet patterns for K = 2 through 10.

b)

1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Fig. 1. A multi-droplet pattern: density plot of u in a local minimizer of E[u] with W (u) = 41 (1−u 2 )2 obtained
Fig. 3. The 100 ζk ’s determined by numerically minimizing F on a unit disc.
numerically for ū = −0.5, ε = 0.025, and Ω = [0, 11.5) × [0, 10), with periodic boundary conditions. Dark
regions correspond
to
u
≈
−1,
and
light
regions
correspond
to
u
≈
1
(from
[14])
Figure 1: Two-dimensional multi-droplet patterns in systems with Coulombic repulsion:a local minimizer of the Ohta-Kawasaki energy on a rectangle with periodic boundary
to an expressionin termsof the interfaces
alone.ofInthe[13,14],
reduction was
perconditions;a localminimizer
sumsuchof atwo-point
Coulombic
potentials on a disk withformed for E using formal asymptotic techniques (see also [30,35,40,41]) and leads to
Neumannboundary
conditions.
Taken
from
[37,
45].
the following reduced energy (for simplicity of notation, we choose the normalizations insuch a way that the parameter ε is, in fact, the domain wall energy, see Sec. 4 for details):!
! !
ε the first 1term penalizes the perimeter of the interface between the two phases,uE[u]= =−1;|∇u| d x +(u(x) − ū)G(x, y)(u(y) − ū) d x d y.
(1.4)
2 Ωopposes too
2 ΩmuchΩand thus
spreading and oscillation. The competition between these two
Here thefunctionau length
takes on values
throughout
and the kernel
the screened
selects
scale,±1whichis aΩ,functionof G
ε.is In
the diffuse interface version (1.1), theCoulomb kernel, i.e., it solves the Neumann problem for

sharp transitions between {u = +1} and {u = −1} are replaced by smooth transitions at2
the scale ε >−0∆G(x,as soonεy) + κas
G(x,y) 1.
= δ(x − y),
(1.5)
In
one
space
dimension
and
in
the
particular
case
ū
=
0 (symmetric phases) the behavior
with some κ > 0. The constant κ has the physical meaning of the inverse of the Debyeof length
the energy
fromtheEworkof unscreenedMüller [35]: the minimizer u is periodic
screening[13,14]. canNote bethat understoodthe sharp interface
energywith theCoulomb
kernel(i.e. with κ between= 0) was derived
and Weithe Γ at-limitof theε1/3 (for other one-dimensional
andalternatesu = by+1Renandu =as −1
scale
diffuse interface energy E under assumptions of weak non-local coupling (i.e., with anresults,
46, 58]).
patterns of minimizers are much
extra factorof ε insee
frontalsoof the[44,Coulomb
kernel)Inandhigherū ∈ (−1,dimensions
1) independent theof ε, asε → 0 more
[30] (seealso [35,37];
note
thatnot
this wellcase isunderstood.
also equivalent toTheconsidering
E on depends on the volume fraction
complexandarebehavior
the domain of size O(ε1/3 )). At the same time, screening becomes important near the
between the phases, i.e. on the constant ū chosen, and also on the dimension. When ū < 0,transition between the uniform and the patterned states which occurs near |ū| = 1,
call uin=the−1
the paper
majorityandu presence= +1 theminority phase, and conversely when
the caseweof interest
present[13,14].phase
Note thatin theof screening
the neutralityin (1.3)
is relaxed.
ū > condition
0. In two
dimensions,numerical simulations lead to expecting round “droplets” ofIn this paper, we rigorously establish the relation between the sharp interface energy
thediffuse
minority
phase
“sea”
of theofmajority
E and theinterfaceenergysurrounded
E, and analyzeby
the aprecise
behavior
minimizers phase (see Fig. 1) for sufficientof the sharp
interface energy
E for theε ' majority
1 in the vicinity
theminoritytransition fromthe (i.e., for ū sufficiently far away
asymmetries
betweenand ofthephasestrivial minimizer
to patternedstatesnear situation|ū| = 1. Weisnotethatcleardespitefor
the ū close to zero, although it isfrom zero)[36, 37,
42,occurring45]. The
less
apparent simplicity of the expression for E, the minimizers of E exhibit quite an intricommonlyin 1this
cate dependence
on thebelieved
parametersthatfor ε '
and |caseū| ( 1.theOur minimizers
analysis in this are
paperone-dimensional stripe patterns
will be[12,restricted
to the36, 37,42].case d = 2. While a number of our results can be extended
In all cases, minimizers are intuitively expected to be periodic. However, at the moment
this seems to be very difficult to prove. The only general result in that direction to dateis that of Alberti, Choksi and Otto [1], which proves that the energy of minimizers of the
sharp interface energy from (1.4) with no screening (with κ = 0 and the neutrality condition

3

from (1.3)) is uniformly distributed in the limit where the size of the domain Ω goes toinfinity (see also [7, 51]). Their results, however, do not provide any further information
about the structure of the energy-minimizing patterns. Note in passing that the questionof proving any periodicity of minimizers for multi-dimensional energies is unsolved evenfor systems of point particles forming simple crystals (see e.g. [30, 49]), with a notable
exception of certain two-dimensional particle systems with short-range interactions whichsomehow reduce to packing problems [43, 54, 56]. Naturally, the situation can be expected
to be more complicated for pattern forming systems in which the constitutive elements are“soft” objects, such as, e.g., droplets of the minority phase in the matrix of the majority
phase in the Ohta-Kawasaki model.
Here we are going to focus on the two-dimensional case and the situation where one
phase is in strong majority with respect to the other, which is imposed by taking ū very closeto −1 as ε → 0. Thus we can expect a distribution of small droplets of u = +1 surrounded
by a sea of u = −1. In this regime, Choksi and Peletier analyzed the asymptotic propertiesof a suitably rescaled version of the sharp interface energy (1.4) with no screening in [13],as well as (1.1) in [14]. They work in the setting of a fixed domain Ω, and in a regime where
the number of droplets remains finite as ε → 0. They showed that the energy minimizing
patterns concentrate to a finite number of point masses, whose magnitudes and locations
are determined via a Γ-expansion of the energy [5]. Here, in contrast, we work in a regime
where the number of droplets is divergent as ε → 0. We note that Γ-convergence of (1.1)
to the functional (1.4) with no screening and for fixed volume fractions was established by
Ren and Wei in [46], who also analyzed local minimizers of the sharp interface energy inthe strong asymmetry regime in two space dimensions [45].
All these works are in the finite domain Ω setting, while we are generally interested in
the large volume (macroscopic) limit, i.e., the regime when the number of droplets tends toinfinity. A rather detailed study of the behavior of the minimizers for the Ohta-Kawasaki
energy in macroscopically large domains was recently performed in [38], still in the regimeof ū close to −1. There the two-dimensional Ohta-Kawasaki energy was considered in thecase when Ω is a unit square with periodic boundary conditions. The interesting regime
corresponds to the parameters ε  1 and 1 + ū = O(ε2/3 | ln ε|1/3 )  1. It is shown in [38]that under these assumptions on the parameters and some technical assumptions on W ,(1.4) gives the correct asymptotic limit of the minimal energy in (1.1). Moreover, it is shownthat when δ̄ := ε−2/3 | ln ε|−1/3 (1 + ū) becomes greater than a certain critical constant δ̄c ,the minimizers of E in (1.4) consist of O(| ln ε|) simply connected, nearly round droplets ofradius ' 31/3 ε1/3 | ln ε|−1/3 , and uniformly distributed throughout the domain [38]. Thus,
the following hierarchy of length scales is established in the considered regime:ε  ε1/3 | ln ε|−1/3  | ln ε|−1/2  1,

(1.6)

where the scales above correspond to the width of the interface, the radius of the droplets,
the average distance between the droplets, and the screening length, respectively. The
multiscale nature of the energy minimizing pattern is readily apparent from (1.6).
4

