Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions
DDroplet phases in non-localGinzburg-Landau models with Coulombrepulsion in two dimensions
Cyrill B. Muratov ∗ October 31, 2018
Abstract
We establish the behavior of the energy of minimizers of non-localGinzburg-Landau energies with Coulomb repulsion in two space dimen-sions near the onset of multi-droplet patterns. Under suitable scaling ofthe background charge density with vanishing surface tension the non-local Ginzburg-Landau energy becomes asymptotically equivalent to asharp interface energy with screened Coulomb interaction. Near the on-set the minimizers of the sharp interface energy consist of nearly identicalcircular droplets of small size separated by large distances. In the limitthe droplets become uniformly distributed throughout the domain. Theprecise asymptotic limits of the bifurcation threshold, the minimal energy,the droplet radii, and the droplet density are obtained.
Spatial patterns are often a result of the competition between thermodynamicforces operating on different length scales. When short-range attractive interac-tions are present in a system, phase separation phenomena can be observed,resulting in aggregation of particles or formation of droplets of new phase,which evolve into macroscopically large domains via coarsening or nucleationand growth (see e.g. [1]). This process, however, can be frustrated in the pres-ence of long-range repulsive forces. As the droplets grow, the contribution ofthe long-range interaction may overcome the short-range forces, whereby sup-pressing further growth. This mechanism was identified in many energy-drivenpattern forming systems of different physical nature, such as various types offerromagnetic systems, type-I superconductors, Langmuir layers, multiple poly-mer systems, etc., just to name a few [2–11]. Remarkably, these systems oftenexhibit very similar pattern formation behaviors [10, 12]. ∗ Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ07102, USA a r X i v : . [ n li n . PS ] M a y ne important class of systems with competing interactions are systems inwhich the long-range repulsive forces are of Coulomb type (for an overview,see [13, 14] and references therein). The nature of the Coulombic forces may bevery different from system to system. For example, these forces may arise whenparticles undergoing phase separation carry net electric charge [15–18], or theymay be a consequence of entropic effects associated with chain conformationsin polymer systems [19–23]. Coulomb interactions may also arise indirectlyas a result of diffusion-mediated processes [4, 24, 25]. All this makes systemswith repulsive Coulombic interactions a ubiquitous example of pattern formingsystems.Studies of systems with competing short-range attractive interactions andlong-range repulsive Coulomb interactions go back to the work of Ohta andKawasaki, who proposed a non-local extension of the Ginzburg-Landau energyin the context of diblock copolymer systems [19]. Even though its validity fordiblock copolymer systems may be questioned [21, 26–28], the Ohta-Kawasakimodel is applicable to a great number of physical problems of different ori-gin [14]. On the other hand, mathematically Ohta-Kawasaki model presents aparadigm of energy-driven pattern forming systems which has been receiving agrowing degree of attention [9, 29–36].The Ohta-Kawasaki energy is a functional of the form [13, 14, 19, 24, 37]: E [ u ] = (cid:90) Ω (cid:18) ε |∇ u | + W ( u ) (cid:19) dx + 12 (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x, y )( u ( y ) − ¯ u ) dx dy. (1.1)Here, u : Ω → R is a scalar quantity denoting the “order parameter” in abounded domain Ω ⊂ R d . Different terms of the energy are as follows: thefirst term penalizes spatial variations of u on the scales shorter than ε , thesecond term, in which W is a symmetric double-well potential drives local phaseseparation towards the minima of W at u = ±
1, and the last term is the long-range interaction, whose Coulombic nature comes from the fact that the kernel G solves the Neumann problem for − ∆ G ( x, y ) = δ ( x − y ) − | Ω | , (cid:90) Ω G ( x, y ) dx = 0 , (1.2)where ∆ is the Laplacian in x and δ ( x ) is the Dirac delta-function. The pa-rameter ¯ u denotes the prescribed uniform background charge, and the overall“charge neutrality” is ensured via the constraint1 | Ω | (cid:90) u dx = ¯ u. (1.3)It is important to note that the kernel G solves (1.2) in the space of the samedimensionality as the order parameter u (not to be confused with the case inwhich the kernel solves the Laplace’s equation in the space of higher spatial di-mensionality, as is common in many other systems with competing interactions,see e.g. [7, 16]). 2 ) t = t = t = t = Figure 1: A multi-droplet pattern: density plot of u in a local minimizer of E [ u ] with W ( u ) = (1 − u ) obtained numerically for ¯ u = − . ε = 0 . , . × [0 , u ≈ −
1, and light regions correspond to u ≈ ε > u when ε is sufficiently small. In fact, it is knownthat no patterns can form in the system if ε is sufficiently large [13, 14, 38].On the other hand, when ε (cid:28)
1, the first term in the functional E becomesa singular perturbation, giving rise to “domain structures” (see Fig. 1), whichare of particular physical interest. These patterns consist of extended regionsin which u is close to one of the minima of the potential W , separated bynarrow domain walls. In this situation one can reduce the energy functionalappearing in (1.1) to an expression in terms of the interfaces alone. In [13, 14],such a reduction was performed for E using formal asymptotic techniques (seealso [30, 35, 39, 40]) and leads to the following reduced energy (for simplicity ofnotation, 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 (cid:90) Ω |∇ u | dx + 12 (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x, y )( u ( y ) − ¯ u ) dx dy. (1.4)Here the function u takes on values ± G is the screened Coulomb kernel, i.e., it solves the Neumann problem for − ∆ G ( x, y ) + κ G ( x, y ) = δ ( x − y ) , (1.5)with some κ >
0. The constant κ has the physical meaning of the inverse of theDebye screening length [13,14]. Note that the sharp interface energy E with the3nscreened Coulomb kernel (i.e. with κ = 0) was derived by Ren and Wei as theΓ-limit of the diffuse interface energy E under assumptions of weak non-localcoupling (i.e., with an extra factor of ε in front of the Coulomb kernel) and¯ u ∈ ( − ,
1) independent of ε , as ε → E on the domain of size O ( ε / )). At the sametime, screening becomes important near the transition between the uniform andthe patterned states which occurs near | ¯ u | = 1, the case of interest in the presentpaper [13, 14]. Note that in the presence of screening the neutrality conditionin (1.3) is relaxed.In this paper, we rigorously establish the relation between the sharp interfaceenergy E and the diffuse interface energy E , and analyze the precise behaviorof minimizers of the sharp interface energy E for ε (cid:28) | ¯ u | = 1. We note that despite the apparent simplicity of the expression for E ,the minimizers of E exhibit quite an intricate dependence on the parametersfor ε (cid:28) | ¯ u | (cid:39)
