Dorian Goldman
email: dgoldman@cims.nyu.edu

Cyrill B Muratov
email: muratov@njit.edu

Sylvia Serfaty
email: serfaty@ann.jussieu.fr

The Γ-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density

This is the first in a series of two papers in which we derive a Γ-expansion for a 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 considered parameter regime is via weak convergence of the suitably normalized charge density in the 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 is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. 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 a characterization of the limit charge density for the energy minimizers in the diffuse interface model.

Introduction

In the studies of energy-driven pattern formation, one often encounters variational problems with competing terms operating on different spatial scales [START_REF] Hubert | Magnetic domains[END_REF][START_REF] Huebener | Magnetic flux structures in superconductors[END_REF][START_REF]Theory of inhomogeneous electron gas[END_REF][START_REF] Muthukumar | Competing interactions and levels of ordering in self-organizing polymeric materials[END_REF][START_REF] Seul | Domain shapes and patterns: the phenomenology of modulated phases[END_REF][START_REF] Strukov | Ferroelectric Phenomena in Crystals: Physical Foundations[END_REF][START_REF] Vedmedenko | Competing Interactions and Pattern Formation in Nanoworld[END_REF]. Despite the fundamental importance of these problems to a multitude of physical systems, their detailed mathematical studies are fairly recent (see e.g. [START_REF] Choksi | Scaling laws in microphase separation of diblock copolymers[END_REF][START_REF] Choksi | Ground state energy scaling laws during the onset and destruction of the intermediate state in a Type-I superconductor[END_REF][START_REF] Choksi | Bounds on the micromagnetic energy of a uniaxial ferromagnet[END_REF][START_REF] Choksi | Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy[END_REF][START_REF] Choksi | Energy minimization and flux domain structure in the intermediate state of a Type-I superconductor[END_REF][START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence[END_REF][START_REF] Kohn | Energy-driven pattern formation[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF]). 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 [START_REF] Choksi | Ground state energy scaling laws during the onset and destruction of the intermediate state in a Type-I superconductor[END_REF][START_REF] Choksi | Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy[END_REF][START_REF] Desimone | Magnetic microstructures-a paradigm of multiscale problems[END_REF][START_REF] Bris | From atoms to crystals: a mathematical journey[END_REF][START_REF] Müller | Singular perturbations as a selection criterion for periodic minimizing sequences[END_REF]. In particular, the popular approach of Γ-convergence [START_REF] Braides | Γ-convergence for beginners[END_REF] is rendered difficult due to the emergence of more than two well-separated spatial scales in suitable asymptotic limits (see e.g. [9-12, 17, 30, 33, 46]).

These issues can be readily seen in the case of the Ohta-Kawasaki model, a canonical mathematical model in the studies of energy-driven pattern forming systems. This model, 1 originally proposed in [START_REF] Ohta | Equilibrium morphologies of block copolymer melts[END_REF] to describe different morphologies observed in diblock copolymer melts (see e.g. [START_REF] Bates | Block copolymers -designer soft materials[END_REF]) is defined (up to a choice of scales) by the energy functional

E[u] = Ω ε 2 2 |∇u| 2 + W (u) dx + 1 2 Ω Ω (u(x) -ū)G 0 (x, y)(u(y) -ū) dx dy, (1.1)
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) = 1 4 (1u 2 ) 2 , ε > 0 is a parameter characterizing interfacial thickness, ū ∈ (-1, 1) is the background charge density, and G 0 is the Neumann Green's function of the Laplacian, i.e., G 0 solves

-∆G 0 (x, y) = δ(x -y) - 1 |Ω| , Ω G 0 (x, y) dx = 0, (1.2)
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" condition

1 |Ω| Ω u dx = ū. (1.3)
Let us point out that in addition to a number of polymer systems [START_REF] De Gennes | Effect of cross-links on a mixture of polymers[END_REF][START_REF] Nyrkova | Microdomain structures in polyelectrolyte systems: calculation of the phase diagrams by direct minimization of the free energy[END_REF][START_REF] Stillinger | Variational model for micelle structure[END_REF], this model is also applicable to many other physical systems due to the fundamental nature of the Coulombic non-local term in (1.1) [START_REF] Chen | Dynamics of simultaneous ordering and phase separation and effect of long-range Coulomb interactions[END_REF][START_REF] Emery | Frustrated electronic phase-separation and hightemperature superconductors[END_REF][START_REF] Glotzer | Reaction-controlled morphology of phase-separating mixtures[END_REF][START_REF]Theory of inhomogeneous electron gas[END_REF][START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulomb type[END_REF][START_REF] Nagaev | Phase separation in high-temperature superconductors and related magnetic systems[END_REF]. 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:

E[u] = ε 2 Ω |∇u| dx + 1
2 Ω Ω (u(x)ū)G(x, y)(u(y)ū) dx dy, (1.4) where G(x, y) is the screened Green's function of the Laplacian, i.e., it solves the Neumann problem for the equation

-∆G + κ 2 G = δ(x -y), (1.5)
where κ := 1/ W (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 no longer 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 be constant equal to its average ū, but since u is valued in {+1, -1} this means in effect that it is advantageous for u to oscillate rapidly between the two phases u = +1 and u = -1; 4 (1-u 2 ) 2 obtained 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 [START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional[END_REF]) to an expression in terms of the interfaces alone. In [START_REF] Choksi | 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions[END_REF][START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional[END_REF], such a reduction was performed for E using formal asymptotic techniques (see also [START_REF] Bris | From atoms to crystals: a mathematical journey[END_REF][START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulomb type[END_REF][START_REF] Ohta | Equilibrium morphologies of block copolymer melts[END_REF][START_REF] Radin | The ground state for soft disks[END_REF]) and leads to the following reduced energy (for simplicity of notation, we choose the normalizations in such a way that the parameter ε is, in fact, the domain wall energy, see Sec. 4 for details):

E[u] = ε 2 Ω |∇u| dx + 1
2 Ω Ω (u(x) -ū)G(x, y)(u(y) -ū) dx dy.

(1.4)

Here the function u takes on values ±1 throughout Ω, and the kernel G is the screened Coulomb kernel, i.e., it solves the Neumann problem for

-∆G(x, y) + κ 2 G(x, y) = δ(x -y), (1.5)
with some κ > 0. The constant κ has the physical meaning of the inverse of the Debye screening length [START_REF] Choksi | 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions[END_REF][START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional[END_REF]. Note that the sharp interface energy E with the unscreened Coulomb kernel (i.e. with κ = 0) was derived by Ren and Wei as the Γ -limit of the diffuse interface energy E under assumptions of weak non-local coupling (i.e., with an extra factor of ε in front of the Coulomb kernel) and ū ∈ (-1, 1) independent of ε, as ε → 0 [START_REF] Bris | From atoms to crystals: a mathematical journey[END_REF] (see also [START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulomb type[END_REF][START_REF] Muthukumar | Competing interactions and levels of ordering in self-organizing polymeric materials[END_REF]; note that this case is also equivalent to considering E on the domain of size O(ε 1/3 )). At the same time, screening becomes important near the transition between the uniform and the patterned states which occurs near | ū| = 1, the case of interest in the present paper [START_REF] Choksi | 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions[END_REF][START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional[END_REF]. Note that in the presence of screening the neutrality condition in (1.3) is relaxed.