The analysis of [38] makes heavy use of the minimality condition for (1.4) and, in particular, the Euler-Lagrange equation associated with the energy. One is thus naturally led toasking whether the qualitative properties of the minimizers established in [38] (roundnessof the droplets, identical radii, uniform distribution) carry over to, e.g., almost minimizersof E, for which no Euler-Lagrange equation is available. More broadly, it is natural to ask
how robust the properties of the energy minimizing patterns are with respect to various
perturbations of the energy, for example, how the picture presented above is affected when
the charge density ū is spatially modulated. A natural way to approach these questions is
via Γ-convergence. However, for a multiscale problem such as the one we are considering
the proper setting for studying Γ-limits of the functionals in (1.1) or (1.4) is presently
lacking. The purpose of this paper is to formulate such a setting and extract the leading
order term in the Γ-expansion of the energy in (1.1). In our forthcoming paper [23], we
obtain the next order term in the Γ-expansion, using the method of “lower bounds for
2-scale energies” via Γ-convergence introduced in [49].
The main question for setting up the Γ-limit in the present context is to choose a suitable
metric for Γ-convergence. This metric turns out to be similar to the one used for the analysis
of vortices in the two-dimensional magnetic Ginzburg-Landau model from the theory ofsuperconductivity [48]. In fact, the problem under consideration and its mathematicaltreatment (here as well as in [23]) share several important features with the latter [48].
In the theory of superconductivity the role of droplets is played by the Ginzburg-Landauvortices, which in the appropriate limits also become uniformly distributed throughout thedomain [47]. We note, however, that the approach developed in [47, 48] cannot be carriedover directly to the problem under consideration, since the vortices are more rigid thantheir droplet counterparts: the topological degrees of the vortices are quantized and can
only take integer values, while the droplet volumes are not. Thus we also have to consider
the possibility of many very small droplets. Developing a control on the droplet volumes
from above and below is one of the key ingredient of the proofs presented below, and relieson the control of their perimeter via the energy.
For simplicity, as in [38] we consider the energy defined on a flat torus (a square withperiodic boundary conditions). The metric we consider is the weak convergence of measures
for a suitably rescaled sequence of characteristic functions associated with droplets (see thenext section for precise definitions and statements of theorems). Then, up to a rescaling,
we show that both the energy E from (1.1) and E from (1.4) Γ-converge to a quadratic
functional in terms of the limit measure, with the quadratic term generated by the screened
Coulomb kernel from (1.5) and the linear term depending explicitly on δ̄ and κ. To be
more precise, we will see that in the regime we study, there are two contributions to theenergy which operate at leading order: one contribution is linear in the density of thedroplets and corresponds to the “self-interaction energy” of each droplet coming from boththe perimeter term and self-interaction part of the double integral in (1.4), and the otheris a quadratic term corresponding to the interaction between the droplets, i.e. the rest of
the contribution of the double-integral term in (1.4). This setting, where both terms are5

of the same order of magnitude is very similar to the regime of [47] and [48, Chap. 7] inthe context of the magnetic Ginzburg-Landau energy.
We note that the obtained limit variational problem is strictly convex and its uniqueminimizer is a measure with constant density across the domain Ω. In particular, this
implies equidistribution of mass and energy for the minimizers of the diffuse interface
energy E in (1.1) in the considered regime. In our companion paper [23], we further
address the mutual arrangement of the droplets in the energy minimizing patterns, using
the formalism developed recently for Ginzburg-Landau vortices [49]. We also obtain acharacterization of the droplet shapes for almost minimizers of the sharp interface energy
E, which, in turn, allows us to make the same conclusions about minimizers of the diffuse
interface energy E for ε  1, which is a new result. The reason we can characterize thedroplets at the diffuse interface level is because the difference between the zero superlevelset of the minimizers at the diffuse interface level and the jump set of almost minimizers
at the sharp interface level occurs essentially on the length scale ε (interfacial thickness),which is much smaller than the characteristic length scale ε1/3 | ln ε|−1/3 of the droplets.
Let us mention other closely related systems from the studies of ferromagnetism andsuperconductivity, where the role of droplets is played by the slender needle-like domainsof opposite magnetization in a three-dimensional ferromagnetic slab at the onset of magnetization reversal [27], or superconducting tunnels in a slab of type-I superconducting
material near the critical field [8, 11]. It may be possible to obtain similar Γ-convergence
results with respect to convergence of measures in the plane for those problems. At thesame time, we point out that extending our results to higher dimensions meets with serious
difficulties, since in the suitable limit the droplets in higher-dimensional problems are expected to solve a non-local isoperimetric problem whose solution is not well characterized
at present [28].
Our paper is organized as follows. In Sec. 2, we introduce the considered scaling
regime and state our main results; in Sec. 4 we prove the Γ-convergence result in the
sharp interface setting; in Sec. 5 we prove the results on the characterization of almost
minimizers of sharp interface energy; and in Sec. 6 we treat the Γ-limit for the case of the
diffuse interface energy.
Some notations. We use the notation (uε ) ∈ A to denote sequences of functions uε ∈ A
as ε = εn → 0, where A is an admissible class. For a measurable set E, we use |E| to denoteits Lebesgue measure and |∂E| to denote its perimeter (in the sense of De Giorgi). We also
use the notation µ ∈ M+ (Ω) to denote a non-negative Radon measure µ on the domain Ω.
With a slight abuse, we will often speak of µ as the “density” on Ω. The symbols H 1 (Ω),
BV (Ω), C(Ω) and H −1 (Ω) denote the usual Sobolev space, space of functions of boundedvariation, space of continuous functions, and the dual of H 1 (Ω), respectively.

62Statement of results

Throughout the rest of the paper the parameters κ > 0, δ̄ > 0 and ` > 0 are assumed tobe fixed, and the domain Ω is assumed to be a flat two-dimensional torus of side length `,i.e., Ω = T2` = [0, `)2 , with periodic boundary conditions. For every ε > 0 we defineūε := −1 + ε2/3 | ln ε|1/3 δ̄.

(2.1)

Under this scaling assumption the sharp interface version of the Ohta-Kawasaki energy (cf.
(1.4)) can be written as
Z
Z
ε
1
ε
E [u] =|∇u| dx +(u − ūε )(−∆ + κ2 )−1 (u − ūε ) dx,
(2.2)
2 T2`
2 T2`
for all u ∈ A, whereA := BV (T2` ; {−1, 1}).

(2.3)

We wish to understand the asymptotic properties of the energy E ε in (2.2) as ε → 0 when
all other parameters are fixed. We then relate our conclusions based on the study of thisenergy to its diffuse interface version, which under the same scaling assumptions takes the
form

Z  2
ε
1
ε
2
ε
−1
ε
E [u] =|∇u| + W (u) + (u − ū )(−∆) (u − ū ) dx,
(2.4)
2
2
T2`
with u ∈ Aε , where
(
Aε :=

1
u ∈ H 1 (T2` ) : 2`

Z)u dx = ūε

.

(2.5)

T2`

Here the symmetric double-well potential W ≥ 0 needs to satisfy
Z 1p
1
00
W (1) = 0,
W (1) = 2 ,2W (u) du = 1,
κ
−1

(2.6)

in order for E ε to be compatible with E ε (see further discussion at the beginning of Sec.
3 and [38, Sec. 4] for precise assumptions on W ). We note that the relation between E εand E ε does not amount to a straightforward application of the standard Modica-Mortola
argument [33, 34], as will be explained in more detail in Sec. 2.2. A formal application of
the latter to (2.4) would result in an energy of the type in (2.2), but with the same (i.e.,
unscreened) Coulomb kernel as in (2.4), which is not Γ-equivalent to E ε . We also notethat at the level of the energy minimizers the relation between the two functionals wasestablished in [38].
7

2.1

Sharp interface energy

The sharp interface energy in (2.2) is most conveniently expressed in terms of droplets, i.e.,+the connected components Ω+
i of the set Ω := {u = +1} (see Lemma 3.1 for technicaldetails). InsertingX(2.7)
u = −1 + 2
χΩ+ ,
i

i

into (2.2), where χΩ+ are the characteristic functions of Ω+i , expressing the result via Githat solves−∆G(x) + κ2 G(x) = δ(x)in T2` ,(2.8)
R
expanding all the terms and using the fact that T2 G(x)dx = κ−2 , we arrive at (see`also [38])`2 (1 + ūε )2
2κ2
o
Xn
XZ
+
+
−2
ε
+
ε|∂Ωi | − 2κ (1 + ū )|Ωi | + 2

E ε [u] =

i

i,j

Ω+
i

Z
Ω+
j

G(x − y) dx dy,(2.9)

where we took into account the translational symmetry of the problem in T2` . Moreover,since the optimal configurations for Ω+
i are expected to consist of droplets of size of order
1/3−1/3
ε | ln ε|(see (1.6) and the discussion around), it is convenient to introduce the rescaled
area and perimeter of each droplet:
Ai := ε−2/3 | ln ε|2/3 |Ω+i |,Pi := ε−1/3 | ln ε|1/3 |∂Ω+
i |.