1. Our analysis in this paper will be restricted to thecase d = 2. While a number of our results can be extended to arbitrary spacedimensions, our methods to obtain sharp estimates for the energy of minimizersrely critically on the properties of minimal curves in two dimensions and thelogarithmic behavior of the Green’s function of the two-dimensional Laplaciannear the singularity. Therefore, they cannot be readily extended to other spatialdimensionalities, and, indeed, one would expect certain important differencesbetween these cases and the case of two space dimensions. At the same time,we will show that in the case d = 2 it is possible to obtain rather detailedinformation about the structure of the transition near | ¯ u | = 1 in terms of energy.Let us note that, since the case d = 1 is now well-understood [29–31, 41], theremaining open case of physical interest is that of d = 3.Before turning to the analysis, let us briefly mention a perfect example ofan experimental system in which the regimes studied by us could be easilyrealized, which is inspired by the beautiful Nobel Lecture of Prof. G. Ertl[42, 43]. Consider molecules which undergo adsorption and desorption to andfrom a crystalline surface. On the surface, the atoms may hop around andreversibly stick to each other to form monolayer aggregates [44]. Then, withinthe framework of phase field models, this process may be described by thefollowing evolution equation for the adsorbate density fraction φ [25]: φ t = M ∆( W (cid:48) ( φ ) − g ∆ φ ) + k + (1 − φ ) − k − φ, (1.6)where W is a double-well potential with two minima between φ = 0 and φ = 1, g is the short-range coupling constant, M is a kinetic coefficient, and k ± are theadsorption and desorption rates, respectively. Note that this equation can berewritten as φ t = M ∆ { W (cid:48) ( φ ) − g ∆ φ + kG ∗ ( φ − ¯ φ ) } , (1.7)where k = ( k + + k − ) /M , ¯ φ = k + / ( k + + k − ), and “ ∗ ” denotes convolution inspace, with G given by (1.2), provided the spatial average of the initial data is ¯ φ .4pon suitable rescaling, this is precisely the H − gradient flow for the energy E , i.e., we have u t = ∆( δ E /δu ), where u is a rescaling of φ . In particular,minimizers of E are ground states of the considered system in equilibrium inthe mean-field limit. We note that the adsorption and desorption rates k ± canbe quite small compared to the hopping rate, resulting in very small values of ε ∼ k / . Therefore, one can achieve a very good scale separation between theinterfacial thickness (atomic scales) and the size of adsorbate clusters (micro-scale) in this experimental setup.Our paper is organized as follows. In Sec. 2, we present heuristic argumentsand give the statements of main results, in Sec. 3, we perform a detailed analysisof the sharp interface energy E , in Sec. 4 we establish a connection between thesharp interface energy E and the diffuse interface energy E . Finally, in Sec. 5we conclude the proofs of the theorems.Throughout the paper, the symbols L p , H k , W k,p , C k,α , BV denote theusual function spaces, | · | denotes the d -dimensional Lebesgue measure or the( d − C , c , etc., denote generic positive constants that can change from line to line.The symbols O (1) and o (1) denote, as usual, uniformly bounded and uniformlysmall quantities, respectively, in the limit ε →
0, etc. Finally, we will say that astatement holds for ε (cid:28)
1, etc., if there exists ε > < ε ≤ ε . For simplicity of notation, the subscript ε is omittedfor all quantities depending on ε . Let us begin our investigation by setting d = 2 and making a simplifying assump-tion that the domain Ω is a torus: Ω = [0 , . Let us also specify the domainsof definition for the functionals E and E . Formally, the diffuse interface energy E [ u ] will be defined for all u ∈ H (Ω) subject to (cid:82) Ω u dx = ¯ u , whereas the sharpinterface energy E [ u ] will be defined for all u ∈ BV (Ω; {− , } ).The assumption that Ω is a torus, which is common in the considered classof problems, eliminates the need to deal with the boundary effects and, evenmore importantly, restores the translational invariance inherent in the problemon the whole of R d (note that the choice of the size of Ω is inconsequential,the obtained energy of the minimizers scales linearly with | Ω | ). As a result, thekernel of the non-local part of the energy becomes a function of x − y only. Witha slight abuse of notation, in the following we will, therefore, replace G ( x, y )with G ( x − y ) everywhere below.On heuristic grounds one would expect that the minimizers of E at ε (cid:28) R ∼ ε / , whenever | ¯ u | < | ¯ u | is not too closeto 1 [9, 13, 14, 19]. A simple scaling analysis shows that in this case E ∼ ε / as ε → u fixed. Our first result gives a justification for this energy scalingwithout any assumptions about the minimizers (for statements about existenceand regularity of minimizers, see the following sections).5 heorem 2.1. Let W satisfy the assumptions (i)–(iv) at the beginning of Sec.4, and let ¯ u ∈ ( − , be fixed. Then there exist ε > and C > c > , suchthat cε / ≤ min E, min E ≤ Cε / (2.1) for all ε ≤ ε . Observe also that for E this result still holds when Ω = [0 , d for any d , whilefor E it holds at least for d < u = 0 and | u | ≤ E (with κ = 0), Alberti, Choksi and Otto recently proved, amongmany other interesting results, a stronger statement that in the limit ε →
0, theconstants in the upper and lower bounds in (2.1) can be chosen to be arbitrarilyclose to each other [36]. We note that the case κ = 0 and ¯ u ∈ ( − ,
1) fixedcan be treated as the limit of energy E considered by us as κ →
0, when theconstraint (cid:82) Ω u dx = ¯ u gets automatically enforced (see (5.2)).Thus, when ¯ u ∈ ( − ,
1) is fixed, the energy E admits a non-trivial minimizer,whose energy scales as in (2.1) when ε (cid:28)
1. What about the case | ¯ u | >
1? Here,in fact, it is easy to see that the only minimizers admitted by E are the trivialones. Consider, for example, the case ¯ u < −
1, the other case is equivalentby symmetry. In this case the problem admits the unique global minimizer u = −
1. To see this, let us introduce the characteristic function χ Ω + of the setΩ + = { u = +1 } for a given u ∈ BV (Ω; {− , } ). Then u = 2 χ Ω + −
1, and by astraightforward computation E [ u ] ≥ (cid:90) Ω (cid:90) Ω (2 χ Ω + ( x ) − − ¯ u ) G ( x − y )(2 χ Ω + ( y ) − − ¯ u ) dxdy ≥ (1 + ¯ u ) κ − u ) κ | Ω + | . (2.2)Thus, when ¯ u < −
1, the second term in the last inequality in (2.2) is positive,hence, is minimized by | Ω + | = 0. But in this case u = − u = − | ¯ u | >
1, non-trivial minimizersof E do not exist, and, therefore, at | ¯ u | = 1 a bifurcation occurs in the limit ε → E and E that occurs in the neighborhoodof | ¯ u | = 1 for ε (cid:28)
1. The energy E captures most of the difficulty associatedwith the considered problem. Therefore, we will spend most of our effort in thispaper to the studies of E (see Sec. 3). At the same time, as we show later(see Sec. 4), the statements about the behavior of min E also extend to that ofmin E for ε (cid:28) , , the kernel G has an explicit representation G ( x ) = 12 π (cid:88) n ∈ Z K ( κ | x − n | ) , (2.3)6here K is the modified Bessel function of the first kind. In particular, G > K [45]: G ( x ) = − π ln(¯ κ | x | ) + O ( | x | ) , | x | (cid:28) , (2.4)where ¯ κ = κ exp γ − (cid:88) n ∈ Z \{ } K ( κ | n | ) , (2.5)and γ ≈ . G ( x ) bounded whenever | x | > δ , for any δ >
0, and (2.4) can be used to estimate derivatives of G to O ( | x | ln | x | ) as well.Consider the case in which the value of ¯ u approaches ¯ u = − ε (cid:28) u = 1 on the minimizer goes to zero as ¯ u → − u = +1 of small size inthe background where u = −
1. Moreover, since on the scale of a droplet theinterfacial energy will give a dominant contribution, these droplets are expectedto be nearly circular. This motivates an introduction of the following reducedenergy: E N ( { r i } , { x i } ) = N (cid:88) i =1 (cid:16) πεr i − π (1 + ¯ u ) κ − r i − πr i (ln ¯ κr i − ) (cid:17) + 4 π N − (cid:88) i =1 N (cid:88) j = i +1 G ( x i − x j ) r i r j . (2.6)which describes the energy of interaction of N well separated disk-shaped dropletsof radius r i centered at x i , to the leading order. More precisely, the first term(2.6) stands for the interfacial energy of all the droplets, the second term is theenergy of interactions between the droplets and the background, the third termis the self-interaction energy of each droplet, and the last term is the interactionenergy of each droplet pair (for the case of a single droplet in R , see [14]).We can use the reduced energy in (2.6) to obtain the leading order scal-ing of various quantities for ε (cid:28) r i = O ( ε / | ln ε | − / ). Balancing this with the second term leads, in turn, to¯ δ = ε − / | ln ε | − / (1 + ¯ u ) (2.7)being an O (1) quantity. Similarly, balancing the last term with the first threeleads to N = O ( | ln ε | ), and the expected behavior of min E N = O ( ε / | ln ε | / ).7ne would also expect that, since the droplets repel each other, in a minimumenergy configuration they would become uniformly distributed throughout Ω.Our main result proves and further quantifies this heuristic picture on thelevel of the sharp interface energy E . Theorem 2.2.
