The limit of the twodimensional OhtaKawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy.
Abstract
This is the second in a series of papers in which we derive a expansion for the twodimensional nonlocal GinzburgLandau energy with Coulomb repulsion known as the OhtaKawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in 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 our previous paper, we computed the limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic GinzburgLandau model. The derivation is based on the abstract scheme of SandierSerfaty that serves to obtain lower bounds for 2scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the EulerLagrange equation, we establish for all configurations which have “almost minimal energy” the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of equivalence, the obtained results also yield an expansion of the minimal energy for the original OhtaKawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers.
1 Introduction
This is our second paper devoted to the convergence study of the twodimensional OhtaKawasaki energy functional[28] in two space dimensions in the regime near the onset of nontrivial minimizers. The energy functional has the following form:
(1.1) 
where is the domain occupied by the material, is the scalar order parameter, is a symmetric doublewell potential with minima at , such as the usual GinzburgLandau potential (for simplicity, the overall coefficient in is chosen to make the associated surface tension constant to be equal to , i.e., we have ), is a parameter characterizing interfacial thickness, is the background charge density, and is the Neumann Green’s function of the Laplacian, i.e., solves
(1.2) 
where is the Laplacian in and is the Dirac deltafunction, with Neumann boundary conditions. Note that is also assumed to satisfy the “charge neutrality” condition
(1.3) 
For a discussion of the motivation and the main quantitative features of this model, see our first paper [19], as well as [26, 25]. For specific applications to physical systems, we refer the reader to [16, 40, 28, 27, 18, 22, 25, 24].
In our first paper [19], we established the leading order term in the expansion of the energy in (1.1) in the scaling regime corresponding to the threshold between trivial and nontrivial minimizers. More precisely, we studied the behavior of the energy as when
(1.4) 
for some fixed and when is a flat twodimensional torus of side length , i.e., when , with periodic boundary conditions. As follows from [19, Theorem 2] and the arguments in the proof of [19, Theorem 3], in this regime minimizers of consist of many small “droplets” (regions where ) and their number blows up as . We showed that, after a suitable rescaling the energy functional in (1.1) converges in the sense of convergence of the (suitably normalized) droplet densities, to the limit functional defined for all densities by:
(1.5) 
where is the screened Green’s function of the Laplacian, i.e., it solves the periodic problem for the equation
(1.6) 
and . Here we noted that the double integral in (1.5) is well defined in the sense , where the latter is interpreted as the HahnBanach extension of the corresponding linear functional defined by the integral on smooth test functions (see also [34, Sec. 7.3.1] and [9] for further discussion). Indeed, is the convolution understood distributionally, i.e., for every and, hence, by elliptic regularity for some , so .
In particular, for , where
(1.7) 
the limit energy is minimized by , where
(1.8) 
When , the limit energy is minimized by , with . The value of thus serves as the threshold separating the trivial and the nontrivial minimizers of the energy in (1.1) together with (1.4) for sufficiently small . Above that threshold, the droplet density of energyminimizers converges to the uniform density .
The key point that enables the analysis above is a kind of equivalence between the energy functional in (1.1) and its screened sharp interface analog (for general notions of equivalence or variational equivalence, see [8, 3]):
(1.9) 
Here, is the screened potential as in (1.6), and , where
(1.10) 
and we note that on the level of the neutrality condition in (1.3) has been removed. As we showed in [19], following the approach of [26], for given by (1.1) in which and is defined in (1.4), we have
(1.11) 
for some . Therefore, in order to understand the leading order asymptotic expansion of the minimal energy in terms of , it is sufficient to obtain such an expansion for . This is precisely what we will do in the present paper.
In view of the discussion above, in this paper we concentrate our efforts on the analysis of the sharp interface energy in (1.9). An extension of our results to the original diffuse interface energy would lead to further technical complications that lie beyond the scope of the present paper and will be treated elsewhere. Here we wish to extract the next order nontrivial term in the expansion of the sharp interface energy after (1.5). In contrast to [26], we will not use the EulerLagrange equation associated to (1.9), so our results about minimizers will also be valid for “almost minimizers” (cf. Theorem 2).