(2.10)

Similarly, let us introduce the suitably rescaled measure µ associated with the droplets:dµ(x) := ε−2/3 | ln ε|−1/3Xi1χΩ+ (x)dx = ε−2/3 | ln ε|−1/3 (1 + u) dx.
i2Note that by the definitions in (2.10) and (2.11) we have
Z
1 X
Ai =
dµ,
| ln ε|T2`(2.11)(2.12)iand the energy E ε [u] may be rewritten asε

E [u] = ε

4/3

2/3

| ln ε|

8

 2 2

δ̄ `
ε
+ Ē [u] ,2κ2(2.13)

where


Z Z
2δ̄
1 X
Pi − 2 Ai + 2
Ē [u] :=G(x − y)dµ(x)dµ(y).
| ln ε|κT2` T2`ε(2.14)i

We now state our Γ-convergence result, which is obtained for configurations (uε ) thatobey the optimal energy scaling, i.e. when Ē ε [uε ] remains bounded as ε → 0. The result isobtained with the help of the framework established in [47], where an analogous result for
the Ginzburg-Landau functional of superconductivity was obtained. What we show is thatthe limit functional E 0 depends only on the limit density µ of the droplets (more precisely,
on a limit measure µ ∈ M+ (T2` ) ∩ H −1 (T2` ), see Lemma 3.2 for technical details about suchmeasures). In passing to the limit the second term in (2.14) remains unchanged, while
the first term is converted into a term proportional to the integral of the measure. The
proportionality constant is non-trivially determined by the optimal droplet profile that will
be discussed later on. We give the statement of the result in terms of the original screened
sharp interface energy E ε , which is defined in terms of u ∈ A. In the proof, we work insteadwith the equivalent energy Ē ε , which is defined through {Aεi }, {Piε } and µε corresponding
to u = uε (cf. (2.13) and (2.14)).
Theorem 1. (Γ-convergence of E ε ) Let E ε be defined by (2.2) with ūε given by (2.1).
Then, as ε → 0 we have thatZ
Z Zδ̄ 2 `22δ̄−4/3−2/3 ε Γ02/3
ε| ln ε|E → E [µ] :=+ 3 − 2dµ + 2G(x − y)dµ(x)dµ(y),
2κ2
κ
T2`
T2` T2`where µ ∈ M+ (T2` ) ∩ H −1 (T2` ). More precisely, we havei) (Lower Bound) Let (uε ) ∈ A be such thatlim sup ε−4/3 | ln ε|−2/3 E ε [uε ] < +∞,(2.15)dµε (x) := 12 ε−2/3 | ln ε|−1/3 (1 + uε (x))dx,(2.16)

ε→0

let

and let v ε satisfy−∆v ε + κ2 v ε = µε

in

T2` .

(2.17)

Then, up to extraction of a subsequence, we haveµε * µ in (C(T2` ))∗ , v ε * v in H 1 (T2` ),as ε → 0, where µ ∈ M+ (T2` ) ∩ H −1 (T2` ) and v ∈ H 1 (T2` ) satisfy−∆v + κ2 v = µ

in

T2` .

Moreover, we have
lim inf ε−4/3 | ln ε|−2/3 E ε [uε ] ≥ E 0 [µ].
ε→09(2.18)

ii) (Upper Bound) Conversely, given µ ∈ M+ (T2` ) ∩ H −1 (T2` ) and v ∈ H 1 (T2` ) solving(2.18), there exist (uε ) ∈ A such that for the corresponding µε , v ε as in (2.16) and(2.17) we haveµε * µ in (C(T2` ))∗ , v ε * v in H 1 (T2` ),as ε → 0, andlim sup ε−4/3 | ln ε|−2/3 E ε [uε ] ≤ E 0 [µ].
ε→0We note that the limit energy E 0 obtained in Theorem 1 may be viewed as the homogenized (or mean-field) version of the non-local part of the energy in the definition of E εassociated with the limit charge density µ of the droplets, plus a term associated with theself-energy of the droplets. The functional E 0 is strictly convex, so there exists a uniqueminimizer µ̄ ∈ M+ (T2` ) ∩ H −1 (T2` ) of E 0 , which is easily seen to be either µ̄ = 0 forδ̄ ≤ 21 32/3 κ2 or µ̄ = 12 (δ̄ − 21 32/3 κ2 ) otherwise. The latter can also be seen immediatelyfrom Remark 2.1 below, which gives a local characterization of the limit energy E 0 (seeLemma 3.2).
Remark 2.1. The limit energy E 0 in Theorem 1 becomes local when written in terms of
the limit potential v defined in (2.18):
Z 
Zδ̄ 2 `2  2/3 2
2
2 2
E 0 [µ] =
+
3
κ
−
2
δ̄
dx.
(2.19)
v
dx
+
2
|∇v|
+
κ
v
2κ2
T2`
T2`
Also, by the usual properties of Γ-convergence [4], the optimal density µ̄ above isexhibited by the minimizers of E ε in the limit ε → 0, in agreement with [38, Theorem 2.2]:
Corollary 2.2. Let ūε be given by (2.1) and let (uε ) ∈ A be minimizers of E ε defined in(2.2). Then, letting δ̄c := 21 32/3 κ2 , if µε is given by (2.16), as ε → 0 we have(i) If δ̄ ≤ δ̄c , thenµε * 0 in (C(T2` ))∗and

ε−4/3 | ln ε|−2/3 `−2 min E ε →

δ̄ 2
.
2κ2

(2.20)(ii) If δ̄ > δ̄c , thenµε * 21 (δ̄ − δ̄c ) in (C(T2` ))∗

and

δ̄c
ε−4/3 | ln ε|−2/3 `−2 min E ε → 2κ
2 (2δ̄ − δ̄c ).
(2.21)

In particular, since the minimal energy scales with the area of T2` , it is an extensive quantity.
We next give the definition of almost minimizers with prescribed limit density, forwhich a number of further results may be obtained. These can be viewed, e.g., as almost
10minimizers of E ε in the presence of anRexternal potential. We note that in view of the strict
convexity of E 0 , minimizing E 0 [µ] + T2 ϕ(x)dµ(x) for a given ϕ ∈ H 1 (T2` ) one obtains a`
one-to-one correspondence between the minimizing density µ and the potential ϕ. It thenmakes sense to talk about almost minimizers of theE ε with prescribed limit density
R energyεεεµ by viewing them as almost minimizers of E + T2 ϕ dµ , where ϕε = ε2/3 | ln ε|1/3 ϕ. Also,`
observe that almost minimizers with the particular prescribed density µ̄ from Corollary 2.2
are simply almost minimizers of E ε . Below we give a precise definition.
Definition 2.3. For a given µ ∈ M+ (T2` ) ∩ H −1 (T2` ), we will call every recovery sequence(uε ) ∈ A in Theorem 1(ii) almost minimizers of E ε with prescribed limit density µ.
For almost minimizers with prescribed limit density, we show that in the limit ε → 0
most of the droplets, with the exception of possibly many tiny droplets comprising a vanishing fraction of the total droplet area, converge to disks of radius r = 31/3 ε1/3 | ln ε|−1/3 .
More precisely, we have the following result.
Theorem 2. Let (uε ) ∈ A be a sequence of almost minimizers of E ε with prescribed limit
density µ. For every γ ∈ (0, 1) define the set Iγε := {i ∈ N : 32/3 πγ ≤ Aεi ≤ 32/3 πγ −1 }.
Then1 X ε plim(2.22)Pi − 4πAεi = 0,ε→0 | ln ε|i2
1 X εlim
Ai − 32/3 π = 0,(2.23)
ε→0 | ln ε|
ε
i∈Iγ

1 X ε
Ai = 0,ε→0 | ln ε|εlim(2.24)

i6∈Iγ

where {Aεi } and {Piε } are given by (2.10) with u = uε .
Note that we may use the isoperimetric deficit terms present in (2.22) to control the
Fraenkel asymmetry of the droplets. The Fraenkel asymmetry measures the deviation of
the set E from the ball of the same area that best approximates E and is defined for any
Borel set E ⊂ R2 byα(E) = min

|E4B|
,
|E|

(2.25)

where the minimum is taken over all balls B ⊂ R2 with |B| = |E|, and 4 denotes thesymmetric difference between sets. Note that the following sharp quantitative isoperimetricinequality holds for α(E) [20]:pp|∂E| − 4π|E| ≥ Cα2 (E) |E|,
(2.26)
with some universal constant C > 0. As a direct consequence of Theorem 2 and (2.12), wethen have the following result.
11

Corollary 2.4. Under the assumptions of Theorem 2, when
32/3 π|Iγε |lim=
ε→0 | ln ε|RT2` dµ > 0 we have

1 X
α(Ω+
i ) = 0,ε→0 |Iγε |
ε

Z

lim

dµ,
T2`

(2.27)i∈Iγ

where |Iγε | denotes the cardinality of Iγε .
This result generalizes the one in [38], where it was found that in the case of the minimizers
all the droplets are uniformly close to disks of the optimal radius r = 31/3 ε1/3 | ln ε|−1/3 .
What we showed here is that this result holds for almost all droplets in the case of almost
minimizers, in the sense that in the limit almost all the mass concentrates in the dropletsof optimal area and vanishing isoperimetric deficit. We note that the density µ is also
the limit of the number density of the droplets, up to a normalization constant, once thedroplets of vanishing area have been discarded.
The result that almost all droplets in almost minimizers with prescribed limit density
have asymptotically the same size, even if the limit density is not constant in T2` appears
to be quite surprising, since in this regime the self-interaction energy, which governs thedroplet shapes and partly their sizes is exactly of the same order as the droplet mutualinteraction energy, as was already mentioned at the end of Sec. 1. In addition, the otherterms governing the droplets extracted in (2.14) (the perimeter and interaction with thebackground uniform charge) are equally strong. This result would hold, for example,
for minimizersR of the energy in the presence of a non-uniform potential, i.e., with a term
1 2/3
| ln ε|1/3 T2 ϕ(x)u(x) dx added to E ε in (2.2) (see also the paragraph before Definition
2ε`2.3). It means that while the density of the energy minimizing droplets would be dependenton ϕ, their radii would not. We note that this observation is consistent with the expectationthat quantum mechanical charged particle systems form Wigner crystals at low particle
densities [24,32,57]. Let us point out that the Ohta-Kawasaki energy E ε bears resemblancewith the classical Thomas-Fermi-Dirac-Von Weizsäcker model arising in the context ofdensity functional theory of quantum systems (see e.g. [30–32]).