Let ¯ u = − ε / | ln ε | / ¯ δ , with some ¯ δ > fixed. Then forany σ > sufficiently small there exists ε > such that for all ε ≤ ε :(i) If ¯ δ < √ κ , then u = − is the unique global minimizer of E , with ε − / | ln ε | − / min E = κ − ¯ δ .(ii) If ¯ δ > √ κ , there exists a non-trivial minimizer of E . The minimizeris u ( x ) = − N (cid:88) i =1 χ Ω + i ( x ) , (2.8) where χ Ω + i are characteristic functions of N disjoint simply connected sets Ω + i ⊂ Ω with boundary of class C , and N = O ( | ln ε | ) . The boundary ofeach set Ω + i is O ( ε / − σ ) -close (in Hausdorff sense) to a circle of radius r i centered at x i . Furthermore, min E = ε / | ln ε | / κ − ¯ δ + E N ( { r i } , { x i } ) + O ( ε / − σ ) , (2.9) with E N = O ( ε / | ln ε | / ) , r i = O ( ε / | ln ε | − / ) , and | x i − x j | > ε σ , ∀ j (cid:54) = i. (2.10) (iii) If ¯ δ > √ κ , in the limit ε → we have ε − / | ln ε | / r i → √ uniformly, | ln ε | N (cid:88) i =1 δ ( x − x i ) → π √ (cid:32) ¯ δ − √ κ (cid:33) , (2.12) weakly in the sense of measures, and ε − / | ln ε | − / min E → √ (cid:32) ¯ δ − √ κ (cid:33) . (2.13)Note that a more detailed result on the structure of the transition occurringnear ¯ δ = √ κ is presented in Proposition 3.20.Let us make a few remarks related to the statements of Theorem 2.2. Forsmall, but finite values of ε this theorem establishes an equivalence between thesharp interface energy E and the energy of N interacting droplets E N , in the8 = 2 N = 3 N = 4 N = 5 Figure 2: “Coulombic dice”: Minimizers of E N with r i = √ ε / | ln ε | − / for κ = 2 and N = 2 , , ,
5, obtained using random search algorithm.sense that the minimizers of E are close to “almost” minimizers of E N , i.e., wehave E N < min E N + O ( ε / − σ ). Nevertheless, to prove closeness of minimizersof E to those of E N we also need some coercivity of the energy E N . This problemhas to do with the properties of the minimizers of the pairwise interaction of thedroplets, i.e. the choice of x i which minimize E N with fixed r i . This becomesa difficult problem in the case of interest, since we generally expect N (cid:29) N and κ = 2 see Fig. 2). It would benatural to conjecture that at small enough ε the minimizing droplets will arrangethemselves into a periodic lattice close to a hexagonal (close-packed) lattice.Proving this kind of result, however, is a major challenge (see [46] for a recentproof for a certain class of pair interactions), which is one of the open questionsalso in many other problems, such as the problem of characterizing Abrikosovvortex lattices, for example [47]. Let us mention here a recent result by Chenand Oshita, who proved that in the case κ = 0 the hexagonal arrangement ofdisks is energetically the best among simple periodic lattices [48]. Yet, it is notknown if the same result also holds for more general arrangements of droplets.Here we prove a weaker result that the number density of droplets becomesasymptotically uniform as ε →
0, leading also to uniform distribution of energy(compare with [36]). Moreover, we identify the precise asymptotic behavior ofthe minimal energy and the size of the minimizing droplets as ε → E (which, of course, agrees with the result for the sharpinterface energy). Theorem 2.3.
Let W satisfy assumptions (i)–(iv) at the beginning of Sec. 4,let ¯ u = − ε / | ln ε | / ¯ δ , with some ¯ δ > fixed, and let κ be given by (4.10).Then(i) If ¯ δ ≤ √ κ , then ε − / | ln ε | − / min E → κ − ¯ δ ,(ii) If ¯ δ > √ κ , then ε − / | ln ε | − / min E → √ (cid:16) ¯ δ − √ κ (cid:17) ,as ε → . The proofs of Theorems 2.1–2.3 are based on a number of propositions es-tablished in Secs. 3 and 4, and are completed in Sec. 5.9
Analysis of the sharp interface problem
Our plan for the analysis of the sharp interface problem consists of a number ofsteps which we list below:1. Introduce a suitably rescaled energy ¯ E and domain ¯Ω.2. Establish existence and regularity of the minimizers of ¯ E (subsets of ¯Ωwhere u = 1).3. Establish some a priori estimates for the geometry of the minimizers of ¯ E and uniform bounds on the induced long-range potential.4. Establish that different connected components of minimizers of ¯ E are sep-arated by large distances in ¯Ω.5. Establish that each connected component of a minimizer of ¯ E is close toa disk (hence the term “droplet”).6. Establish equivalence between min ¯ E and min ¯ E N (the suitably rescaledversion of E N ).7. Improve the estimate for the separation distance between different droplets.8. Prove uniform convergence of the rescaled droplet radii to a universalconstant.9. Prove convergence of min ¯ E to a limit and convergence of the normalizeddroplet density in the original, unscaled domain Ω to a limit, as ε → We begin by introducing a suitable rescaling, in which the main quantities ofinterest become O (1) quantities in the limit ε →
0. Motivated by the discussionof Sec. 2, we define the rescaled energy ¯ E (with the energy of the uniform state u = − x ∈ ¯Ω = [0 , ε / | ln ε | − / ) , where ¯Ωis a two-dimensional torus with period ε / | ln ε | − / : E = ε / | ln ε | / ( κ − ¯ δ + ¯ E ) , x = ε / | ln ε | / ¯ x. (3.1)The energy ¯ E can be conveniently expressed in term of the set ¯Ω + ⊂ ¯Ω in which u = 1: ¯ E = | ln ε | − (cid:0) | ∂ ¯Ω + | − δκ − | ¯Ω + | (cid:1) + 2 | ln ε | − (cid:90) ¯Ω + (cid:90) ¯Ω + G (cid:0) ε / | ln ε | − / (¯ x − ¯ y ) (cid:1) d ¯ x d ¯ y. (3.2)10e also need an expression for the rescaled energy ¯ E N of a system of in-teracting droplets. With the help of (3.1), we can write the rescaling of (2.6)as¯ E N = 2 π | ln ε | N (cid:88) i =1 (cid:110) ¯ r i − ¯ δκ − ¯ r i − | ln ε | − ¯ r i (cid:16) ln( ε / | ln ε | − / ¯ κ ¯ r i ) − (cid:17)(cid:111) + 4 π | ln ε | N − (cid:88) i =1 N (cid:88) j = i +1 G ( ε / | ln ε | − / (¯ x i − ¯ x j ))¯ r i ¯ r j , (3.3)where ¯ r i and ¯ x i are the radii and the centers of the droplets, respectively. Let us begin with the statement of a result on the existence and regularity ofminimizers of ¯ E (or, equivalently, of E ), which is obtained by straightforwardlyadapting the results of [49] for sets of prescribed mean curvature. Proposition 3.1.
There exists a set ¯Ω + of finite perimeter which minimizes ¯ E in (3.2). The boundary ∂ ¯Ω + of this set is a curve of class C ,α for some α ∈ (0 , . In view of this, in the following we will always assume that minimizers ¯Ω + of ¯ E are closed sets. We also note that v (¯ x ) = | ln ε | − (cid:90) ¯Ω + G ( ε / | ln ε | − / (¯ x − ¯ y )) d ¯ y (3.4)is in W ,p ( ¯Ω), with any p >
1, and, hence, in C ,α ( ¯Ω) for any α ∈ (0 , v solves the equation − ∆ v + ε / | ln ε | − / κ v = | ln ε | − χ ¯Ω + , (3.5)where χ ¯Ω + is the characteristic function of ¯Ω + , in ¯Ω, and so the result followsby standard elliptic regularity theory [50]. As a consequence, we have a higherregularity for the boundary of the minimizer ¯Ω + of ¯ E ( [51], see also [35]): Corollary 3.2.