In this paper, we rigorously establish the relation between the sharp interface energy E and the diffuse interface energy E, and analyze the precise behavior of minimizers of the sharp interface energy E for ε 1 in the vicinity of the transition from the trivial minimizer to patterned states occurring near | ū| = 1. We note that despite the apparent simplicity of the expression for E, the minimizers of E exhibit quite an intricate dependence on the parameters for ε 1 and | ū| 1. Our analysis in this paper will be restricted to the case d = 2. While a number of our results can be extended the first term penalizes the perimeter of the interface between the two phases, and thus opposes too much spreading and oscillation. The competition between these two selects a length scale, which is a function of ε. In the diffuse interface version (1.1), the sharp transitions between {u = +1} and {u = -1} are replaced by smooth transitions at the scale ε > 0 as soon as ε 1. In one space dimension and in the particular case ū = 0 (symmetric phases) the behavior of the energy can be understood from the work of Müller [START_REF] Müller | Singular perturbations as a selection criterion for periodic minimizing sequences[END_REF]: the minimizer u is periodic and alternates between u = +1 and u = -1 at scale ε 1/3 (for other one-dimensional results, see also [START_REF] Ren | On the multiplicity of solutions of two nonlocal variational problems[END_REF][START_REF] Yip | Structure of stable solutions of a one-dimensional variational problem[END_REF]). In higher dimensions the patterns of minimizers are much more complex and are not well understood. The behavior depends on the volume fraction between the phases, i.e. on the constant ū chosen, and also on the dimension. When ū < 0, we call u = -1 the majority phase and u = +1 the minority phase, and conversely when ū > 0. In two dimensions, numerical simulations lead to expecting round "droplets" of the minority phase surrounded by a "sea" of the majority phase (see Fig. 1) for sufficient asymmetries between the majority and the minority phases (i.e., for ū sufficiently far away from zero) [START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulombic type[END_REF][START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulomb type[END_REF][START_REF] Ohta | Equilibrium morphologies of block copolymer melts[END_REF][START_REF] Ren | Many droplet pattern in the cylindrical phase of diblock copolymer morphology[END_REF]. The situation is less clear for ū close to zero, although it is commonly believed that in this case the minimizers are one-dimensional stripe patterns [START_REF] Choksi | 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions[END_REF][START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulombic type[END_REF][START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulomb type[END_REF][START_REF] Ohta | Equilibrium morphologies of block copolymer melts[END_REF].

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 date is that of Alberti, Choksi and Otto [START_REF] Alberti | Uniform energy distribution for an isoperimetric problem with long-range interactions[END_REF], 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 from (1.3)) is uniformly distributed in the limit where the size of the domain Ω goes to infinity (see also [START_REF] Choksi | Scaling laws in microphase separation of diblock copolymers[END_REF][START_REF] Spadaro | Uniform energy and density distribution: diblock copolymers' functional[END_REF]). Their results, however, do not provide any further information about the structure of the energy-minimizing patterns. Note in passing that the question of proving any periodicity of minimizers for multi-dimensional energies is unsolved even for systems of point particles forming simple crystals (see e.g. [START_REF] Bris | From atoms to crystals: a mathematical journey[END_REF][START_REF] Sandier | From the Ginbzurg-Landau model to vortex lattice problems[END_REF]), with a notable exception of certain two-dimensional particle systems with short-range interactions which somehow reduce to packing problems [START_REF] Radin | The ground state for soft disks[END_REF][START_REF] Theil | A proof of crystallization in two dimensions[END_REF][START_REF] Wagner | Crystallinity in two dimensions: a note on a paper of C. Radin: "The ground state for soft disks[END_REF]. 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 close to -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 properties of a suitably rescaled version of the sharp interface energy (1.4) with no screening in [START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional[END_REF], as well as (1.1) in [START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: II. Diffuse interface functional[END_REF]. 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 [START_REF] Braides | Asymptotic expansions by Γ-convergence[END_REF]. 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 [START_REF] Ren | On the multiplicity of solutions of two nonlocal variational problems[END_REF], who also analyzed local minimizers of the sharp interface energy in the strong asymmetry regime in two space dimensions [START_REF] Ren | Many droplet pattern in the cylindrical phase of diblock copolymer morphology[END_REF].

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 to infinity. A rather comprehensive study of the behavior of the minimizers for the Ohta-Kawasaki energy in macroscopically large domains was recently performed in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF], still in the regime ū close to -1. There the two-dimensional Ohta-Kawasaki energy was considered in the case when Ω is a unit square with periodic boundary conditions. The interesting regime corresponds to the parameters ε 1 and 1 + ū = O(ε 2/3 | ln ε| 1/3 ) 1. It is shown in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] 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 shown that when δ := ε -2/3 | ln ε| -1/3 (1 + ū) becomes greater than a certain critical constant δc , the minimizers of E in (1.4) consist of O(| ln ε|) simply connected, nearly round droplets of radius 3 1/3 ε 1/3 | ln ε| -1/3 , and uniformly distributed throughout the domain [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]. Thus, the following hierarchy of length scales is established in the considered regime:

ε ε 1/3 | 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).

The analysis of [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] 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 to asking whether the qualitative properties of the minimizers established in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] (roundness of the droplets, identical radii, uniform distribution) carry over to, e.g., almost minimizers of 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 [START_REF] Goldman | The Γ-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement via the renormalized energy[END_REF], 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 [START_REF] Sandier | From the Ginbzurg-Landau model to vortex lattice problems[END_REF].

For simplicity, as in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] we consider the energy defined on a torus (a flat square with periodic boundary conditions). The main question for setting up the Γ-limit in the present context is to choose a suitable metric for Γ-convergence. In this paper we show that the right metric to consider is the weak convergence of measures for a suitably rescaled sequence of characteristic functions associated with droplets (see the next section for precise definitions and statements of theorems). Then, up to a rescaling, 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 δ and κ.