We recall that for the energy minimizers for and consist of nearly circular droplets of radius uniformly distributed throughout [26, Theorem 2.2]. This is in contrast with the study of [12, 13] for a closely related energy, where the number of droplets remains bounded as , and the authors extract a limiting interaction energy for a finite number of points.
By convergence, we obtained in [19, Theorem 1] the convergence of the droplet density of almost minimizers of :
(1.12) 
to the uniform density defined in (1.8). However, this result does not say anything about the microscopic placement of droplets in the limit . In order to understand the asymptotic arrangement of droplets in an energy minimizer, our plan is to blowup the coordinates by a factor of , which is the inverse of the scale of the typical interdroplet distance, and to extract the next order term in the expansion of the energy in terms of the limits as of the blownup configurations (which will consist of an infinite number of point charges in the plane with identical charge).
We will show that the arrangement of the limit point configurations is governed by the Coulombic renormalized energy , which was introduced in [34]. That energy was already derived as a next order limit for the magnetic GinzburgLandau model of superconductivity [35, 34], and also for twodimensional Coulomb gases [37]. Our results here follow the same method of [35], and yield almost identical conclusions.
The “Coulombic renormalized energy” is a way of computing a total Coulomb interaction between an infinite number of point charges in the plane, neutralized by a uniform background charge (for more details see Section 2). It is shown in [35] that its minimum is achieved. It is also shown there that the minimum among simple lattice patterns (of fixed volume) is uniquely achieved by the triangular lattice (for a closely related result, see [10]), and it is conjectured that the triangular lattice is also a global minimizer. This triangular lattice is called “Abrikosov lattice” in the context of superconductivity and is observed in experiments in superconductors [41].
The next order limit of that we shall derive below is in fact the average of the energy over all limits of blownup configurations (i.e. average with respect to the blow up center). Our result says that limits of blowups of (almost) minimizers should minimize this average of . This permits one to distinguish between different patterns at the microscopic scale and it leads, in view of the conjecture above, to expecting to see triangular lattices of droplets (in the limit ), around almost every blowup center (possibly with defects). Note that the selection of triangular lattices was also considered in the context of the OhtaKawasaki energy by Chen and Oshita [10], but there they were only obtained as minimizers among simple lattice configurations consisting of nonoverlapping ideally circular droplets.
It is somewhat expected that minimizers of the OhtaKawasaki energy in the macroscopic setting are periodic patterns in all space dimensions (in fact in the original paper [28] only periodic patterns are considered as candidates for minimizers). This fact has never been proved rigorously, except in one dimension by Müller [23] (see also [31, 42]), and at the moment seems very difficult. For higherdimensional problems, some recent results in this direction were obtained in [2, 38, 26] establishing equidistribution of energy in various versions of the OhtaKawasaki model on macroscopically large domains. Several other results [12, 13, 39, 15] were also obtained to characterize the geometry of minimizers on smaller domains. The results we obtain here, in the regime of small volume fraction and in dimension two, provide more quantitative and qualitative information (since we are able to distinguish between the cost of various patterns, and have an idea of what the minimizers should be like) and a first setting where periodicity can be expected to be proved.
The OhtaKawasaki setting differs from that of the magnetic GinzburgLandau model in the fact that the droplet “charges” (i.e., their volume) are all positive, in contrast with the vortex degrees in GinzburgLandau, which play an analogous role and can be both positive and negative integers. It also differs in the fact that the droplet volumes are not quantized, contrary to the degrees in the GinzburgLandau model. This creates difficulties and the major difference in the proofs. In particular we have to account for the possibility of many very small droplets, and we have to show that the isoperimetric terms in the energy suffice to force (almost) all the droplets to be round and of fixed volume. This has to be done at the same time as the lower bound for the other term in the energy, for example an adapted “ball construction” for nonquantized quantities has to be reimplemented, and the interplay between these two effects turns out to be delicate.