The boundary ∂ ¯Ω + of a minimizer ¯Ω + of ¯ E is of class C ,α . Note that this regularity result also holds more generally for local minimizers of¯ E in dimensions d ≤ E in d ≤ ρ ∈ C ( ∂ ¯Ω), a >
0, and ¯Ω a is the set obtainedby displacing ∂ ¯Ω + by aρ in the outward normal direction, then a (cid:55)→ ¯ E ( ¯Ω + a ) istwice continuously differentiable at a = 0, and we have [14] (for the reader’s11onvenience, the computation is reproduced in Appendix C): | ln ε | d ¯ E ( ¯Ω + a ) da (cid:12)(cid:12)(cid:12)(cid:12) a =0 = (cid:90) ∂ ¯Ω + ( K (¯ x ) − δκ − + 4 v (¯ x )) ρ (¯ x ) d H (¯ x ) , (3.6) | ln ε | d ¯ E ( ¯Ω + a ) da (cid:12)(cid:12)(cid:12)(cid:12) a =0 = (cid:90) ∂ ¯Ω + (cid:0) |∇ ρ (¯ x ) | + 4 ν (¯ x ) · ∇ v (¯ x ) ρ (¯ x ) (cid:1) d H (¯ x )+ (cid:90) ∂ ¯Ω + (4 v (¯ x ) − δκ − ) K (¯ x ) ρ (¯ x ) d H (¯ x )+4 | ln ε | − (cid:90) ∂ ¯Ω + (cid:90) ∂ ¯Ω + G ( ε / | ln ε | − / (¯ x − ¯ y )) ρ (¯ x ) ρ (¯ y ) d H (¯ x ) d H (¯ y ) . (3.7)where K (¯ x ) is the curvature at point ¯ x ∈ ∂ ¯Ω + , with the sign convention that K > + is convex, and ν (¯ x ) is the outward unit normal to ∂ ¯Ω + at thatpoint. The associated Euler-Lagrange equation for ∂ ¯Ω + reads K (¯ x ) = 2¯ δκ − − v (¯ x ) , (3.8)which also allows to simplify the expression in (3.7) evaluated on a minimizerto | ln ε | d ¯ E ( ¯Ω + a ) da (cid:12)(cid:12)(cid:12)(cid:12) a =0 = (cid:90) ∂ ¯Ω + (cid:0) |∇ ρ (¯ x ) | + 4 ν (¯ x ) · ∇ v (¯ x ) ρ (¯ x ) − K (¯ x ) ρ (¯ x ) (cid:1) d H (¯ x )+4 | ln ε | − (cid:90) ∂ ¯Ω + (cid:90) ∂ ¯Ω + G ( ε / | ln ε | − / (¯ x − ¯ y )) ρ (¯ x ) ρ (¯ y ) d H (¯ x ) d H (¯ y ) . (3.9)We will use these equations later on to establish some properties of theminimizers for ε (cid:28)
1. Meanwhile, let us begin our analysis with some basicestimates.
Lemma 3.3.
Let ¯Ω + be a minimizer of ¯ E . Then there exists C > such that | ¯Ω + | ≤ C | ln ε | , (3.10) | ∂ ¯Ω + | ≤ C | ln ε | . (3.11) for ε (cid:28) .Proof. First of all, by representation (2.3) we have G ( x − y ) ≥ c > x, y ∈ Ω. Therefore, in view of the fact that min ¯ E ≤ E = 0 if ¯Ω + = ∅ ),from (3.2) we have0 ≥ | ln ε | ¯ E ≥ | ln ε | − (cid:90) ¯Ω + (cid:90) ¯Ω + G (cid:0) ε / | ln ε | − / (¯ x − ¯ y ) (cid:1) d ¯ x − δκ − | ¯Ω + |≥ c | ln ε | − | ¯Ω + | − δκ − | ¯Ω + | , (3.12)which gives (3.10). On the other hand, we also have | ∂ ¯Ω + | ≤ δκ − | ¯Ω + | . (3.13)Therefore, from (3.10) we immediately obtain (3.11).12s a corollary, it follows from (3.11) that the diameter of each connectedsubset ¯Ω + i of ¯Ω + is bounded by O ( | ln ε | )diam( ¯Ω + i ) ≤ C | ln ε | , (3.14)for some C > ε (cid:28) + (cid:54) = ∅ is uniformly bounded above and below independently of ε . Lemma 3.4.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i arethe disjoint connected components of ¯Ω + . Then, there exist C > c > such that c ≤ | ¯Ω + i | , | ∂ ¯Ω + i | ≤ C, diam( ¯Ω + i ) ≤ C, (3.15) for ε (cid:28) .Proof. First, note that since by Corollary 3.2 the set ∂ ¯Ω + is of class C ,α wehave N < ∞ . To see that (3.15) holds, we first write ¯ E as | ln ε | ¯ E = N (cid:88) i =1 (cid:32) | ∂ ¯Ω + i | − δκ − | ¯Ω + i | + 2 | ln ε | − (cid:90) ¯Ω + i (cid:90) ¯Ω + i G ( ε / | ln ε | − / (¯ x − ¯ y )) d ¯ xd ¯ y + 2 | ln ε | − (cid:88) j (cid:54) = i (cid:90) ¯Ω + i (cid:90) ¯Ω + j G ( ε / | ln ε | − / (¯ x − ¯ y )) d ¯ xd ¯ y (cid:33) . (3.16)In view of (3.14) and (2.4), the integral in the second line in (3.16) is boundedfrom below by π (1 − δ ) | ln ε | | ¯Ω + i | for any δ >
0, provided ε is small enough.Therefore, removing the set ¯Ω + i from ¯Ω + will result in the change of energy ∆ ¯ E estimated as | ln ε | ∆ ¯ E ≤ − (cid:18) | ∂ ¯Ω + i | − δκ − | ¯Ω + i | + π (1 − δ ) | ¯Ω + i | (cid:19) ≤ − (cid:18) √ π | ¯Ω + i | / − δκ − | ¯Ω + i | + π (1 − δ ) | ¯Ω + i | (cid:19) , (3.17)where in the first line we took into account that G >
E <
0, contradicting minimality of ¯ E on ¯Ω + , unless c ≤ | ¯Ω + i | ≤ C forsome C > c >
0, independently of ε (cid:28)
1. Finally, the lower bound for | ∂ ¯Ω + i | follows from the isoperimetric inequality, and the upper bound is obtained byapplying the previous argument to the first line in (3.17).Following the same arguments, we also immediately arrive at the followingnon-existence result: Proposition 3.5.
Let ¯ δ < √ κ be fixed. Then the unique minimizer of ¯ E is ¯Ω + = ∅ for ε (cid:28) . (cid:60) Κ ∆ (cid:61) Κ ∆ (cid:62) Κ r (cid:45) V Figure 3: The graph of V (¯ r ) from (3.18) for different values of ¯ δ . Proof.
Let us introduce the function V v : [0 , ∞ ) → R , defined as V v (¯ r ) = 2 π (cid:0) ¯ r + (2 v − ¯ δκ − )¯ r + ¯ r (cid:1) , (3.18)whose graph at v = 0 and several values of ¯ δ is shown in Fig. 3. If ¯Ω + i is aconnected component of ¯Ω + and ¯ r i = ( π | ¯Ω + i | ) / , then by the same argumentsas in Lemma 3.4, the energy gained by removing ¯Ω + i from ¯Ω + is bounded belowby | ln ε | − ( V (¯ r i ) + o (1)), as long as ε (cid:28)
1. Then, by direct inspection V (¯ r )is always positive under the assumptions of the proposition, making ¯Ω + = ∅ energetically preferred.Note that the asymptotic value of the threshold of ¯ δ in Proposition 3.5 belowwhich no non-trivial minimizers are present was computed in [14]. Anothersimple corollary to Proposition 3.4 is the following Lemma 3.6.