We note that the obtained limit variational problem is strictly convex and its unique minimizer 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, which is a new result. In our companion paper [START_REF] Goldman | The Γ-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement via the renormalized energy[END_REF], we further address the mutual arrangement of the droplets in the energy minimizing patterns, using the formalism developed recently for Ginzburg-Landau vortices [START_REF] Sandier | From the Ginbzurg-Landau model to vortex lattice problems[END_REF]. In fact, the problem under consideration, and its mathematical treatment (here as well as in [START_REF] Goldman | The Γ-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement via the renormalized energy[END_REF]), share several important features with the two-dimensional magnetic Ginzburg-Landau model from the theory of superconductivity [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF]. There the role of droplets is played by the Ginzburg-Landau vortices, which in the appropriate limits also become uniformly distributed throughout the domain [START_REF] Sandier | A rigorous derivation of a free-boundary problem arising in superconductivity[END_REF]. We note, however, that the approach developed in [START_REF] Sandier | A rigorous derivation of a free-boundary problem arising in superconductivity[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF] cannot be carried over directly to the problem under consideration, since the vortices are more rigid than their droplet counterparts: the topological degrees of the vortices are quantized and can only take integer values, while their counterparts here, which are 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 relies on the control of their perimeter via the energy. Let us also mention yet another closely related system from the study of ferromagnetism, where the role of vortices is played by the slender needle-like domains of opposite magnetization at the onset magnetization reversal [START_REF] Knüpfer | Domain structure of bulk ferromagnetic crystals in applied fields near saturation[END_REF].

Our paper is organized as follows. In Sec. 2, we introduce the considered scaling regime and state our main results; in Sec. 3 we prove the Γ-convergence result in the sharp interface setting; in Sec. 4 we prove the results on the characterization of almost minimizers of sharp interface energy; and in Sec. 5 we treat the Γ-limit for the case of the diffuse interface energy. Some notations. We use the notation (u ε ) ∈ A to denote sequences of functions u ε ∈ A as ε = ε n → 0, where A is an admissible class. We also use the notation µ ∈ M(Ω) to denote a positive finite Radon measure dµ on the domain Ω. With a slight abuse of notation, 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 bounded variation, space of continuous functions, and the dual of H 1 (Ω), respectively.

Statement of results

Throughout the rest of the paper the domain Ω is assumed to be a flat two-dimensional torus of side length , i.e., when Ω = T 2 = [0, ) 2 , with periodic boundary conditions. Given the parameters κ > 0, δ > 0 and > 0, for every ε > 0 we define

ūε := -1 + ε 2/3 | ln ε| 1/3 δ. ( 2.1)
Then the sharp interface version of the Ohta-Kawasaki energy (cf. (1.4)) can be written as

E ε [u] = ε 2 T 2 |∇u| dx + 1 2 T 2 (u -ūε )(-∆ + κ 2 ) -1 (u -ūε ) dx, (2.2)
for all u ∈ A, where

A := BV (T 2 ; {-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 this energy to its diffuse interface version, which under the same scaling assumptions takes the form

E ε [u] = T 2 ε 2 2 |∇u| 2 + W (u) + 1 2 (u -ūε )(-∆) -1 (u -ūε ) dx, (2.4)
for all u ∈ A ε , where

A ε := u ∈ H 1 (T 2 ) : 1 2 T 2 u dx = ūε . (2.5)
We note that at the level of the energy minimizers the relation between the two functionals was established in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF].

The energy E ε may alternatively be written in terms of the level sets of u. Indeed, the set Ω + := {u = +1} is a set of finite perimeter (for precise definitions, see e.g. [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]), and when |∂Ω + | is sufficiently small, the set Ω + may be uniquely decomposed into an at most countable union of connected components Ω + i , where the boundaries ∂Ω + i of each connected component are Jordan curves which are essentially disjoint (after an extension to the whole of R 2 and up to negligible sets, see e.g. [2, Corollary 1 and Theorem 8]). In the context of Γ-convergence the sets Ω + i may be viewed as a suitable generalization of the droplets introduced earlier in the studies of energy minimizing patterns [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]. Note, however, that the sets Ω + i lack the regularity properties of the energy minimizers in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] and may 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 associated with E ε and, in particular, will be shown to be close, in some average sense, to disks of prescribed radii for almost minimizers of energy.

In terms of the droplets, from the above discussion we have

u = -1 + 2 i χ Ω + i , (2.6)
where χ Ω + i are the characteristic functions of Ω + i . Inserting this into (2.2), expressing the result via G that solves

-∆G(x) + κ 2 G(x) = δ(x) in T 2 , (2.7)
expanding all the terms and using the fact that T 2 G(x)dx = κ -2 , we arrive at (see also [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF])

E ε [u] = 2 (1 + ūε ) 2 2κ 2 + i ε|∂Ω + i | -2κ -2 (1 + ūε )|Ω + i | + 2 i,j Ω + i Ω + j G(x -y) dx dy, (2.8)
where we took into account the translational symmetry of the problem in T 2 . Moreover, since the optimal configurations for Ω + i are expected to consist of droplets of size of order ε 1/3 | ln ε| -1/3 (see (1.6) and the discussion around), it is convenient to introduce the rescaled area and perimeter of each droplet:

A i := ε -2/3 | ln ε| 2/3 |Ω + i |, P i := ε -1/3 | ln ε| 1/3 |∂Ω + i |. (2.9)
Similarly, let us introduce the suitably rescaled measure dµ associated with the droplets:

dµ(x) := ε -2/3 | ln ε| -1/3 i χ Ω + i (x)dx = 1 2 ε -2/3 | ln ε| -1/3 (1 + u) dx. (2.10)
Then the energy E ε [u] may be rewritten as

E ε [u] = ε 4/3 | ln ε| 2/3 δ2 2 2κ 2 + Ēε [u] , (2.11)
where

Ēε [u] := 1 | ln ε| i P i - 2 δ κ 2 A ε i + 2 T 2 T 2
G(xy)dµ(x)dµ(y).

(2.12)

Observe that for the nontrivial minimizers we know from [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] that Ēε = O(1), A i = O(1) and P i = O(1) (and even more precisely A i 3 2/3 π and P i 2 • 3 1/3 π), the number of droplets 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 . Observe also that if (u ε ) ∈ A is such that lim sup ε→0 Ēε [u ε ] < +∞, we have lim sup ε→0 1 | ln ε| i P ε i < +∞, lim sup ε→0 1 | ln ε| i A ε i < +∞, (2.13)
where {A ε i }, {P ε i } and µ ε are associated with u ε . Indeed, by (2.9) and (2.10) we have

1 | ln ε| i A ε i = T 2 dµ ε , (2.14)
and thus, from (2.12) we obtain

Ēε [u] ≥ - 2 δ κ 2 T 2 dµ ε + 2 T 2 T 2 G(x -y)dµ ε (x)dµ ε (y) ≥ - 2 δ κ 2 T 2 dµ ε + 2 κ 2 2 T 2 dµ ε 2 , (2.15)
where we used the well-known fact that for any u ∈ L 2 (T 2 ) we have

T 2 T 2 G(x -y)u(x)u(y)dx dy ≥ 1 2 T 2 G(x) dx T 2 u(x) dx 2 , (2.16)
which is readily obtained by Fourier transform. Then it follows that

T 2 dµ ε ≤ δ 2 + κ max(0, E ε [u ε ]) 1/2 , (2.17)
implying both statements in (2.13).