2.2

Diffuse interface energy

We now turn to relating the results obtained so far for the screened sharp interface energy
E ε to the original diffuse interface energy E ε . On the level of the minimal energy, theasymptotic equivalence of the energies in the considered regime, namely, that for every
δ>0(1 − δ) min E ε ≤ min E ε ≤ (1 + δ) min E ε(2.28)for ε  1 was established in [38, Theorem 2.3]. The main idea of the proof in [38] isfor a given function uε ∈ Aε to establish an approximate lower bound for E ε [uε ] in termsof (1 − δ)E ε [ũε ] for some ũε ∈ A, with δ > 0 which can be chosen arbitrarily small for
12

u
1.0

0.5

r

-0.5

-1.0

Figure 2: A qualitative form of the u-profile for a single droplet from the Euler-Lagrange
equation associated with E. The horizontal line shows the level corresponding to ū. Charge
is transferred from the region where u < ū (depletion shown in green) to the region whereu > ū (excess shown in orange). At the sharp interface level the corresponding profile isgiven by sgn(u), whose average charge is not equal to ū.
ε  1. The matching approximate upper bound is then obtained by a suitable lifting of
the minimizer uε ∈ A of E ε into Aε .
Here we show that the procedure outlined above may also be applied to almost minimizers of E ε in a suitably modified version of Definition 2.3 involving E ε , using almost
minimizers of E ε for comparisons. We note right away, however, that it is not possible tosimply replace E ε with E ε in Definition 2.3. The reason for this is the presence of the massconstraint in the definition of the admissible class Aε for E ε . This implies,R for example,
that any sequence of almost minimizers (uε ) ∈ Aε of E ε must satisfy `−2 T2 dµε = 21 δ̄,`
while, according
to Corollary 2.2, for sequences of almost minimizers (uε ) ∈ A of E ε weRhave `−2 T2 dµε → µ̄ 6= 12 δ̄. This phenomenon is intimately related to the effect of screen`
ing of the Coulombic potential from the droplets by the compensating charges that moveinto their vicinity [37]. For a single radially symmetric droplet the solution of the EulerLagrange equation associated with E ε has the form shown in Fig. 2, which illustrates thegap between the “prescribed” total charge at the diffuse interface level and the total charge
at the sharp interface level.
In order to be able to extract the limit behavior of the energy, we need to take into
consideration the redistribution of charge discussed above and define almost minimizers
with prescribed limit density that belong to Aε and for which the screening charges are removed from the consideration of convergence to the limit density. Hence, given a candidate
function uε ∈ Aε , we define a new function(+1, uε (x) > 0,εu0 (x) :=(2.29)−1, uε (x) ≤ 0,13

whose jump set coincides with the zero level set of uε . This introduces a nonlinear filtering
operation that eliminates the effect of the small deviations of uε from ±1 in almost minimizers on the limit density (compare also with [27]). The measure µε0 associated with the
droplets is now defined via
dµε0 := 12 ε−2/3 | ln ε|−1/3 (1 + uε0 (x))dx.

(2.30)

We can follow the ideas of [38] to establish an analog of Theorem 1 for the diffuse
interface energy. To avoid many technical assumptions,
we formulate the result for ap
9
specific choice of W (u) = 32
(1 − u2 )2 and κ = 1/ W 00 (1) = 23 (see the discussion at
the beginning of Sec. 3). A general result may easily be reconstructed. Also, we make atechnical assumption to avoid dealing with the case lim supε→0 kuε kL∞ (T2 ) > 1, when spiky`
configurations in which |uε | significantly exceeds 1 in regions of vanishing size may appear.
We note that this condition is satisfied by the minimizers of E ε [38, Proposition 4.1].
9Theorem 3. (Γ-convergence of E ε ) Let E ε be defined by (2.4) with W (u) = 32
(1 − u2 )2and ūε given by (2.1). Then, as ε → 0 we have that

Z
Z Z
2δ̄
δ̄ 2 `2−4/3−2/3 ε Γ
2/30G(x − y)dµ(x)dµ(y),dµ + 2ε| ln ε|E → E [µ] :=+ 3 − 22κ2
κT2` T2`
T2`

where µ ∈ M+ (T2` ) ∩ H −1 (T2` ) and κ = 23 . More precisely, we havei) (Lower Bound) Let (uε ) ∈ Aε be such that lim supε→0 kuε kL∞ (T2 ) ≤ 1 and`lim sup ε−4/3 | ln ε|−2/3 E ε [uε ] < +∞,(2.31)

ε→0

and let µε0 (x) be defined by (2.29) and (2.30).
Then, up to extraction of subsequences, we have
µε0 * µ in (C(T2` ))∗ ,as ε → 0, where µ ∈ M+ (T2` )∩H −1 (T2` ). Moreover, we have lim supε→0 kuε kL∞ (T2 ) =`1 and
lim inf ε−4/3 | ln ε|−2/3 E ε [uε ] ≥ E 0 [µ].
ε→0ii) (Upper Bound) Conversely, given µ ∈ M+ (T2` ) ∩ H −1 (T2` ), there exist (uε ) ∈ Aε suchthat lim supε→0 kuε kL∞ (T2 ) = 1 and for µε0 defined by (2.29) and (2.30) we have`

µε0 * µ in (C(T2` ))∗ ,as ε → 0, andlim sup ε−4/3 | ln ε|−2/3 E ε [uε ] ≤ E 0 [µ].
ε→014Based on the result of Theorem 3, we have the following analog of Corollary 2.2 for the
diffuse interface energy E ε .
Corollary 2.5. Let ūε be given by (2.1) and let (uε ) ∈ Aε be minimizers of E ε defined in
9(2.4) with W (u) = 32
(1 − u2 )2 . Then, letting κ = 32 and δ̄c := 12 32/3 κ2 , if uε0 and µε0 are
defined via (2.29) and (2.30), respectively, as ε → 0 we have(i) If δ̄ ≤ δ̄c , then
µε0 * 0 in (C(T2` ))∗ ,and

ε−4/3 | ln ε|−2/3 `−2 min E ε →

δ̄ 2
.
2κ2

(2.32)(ii) If δ̄ > δ̄c , thenµε * 21 (δ̄ − δ̄c ) in (C(T2` ))∗ ,

and

δ̄c
ε−4/3 | ln ε|−2/3 `−2 min E ε → 2κ
2 (2δ̄ − δ̄c ).
(2.33)In addition, we have the following analog of Theorem 2, which, in particular, applies tominimizers of the diffuse interface energy E ε .
R
Theorem 4. Let (uε ) ∈ Aε be a recovery sequence as in Theorem 3(ii) and let T2 dµ > 0.
`

Then there exists a set of finite perimeter Ω+ such that if Ω+
i are its connected components,
then the conclusion of Theorem 2 holds with {Aεi } and {Piε } given by (2.10), and|Ω+ 4{uε > 0}|
= 0.
ε→0
|Ω+ |
lim

(2.34)

Theorem 4 essentially says that the zero superlevel set of uε from every recovery sequence of Theorem 3 may be well approximated in L1 sense by a union of of droplets thatare, in turn, close to disks of radius r = 31/3 ε1/3 | ln ε|−1/3 for ε  1. The L1 error arises,
since we do not have control on the perimeter of every superlevel set of uε . At the sametime, the choice of the zero superlevel set of uε in the definition of the truncated version
uε0 of uε in (2.29) was arbitrary. We could equivalently use the superlevel set {uε > c} forany c ∈ (−1, 1) fixed. Also, we point out that the conclusions of Corollary 2.4 remain truefor Ω+ in Theorem 4 under the assumptions of Theorem 3.

3Some auxiliary lemmas

In this section we collect some technical results that are needed in the proofs of our theorems. Before proceeding to those results, however, let us first show that the assumptionin (2.6) that needs to be imposed on W in order to have Γ-equivalence between E ε and15

E ε defined in (2.2) and (2.4), respectively, and, hence, the conclusion of Theorem 3 (seealso [38]), is not restrictive. Indeed, given the definition of E ε in (2.4), introduce a rescaling:f,
W = λ2 W˜` = λ`,

ε = λ2 ε̃.