Let ¯Ω + be a non-trivial minimizer of ¯ E and let N be the numberof disjoint connected components of ¯Ω + . Then there exists C > such that N ≤ C | ln ε | , (3.19) for ε (cid:28) . Let us now establish a uniform bound on the potential v . Note that a versionof this result is also an important component in the proofs of [36]. Lemma 3.7.
Let ¯Ω + be a non-trivial minimizer of ¯ E . Then for any α ∈ (0 , we have < v ≤ C, || v || C ,α (¯Ω) ≤ C, (3.20) where v is given by (3.4), for some C > independent of ε (cid:28) . roof. We start by noting that v > G . Let us nowestimate the gradient of v . Using (2.4) and Lemmas 3.4 and 3.6, we get |∇ v (¯ x ) | ≤ | ln ε | − (cid:90) ¯Ω + |∇ G ( ε / | ln ε | − / | ¯ x − ¯ y | ) | d ¯ y ≤ | ln ε | − (cid:90) B ¯ r (¯ x ) |∇ G ( ε / | ln ε | − / | ¯ x − ¯ y | ) | d ¯ y + | ln ε | − (cid:90) ¯Ω + \ B ¯ r (¯ x ) |∇ G ( ε / | ln ε | − / | ¯ x − ¯ y | ) | d ¯ y ≤ C ( | ln ε | − ¯ r + ¯ r − ) ≤ C | ln ε | − / , (3.21)for some C >
0, where B ¯ r (¯ x ) is a disk of radius ¯ r centered at ¯ x , and the lastinequality is obtained by choosing ¯ r = | ln ε | / . Therefore, by the results ofLemma 3.4, we see that osc ¯ x ∈ ¯Ω + i v (¯ x ) = o (1) , (3.22)for each connected component ¯Ω + i of ¯Ω + . To see that this implies the conclusionof the lemma, suppose that, to the contrary, we have max v = M (cid:29)
1. Sinceby (3.5) the function v is subharmonic in ¯Ω \ ¯Ω + , it achieves its maximum in theclosure of some ¯Ω + i . Therefore, in view of (3.22) we have v ≥ M in ¯Ω + i . Then,following the same arguments as in the proof of Lemma 3.4, for large enough M we can lower the energy by removing ¯Ω + i from ¯Ω + .Finally, by [50, Theorem 9.11] we have || v || W ,p ( B (¯ x )) ≤ C , where B (¯ x ) is adisk of radius 1 centered at ¯ x ∈ ¯Ω, for some C > p >
2, independentlyof ¯ x and ε (cid:28)
1. Hence, the uniform H¨older estimate on the gradient follows bySobolev imbedding.We can also immediately conclude from (3.8) and (3.22) that the curvatureof ∂ ¯Ω + is uniformly bounded both from above and below by positive constants,implying that each ¯Ω + i is convex. Note that this result justifies the terminology“droplet” for each ¯Ω + i which we will be using from now on. Lemma 3.8.
Let ∂ ¯Ω + be the boundary of a minimizer ¯Ω + of ¯ E . Then we have c ≤ K (¯ x ) ≤ C, (3.23) for all ¯ x ∈ ∂ ¯Ω + , with some C > c > independent of ε (cid:28) . In particular, when ε (cid:28) , each connected component ¯Ω + i of ¯Ω + is convex and simply connected.Proof. The upper bound is an immediate consequence of (3.8) and positivity of v . To obtain the lower bound, let us note that by the results of Lemma 3.4, forevery connected component ¯Ω + i there exists a disk B ¯ r i (¯ x i ), with ¯ r i = O (1), suchthat ¯Ω + i ⊂ B ¯ r i (¯ x i ). Therefore, translating B ¯ r i (¯ x i ) until its boundary touches ∂ ¯Ω + i , we obtain a point ¯ x (cid:48) i ∈ ∂ ¯Ω + i , such that K (¯ x (cid:48) i ) ≥ ¯ r − i ≥ c , for some c > ε (cid:28)
1. Now, by (3.8) we have v (¯ x (cid:48) i ) ≤ (¯ δκ − − c ). At the sametime, by (3.22) this implies that v (¯ x ) ≤ (2¯ δκ − − c ) for all ¯ x ∈ ∂ ¯Ω + i , which,again, by (3.8) gives the statement. 15e now show that different connected components of ¯Ω + cannot come tooclose to each other when ε (cid:28) Lemma 3.9.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + , and let N ≥ . Then, there exists C > such that dist( ¯Ω + i , ¯Ω + j ) ≥ C ∀ i (cid:54) = j, (3.24) for ε (cid:28) .Proof. Let ¯ x i ∈ ¯Ω + i and ¯ x j ∈ ¯Ω + j be such that r = | ¯ x i − ¯ x j | = dist( ¯Ω + i , ¯Ω + j ) > B centered at (¯ x + ¯ x ) with radius R = 2 r and a rectangle Q inscribed into B which is shown by the thick solid lines in Fig. 4. In view ofthe uniform bound on the curvature of ∂ ¯Ω + obtained in Lemma 3.8, the curvesegments ∂ ¯Ω + i ∩ Q and ∂ ¯Ω + j ∩ Q passing through ¯ x i and ¯ x j , respectively, intersect ∂Q transversally as in Fig. 4 when r (cid:28)
1. Furthermore, we have dist( ∂ ¯Ω + i ∩ ∂Q + , ∂ ¯Ω + j ∩ ∂Q + ) ≤ r and dist( ∂ ¯Ω + i ∩ ∂Q − , ∂ ¯Ω + j ∩ ∂Q − ) ≤ r , where ∂Q + and ∂Q − are the right and the left side of the boundary of the rectangle relative tothe line through ¯ x and ¯ x , respectively, for sufficiently small r independent of ε (cid:28) | ∂ ¯Ω + i ∩ Q | + | ∂ ¯Ω + j ∩ Q | ≥ r √ ∂ ¯Ω + i ∩ ∂Q + with ∂ ¯Ω + j ∩ ∂Q + , and ∂ ¯Ω + i ∩ ∂Q − with ∂ ¯Ω + j ∩ ∂Q − by straight lines and including the region between them into¯Ω, we will decrease | ∂ ¯Ω + | by at least 4( √ − r . Thus, the change ∆ ¯ E in thetotal energy is estimated to be | ln ε | ∆ ¯ E ≤ − √ − r + 4 (cid:90) Q v (¯ x ) d ¯ x ++ 2 | ln ε | − (cid:90) Q (cid:90) Q G ( ε / | ln ε | − / (¯ x − ¯ y )) d ¯ xd ¯ y. (3.25)Finally, in view of Lemma 3.7 and (2.4), the the right-hand side of (3.25) isbounded above by − C r + C r , with C , > ε (cid:28)
1. Hence,the energy of such a rearrangement will be lower if r is sufficiently small, for all ε (cid:28) Lemma 3.10.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + , and let N ≥ . Then there exists α > such that dist( ¯Ω + i , ¯Ω + j ) > ε − α ∀ i (cid:54) = j, (3.26)16 ) b) rr r 3r 2r Figure 4: Schematics of the rearrangement argument of Lemma 3.9. In (a), theset ¯Ω + is shown in gray, solid arcs show the bounds on the location of ∂ ¯Ω + ,the thick solid lines show the rectangle Q . In (b), the gray region shows therearranged ¯Ω + . for ε (cid:28) .Proof. Consider the second variation of ¯ E with respect to the perturbation, inwhich the boundary of each connected component ¯Ω + i is expanded uniformly bya distance ac i in the normal direction, i.e., we have ρ (¯ x ) = c i for all ¯ x ∈ ∂ ¯Ω + i .By (3.9), we have d ¯ E ( ¯Ω + a ) da (cid:12)(cid:12)(cid:12)(cid:12) a =0 = | ln ε | − (cid:88) i,j Q ij c i c j , (3.27)where the coefficients Q ij of the quadratic form Q can be estimated as Q ii = − (cid:90) ∂ ¯Ω + i K (¯ x ) d H (¯ x ) + 23 π | ∂ ¯Ω + i | + o (1) , (3.28)(3.29)where we took into account that by (3.5) and Gauss’s theorem (cid:82) ∂ ¯Ω + i ν (¯ x ) ·∇ v (¯ x ) d H (¯ x ) = −| ln ε | − | ¯Ω + i | + O ( ε / | ln ε | − / ) and used the expansion in(2.4) together with Lemmas 3.8, 3.7 and 3.4, for ε (cid:28)