Our paper is organized as follows. In Section 2 we formulate the problem and state our main results concerning the limit of the next order term in the energy (1.9) after the zeroth order energy derived in [19] is subtracted off. In Section 3, we derive a lower bound on this next order energy via an energy expansion as done in [19] however isolating lower order terms obtained via the process. We then proceed via a ball construction as in [20, 33, 34] to obtain lower bounds on this energy in Section 4 and consequently obtain an energy density bounded from below with almost the same energy via energy displacement as in [35] in Section 5. In Section 6 we obtain explicit lower bounds on this density on bounded sets in the plane in terms of the renormalized energy for a finite number of points. We are then in the appropriate setting to apply the multiparameter ergodic theorem as in [35] to extend the lower bounds obtained to global bounds, which we present at the end of Section 6. Finally the corresponding upper bound (cf. Part (ii) of Theorem 1) is presented in Section 7.
Some notations.
We use the notation to denote sequences of functions as , where is an admissible class. We also use the notation to denote a positive finite Radon measure on the domain . With a slight abuse of notation, we will often speak of as the “density” on and set whenever . With some more abuse of notation, for a measurable set we use to denote its Lebesgue measure, to denote its perimeter (in the sense of De Giorgi), and to denote . The symbols , , and denote the usual Sobolev space, the space of functions of bounded variation, the space of times continuously differentiable functions, and the dual of , respectively. The symbol stands for the quantities that tend to zero as with the rate of convergence depending only on , and .
2 Problem formulation and main results
In the following, we fix the parameters , and , and work with the energy in (1.9), which can be equivalently rewritten in terms of the connected components of the family of sets of finite perimeter , where are almost minimizers of , for sufficiently small (cf. the discussion at the beginning of Sec. 2 in [19]). The sets can be decomposed into countable unions of connected disjoint sets, i.e., , whose boundaries are rectifiable and can be decomposed (up to negligible sets) into countable unions of disjoint simple closed curves. Then the density in (1.12) can be rewritten as
(2.1) 
where are the characteristic functions of . Motivated by the scaling analysis in the discussion preceding equation (1.12), we define the rescaled areas and perimeters of the droplets:
(2.2) 
Using these definitions, we obtain (see [19, 26]) the following equivalent definition of the energy of the family :
(2.3) 
where
(2.4) 
Also note the relation
(2.5) 
As was shown in [26, 19], in the limit the minimizers of are nontrivial if and only if , and we have asymptotically
(2.6) 
Furthermore, if is as in (2.1) and we let be the unique solution of
(2.7) 
for any , then we have
(2.8) 
To extract the next order terms in the expansion of we, therefore, subtract this contribution from to define a new rescaled energy (per unit area):
(2.9) 
Note that we also added the third term into the bracket in the righthand side of (2.9) to subtract the nexttoleading order contribution of the droplet selfenergy, and we have scaled in a way that allows to extract a nontrivial contribution to the minimal energy (see details in Section 3). The main result of this paper in fact is to establish convergence of to the renormalized energy which we now define.
In [35], the renormalized energy was introduced and defined in terms of the superconducting current , which is particularly convenient for the studies of the magnetic GinzburgLandau model of superconductivity. Here, instead, we give an equivalent definition, which is expressed in terms of the limiting electrostatic potential of the charged droplets, after blowup, which is the limit of some proper rescaling of (see below). However, this limiting electrostatic potential will only be known up to additive constants, due to the fact that we will take limits over larger and larger tori. This issue can be dealt with in a natural way by considering equivalence classes of potentials, whereby two potentials differing by a constant are not distinguished:
(2.10) 
This definition turns the homogeneous spaces into Banach spaces of equivalence classes of functions in defined in (2.10) (see, e.g., [29]). Here we similarly define the local analog of the homogeneous Sobolev spaces as
(2.11) 
with the notion of convergence to be that of the convergence of gradients. In the following, we will omit the brackets in to simplify the notation and will write to imply that is any member of the equivalence class in (2.10).
We define the admissible class of the renormalized energy as follows :
Definition 2.1.
For given and , we say that belongs to the admissible class , if and solves distributionally
(2.12) 
where is a discrete set and
(2.13) 
Remark 2.2.