(3.1)

Then it is easy to see that if ũ(x) := u(λx), then E ε [u] = λ4 Ẽ ε̃ [ũ], where Ẽ ε̃ is obtained from
(2.4) by replacing all the quantities with their tilde equivalents. In particular, choosing√
f (u) =
λ = 3/(2 2) we can relate the original Ohta-Kawasaki energy Ẽ ε̃ , which has W
1
2
2
4 (1 − u ) [42], to the energy appearing in the statement of Theorem 3. The choice of W
satisfying (2.6) simply avoids many extra constants appearing in the statements of results.
As was already mentioned, the energy E ε may be alternatively written in terms of thelevel sets of u. Indeed, when E ε [u] < +∞, the set Ω+ := {u = +1} is a set of finite
perimeter (for precise definitions and the terminology used below, see [2]). We then havethe following result about decomposing Ω+ into measure theoretic connected componentsΩ+i , which in view of the scaling of the upper bound on energy will be shown to hold forall sufficiently small ε > 0. Note that the latter assumption implies that each connectedcomponent on the torus has the same geometric structure as connected components of setsof finite perimeter in the whole plane, thus excluding a possibility of stripe-like components
winding around the torus and, hence, justifying the use of the word “droplet”. We will
also make repeated use of the basic fact that the diameter of a connected component isessentially controlled by its perimeter (i.e., modulo a set of measure zero).
1 2
Lemma 3.1. Let Ω+ ⊂ T2` be a set of finite perimeter, and assume that |Ω+ | ≤ 160
` and
1
+
+
|∂Ω | ≤ 10 `. Then Ω may be uniquely decomposed (up to negligible sets) into an at mostcountable union of connected sets Ω+
i of positive measure, which, after a suitable translation2and extension to R , are essentially bounded and whose essential boundaries ∂ M Ω+
i are (upto negligible sets) at most countable unions of Jordan curves that are essentially disjoint.
Furthermore, we have1+
ess diam Ω+(3.2)
i ≤ |∂Ωi |.
2
2
2
+
2
Proof. Let Ω+# be the periodic extension of Ω from T` to R , and let KR := (−R, R) .
2
Then for every R ∈ (`, 23 `) the set Ω+# ∩ KR ⊂ R is a set of finite perimeter, and we have
+
+
1
|∂(Ω+
# ∩ KR )| ≤ 9|∂Ω | + H (Ω̊# ∩ ∂KR ).

On the other hand, by the co-area formula we have
Z 3`
2
+
+
H1 (Ω̊+# ∩ ∂Kt )dt = |Ω# ∩ K 3 ` \K` | ≤ 8|Ω |.
2

`

(3.3)

(3.4)

−1 +
Therefore, there exists R ∈ (`, 32 `) such that H1 (Ω̊+# ∩ ∂KR ) ≤ 16` |Ω |. Using the1
assumptions of the Lemma, we then conclude that H1 (Ω̊+# ∩ ∂KR ) ≤ 10 ` and by (3.3) wehave |∂(Ω+# ∩ KR )| ≤ `.

16

We now apply the results of [2, Corollary 1 and Theorem 8] to the set Ω+# ∩ KR to+
obtain its decomposition into connected components and denote by Ωi those components
for which |Ω+i ∩ K 21 ` | > 0. In turn, by [2, Theorem 7 and Lemma 4] and noting that in viewof [2, Proposition 6(ii)] it is sufficient to consider only simple sets (see [2, Definition 3]), we+
have that Ω+i satisfy (3.2). Therefore, from our estimate on |∂(Ω# ∩ KR )| we conclude that+|Ω+i ∩ K 3 ` \K` | = 0, and so |∂Ωi | does not have contributions from ∂KR . Together with2

the assumptions of the Lemma, this then implies that each Ω+i is essentially contained,
after a suitable translation, in K 1 ` . Finally, identifying all translates of Ω+i by ±` in either
4

coordinate direction with the connected components of Ω+ in T2` , we obtain the desired
decomposition of Ω+ ⊂ T2` for which (3.2) also holds in the case of the perimeter relative
to T2` .

In the context of Γ-convergence the sets Ω+
i may be viewed as a suitable generalizationof the droplets introduced earlier in the studies of energy minimizing patterns [38]. Note,however, that the sets Ω+i lack the regularity properties of the energy minimizers in [38] andmay in general be fairly ill-behaved (in particular, they do not have to be simply connected).
Nevertheless, they are fundamental for the description of the low energy states associatedwith E ε and, in particular, will be shown to be close, in some average sense, to disks ofprescribed radii for almost minimizers of energy.
We now discuss the precise nature of the limit measures appearing in our analysis.
We say that µ ∈ M+ (T2` ) ∩ H −1 (T2` ), if the non-negative Radon measure µ has bounded
Coulombic energy, i.e., if
Z Z
G(x − y) dµ(x) dµ(y) < ∞.
(3.5)
T2`

T2`

Our notation is justified by the following fundamental properties of such measures.
Lemma 3.2. Let µ ∈ M+ (T2` ) and let (3.5) hold. Then(i) µ can be extended to a bounded linear functional over H 1 (T2` ).

(ii) IfZv(x) :=T2`

G(x − y) dµ(y),(3.6)

then v ∈ H 1 (T2` ). Furthermore, v solves−∆v + κ2 v = µ,weakly in H 1 (T2` ), and∇v(x) =Z
T2`

∇G(x − y) dµ(y),in the sense of distributions.
17

(3.7)

(3.8)

R
R
(iii) If v is as in (ii), we have κ2 T2 v dx = T2 dµ and
`

Z
T2`

Z
T2`

`

G(x − y) dµ(x) dµ(y) =

Z
T2`


|∇v|2 + κ2 v 2 dx.

(3.9)

Proof. We first show that v defined in (3.6) has distributional first derivatives in L2 (T2` ).
Introduce H(x) > 0 defined for all x ∈ T2` by
1 X e−κ|x−n`|,2π|x − n`|2H(x) :=(3.10)

n∈Z

whose Fourier coefficients are easily seen to beZ1beik·x H(x) dx = p
,
H(k)
:=
2
2
κ + |k|2T`k ∈ 2π`−1 Z2 .

e
Indeed, H(x) may be viewed as the trace H(x,0) of the solution ofXee=2δ(x − n`),x ∈ R3 ,−∆H(x)+ κ2 H(x)(3.11)(3.12)n∈Z2 ×{0}e k (z) the Fourier coefficients
which is given by the same formula as in (3.10). Denoting by H
2eeof H(x, z) in x ∈ T` , from (3.12) one obtains that Hk (z) solvese k (z) = 2δ(z),e 00 (z) + (κ2 + |k|2 )H
−H
k

√
p
e k (z) = e−z κ2 +|k|2 / κ2 + |k|2 .
whose explicit solution is H
From (3.11) and the equation satisfied by G one immediately concludes thatZ
H(x − y)H(y) dy.
G(x) =(3.13)(3.14)

T2`

Furthermore, by direct inspection one can see that|∇G(x)| ≤ CH(x)∀x ∈ T2` ,(3.15)

for some C > 0. In addition, definingZb(x) :=T2`

H(x − y) dµ(y),18(3.16)

by Tonelli’s theorem and (3.14) we have
Z
Z Z Z
2
b dx =
H(x − z)H(y − z) dµ(x) dµ(y) dz
T2`

T2`

T2`

T2`

Z

Z

=
T2`

T2`

G(x − y) dµ(x) dµ(y),(3.17)

and, hence, by (3.5) we have b ∈ L2 (T2` ). Therefore, ifZh(x) :=∇G(x − y) dµ(y),(3.18)

T2`

then by (3.15) and (3.16) it is well defined, and we have h ∈ L2 (T2` ; R2 ) as well.
Now, testing (3.6) with ∇ϕ, where ϕ ∈ C ∞ (T2` ), yields
Z
Z Z
Z
−
∇ϕ(x)v(x) dx = −∇ϕ(x)G(x − y)dµ(y) =ϕ(x)h(x) dx,
T2`

T2`

T2`

(3.19)

T2`

which is justified by Fubini’s theorem, in view of the fact that h ∈ L2 (T2` ; R2 ). Hence∇v = h ∈ L2 (T2` ; R2 ) distributionally, proving (3.8). To prove that v ∈ H 1 (T2` ), observethat by Tonelli’s theorem
!2
Z
Z
Z Z Z
v 2 dx =

T2`

T2`

T2`

T2`

G(x − z)G(y − z) dµ(x) dµ(y) dz ≤ C

dµ,(3.20)

T2`

for some C > 0. On the other hand, since by maximum principle G(x) ≥ c > 0 for allx ∈ T2` , we conclude that
!2 Z Z
Z
dµ

c
T2`

≤

T2`

T2`

G(x − y) dµ(x) dµ(y).