1. Furthermore, since byconvexity of ¯Ω + i (see Lemma 3.8) the boundary of each ¯Ω + i is a closed curve, byCauchy-Schwarz inequality we have4 π = (cid:32)(cid:90) ∂ ¯Ω + i K (¯ x ) d H (¯ x ) (cid:33) ≤ | ∂ ¯Ω + i | (cid:90) ∂ ¯Ω + i K (¯ x ) d H (¯ x ) . (3.30)Therefore, the diagonal elements of Q can be further estimated as Q ii ≤ π | ∂ ¯Ω + i | − π | ∂ ¯Ω + i | + o (1) . (3.31)On the other hand, define α = | ln ε | − ln(dist( ¯Ω + i , ¯Ω + j )), and suppose, to thecontrary of the statement of the proposition, that α is sufficiently small for some17air of indices for a sequence of ε →
0. Then, with the help of Lemma 3.9 and(2.4) we can estimate Q ij = 23 π (1 − α ) | ∂ ¯Ω + i | | ∂ ¯Ω + j | + o (1) . (3.32)Now, for the index pair ( i, j ) above let us choose c i = | ∂ ¯Ω + j | , c j = −| ∂ ¯Ω + i | ,and let us set c k = 0 for all other indices k . A simple calculation of the sum in(3.27) then shows that for this choice of c ’s we have | ln ε | d ¯ E ( ¯Ω + a ) da (cid:12)(cid:12)(cid:12)(cid:12) a =0 ≤ | ∂ ¯Ω + i | | ∂ ¯Ω + j | π (cid:32) α − π | ∂ ¯Ω + i | − π | ∂ ¯Ω + j | (cid:33) + o (1) , (3.33)where we took into account Lemma 3.4. This expression is negative for ε (cid:28) α < π min {| ∂ ¯Ω + i | − , | ∂ ¯Ω + j | − } , (3.34)which, in view of Lemma 3.4, contradicts minimality of ¯ E for small enough ε . Let us also point out that the proof of Proposition 3.10 gives a universal lowerbound for the perimeter of the connected portions of the minimizers. Indeed,the quadratic form Q has a negative eigenvalue, if π | ∂ ¯Ω + i | − π | ∂ ¯Ω + i | − < ε (cid:28)
1, which implies the following result:
Proposition 3.11.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + . Then, for every δ > | ∂ ¯Ω + i | ≥ π √ − δ, (3.35) for ε (cid:28) . Note that this criterion in the radially-symmetric case was obtained in [13,14,55]and is also applicable to all local minimizers (for global minimizers, a betterbound will be obtained below). We also derive another quantitative estimate on v and the geometry of ¯Ω + i that remains valid for local minimizers of low energy. Proposition 3.12.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + . Then < v < ¯ δ κ , | ¯Ω + i | < π ¯ δκ , (3.36) for ε (cid:28) .Proof. Let ¯ x ∈ ∂ ¯Ω + i . Then, using Lemma 3.8, (3.8), (2.4), and positivity of G ,for some c > ε (cid:28) < c ≤ K (¯ x ) = 2¯ δκ − − v (¯ x ) ≤ (cid:32) δκ − − | ln ε | − (cid:90) ¯Ω + i G ( ε / | ln ε | − / (¯ x − ¯ y )) d ¯ y (cid:33) ≤ δκ − − π | ¯Ω + i | + o (1) , (3.37)18hich, together with (3.22) and the fact that v reaches its maximum in theclosure of ¯Ω + , yields the statement.We now prove that for ε (cid:28) G at small distances the potential v inside each droplet is approximatelyconstant. Therefore, the shape of the droplet approximately minimizes theusual isoperimetric problem, and the size of the droplet is determined by thebalance of surface tension and the pressure due to non-local forces inside thedroplet [13, 14].If the droplet ¯Ω + i were exactly a disk B ¯ r i (¯ x i ) of radius ¯ r i centered at ¯ x i ,then the potential v would be given by v ∗ (¯ x ) = v ∗ i (¯ x ) + v i (¯ x ) , (3.38)where v ∗ i (¯ x ) = | ln ε | − (cid:88) n ∈ Z v B ( | ¯ x − ¯ x i − n | , ¯ r i , ε / | ln ε | − / κ ) , (3.39)with the function v B ( ρ, r, κ ) being the solution of − ∆ v B + κ v B = χ B r (0) in R , given explicitly in terms of the modified Bessel functions: v B ( ρ, r, κ ) = (cid:40) κ − − κ − rK ( κr ) I ( κρ ) , ρ ≤ r,κ − rI ( κr ) K ( κρ ) , ρ ≥ r, (3.40)and v i (¯ x ) = | ln ε | − (cid:88) j (cid:54) = i (cid:90) ¯Ω + j G ( ε / | ln ε | − / (¯ x − ¯ y )) d ¯ y. (3.41)Note that in view of Lemmas 3.10 and 3.4, and (2.4), we have |∇ v i | ≤ Cε α , (3.42)for some C > α >
0, in any disk of O (1) radius containing ¯Ω + i for ε (cid:28) x ∈ ∂B ¯ r i (¯ x i ), by Taylor-expanding the Bessel functions [45] wehave for any α < v ∗ (¯ x ) = ¯ v i + O ( ε α ) , |∇ v ∗ (¯ x ) | = O ( | ln ε | − ) , (3.43)where the constant ¯ v i is given by¯ v i = − | ln ε | − ¯ r i ln( ε / | ln ε | − / ¯ κ ¯ r i )+ π | ln ε | − (cid:88) j (cid:54) = i ¯ r j G ( ε / | ln ε | − / κ (¯ x i − ¯ x j )) , ¯ r j = ( π | ¯Ω + j | ) / . (3.44)In the following, we will show that v (¯ x ) on ∂ ¯Ω + i also coincides with ¯ v i to O ( ε α ),giving the balance of forces at the interface. We are now ready to state ourresult: 19 roposition 3.13. Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + . Then there exists a constant α > such that for all ε (cid:28) :(i) For each ¯Ω + i there exists a point ¯ x i ∈ ¯Ω + i , such that B ¯ r i − ε α (¯ x i ) ⊂ ¯Ω + i ⊂ B ¯ r i + ε α (¯ x i ) , (3.45) where ¯ r i = ( π | ¯Ω + i | ) / and B ¯ r (¯ x ) is a disk of radius ¯ r centered at ¯ x ;(ii) The values of ¯ r i satisfy ¯ r − i − δκ − + 4¯ v i = O ( ε α ) , (3.46) where ¯ v i are given by (3.44).Proof. Let us pick a point ¯ x (cid:48) i ∈ ¯Ω + i , then ¯Ω + i ⊂ B | ∂ ¯Ω + i | (¯ x (cid:48) i ). Let us then replace¯Ω + i with a disk of the same area centered at ¯ x (cid:48) i . By Lemmas 3.10 and 3.4, theresulting set B ¯ r i (¯ x (cid:48) ) still satisfies the bound in (3.26), and the change of energy∆ ¯ E under this rearrangement can be estimated as | ln ε | ∆ ¯ E = 2 √ π | ¯Ω + i | / − | ∂ ¯Ω + i | + O (cid:0) | ln ε | − (cid:1) , (3.47)where we used (2.4), (3.42) and Lemma 3.4. Thus, the energy will decreaseunder this rearrangement, contradicting minimality of ¯ E , unless for some C > + i D ( ¯Ω + i ) = | ∂ ¯Ω + i | √ π | ¯Ω + i | / − ≤ C | ln ε | − , (3.48)for ε (cid:28)