We now state our Γ-convergence result, which is obtained for configurations (u ε ) that obey the optimal energy scaling, i.e. when Ēε [u ε ] remains bounded as ε → 0. The result is obtained with the help of the framework established in [START_REF] Sandier | A rigorous derivation of a free-boundary problem arising in superconductivity[END_REF], where an analogous result for the Ginzburg-Landau functional of superconductivity was obtained. What we show is that the limit functional E 0 depends only on the limit density µ of the droplets (more precisely, on a general positive measure dµ). In passing to the limit the second term in (2.12) 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 instead with the equivalent energy Ēε , which is defined through {A ε i }, {P ε i } and µ ε (cf. (2.11) and (2.12)).

Theorem 1. (Γ-convergence of E ε ) Fix δ > 0, κ > 0 and > 0, and let E ε be defined by (2.2) with ūε given by (2.1). Then, as ε → 0 we have that

ε -4/3 | ln ε| -2/3 E ε Γ → E 0 [µ] := δ2 2 2κ 2 + 3 2/3 - 2 δ κ 2 T 2 dµ + 2 T 2 T 2 G(x -y)dµ(x)dµ(y), where µ ∈ M(T 2 ) ∩ H -1 (T 2 ). More precisely, we have i) (Lower Bound) Let (u ε ) ∈ A be such that lim sup ε→0 ε -4/3 | ln ε| -2/3 E ε [u ε ] < +∞, (2.18)
let dµ ε (x) := 1 2 ε -2/3 | ln ε| -1/3 (1 + u ε (x))dx, (2.19
)

and let v ε satisfy -∆v ε + κ 2 v ε = µ ε in T 2 .
(2.20)

Then, up to extraction of a subsequence, we have

µ ε µ in (C(T 2 )) * , v ε v in H 1 (T 2 ), as ε → 0, where µ ∈ M(T 2 ) ∩ H -1 (T 2 ) and v ∈ H 1 (T 2 ) satisfy -∆v + κ 2 v = µ in T 2 .
(2.21)

Moreover, we have lim inf ε→0 ε -4/3 | ln ε| -2/3 E ε [u ε ] ≥ E 0 [µ].
ii) (Upper Bound) Conversely, given µ ∈ M(T 2 ) ∩ H -1 (T 2 ) and v ∈ H 1 (T 2 ) solving (2.21), there exist (u ε ) ∈ A such that for the corresponding µ ε , v ε as in (2. [START_REF] Tóth | Über einen geometrischen Satz[END_REF]) and (2.20) we have

µ ε µ in (C(T 2 )) * , v ε v in H 1 (T 2 ),
as ε → 0, and lim sup

ε→0 ε -4/3 | ln ε| -2/3 E ε [u ε ] ≤ E 0 [µ].
We 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 ε associated with the limit charge density µ of the droplets, plus a term associated with the self-energy of the droplets. The functional E 0 is strictly convex, so there exists a unique minimizer μ ∈ M(T 2 ) ∩ H -1 (T 2 ) of E 0 , which is easily seen to be either μ = 0 for δ ≤ 1 2 3 2/3 κ 2 or μ = 1 2 ( δ -1 2 3 2/3 κ 2 ) otherwise. The latter can also be seen immediately from Remark 2.1 below, which gives a local characterization of the limit energy E 0 .

Remark 2.1. The potential v from (2.21) can also be expressed as v(x) = T 2 G(xy) dµ(y). The limit energy E 0 in Theorem 1 becomes local when written in terms of that limit potential v:

E 0 [µ] = δ2 2 2κ 2 + 3 2/3 κ 2 -2 δ T 2 v dx + 2 T 2 |∇v| 2 + κ 2 v 2 dx. (2.22)
Also, by the usual properties of Γ-convergence [START_REF] Braides | Γ-convergence for beginners[END_REF], the optimal density μ above is exhibited by the minimizers of E ε in the limit ε → 0, in agreement with [36, Theorem 2.2]: Corollary 2.2. For given δ > 0, κ > 0 and > 0, let ūε be given by (2.1) and let (u ε ) ∈ A be minimizers of E ε . Then, if µ ε is given by (2.19), as ε → 0 we have

µ ε 0 1 2 ( δ -δc ) in (C(T 2 )) * , ε -4/3 | ln ε| -2/3 -2 min E ε → δ2 2κ 2 δc 2κ 2 (2 δ -δc ), (2.23)
when δ ≤ δc or δ > δc , respectively, with δc :

= 1 2 3 2/3 κ 2 .
In particular, since the minimal energy scales with the area of T 2 , it is an extensive quantity.

We next give the definition of almost minimizers with prescribed limit density, for which a number of further results may be obtained. These can be viewed, e.g., as almost minimizers of E ε in the presence of an external potential. We note that in view of strict convexity of E 0 , minimizing E 0 [µ] -T 2 ϕ(x)dµ(x) for a given ϕ ∈ H 1 (T 2 ) one obtains a one-to-one correspondence between the minimizing density µ and the potential ϕ. It then makes sense to talk about almost minimizers of the energy E ε with prescribed limit density µ by viewing them as almost minimizers of E ε -T 2 ϕdµ ε . Also, observe that almost minimizers with the particular prescribed density μ from Corollary 2.2 are simply almost minimizers of E ε . Below we give a precise definition. Definition 2.3. For given δ > 0, κ > 0, > 0 and a given µ ∈ M(T 2 ) ∩ H -1 (T 2 ), we will call every recovery sequence (u ε ) ∈ 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 = 3 1/3 ε 1/3 | ln ε| -1/3 . More precisely, we have the following result.

Theorem 2. Let (u ε ) ∈ A be a sequence of almost minimizers of E ε with prescribed limit density µ. For every γ ∈ (0, 1) define the set

I ε γ := {i ∈ N : 3 2/3 πγ ≤ A ε i ≤ 3 2/3 πγ -1 }. Then lim ε→0 1 | ln ε| i P ε i -4πA ε i = 0, (2.24) lim ε→0 1 | ln ε| i∈I ε γ A ε i -3 2/3 π 2 = 0, (2.25)
lim ε→0 1 | ln ε| i ∈I ε γ A ε i = 0, (2.26)
where {A ε i } and {P ε i } are given by (2.9) with u = u ε .

Note that we may use the isoperimetric deficit terms present in (2.24) 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 ⊂ R 2 by

α(E) = min |E B| |E| , (2.27)
where the minimum is taken over all balls B ⊂ R 2 with |B| = |E|, and denotes the symmetric difference between sets. Note that the following sharp quantitative isoperimetric inequality holds for α(E) [START_REF] Fusco | The sharp quantitative isoperimetric inequality[END_REF]:

per(E) -4π|E| ≥ Cα(E) |E|, (2.28)
with some universal constant C > 0. As a direct consequence of Theorem 2 and (2.14), we then have the following result.

Corollary 2.4. Under the assumptions of Theorem 2, when

T 2 dµ > 0 we have lim ε→0 3 2/3 π|I ε γ | | ln ε| = T 2 dµ, lim ε→0 1 |I ε γ | i∈I ε γ α(Ω + i ) = 0, (2.29)
where |I ε γ | denotes the cardinality of I ε γ .