Observe that if , then for every we have
(2.14) 
where is a finite set of distinct points and is analytic in . In particular, the definition of is independent of .
We next define the renormalized energy.
Definition 2.3.
For a given , the renormalized energy of is defined as
(2.15) 
where , is a smooth cutoff function with the properties that , in , for all , for all , and for some independent of .
Various properties of are established in [35], we refer the reader to that paper. The most relevant to us here are

is achieved for each .

If and , then and
(2.16) hence

is minimized over potentials in generated by charge configurations consisting of simple lattices by the potential of a triangular lattice, i.e. [35, Theorem 2 and Remark 1.5],
where , is the Dedekind eta function, , and are real numbers such that is the dual lattice to a triangular lattice whose unit cell has area , and solves (2.12) with .
In particular, from property 2 above it is easy to see that the role of in the definition of is inconsequential.
We are now ready to state our main result. Let . For a given , we then introduce the potential (recall that is a representative in the equivalence class defined in (2.10))
(2.17) 
where is a periodic extension of from to the whole of . We also define to be the family of translationinvariant probability measures on concentrated on with .
Theorem 1.
(convergence of ) Fix , , and , and let be defined by (2.9). Then, as we have
(2.18) 
where . More precisely:

(Lower Bound) Let be such that
(2.19) and let be the probability measure on which is the pushforward of the normalized uniform measure on by the map , where is as in (2.17). Then, upon extraction of a subsequence, converges weakly to some , in the sense of measures on and
(2.20) 
(Upper Bound) Conversely, for any probability measure , letting be its pushforward under , there exists such that letting be the pushforward of the normalized Lebesgue measure on by , where is as in (2.17), we have , in the sense of measures on , and
(2.21) as .
We will prove that the minimum of is achieved. Moreover, it is achieved for any which is concentrated on minimizers of with .
Remark 2.4.
The phrasing of the theorem does not exactly fit the framework of convergence, since the lower bound result and the upper bound result are not expressed with the same notion of convergence. However, since weak convergence of to implies weak convergence of to , the theorem implies a result of convergence where the sense of convergence from to is taken to be the weak convergence of their pushforwards to the corresponding .
The next theorem expresses the consequence of Theorem 1 for almost minimizers:
Theorem 2.
Let and let be a family of almost minimizers of , i.e., let
Then, if is the limit measure from Theorem 1, almost every minimizes over . In addition
(2.22) 
Note that the formula in (2.22) is not totally obvious, since the probability measure concentrated on a single minimizer of does not belong to .
The result in Theorem 2 allows us to establish the expansion of the minimal value of the original energy by combining it with (2.9) and (1.11).
Theorem 3.
As mentioned above, the limit in Theorem 1 cannot be expressed in terms of a single limiting function , but rather it effectively averages over all the blownup limits of , with respect to all the possible blowup centers. Consequently, for almost minimizers of the energy, we cannot guarantee that each blownup potential converges to a minimizer of , but only that this is true after blowup except around points that belong to a set with asymptotically vanishing volume fraction. Indeed, one could easily imagine a configuration with some small regions where the configuration does not ressemble any minimizer of , and this would not contradict the fact of being an almost minimizer since these regions would contribute only a negligible fraction to the energy. Near all the good blowup centers, we will know some more about the droplets: it will be shown in Theorem 4 that they are asymptotically round and of optimal radii.