1. Choosing ¯ x i ∈ B | ∂ ¯Ω + i | (¯ x (cid:48) i ) to minimize | ¯Ω + i ∆ B ¯ r i (¯ x i ) | , where ¯Ω + i ∆ B ¯ r i (¯ x i )denotes the symmetric difference of sets ¯Ω + i and B ¯ r i (¯ x i ), by the results of[56] we have | ¯Ω + i ∆ B ¯ r i (¯ x i ) | ≤ C (cid:48) | ln ε | − / , and C (cid:48) > ε (cid:28)
1. Then, by Lemma 3.8 the set ∂ ¯Ω + i is uniformly closeto ∂B ¯ r i (¯ x i ), and ¯ x i ∈ ¯Ω + i (if not, then by convexity of ¯Ω + i we would have | ¯Ω + i ∆ B ¯ r i (¯ x i ) | ≥ | B ¯ r i (¯ x i ) | ), giving (i) to o (1).To obtain the O ( ε α ) bound in (i), let ρ : ∂B ¯ r i (¯ x i ) → R be the signeddistance from a given point on ∂B ¯ r i (¯ x i ) to ∂ ¯Ω + i along the outward normal to ∂B ¯ r i (¯ x i ). Note that by convexity of ¯Ω + i the function ρ defines a one-to-onemap between ∂ ¯Ω + i and ∂B ¯ r i (¯ x i ). Furthermore, if || ρ || L ∞ ( ∂B ¯ ri (¯ x i )) = δ , we have ||∇ ρ || L ∞ ( ∂B ¯ ri (¯ x i )) ≤ Cδ / for some C > ε (cid:28) δ →
0, as ε →
0. Also, by Corollary 3.2 we have ρ ∈ C ( ∂B ¯ r i (¯ x i )).Treating ¯Ω + as a perturbation of the set ¯Ω ∗ = B ¯ r i (¯ x i ) ∪ ( ¯Ω + \ ¯Ω + i ) and20xpanding as in Lemma C.1, we can write | ln ε | ( ¯ E ( ¯Ω + ) − ¯ E ( ¯Ω ∗ )) = (cid:90) ∂B ¯ ri (¯ x i ) (cid:0) ¯ r − i − δκ − + 4 v ∗ (¯ x ) (cid:1) ρ (¯ x ) d H (¯ x )+ 12 (cid:90) ∂B ¯ ri (¯ x i ) (cid:0) |∇ ρ (¯ x ) | + 4 ν (¯ x ) · ∇ v ∗ (¯ x ) ρ (¯ x ) (cid:1) d H (¯ x )+ 12¯ r i (cid:90) ∂B ¯ ri (¯ x i ) (4 v ∗ (¯ x ) − δκ − ) ρ (¯ x ) d H (¯ x )+ 2 | ln ε | (cid:90) ∂B ¯ ri (¯ x i ) (cid:90) ∂B ¯ ri (¯ x i ) G ( ε / | ln ε | − / (¯ x − ¯ y ) ρ (¯ x ) ρ (¯ y ) d H (¯ x ) d H (¯ y )+ O ( δ α ) , (3.49)for any α ∈ (0 , ε (cid:28) | ¯Ω + i | = | B ¯ r i (¯ x i ) | , we have0 = (cid:90) ∂B ¯ ri (¯ x i ) (cid:90) ρ (¯ x )0 (1 + ¯ r − i r ) dr d H (¯ x )= (cid:90) ∂B ¯ ri (¯ x i ) ρ (¯ x ) d H (¯ x ) + 12¯ r i (cid:90) ∂B ¯ ri (¯ x i ) ρ (¯ x ) d H (¯ x ) . (3.50)Therefore, using the estimate in (3.43) we can rewrite (3.49) as | ln ε | ( ¯ E ( ¯Ω + ) − ¯ E ( ¯Ω ∗ )) = 12 (cid:90) ∂B ¯ ri (¯ x i ) (cid:0) |∇ ρ | − ¯ r − i ρ (¯ x ) (cid:1) d H (¯ x )+ 2 | ln ε | (cid:90) ∂B ¯ ri (¯ x i ) (cid:90) ∂B ¯ ri (¯ x i ) G ( ε / | ln ε | − / (¯ x − ¯ y ) ρ (¯ x ) ρ (¯ y ) d H (¯ x ) d H (¯ y )+ O ( | ln ε | − || ρ || L B ¯ ri (¯ xi ) ) + O ( ε α || ρ || L B ¯ ri (¯ xi ) ) + O ( δ α || ρ || H B ¯ ri (¯ xi ) ) , (3.51)where we took into account that δ ≤ C || ρ || H B ¯ ri (¯ xi ) for some C >
0. Furtherestimating the double integral in (3.51), using (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂B ¯ ri (¯ x i ) (cid:18) | ln ε | G ( ε / | ln ε | − / (¯ x − ¯ y )) − π (cid:19) ρ (¯ y ) d H (¯ y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ ln | ln ε || ln ε | , (3.52)we have | ln ε | ( ¯ E ( ¯Ω + ) − ¯ E ( ¯Ω ∗ )) =12 (cid:90) ∂B ¯ ri (¯ x i ) (cid:0) |∇ ρ | − ¯ r − i ρ (¯ x ) (cid:1) d H (¯ x ) + 13 π (cid:32)(cid:90) ∂B ¯ ri (¯ x i ) ρ (¯ x ) d H (¯ x ) (cid:33) + O ( ε α || ρ || L B ¯ ri (¯ xi ) ) + o ( || ρ || L B ¯ ri (¯ xi ) ) + o ( || ρ || H B ¯ ri (¯ xi ) ) . (3.53)21ow, write ρ as ρ = ρ + ρ + ρ , where ρ = π ¯ r i (cid:82) ∂B ¯ ri (¯ x i ) ρ (¯ x ) d H (¯ x ), ρ (¯ x ) = ¯ x − ¯ x i | ¯ x − ¯ x i | · b , for some vector b ∈ R , and ρ orthogonal to ρ and ρ in L ( ∂B ¯ r i (¯ x i )). By (3.50) we have | ρ | = O ( || ρ || L ( ∂B ¯ ri (¯ x i )) ), which is, therefore,negligibly small compared to | b | and || ρ || L ( ∂B ¯ ri (¯ x i )) in all the arguments below.Then, using Poincar´e’s inequality, we find that¯ E ( ¯Ω + ) ≥ ¯ E ( ¯Ω ∗ )+ 14 (cid:90) ∂B ¯ ri (¯ x i ) |∇ ρ | (¯ x ) d H (¯ x ) − Cε α || ρ || L B ¯ ri (¯ xi ) − c | b | , (3.54)for some C > < c (cid:28)
1, whenever ε (cid:28)
1. This implies that || ρ || H B ¯ ri (¯ xi ) ≤ Cε α || ρ || L B ¯ ri (¯ xi ) + 4 c | b | , (3.55)for ε (cid:28)
1, otherwise replacing ¯Ω + i with B ¯ r i (¯ x i ) lowers the energy. On the otherhand, we also have | b | = O ( || ρ || H B ¯ ri (¯ xi ) ). If not, then ∂ ¯Ω + i will be o ( | b | ) close to ∂B ¯ r i (¯ x i + b ) for ε (cid:28)
1. This, however, contradicts the choice of ¯ x i to minimize | ¯Ω + i ∆ B ¯ r i (¯ x i ) | . Therefore, we have || ρ || H B ¯ ri (¯ xi ) ≤ Cε α || ρ || H B ¯ ri (¯ xi ) + c || ρ || H B ¯ ri (¯ xi ) , (3.56)for some C > < c (cid:28)
1, implying || ρ || H B ¯ ri (¯ xi ) = O ( ε α ) and, hence, δ = O ( ε α ). This gives part (i) of the statement of the proposition.Finally, to prove part (ii) of the statement, let ¯Ω + a be obtained from ¯Ω + byexpanding ¯Ω + i by an amount a >
0, i.e., let us change ρ (¯ x ) → ρ (¯ x ) + a for every¯ x ∈ ∂B ¯ r i (¯ x i ). By (3.49), the change of energy can be estimated as | ln ε | ( ¯ E ( ¯Ω + a ) − ¯ E ( ¯Ω + )) = 2 πa ¯ r i (¯ r − i − δκ − + 4¯ v i ) + O ( aδ ) + O ( a ) , (3.57)where we took into account (3.43). Then, since ¯Ω + is a minimizer, the right-hand side of (3.57) should vanish to O ( a ). Therefore, by previous result weobtain the statement.Also, from the proof of Proposition 3.13 we obtain the following universallower bound on | ¯Ω + i | : Proposition 3.14.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + . Then, for every δ > | ¯Ω + i | ≥ π √ − δ, (3.58) for ε (cid:28) .Proof. By Proposition 3.13 each value of ¯ r i = ( π | ¯Ω + i | ) / satisfies (3.46) and,hence, is close to a critical point of V v i (¯ x i ) defined in (3.18), which can be seenfrom (3.44) by Taylor expansion. Furthermore, we must have 2¯ v i (¯ x i ) − ¯ δκ − ≤ √ o (1), so that V v i (¯ x i ) (¯ r i ) ≤ o (1) for ε (cid:28)