This result generalizes the one in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF], where it was found that in the case of the minimizers all the droplets are uniformly close to disks of the optimal radius r = 3 1/3 ε 1/3 | 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 droplets of 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 the droplets 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 T 2 appears to be quite surprising. In fact, this observation is consistent with the expectation that quantum mechanical charged particle systems form Wigner crystals at low particle densities [START_REF] Grimes | Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons[END_REF][START_REF]Theory of inhomogeneous electron gas[END_REF][START_REF] Wigner | On the interaction of electrons in metals[END_REF]. Let us point out that the Ohta-Kawasaki energy E ε bares a number of similarities with the classical Thomas-Fermi-Dirac-Von Weizsäcker model arising in the context of density functional theory of quantum systems (see e.g. [START_REF] Bris | From atoms to crystals: a mathematical journey[END_REF][START_REF] Lieb | Thomas-Fermi and related theories of atoms and molecules[END_REF][START_REF]Theory of inhomogeneous electron gas[END_REF]).

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, the asymptotic 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 (2.30)
for ε 1 was established in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]Theorem 2.3]. The main idea of the proof in [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] is for a given function u ε ∈ A ε to establish an approximate lower bound for E ε [u ε ] in terms of (1δ)E ε [ũ ε ] for some ũε ∈ A, with δ > 0 which can be chosen arbitrarily small for ε 1. The matching approximate upper bound is then obtained by a suitable lifting of the minimizer

u ε ∈ A of E ε into A ε .
Here we show that the procedure outlined above may also be applied to almost minimizers of E 0 in a suitably modified version of Definition 2.3 involving E ε . We note right away, however, that it is not possible to simply replace E ε with E ε in Definition 2.3. The reason for this is the presence of the mass constraint in the definition of the admissible class A ε for E ε . This implies, for example, that any sequence of almost minimizers (u ε ) ∈ A ε of E ε must satisfy -2 T 2 dµ ε = 1 2 δ, while, according to Corollary 2.2, for sequences of almost minimizers (u ε ) ∈ A of E ε we have -2

T 2 dµ ε → μ = 1 2 δ. This phenomenon is intimately related to the effect of screening of the Coulombic field from the droplets by the compensating charges that move into their vicinity [START_REF] Muratov | Theory of domain patterns in systems with long-range interactions of Coulomb type[END_REF]. For a single radially symmetric droplet the solution of the Euler-Lagrange equation associated with E ε will have the form shown in Fig. 2.

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 A ε and for which the screening charges are removed from the consideration of convergence to the limit density. Hence, given a candidate function u ε ∈ A ε , we define a new function

u ε 0 (x) := +1, u ε (x) > 0, -1, u ε (x) ≤ 0, (2.31)
whose jump set coincides with the zero level set of u ε . This introduces a nonlinear filtering operation that eliminates the effect of the small deviations of u ε from ±1 in almost minimizers on the limit density (compare also with [START_REF] Knüpfer | Domain structure of bulk ferromagnetic crystals in applied fields near saturation[END_REF]). The density is now defined as

dµ ε 0 := 1 2 ε -2/3 | ln ε| -1/3 (1 + u ε 0 (x))dx. (2.32)
We can follow the ideas of [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] to establish an analog of Theorem 1 for the diffuse interface energy. To avoid many technical assumptions, we formulate the result for a specific choice of W . A general result may easily be reconstructed. Also, we make a technical assumption to avoid dealing with the case lim sup ε→0 u ε L ∞ (T 2 ) > 1, when spiky configurations in which |u ε | significantly exceeds 1 in regions of vanishing size may appear. We note that this condition is satisfied by the minimizers of E ε [36, Proposition 4.1].

Theorem 3. (Γ-convergence of E ε ) Fix δ > 0 and > 0, and let E ε be defined by (2.4) with W (u) = 9 32 (1u 2 ) 2 and ūε given by (2.1). Then, as ε → 0 we have that

ε -4/3 | ln ε| -2/3 E ε Γ → E 0 [µ] := δ2 2 2κ 2 + 3 2/3 - 2 δ κ 2 T 2 dµ + 2 T 2 T 2 G(x -y)dµ(x)dµ(y),
where µ ∈ M(T 2 ) ∩ H -1 (T 2 ) and κ = 2 3 . More precisely, we have i) (Lower Bound) Let (u ε ) ∈ A ε be such that lim sup ε→0 u ε L ∞ (T 2 ) ≤ 1 and

lim sup ε→0 ε -4/3 | ln ε| -2/3 E ε [u ε ] < +∞, (2.33)
and let µ ε 0 (x) be defined by (2.31) and (2.32). Then, up to extraction of subsequences, we have

µ ε 0 µ in (C(T 2 )) * , as ε → 0, where µ ∈ M(T 2 )∩H -1 (T 2 ). Moreover, we have lim sup ε→0 u ε L ∞ (T 2 ) = 1 and lim inf ε→0 ε -4/3 | ln ε| -2/3 E ε [u ε ] ≥ E 0 [µ]. ii) (Upper Bound) Conversely, given µ ∈ M(T 2 ) ∩ H -1 (T 2 ), there exist (u ε ) ∈ A ε such that lim sup ε→0 u ε L ∞ (T 2 )
= 1 and for µ ε 0 defined by (2.31) and (2.32) we have

µ ε 0 µ in (C(T 2 )) * ,
as ε → 0, and lim sup

ε→0 ε -4/3 | ln ε| -2/3 E ε [u ε ] ≤ E 0 [µ].
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. For given δ > 0 and > 0, let ūε be given by (2.1) and let (u ε ) ∈ A ε be minimizers of E ε defined in (2.4) with W (u) = 9 32 (1u 2 ) 2 . Then, if u ε 0 and µ ε 0 are defined via (2.31) and (2.32), respectively, as ε → 0 we have

µ ε 0 0 1 2 ( δ -δc ) in (C(T 2 )) * , ε -4/3 | ln ε| -2/3 -2 min E ε → δ2 2κ 2 , δc 2κ 2 (2 δ -δc ), (2.34)
when δ ≤ δc or δ > δc , respectively, with δc := 1 2 3 2/3 κ 2 and κ = 2 3 . Remark 2.6. The choice of the zero level set of u ε in the definition of the truncated version u ε 0 of u ε in (2.31) was arbitrary. We could equivalently use the level set {u ε = c} for any c ∈ (-1, 1) fixed.

Proof of Theorem 1

Throughout all the proofs below, the values of A ε i and P ε i are always the rescaled areas and perimeters, defined in (2.9), of the connected components Ω + i of Ω + = {u = +1} for a given u = u ε , as in Sec. 2. The presentation is clarified by working with the rescaled energy Ēε defined by (2.12) rather than E ε directly. We begin by proving Part i) of Theorem 1, the lower bound.

Proof of lower bound, Theorem 1 i)

Step 1: Estimate of Ēε in terms of A ε i and P ε i .