(3.21)

Therefore, by (3.5) we have that µ is bounded in the sense of measures, and so from (3.20)
follows that v ∈ L2 (T2` ) as well.
We may next show that (3.7) holds distributionally by testing v in (3.6) with −∆ϕ +κ2 ϕ ∈ C ∞ (T2` ) and integrating by parts. Then, to conclude the proof of the lemma, wetest (3.7) with ϕ ∈ C ∞ (T2` ) and apply the Cauchy-Schwarz inequality to obtainZ

Z
ϕ dµ =T2`

T2`


∇ϕ · ∇v + κ2 ϕv dx ≤ CkvkH 1 (T2 ) kϕkH 1 (T2 ) ,``(3.22)

for some C > 0. This yields (i), and, hence, (3.7) also holds weakly in H 1 (T2` ). Finally, toobtain (iii), we interpret µ in (3.7) as an element of H −1 (T2` ) and test (3.7) with either 1or v itself.
19

Remark 3.3. It is not difficult to extend the proof of Lemma 3.2 to the case of measureswith finite Coulombic energy defined on a sufficiently regular domain Ω with either Dirichlet
or Neumann boundary conditions for the potential. In this case the role of H would beplayed by the kernel of the Neumann-to-Dirichlet map for the operator −∆ + κ2 extended
to Ω × R+ .
Observe that for the nontrivial minimizers we know from [38] that Ē ε = O(1), Ai =O(1) and Pi = O(1) (and even more precisely Ai ' 32/3 π and Pi ' 2 · 31/3 π), the number ofdroplets is N = O(| ln ε|), and µ closely approximates the sum of Dirac masses at the droplet
centers with weights of order | ln ε|−1 . If, on the other hand, the considered configurations
only obey an energy bound under the optimal scaling, then the same estimates turn out
to hold for the droplets on average. The precise result is stated in the following lemma.
Lemma 3.4. Let (uε ) ∈ A, let lim supε→0 Ē ε [uε ] < +∞, and let {Aεi }, {Piε } and µε begiven by (2.10) and (2.11) with u = uε . Thenlim supε→0

1 X ε
Pi < +∞,| ln ε|lim sup
ε→0

i

1 X ε
Ai < +∞,| ln ε|(3.23)iandZdµε < +∞.

lim sup(3.24)T2`

ε→0

Proof. By (2.12) and the positivity of RPiε , we obtain the result, once we prove (3.24). To
prove the latter, we simply note that if T2 dµε ≥ 2δ̄/(cκ2 ), where c is the same as in (3.21),`

then by (2.12) we have from the definition of Ē ε in (2.14):2δ̄Ē [u] ≥ − 2
κ
ε

Z

ε

!2

Z

dµ + 2c

dµ

T2`

T2`

ε

≥c

!2

Z
dµ

ε

,

(3.25)

T2`

which yields (3.24).

4

Proof of Theorem 1

Throughout all the proofs below, the values of Aεi and Piε are always the rescaled areas+and perimeters, defined in (2.10), of the connected components Ω+
i of Ω = {u = +1}
εfor a given u = u , as in Lemma 3.4. The presentation is clarified by working with therescaled energy Ē ε defined by (2.14) rather than E ε directly. We begin by proving Part i)of Theorem 1, the lower bound.

20

4.1

Proof of lower bound, Theorem 1 i)

Step 1: Estimate of Ē ε in terms of Aεi and Piε .
First, for a fixed γ ∈ (0, 1) we define a truncated rescaled droplet area:(Aεi ,if Aεi < 32/3 πγ −1Ãεi :=(32/3 πγ −1 )1/2 |Aεi |1/2 if Aεi ≥ 32/3 πγ −1 ,(4.1)

and the isoperimetric deficit
ε
Idef
:=


1 X ε p
Pi − 4πAεi ≥ 0,
| ln ε|(4.2)

iwhich will be used throughout the proof. The purpose of defining the truncated droplet
area in (4.1) will become clearPlater.
We start by writing µε = i µεi , withdµεi (x) := ε−2/3 | ln ε|−1/3 χΩ+ (x)dx,
i

(4.3)

+
ε
where Ω+
i are the connected components of Ω = {u = +1}, and the index ε was omitted+
from Ωi to avoid cumbersome notation. For small enough ε this is justified by Lemma
3.1, in view of the fact that for some C > 0 we have|∂Ω+ | ≤ ε−1 E ε [uε ] ≤ Cε1/3 | ln ε|2/3 ,

(4.4)

+
so |∂Ω+
i |  ` whenever ε  1. In particular, (3.2) holds for Ωi when ε is sufficiently small.
For a fixed ρ > 0 we introduce the “far field truncation” Gρ ∈ C ∞ (T2` ) of the Green’s
function G:Gρ (x − y) := G(x − y)φρ (|x − y|)∀(x, y) ∈ T2` × T2` ,(4.5)

where φρ ∈ C ∞ (R) is a monotonically increasing cutoff function such that φρ (t) = 0 for
all t < 12 ρ and φρ (t) = 1 for all t > ρ. Then, for sufficiently small ε we have |∂Ω+
i | ≤ ρ inview of (4.4), and from (2.14) and (3.2) we obtain
!
Xp
2
δ̄
1
ε
Ē ε [uε ] ≥Idef+
4πAεi − 2 Aεi| ln ε|κiZZX
G(x − y)dµεi (x)dµεi (y)
(4.6)
+2
Z iZ

+2

Gρ (x − y)dµε (x)dµε (y),21where we used (3.2) and the positivity of G (cf. e.g. [38]), and here and everywhere below
we omit T2` × T2` as the domain of integration for double integrals to simplify the notation.
We recall that the Green’s function for −∆ + κ2 on T2` can be written as G(x − y) =1
− 2π ln |x − y| + O(|x − y|) [38]. With the help of this fact, together with (4.4) and (3.2),
for ε sufficiently small we have the following estimate for the self-interaction energy:
X ZZ
εĒself := 2
G(x − y)dµεi (x)dµεi (y)
i

1X
≥−
π
i

ZZ

(ln |x − y| + C) dµεi (x)dµεi (y).

X
1
=−
2
π| ln ε|
i

Z

Z
+

Ωi


+

Ωi


ln(ε1/3 | ln ε|2/3 |x − y|) + C dx̄ dȳ,

(4.7)

for some C > 0 independent of ε, where in equation (4.7) we have rescaled coordinates+x̄ = ε−1/3 | ln ε|1/3 x, ȳ = ε−1/3 | ln ε|1/3 and introduced the rescaled versions Ωi of Ω+i .
εcan beExpanding the logarithm in (4.7) and using (3.23) and (3.2), we obtain that Ēselfbounded from below as follows:
!


Z Z
X
1
1
ln
|
ln
ε|
1
ε
≥
Ēself−C−|Aεi |2
ln |x − y| dx dy+| ln ε|3π| ln ε|π|Aεi |2 | ln ε| Ω+
Ωi
i
i




1 X ε2 1
ln | ln ε|
1
ε
≥
−C
−
ln Pi
(4.8)
|Ai |
| ln ε|3π| ln ε|π| ln ε|i

1 X ε2 1
ln | ln ε|
≥
−C
,
|Ai |
| ln ε|3π| ln ε|i

for some C > 0 independent of ε (which changes from line to line).
Now observe that the term in parentheses appearing in the right-hand side of (4.8) is
positive for ε sufficiently small. Using this and the fact that Aεi ≥ Ãεi , from (4.8) we obtain
1 X ε2 1
ln | ln ε|εĒself ≥−C,(4.9)|Ãi || ln ε|3π| ln ε|iwhere C > 0 is a constant independent of ε. It is also clear from the definition of Ãεi that
there exists a constant c > 0 such that|Ãεi |2 ≤ cAεi .

(4.10)

Combining this inequality with (4.9) and choosing any η > 0, for ε small enough we have22η > Cc ln| ln| lnε|2ε| and, therefore, from (4.6) we obtain
ε





1 X p
1 ε2
2δ̄
ε
ε
4πAi −+ η Ai +|Ã || ln ε|
κ2
3π i
i
ZZ
+2
Gρ (x − y)dµε (x)dµε (y).

εεĒ [u ] ≥Idef+(4.11)

Step 2: Optimization over Aεi .
Focusing on the second term in the right-hand side of (4.11), we define√
2 π
1f (x) := √ +x,
3πx(4.12)and observe that f is strictly convex and attains its minimum of 32/3 at x = 32/3 π, with√
3 π
00
(4.13)
f (x) = 5/2 .
2x
We claim that we can bound the second term in the right-hand side of (4.11) from below
by the sum I + II + III of the following three terms:

X
X
1
2δ̄
1
2/3
I=
3 − 2 −ηAεi +32/3 (3−1 γ −1 − 1)Aεi ,(4.14)| ln ε|κ| ln ε| ε 2/3 −1
i

II =

III =

γ 5/2

1
| ln ε| 4π 2 · 32/3
1 γ 7/2| ln ε| 4πXAεi <32/3 πγ

Ai >3

πγ

Aεi (Aεi − 32/3 π)2 ,(4.15)(Aεi − 32/3 π)2 .

(4.16)

X
32/3 πγ≤Aεi ≤32/3 πγ −1

Before proving this, observe that defining

X
ZZ
2δ̄
1
2/3
ε
ε
ε ε
M := Ē [u ] −3 − 2 −ηAi − 2Gρ (x − y)dµε (x)dµε (y),| ln ε|κ(4.17)

iwe have from (4.11) and (4.14)–(4.16) that if Iγε is as in Theorem 2, thenMε ≥c1 X εc2 X εε(A − 32/3 π)2 + Idef≥0
Ai +| ln ε| ε| ln ε| ε i
i∈I
/ γ

i∈Iγ

for some constants c1 , c2 > 0 depending only on γ.
23∀γ ∈ (0, 13 ),(4.18)

We now argue in favor of the lower bound based on (4.14)–(4.16). First observe thatby (4.1) we have for all Aεi ≥ 32/3 πγ −1 :




p
1 ε2
2δ̄
2δ̄
ε
2/3
ε
+ η Ai ≥ 3 − 2 − η Aεi + 32/3 (3−1 γ −1 − 1)Aεi . (4.19)
4πAi +|Ã | −
3π i
κ2κWhen Aεi < 32/3 πγ −1 , which corresponds to both (4.15) and (4.16), we use the convexityof f and (4.13):
!