We finish this section with a short sketch of the proof. Most of the proof consists in proving the lower bound, i.e. Part (i) of Theorem 1. The first step, accomplished in Section 3 is, following the ideas of [26], to extract from some positive terms involving the sizes and shapes of the droplets and which are minimized by round droplets of fixed appropriate radius. These positive terms, gathered in what will be called , can be put aside and will serve to control the discrepancy between the droplets and the ideal round droplets of optimal sizes. We then consider what remains when this is subtracted off from and express it in blownup coordinates . It is then an energy functional, expressed in terms of some rescaling of which has no sign and which ressembles that studied in [35]. Thus we apply to it the strategy of [35]. The main point is to show that, even though the energy density is not bounded below, it can be transformed into one that is by absorbing the negative terms into positive terms in the energy in the sense of energy displacement [35], while making only a small error. In order to prove that this is possible, we first need to establish sharp lower bounds for the energy carried by the droplets (with an error per droplet). These lower bounds contain possible errors which will later be controlled via the term. This is done in Section 4 via a ball construction as in [20, 33, 34]. In Section 5 we use these lower bounds to perform the energy displacement as in [35]. Once the energy density has been replaced this way by an essentially equivalent energy density which is bounded below, we can apply the abstract scheme of [35] that serves to obtain lower bounds for “twoscale energies” which converge at the microscopic scale, via the multiparameter ergodic theorem. This is achieved is Section 6. Prior to this we obtain explicit lower bounds at the microscopic scale in terms of the renormalized energy for a finite number of points. It is then these lower bounds that get integrated out, or averaged out at the macroscopic scale to provide a global lower bound.
Finally, there remains to obtain the corresponding upper bound. This is done via an explicit construction of a periodic testconfiguration, following again the method of [35].
3 Derivation of the leading order energy
In preparation for the proof of Theorem 1, we define
(3.1) 
Recall that to leading order the droplets are expected to be circular with radius . Thus is the expected radius, once we have blown up coordinates by the factor of , which will be done below. Also, we know that the expected normalized area is , but this is only true up to lower order terms which were negligible in [19]; as we show below, a more precise estimate is , so above can be viewed as a “corrected” normalized droplet radius. Since our estimates must be accurate up to per droplet and the selfenergy of a droplet is of order , we can no longer ignore these corrections.
The goal of the next subsection is to obtain an explicit lower bound for defined by (2.9) in terms of the droplet areas and perimeters, which will then be studied in Sections 4 and onward. We follow the analysis of [19], but isolate higher order terms.
3.1 Energy extraction
We begin with the original energy (cf. (2.4)) while adding and subtracting the truncated self interaction: first we define, for , truncated droplet volumes by
(3.2) 
as in [19]. The motivation for this truncation will become clear in the proof of Proposition 5.1, when we obtain lower bounds on the energy on annuli. In [19] the selfinteraction energy of each droplet extracted from was , yielding in the end the leading order energy in (1.5). A more precise calculation of the selfinteraction energy corrects the coefficient of by an term, yielding the following corrected leading order energy for :
(3.3) 
The energy in (3.3) is explicitly minimized by (again a correction to the previously known from (1.8)) where
(3.4) 
and
(3.5) 
Observing that we immediately check that
(3.6) 
and in addition that (3.5) converges to the second expression in (1.8). To obtain the next order term, we Taylorexpand the obtained formulas upon substituting the definition of . After some algebra, we obtain
(3.7) 
Recalling once again the definition of from (2.9), we then find
and in view of the definition of from (2.3), we thus may write
(3.8) 
Thus obtaining a lower bound for the first term in the righthand side of (3.8) implies, up to , a lower bound for . This is how we proceed to prove Lemma 3.1 below.
3.2 Blowup of coordinates
We now rescale the domain by making the change of variables
(3.10)  
Observe that
(3.11) 
where is defined by (2.17). It turns out to be more convenient to work with and rescale only at the end back to .
3.3 Main result
We are now ready to state the main result of this section, which provides an explicit lower bound on . The strategy, in particular for dealing with droplets that are too small or too large is the same as [19], except that we need to go to higher order terms.
Proposition 3.1.
There exist universal constants , , and such that if and with , then for all
(3.12) 
where is defined by
(3.13) 
Remark 3.2.
Defining , by isoperimetric inequality applied to each connected component of separately every term in the first sum in the definition of in (3.13) is nonnegative. In particular, measures the discrepancy between the droplets with and disks of radius .
The proposition will be proved below, but before let us examine some of its further consequences. The result of the proposition implies that our a priori assumption translates into
for some independent of , which, in view of (3.1) is also
(3.14) 
A major goal of the next sections is to obtain the following estimate