First, for a fixed γ ∈ (0, 1) we define a truncated rescaled droplet area:

Ãε i := A ε i , if A ε i < 3 2/3 πγ -1 (3 2/3 πγ -1 ) 1/2 |A ε i | 1/2 if A ε i ≥ 3 2/3 πγ -1 , (3.1)
and the isoperimetric deficit

I ε def := 1 | ln ε| i P ε i -4πA ε i ≥ 0, (3.2)
which will be used throughout the proof. The purpose of defining the truncated droplet area in (3.1) will become clear later. We will make repeated use of the basic fact that the diameter of an indecomposable set of finite perimeter in the plane is essentially controlled by its perimeter (i.e., modulo a set of measure zero):

ess diam ω ≤ 1 2 |∂ω| ∀ω ⊂ R 2 , 0 < |ω| < ∞. (3.3)
The proof follows, e.g., from [2, Theorem 7 and Lemma 4] by noting that in view of [2, Proposition 6(ii)] it is sufficient to consider only simple sets [2, Definition 3]. We start by writing µ ε = i µ ε i , with

dµ ε i (x) := ε -2/3 | ln ε| -1/3 χ Ω + i (x)dx, (3.4)
where Ω + i are the connected components of Ω + = {u ε = +1}, and the index ε was omitted from Ω + i to avoid cumbersome notation. For small enough ε this is justified by the discussion of Sec. 2, in view of the fact that for some C > 0 we have

|∂Ω + | ≤ ε -1 E ε [u ε ] ≤ Cε 1/3 | ln ε| 2/3 , (3.5)
so

|∂Ω + i | whenever ε 1.
In particular, we can apply (3.3) to Ω + i on T 2 to obtain

ess diam Ω + i ≤ 1 2 |∂Ω + i |, (3.6)
when ε is sufficiently small. For a fixed ρ > 0 we introduce the "far field truncation" G ρ ∈ C ∞ (T 2 ) of the Green's function G:

G ρ (x -y) := G(x -y)φ ρ (|x -y|) ∀(x, y) ∈ T 2 × T 2 , (3.7)
where φ ρ ∈ C ∞ (R) is a monotonically increasing cutoff function such that φ ρ (t) = 0 for all t < 1 2 ρ and φ ρ (t) = 1 for all t > ρ. Then, for sufficiently small ε we have |∂Ω + i | ≤ ρ in view of (3.5), and from (2.12) we obtain

Ēε [u ε ] ≥I ε def + 1 | ln ε| i 4πA ε i - 2 δ κ 2 A ε i + 2 i G(x -y)dµ ε i (x)dµ ε i (y) (3.8) + 2 G ρ (x -y)dµ ε (x)dµ ε (y),
where we used (3.6) and the positivity of G (cf. e.g. [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]), and here and everywhere below we omit T 2 × T 2 as the domain of integration for double integrals to simplify the notation. We recall that the Green's function for -∆ + κ 2 on T 2 can be written as G(xy) = -1 2π ln |x -y| + O(|x -y|) [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]. With the help of this fact, together with (3.5) and (3.6), for ε sufficiently small we have the following estimate for the self-interaction energy:

Ēε self := 2 i G(x -y)dµ ε i (x)dµ ε i (y) ≥ - 1 π i (ln |x -y| + C) dµ ε i (x)dµ ε i (y). (3.9) = - 1 π| ln ε| 2 i Ω + i Ω + i ln(|ε 1/3 | ln ε| 2/3 |x -y|) + C dx dȳ,
for some C > 0 independent of ε, where in equation (3.9) we have rescaled coordinates x = ε -1/3 | ln ε| 1/3 x, ȳ = ε -1/3 | ln ε| 1/3 and introduced the rescaled versions Ω + i of Ω + i . Expanding the logarithm in (3.9) and using (2.13) and (3.6), we obtain that Ēε self can be bounded from below as follows:

Ēε self ≥ 1 | ln ε| i |A ε i | 2 1 3π -C ln | ln ε| | ln ε| - 1 π|A ε i | 2 | ln ε| Ω + i Ω + i ln |x -y| dx dy ≥ 1 | ln ε| i |A ε i | 2 1 3π -C ln | ln ε| | ln ε| - 1 π| ln ε| ln P ε i (3.10) ≥ 1 | ln ε| i |A ε i | 2 1 3π -C ln | ln ε| | ln ε| ,
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 (3.10) is positive for ε sufficiently small. Using this and the fact that

A ε i ≥ Ãε i , from (3.10) we obtain Ēε self ≥ 1 | ln ε| i | Ãε i | 2 1 3π -C ln | ln ε| | ln ε| , (3.11)
where C > 0 is a constant independent of ε. It is also clear from the definition of Ãε i that there exists a constant c > 0 such that

| Ãε i | 2 ≤ cA ε i . (3.12)
Combining this inequality with (3.11) and choosing any δ > 0, for ε small enough we have δ > Cc ln | ln ε| | ln ε| 2 and, therefore, from (3.8) we obtain

Ēε [u ε ] ≥I ε def + 1 | ln ε| i 4πA ε i - 2 δ κ 2 + δ A ε i + 1 3π | Ãε i | 2 + 2 G ρ (x -y)dµ ε (x)dµ ε (y). (3.13)
Step 2: Optimization over A ε i .

Focusing on the second term in the right-hand side of (3.13), we define

f (x) := 2 √ π √ x + 1 3π x, (3.14)
and observe that f is strictly convex and attains its minimum of 3 2/3 at x = 3 2/3 π, with

f (x) = 3 √ π 2x 5/2 .
(3.15)

We claim that we can bound the second term in the right-hand side of (3.13) from below by the sum I + II + III of the following three terms:

I = 1 | ln ε| 3 2/3 - 2 δ κ 2 -δ i A ε i + 1 | ln ε| A ε i >3 2/3 πγ -1 3 2/3 (3 -1 γ -1 -1)A ε i , (3.16
)

II = 1 | ln ε| γ 5/2 4π 2 • 3 2/3 A ε i <3 2/3 πγ A ε i (A ε i -3 2/3 π) 2 , (3.17)
III = 1 | ln ε| γ 7/2 4π 3 2/3 πγ≤A ε i ≤3 2/3 πγ -1 (A ε i -3 2/3 π) 2 . (3.18)
Before proving this, observe that defining

M ε := Ēε [u ε ] - 1 | ln ε| 3 2/3 - 2 δ κ 2 -δ i A ε i -2 G ρ (x -y)dµ ε (x)dµ ε (
M ε ≥ c 1 | ln ε| i / ∈I ε γ A ε i + c 2 | ln ε| i∈I ε γ (A ε i -3 2/3 π) 2 + I ε def ≥ 0 ∀γ ∈ (0, 1 3 ), (3.20)
for some constants c 1 , c 2 > 0 depending only on γ. We now argue in favor of the lower bound based on (3.16)- (3.18). First observe that by (3.1) we have for all A ε i ≥ 3 2/3 πγ -1 :

4πA ε i + 1 3π | Ãε i | 2 - 2 δ κ 2 + δ A ε i ≥ 3 2/3 - 2 δ κ 2 -δ A ε i + 3 2/3 (3 -1 γ -1 -1)A ε i . (3.21)
When A ε i < 3 2/3 πγ -1 , which corresponds to both (3.17) and (3.18), we utilize convexity of f and (3.15):

4πA ε i + 1 3π | Ãε i | 2 - 2 δ κ 2 + δ A ε i = A ε i 2 √ π A ε i + 1 3π A ε i - 2 δ κ 2 -δ = A ε i f (A ε i ) - 2 δ κ 2 -δ (3.22) ≥ 3 2/3 - 2 δ κ 2 -δ A ε i + 1 2 A ε i f 3 2/3 πγ -1 (A ε i -3 2/3 π) 2 ,
where the last line follows from the second order Taylor formula for f (x) about x = 3

[u ε ] ≥ 1 | ln ε| 3 2/3 - 2 δ κ 2 -δ i A ε i + 2 G ρ (x -y)dµ ε (x)dµ ε (y). (3.23)
Step 3: Passage to the limit.