√
p
π
1
2
1
2
δ̄
2
δ̄
4πAεi ++ η Aεi = Aεi p ε +−η|Ãε |2 −
Aε −
3π i
κ2
3π i κ2
Ai


2δ̄
ε
ε
= Ai f (Ai ) − 2 − η(4.20)
κ




1
2δ̄
≥ 32/3 − 2 − η Aεi + Aεi f 00 32/3 πγ −1 (Aεi − 32/3 π)2 ,κ2where the last line follows from the second order Taylor formula for f (x) about x = 32/3 π
and the fact that f 00 (x) is decreasing. Combining (4.17), (4.19) and (4.20) yields M ε ≥I + II + III.
Now using (4.17) and (4.18) with γ sufficiently small, we deduce that

X
ZZ
1
2δ̄
Ē ε [uε ] ≥32/3 − 2 − ηAεi + 2Gρ (x − y)dµε (x)dµε (y).
(4.21)| ln ε|κi

Step 3: Passage to the limit.
We may now conclude from (2.13)–(2.15), (2.17), (3.7), (3.9) and (3.23) thatZ
lim sup(|∇v ε |2 + κ2 |v ε |2 )dx < +∞,ε→0(4.22)

T2l

while (µε ) are bounded in the sense of measures from (3.24). Consequently, up to a
subsequencev ε * v in H 1 (T2` ),(4.23)ε ∗µ * µ in C(T2` ),(4.24)−∆v + κ2 v = µ(4.25)

where24

holds in the distributional sense. Now passing to the limit in (4.21) and recalling (2.12),
we obtain

Z
ZZ
2δ̄
lim inf Ē ε [uε ] ≥ 32/3 − 2 − ηdµ + 2Gρ (x − y)dµ(x)dµ(y),(4.26)
ε→0κ
using continuity of Gρ . On the other hand, we have Gρ (x − y) → G(x − y) monotonicallyfrom below for each x 6= y as ρ → 0. Moreover, since µ satisfies (3.5), the set {(x, y) ∈T2` × T2` : x = y} is µ ⊗ µ-negligible. An application of monotone convergence theorem thenyields
Z

ZZ
2δ̄
ε ε
2/3
lim inf Ē [u ] ≥ 3 − 2
dµ + 2G(x − y)dµ(x)dµ(y),(4.27)
ε→0κ
upon sending ρ → 0 and then η → 0.

We now argue in favor of the corresponding upper bound in Theorem 1. The construction resembles quite closely that of the vortex construction in [47] for the two dimensional
Ginzburg-Landau functional and indeed we borrow several ideas from that proof and occasionally refer the reader to that paper for details.

4.2

Proof of the Upper Bound, Theorem 1 ii)

P εε (x) as in (4.3), so that µε =
As
in
the
proof
of
the
lower
bound,
weset
dµµi . Ifi
R
T2` dµ = 0, there is nothing to prove. Otherwise, using a mollification with a strictly
positive mollifier we can always approximate the measure µ by a measure with a smooth
strictly positive density and retrieve a recovery sequence by a standard diagonal argument.
Hence without loss of generality in this section we assume thatdµ(x) = g(x)dx,c ≤ g ≤ C,

(4.28)

for some C > c > 0.
Step 1: Construction of the configuration.
We claim that for ε sufficiently small it is possible to place a total of N (ε) disjoint spherical
droplets, whereN (ε) =

1 | ln ε|µ(T2` ) + o(| ln ε|),π

32/3

(4.29)

with centers {ai } in T2` and radiusr = 31/3 ε1/3 | ln ε|−1/3 ,25(4.30)

and satisfying for all i 6= jCd(ε) := min |ai − aj | ≥ p,
N (ε)(4.31)

for some constant C > 0 depending only on µ. Indeed, given µ satisfying (4.28), for εsufficiently small we can partition T2` into disjoint squares {Ki } of side length ηε > 0(hereafter simply denoted η) satisfying| ln ε|−1/2  η  1.

(4.32)

1 | ln ε|µ(Ki )NKi (ε) = 2/3
π3(4.33)

In each Ki we placepoints ai (here m = bxc denotes the smallest integer m ≤ x) satisfying (4.31) and in
additionXC,
N (ε) :=
NKi .
(4.34)
dist (ai , ∂Ki ) ≥ p
N (ε)
iAs argued in [47], our ability to do this follows from the estimate:cη 2 ≤ µ(Ki ) ≤ Cη 2 ,(4.35)

which follows from (4.28) together with (4.32). We finally define our configuration uε by+εsetting the connected components Ω+
i of Ω = {u = +1} to be balls of radius r from+ε(4.30) centered at ai , i.e. Ωi := Br (ai ). We set u = −1 in the complement of these balls.
With these choices we have
ZZ
2π · 31/3 N (ε) 2π · 32/3 N (ε)δ̄ε εĒ [u ] =−+2
G(x − y)dµε (x)dµε (y)| ln ε|| ln ε|κ2ZZ2δ̄2G(x − y)dµε (x)dµε (y) + o(1).
(4.36)= 1/3 µ(T2` ) − 2 µ(T2` ) + 2
κ3
The
the proof
is to show that the integral term in (4.36) converges
RRmain point of the rest of −1/3RtoG(x − y)dµ(x)dµ(y) + 3
dµ, with the non-trivial last term coming from the selfinteraction of the droplets. To prove that these are the only contributions to the limit
energy, we need to use the fact that the droplets do not concentrate too much as ε → 0.
Step 2: Convergence of the configurations.
Defining µε as before, it is clear from the construction thatµε * µ in (C(T2` ))∗ .
26(4.37)

Fix ρ > 0 sufficiently small (depending only on κ and `) and consider Gρ (x − y) defined asin (4.5). By the continuity of Gρ in T2` we have
ZZ
ZZ
ε
ε
lim
Gρ (x − y)dµ (x)dµ (y) =Gρ (x − y)dµ(x)dµ(y).
(4.38)
ε→0

Now, let Iρ be the collection of indices (i, j) such that 0 < |ai − aj | < ρ. Then for ε smallenough we can writeZZ
G(x − y) − Gρ (x − y) dµε (x)dµε (y)≤N (ε) Z Z

≤

X
i=1

G(x − y)dµεi (x)dµεi (y) +X ZZ(i,j)∈Iρ

G(x − y)dµεi (x)dµεj (y)(4.39)N (ε)
XX
1C ln | ln ε|C0+|Aεi |2 ++Aεi Aεj ln dist (Ω+i , Ωj ) ,2
6π| ln ε|| ln ε|| ln ε|i=1(i,j)∈Iρ

for some C, C 0 > 0 independent of ε or ρ, where Aεi = 32/3 π and we expanded the Green’s
function as in (4.7) in the proof of the lower bound. Now, for k = 1, 2, . . . , Kρ (ε), withKρ (ε) := bρ/d(ε)c, let Iρk ⊂ Iρ be disjoint sets consisting of all indices (i, j) such thatkd(ε) ≤ |ai − aj | < (k + 1)d(ε). Since by the result on optimal packing density of disks inthe plane [18] we have |Iρk | ≤ ckN (ε) for some universal c > 0 (here again |Iρk | denotes the
cardinality of Iρk ), in view of (4.29) it holds thatKρ (ε)X
1
CN (ε) X++ε ε
Ai Aj ln dist (Ωi , Ωj ) ≤k| ln(kd(ε))|| ln ε|2| ln ε|2
k=1(i,j)∈IρZ ρ
2CN (ε)0 | ln d(ε)|
2≤t| ln t|dt ≤ C+ ρ | ln ρ| ≤ 2C 0 ρ2 | ln ρ|,| ln ε|2 d2 (ε) d(ε)| ln ε|(4.40)

for some C, C 0 > 0 independent of ε or ρ, when ε and ρ are sufficiently small. Therefore,
from (4.30) and (4.39) we obtainZZlim supG(x − y) − Gρ (x − y) dµε (x)dµε (y) ≤ 2−1 · 3−1/3 + o(ρ).
(4.41)
ε→0

Finally combining (4.41) with (4.36) and (4.38), upon sending ε → 0, then ρ → 0 andapplying the monotone convergence theorem we have
Z

ZZ
2δ̄
dµ + 2G(x − y)dµ(x)dµ(y),(4.42)
lim Ē ε [uε ] ≤ 32/3 − 2ε→0κas required. The fact that v ε * v follows from (4.37) and the uniform bounds just demonstrated on the terms involving the Green’s function in (4.36), from which it follows that
(2.18) is satisfied distributionally.

27

5

Proof of Theorem 2

In the proof of Sec. 4, we have in fact established Theorem 2, which is clear by (4.18).
Indeed, we have for a sequence of almost minimizers (uε ):lim E ε [uε ] − E0 [µ] = 0.