We may in fact conclude from (2.12), (2.13), (2.18) and (2.20), that lim sup

ε→0 T 2 l |∇v ε | 2 + κ 2 |v ε | 2 < +∞, (3.24)
while (µ ε ) is bounded in the sense of measures from (2.13) and (2.14).

Consequently, up to a subsequence

v ε v in H 1 (T 2 ), (3.25)
µ ε * µ in C(T 2 ), (3.26)
where

-∆v + κ 2 v = µ (3.27)
holds in the distributional sense. Now passing to the limit in (3.23) and recalling (2.14), we obtain

lim inf ε→0 Ēε [u ε ] ≥ 3 2/3 - 2 δ κ 2 -δ dµ + 2 G ρ (x -y)dµ(x)dµ(y), (3.28)
using continuity of G ρ . An application of the monotone convergence theorem then yields

lim inf ε→0 Ēε [u ε ] ≥ 3 2/3 - 2 δ κ 2 dµ + 2 G(x -y)dµ(x)dµ(y), (3.29)
upon sending ρ → 0 and then δ → 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 [START_REF] Sandier | A rigorous derivation of a free-boundary problem arising in superconductivity[END_REF] 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.

Proof of the Upper Bound, Theorem 1 ii)

As in the proof of the lower bound, we set dµ ε i (x) as in (3.4), so that µ ε = µ ε i . Using the approximation argument of Proposition II.2 in [START_REF] Sandier | A rigorous derivation of a free-boundary problem arising in superconductivity[END_REF] we may assume that the limit measure µ satisfies

dµ(x) = g(x)dx, c ≤ g ≤ C, (3.30)
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, where

N (ε) = 1 3 2/3 | ln ε| π µ(T 2 ) + o(| ln ε|), (3.31)
with centers {a i } in T 2 and radius

r = 3 1/3 ε 1/3 | ln ε| -1/3 , (3.32)
and satisfying for all i

= j d(ε) := min |a i -a j | ≥ C N (ε) , (3.33)
for some constant C > 0 depending only on µ. Indeed, given µ satisfying (3.30), for ε sufficiently small we can partition T 2 into disjoint squares {K i } of side length δ ε > 0 (hereafter simply denoted δ) satisfying

| ln ε| -1/2 δ 1. (3.34)
In each K i we place

N K i (ε) = 1 3 2/3 | ln ε| π µ(K i ) (3.35)
points a i (here m = x denotes the smallest integer m ≤ x) satisfying (3.33) and in addition dist

(a i , ∂K i ) ≥ C N (ε) , N (ε) := i N K i . (3.36)
As argued in [START_REF] Sandier | A rigorous derivation of a free-boundary problem arising in superconductivity[END_REF], our ability to do this follows from the estimate:

cδ 2 ≤ µ(K i ) ≤ Cδ 2 , (3.37)
the plane [START_REF] Tóth | Über einen geometrischen Satz[END_REF] we have |I k ρ | ≤ ckN (ε) for some universal c > 0 (here again |I k ρ | denotes the cardinality of I k ρ ), in view of (3.31) it holds that

1 | ln ε| 2 (i,j)∈Iρ A ε i A ε j ln dist (Ω + i , Ω + j ) ≤ CN (ε) | ln ε| 2 Kρ(ε) k=1 k| ln(kd(ε))| ≤ 2CN (ε) | ln ε| 2 d 2 (ε) ρ d(ε) t| ln t|dt ≤ C | ln d(ε)| | ln ε| + ρ 2 | ln ρ| ≤ 2C ρ 2 | ln ρ|, (3.42)
for some C, C > 0 independent of ε or ρ, when ε and ρ are sufficiently small. Therefore, from (3.32) and (3.41) we obtain

lim sup ε→0 G(x -y) -G ρ (x -y) dµ ε (x)dµ ε (y) ≤ 2 -1 • 3 -1/3 + o(ρ). ( 3
[u ε ] ≤ 3 2/3 - 2 δ κ 2 dµ + 2 G(x -y)dµ(x)dµ(y), (3.44)
as required. The fact that v ε v follows from (3.39) and the uniform bounds just demonstrated on the terms involving the Green's function in (3.38), from which it follows that (2.21) is satisfied distributionally.

Proof of Theorem 2

In the proof of Sec. 3, we have in fact established Theorem 2, which is clear by (3.20). Indeed, we have for a sequence of almost minimizers (u ε ):

lim ε→0 E ε [u ε ] -E 0 [µ] = 0. ( 4.1)
Observing that M ε defined in (3.19) does not contribute to E 0 [µ], we have established that M ε → 0 as ε → 0 for any γ < 1 3 and, as a consequence, we obtain (2.24)-(2.26) for, say, γ = 1 6 . Then it is easy to see from the definition of I ε γ that the statement of the Theorem, in fact, holds for any γ ∈ (0, 1).

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 refinement of the ideas of [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]Sec. 4] to establish matching upper and lower bounds for E ε in terms of 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 equivalen form

E ε [u ε ] = T 2 ε 2 2 |∇u ε | 2 + W (u ε ) + 1 2 |∇v ε | 2 dx, -∆v ε = u ε -ūε , T 2
v ε dx = 0.

(5.1)

Fix any δ ∈ (0, 1) and consider a sequence (u ε ) ∈ A ε such that lim sup ε→0 u ε L ∞ (T 2 ) ≤ 1 and E ε [u ε ] ≤ Cε 4/3 | ln ε| 2/3 for some C > 0 independent of ε. Then we claim that lim sup

ε→0 u ε L ∞ (T 2 ) = 1, lim ε→0 v ε L ∞ (T 2 ) = 0. (5.2)
Indeed, for the first statement we have from the definition of E ε in (2.4) that

|Ω δ 0 | ≤ Cε 4/3 | ln ε| 2/3 δ -2 , Ω δ 0 := {-1 + δ ≤ u ε ≤ 1 -δ}, (5.3)
for some C > 0 independent of ε. Hence, in particular, lim sup ε→0 u ε L ∞ (T 2 ) ≥ 1, proving the first statement of (5.2). To prove the second statement in (5.2), we note that by standard elliptic theory [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] we have v ε W 2,p (T 2 ) ≤ C for any p > 2 and some C > 0 independent of ε and, hence, by Sobolev imbedding ∇v ε L ∞ (T 2 ) ≤ C for some C > 0 independent of ε as well. Therefore, applying Poincaré's inequality, we obtain

Cε 4/3 | ln ε| 2/3 ≥ E ε [u ε ] ≥ C T 2 |v ε | 2 dx ≥ C v ε 4 L ∞ (T 2 ) , (5.4)
for some C , C > 0 independent of ε, yielding the claim. In view of (5.2), for small enough ε we have u ε L ∞ (T 2 ) ≤ 1 + δ 3 and v ε L ∞ (T 2 ) ≤ δ 3 , and by the assumption on energy we may further assume that E ε [u ε ] ≤ δ 12 . Therefore, by [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]Proposition 4.2] there exists a function ũε 0 ∈ A such that

E ε [u ε ] ≥ (1 -δ 1/2 )E ε [ũ ε 0 ]. (5.5)
In particular, (ũ ε 0 ) satisfy the assumptions of Theorem 1, and, therefore, upon extraction of subsequences we have με where u ε 0 is given by (2.31), for some C > 0 independent of ε (recall that the jump set of u ε 0 is the zero level set {u ε = 0}). Comparing (5.7) with (5.6), we then see that µ ε 0 µ in (C(T 2 )) * as well. The result of part (i) of Theorem 3 then follows by arbitrariness of δ > 0 via a diagonal process.

Step 2: Approximate upper bound.

We now construct the approximate upper bounds for a suitable lifting of the recovery sequences in the proof of Theorem 1(ii) to A ε . Let (ũ ε 0 ) ∈ A be a recovery sequence constructed in Sec. 3.2. This sequence consists of circular droplets of the optimal radius r = 3 1/3 ε 1/3 | ln ε| -1/3 ε 1/2 and mutual distance d ≥ C| ln ε| -1/2 ε 1/2 , for some C > 0 independent of ε. In addition, since

E ε [ũ ε 0 ] = ε 2 T 2 |∇ũ ε 0 | dx + 1 2 T 2 |∇ṽ ε | 2 + κ 2 |ṽ ε | 2 dx ≤ Cε 4/3 | ln ε| 2/3 , (5.8)
where ṽε (x) = T 2 G(xy)(ũ ε 0 (y)ūε )dy, for some C > 0 independent of ε, by the argument of (5.4) one can see that lim ε→0 ṽε L ∞ (T 2 ) = 0. Therefore, for any δ ∈ (0, 1) and ε > 0 sufficiently small we have ṽε L ∞ (T 2 ) ≤ δ and E ε [ũ ε 0 ] ≤ δ 5/2 . So we can apply [START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF]Proposition 4.3] to obtain a function u ε ∈ A ε such that for some C > 0 independent of ε, where u ε 0 is given by (2.31), and we used (5.8). Hence µ ε 0 µ = lim ε→0 με 0 in (C(T 2 )) * . The result of part (ii) of Theorem 3 again follows by arbitrariness of δ > 0 via a diagonal process.

E ε [u ε ] ≤ (1 + δ 1/2 )E ε [ũ ε 0 ]. ( 5
Remark 5.1. It is possible to chose δ = ε α for α > 0 sufficiently small in the arguments of the proof of Theorem 3. Therefore, given a sequence of almost minimizers (u ε ) ∈ A ε of E ε and the corresponding sequence (u ε 0 ) ∈ A of almost minimizers of E ε , one has

ε -4/3 | ln ε| -2/3 E ε [u ε ] = ε -4/3 | ln ε| -2/3 E ε [u ε 0 ] + O(ε α ), (5.11)
for some α 1, as ε → 0.

Fig. 1 .

 1 Fig. 1. A multi-droplet pattern: density plot of u in a local minimizer of E[u] with W (u) = 1 4 (1-u 2 ) 2 obtained 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])

Fig. 2 .Fig. 3 .Figure 1 :

 231 Fig. 2. The droplet patterns for K = 2 through 10.

Figure 2 :

 2 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 where u > ū (excess shown in orange).

. 43 )

 43 Finally combining (3.43) with (3.38) and (3.40), upon sending ε → 0, then ρ → 0 and applying the monotone convergence theorem we have lim ε→0 Ēε

0µ

  ∈ M(T 2 ) ∩ H -1 (T 2 ) in (C(T 2 )) * , where dμ ε 0 (x) := 1 2 ε -2/3 | ln ε| -1/3 (1 + ũε 0 (x))dx. (5.6)Furthermore, recalling that by construction the jump set of ũε 0 coincides with the level set {u ε = c} for some c ∈ (-1 + δ, 1δ), see the proof of[START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] Lemma 4.1], from (5.3) we have ũε 0u ε 0 L 1 (T 2 ) ≤ Cε 4/3 | ln ε| 2/3 δ -2 , (5.7)

. 9 )

 9 Furthermore, by the construction of u ε (see[START_REF] Muratov | Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions[END_REF] Eqs. (4.31)-(4.33)]) and arbitrariness of δ > 0, we also have lim sup ε→0 u ε L ∞ (T 2 ) = 1, and ũε 0u ε 0 L 1 (T 2 ) ≤ Cε 4/3 | ln ε| 2/3 , (5.10)

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 work of S. S. was supported by the EURYI award. C. B. M. would like to acknowledge valuable discussions with H. Knüpfer and M. Novaga.

which follows from (3.30) together with (3.34). We finally define our configuration u ε by setting the connected components Ω + i of Ω + = {u ε = +1} to be balls of radius r from (3.32) centered at a i , i.e. Ω + i := B r (a i ). We set u ε = -1 in the complement of these balls. With these choices we have

The main point of the rest of the proof is to show that the integral term in (3.38) converges to G(xy)dµ(x)dµ(y) + 3 -1/3 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 cannot concentrate too much as ε → 0.

Step 2: Convergence of the configurations.

Defining µ ε as before, it is clear from the construction that

Fix ρ > 0 sufficiently small (depending only on κ and ) and consider G ρ (xy) defined as in (3.7). By continuity of G ρ in T 2 we have lim ε→0 G ρ (xy)dµ ε (x)dµ ε (y) = G ρ (xy)dµ(x)dµ(y).

(3.40)

Now, let I ρ be the collection of indices (i, j) such that 0 < |a ia j | < ρ. Then for small enough ε we can write

for some C, C > 0 independent of ε or ρ, where A ε i = 3 2/3 π and we expanded the Green's function as in (3.9) in the proof of the lower bound. Now, for k = 1, 2, . . . , K ρ (ε), with K ρ (ε) := ρ/d(ε) , let I k ρ ⊂ I ρ be disjoint sets consisting of all indices (i, j) such that kd(ε) ≤ |a ia j | < (k + 1)d(ε). Since by the result on optimal packing density of disks in