(5.1)

ε→0

Observing that M ε defined in (4.17) does not contribute to E0 [µ], we have established thatM ε → 0 as ε → 0 for any γ < 31 and, as a consequence, we obtain (2.22)–(2.24) for, say,γ = 16 . Then it is easy to see from the definition of Iγε that the statement of the Theorem,
in fact, holds for any γ ∈ (0, 1).
6

Proof of Theorem 3

We now turn to the proof of Theorem 3 extending the result of Theorem 1 for the sharp
interface energy E ε to the diffuse interface energy E ε . The proof proceeds by a refinementof the ideas of [38, Sec. 4] to establish matching upper and lower bounds for E ε in termsof E ε for sequences with bounded energy.
Step 1: Approximate lower bound.
In the following, it is convenient to rewrite the energy (2.4) in an equivalent form

Z  2
Z
ε
1
ε ε
ε 2
ε
ε 2
ε
ε
ε
E [u ] =|∇u | + W (u ) + |∇v | dx,−∆v = u − ū ,v ε dx = 0.
2
2
2
2
T`
T`
(6.1)
Fix any δ ∈ (0, 1) and consider a sequence (uε ) ∈ Aε such that lim supε→0 kuε kL∞ (T2 ) ≤ 1
and E ε [uε ] ≤ Cε4/3 | ln ε|2/3 for some C > 0 independent of ε. Then we claim thatlim sup kuε kL∞ (T2 ) = 1,ε→0lim kv ε kL∞ (T2 ) = 0.

ε→0

`

`

`

(6.2)

Indeed, for the first statement we have from the definition of E ε in (2.4) that|Ωδ0 | ≤ Cε4/3 | ln ε|2/3 δ −2 ,Ωδ0 := {−1 + δ ≤ uε ≤ 1 − δ},(6.3)

for some C > 0 independent of ε. Hence, in particular, lim supε→0 kuε kL∞ (T2 ) ≥ 1, proving`
the first statement of (6.2). To prove the second statement in (6.2), we note that bystandard elliptic theory (see, e.g., [21, Theorem 9.9]) we have kv ε kW 2,p (T2 ) ≤ C 0 for any`
p > 2 and some C 0 > 0 independent of ε and, hence, by Sobolev embedding k∇v ε kL∞ (T2 ) ≤`28

C 00 for some C 00 > 0 independent of ε as well. Therefore, applying Poincaré’s inequality,
we obtain
Z
4/3
2/3
ε ε
0
(6.4)
Cε | ln ε| ≥ E [u ] ≥ C|v ε |2 dx ≥ C 00 kv ε k4L∞ (T2 ) ,`T2`for some C 0 , C 00 > 0 independent of ε, yielding the claim.
In view of (6.2), for small enough ε we have kuε kL∞ (T2 ) ≤ 1 + δ 3 and kv ε kL∞ (T2 ) ≤ δ 3 ,``and by the assumption on energy we may further assume that E ε [uε ] ≤ δ 12 . Therefore,by [38, Proposition 4.2] there exists a function ũε0 ∈ A such thatE ε [uε ] ≥ (1 − δ 1/2 )E ε [ũε0 ].

(6.5)

In particular, (ũε0 ) satisfy the assumptions of Theorem 1, and, therefore, upon extractionof subsequences we have µ̃ε0 * µ ∈ M+ (T2` ) ∩ H −1 (T2` ) in (C(T2` ))∗ , wheredµ̃ε0 (x) := 12 ε−2/3 | ln ε|−1/3 (1 + ũε0 (x))dx.

(6.6)

Furthermore, recalling that by construction the jump set of ũε0 is either contained in Ωδ0 or
empty, see the proof of [38, Lemma 4.1], from (6.3) we havekũε0 − uε0 kL1 (T2 ) ≤ Cε4/3 | ln ε|2/3 δ −2 ,
`

(6.7)

where uε0 is given by (2.29), for some C > 0 independent of ε. Comparing (6.7) with (6.6),
we then see that µε0 * µ in (C(T2` ))∗ as well. The result of part (i) of Theorem 3 thenfollows by the arbitrariness of δ > 0 via a diagonal process.
Step 2: Approximate upper bound.
First note that if µ = 0, we can choose uε = ūε . Indeed, we have ε−4/3 | ln ε|−2/3 E ε [ūε ] =R
2 2`2 W (ūε ) = `2κδ̄2 + o(1) and ūε → −1. On the other hand, if T2 dµ > 0, we can construct`
the approximate upper bounds for a suitable lifting of the recovery sequences in the proofof Theorem 1(ii) to Aε . Let (ũε0 ) ∈ A be a recovery sequence constructed in Sec. 4.2. Thissequence consists of circular droplets of the optimal radius r = 31/3 ε1/3 | ln ε|−1/3  ε1/2
and mutual distance d ≥ C| ln ε|−1/2  ε1/2 , for some C > 0 independent of ε. In addition,
since
Z
Z 

ε
ε ε
ε
E [ũ0 ] =|∇ũ0 | dx + 2|∇ṽ ε |2 + κ2 |ṽ ε |2 dx ≤ Cε4/3 | ln ε|2/3 ,
(6.8)
2 T2`
T2`
R
where ṽ ε (x) = T2 G(x − y)(ũε0 (y) − ūε )dy, for some C > 0 independent of ε, by the`argument of (6.4) one can see that limε→0 kṽ ε kL∞ (T2 ) = 0. Therefore, for any δ ∈ (0, 1)`

29and ε > 0 sufficiently small we have kṽ ε kL∞ (T2 ) ≤ δ and E ε [ũε0 ] ≤ δ 5/2 . We can then`apply [38, Proposition 4.3] to obtain a function uε ∈ Aε such thatE ε [uε ] ≤ (1 + δ 1/2 )E ε [ũε0 ].

(6.9)

Furthermore, by the construction of uε (see [38, Eqs. (4.31)–(4.33)]) and arbitrariness of
δ > 0, we also have lim supε→0 kuε kL∞ (T2 ) = 1, and`kũε0 − uε0 kL1 (T2 ) ≤ Cε4/3 | ln ε|2/3 ,`(6.10)

for some C > 0 independent of ε, where uε0 is given by (2.29), and we used (6.8). Henceµε0 * µ = limε→0 µ̃ε0 in (C(T2` ))∗ . The result of part (ii) of Theorem 3 again follows byarbitrariness of δ > 0 via a diagonal process.
Remark 6.1. It is possible to chose δ = εα for α > 0 sufficiently small in the argumentsof the proof of Theorem 3. Therefore, given a sequence of minimizers (uε ) ∈ Aε of E ε andthe corresponding sequence (uε0 ) ∈ A of minimizers of E ε , one has
ε−4/3 | ln ε|−2/3 E ε [uε ] = ε−4/3 | ln ε|−2/3 E ε [uε0 ] + O(εα ),(6.11)

for some α  1, as ε → 0.

7

Proof of Theorem 4

Let (uε ) be a sequence from Theorem 3(ii). Arguing as in Step 1 of the proof of Theorem
3, for every δ > 0 sufficiently small there exists a sequence (ũε0 ) ∈ A such that (6.5) holds,
the jump set of ũε0 is contained in {−1 + δ ≤ uε ≤ 1 − δ}, and if µ̃ε0 is defined via (6.6), thenµ̃ε0 * µ in (C(T2` ))∗ . On the other hand, applying the result of Theorem 1(i), we obtain
E 0 [µ] ≥ lim sup ε−4/3 | ln ε|−2/3 E ε [uε ] ≥ (1 − δ 1/2 ) lim sup ε−4/3 | ln ε|−2/3 E ε [ũε0 ]
ε→0≥ (1 − δ

1/2) lim inf ε
ε→0

−4/3

ε→0
−2/3

| ln ε|E ε [ũε0 ] ≥ (1 − δ 1/2 )E 0 [µ]. (7.1)

Therefore, in view of arbitrariness of δ > 0 we conclude that (ũε0 ) is a sequence of almost
minimizers of E ε with prescribed density µ by a diagonal process. As a consequence,
Theorem 2 applies to (ũε0 ). Moreover, by (6.7) and the fact that
Z
−2/3
−1/3
ε
lim ε
| ln ε||{u > 0}| =dµ > 0,
(7.2)
ε→0

T2`

we obtain (2.34).
Acknowledgments The research of D. G. was partially supported by an NSERC PGS D
award. The work of C. B. M. was supported, in part, by NSF via grants DMS-0718027
and DMS-0908279. The research of S. S. was supported by the EURYI award. C. B. M.
would like to acknowledge valuable discussions with H. Knüpfer and M. Novaga.
30

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Dorian Goldman
Courant Institute of Mathematical Sciences, New York, NY 10012, USA,
& UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions,34

Paris, F-75005 France ;CNRS, UMR 7598 LJLL, Paris, F-75005 France
dgoldman@cims.nyu.edu
Cyrill B. Muratov
Department of Mathematical Sciences,
New Jersey Institute of Technology,
Newark, NJ 07102, USA
muratov@njit.edu
Sylvia Serfaty
UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions,
Paris, F-75005 France ;CNRS, UMR 7598 LJLL, Paris, F-75005 France& Courant Institute, New York University
251 Mercer st, NY NY 10012, USA
serfaty@ann.jussieu.fr

35