1, otherwise, arguing as inLemma 3.4, we can reduce the energy by removing ¯Ω + i from ¯Ω + . Therefore,¯ r i should be close to the positive minimum of V v i (¯ x i ) . By inspection, in thissituation ¯ r i ≥ √ − δ , for any δ > ε is sufficiently small,hence, the claim.The results of Proposition 3.13 just obtained immediately allow to establishan asymptotic equivalence of the energy ¯ E and the reduced energy ¯ E N on theminimizers for ε (cid:28) Proposition 3.15.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + , and let ¯ r i and ¯ x i be as inProposition 3.13. Then min ¯ E = O (1) , min ¯ E = min ¯ E N + O ( ε α ) , (3.59) for some α > independent of ε (cid:28) .Proof. The first equation in (3.59) is a direct consequence of the definition of¯ E in (3.2), according to which 0 ≤ ¯ δ κ − + min ¯ E ≤ ε − / | ln ε | / E [ −
1] = ¯ δ κ − . The upper bound in the second equation follows by choosing a trialfunction for ¯ E in the form of disks of radius ¯ r i centered at ¯ x i which minimize¯ E N and taking into consideration Lemmas 3.4 and 3.10 and (2.4). On theother hand, by Proposition 3.13(i), we have ¯Ω + i ⊃ B ¯ r i − ε α (¯ x i ) for ε (cid:28)
1, hence, | ∂ ¯Ω + i | > π (¯ r i − ε α ) and | ¯Ω + i | > π (¯ r i − ε α ) . This controls from below all theterms of ¯ E , except the one involving ¯ δ , by the corresponding terms of ¯ E N . Thelatter, however, is controlled by the second inclusion in Proposition 3.13(i).To summarize, for 0 < ε (cid:28) E have the formof well-separated nearly circular droplets. In fact, from Proposition 3.15 oneshould expect that the droplet-droplet interaction part of the energy, whichis given by the last term in the expression (3.3) for ¯ E N , should be close tothe minimum for fixed droplet sizes. Proving this, however, generally requiresinformation about coercivity of the interaction energy, which becomes difficultto establish when N (cid:29)
1, the asymptotic case of interest. Nevertheless, withthe help of Lemma 3.7 we can prove that in the original scaling the dropletsstay away from each other a distance O ( ε β ) in Ω, with an arbitrary β > ε (cid:28)
1, i.e., that the statement of Lemma 3.10 actually holds for any α ∈ (0 , ),provided that ε is small enough. Proposition 3.16.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + , and let ¯ x i be as in Proposition3.13. Then, for any α ∈ (0 , ) we have | ¯ x i − ¯ x j | > ε − α , for all i (cid:54) = j , as longas ε (cid:28) .Proof. First of all, note that by Lemma 3.10 the statement of the Propositionholds for some α >
0. To prove that α could be chosen arbitrarily close to ,suppose that, to the contrary, there exists a sequence of ε → i, j ), depending on ε , such that | ¯ x i − ¯ x j | ≤ ε − α with some 0 < α < . Letus denote by I the set of indices of those droplets whose centers are containedin a disk B centered at ¯ x = (¯ x i + ¯ x j ) with radius ε − α . By assumption wehave | I | ≥
2, where | · | denotes the counting measure. Also, we have | I | < M for some M ∈ N independent of ε (cid:28)
1. Indeed, by Lemmas 3.4 and 3.7, and by(2.4) we have for some c > C ≥ v (¯ x ) ≥ | ln ε | − (cid:88) k ∈ I (cid:90) ¯Ω + k G ( ε / | ln ε | − / (¯ x i − ¯ y )) d ¯ y ≥ (cid:0) − α + o (1) (cid:1) c | I | , (3.60)for ε (cid:28) σ > ε , and consider a sequenceof nested disks B k of radii ε − α (1+ kσ ) centered at ¯ x . By repeating the argumentabove, we also have | I M | ≤ M , as long as ε (cid:28)
1, where | I k | is the countingmeasure of the set I k of indices such that ¯ x l ∈ B k for all l ∈ I k . Therefore,in view of the fact that | I | >
1, we must have | I k +1 | − | I k | = 0 for some1 ≤ k ≤ M −
1, implying that B k +1 \ B k ∩ ¯Ω + = ∅ . Thus, there exists a clusterof droplets, whose indices are denoted by I k , which are within O ( ε − α (1+ kσ ) )distance of ¯ x and are separated from all other droplets by O ( ε − α (1+ σ + kσ ) )distance.Let us show that this contradicts the minimality of ¯ E for small enough ε .Indeed, let us displace the droplets in B k to the new locations ¯ x (cid:48) l = ¯ x l + λ (¯ x l − ¯ x i ),with l ∈ I k , which represents a dilation of B k by a factor of 1 + λ relative to ¯ x i ,keeping all ¯ r i fixed. For 0 < λ (cid:28) E of energy satisfies | ln ε | ∆ ¯ E ≤ − cλ + Cλε σα | ln ε | , (3.61)for some C, c > ε (cid:28)
1, where we used Lemmas 3.3 and 3.4, andthe estimate (2.4), arguing as in the derivation of (3.42). Thus, the consideredrearrangement lowers the energy.As a simple corollary to this result, we actually have the following universal(¯ δ -independent) upper bound on | ¯Ω + i | and, hence, on ¯ r i : Corollary 3.17.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + . Then, for any δ > | ¯Ω + i | ≤ π (cid:16) √ − (cid:17) / + δ, (3.62) when ε is sufficiently small.Proof. If | ¯Ω + i | is bigger, split ¯Ω + i into two disks of equal area and move themapart a distance d = ε − β , with 0 < β < α < . Arguing as before, the energychange ∆ ¯ E upon this manipulation is given by | ln ε | ∆ ¯ E ≤ √ − √ π | ¯Ω + i | / − β π | ¯Ω + i | + o (1) . (3.63)24n view of the arbitrary closeness of β to , the energy change is, therefore,negative for ε (cid:28) λ → We now investigate the limiting behavior of the minimizers of ¯ E as ε →
0, with¯ δ > √ κ fixed, i.e., the situation in which minimizers are non-trivial. Asthe value of ε is decreased, the number of droplets in a minimizer are expectedto grow. What we will show below is that in the limit ε → ε →
0. We have thefollowing result.
Proposition 3.18.
Let ¯Ω + = ∪ Ni =1 ¯Ω + i be a non-trivial minimizer of ¯ E , where ¯Ω + i are the disjoint connected components of ¯Ω + , and let ¯ r i be as in Proposition3.13. Then ¯ r i → √ uniformly as ε → .Proof. First of all, by Proposition 3.14 we already know that ¯ r i ≥ √ − δ for any δ >
0, provided that ε (cid:28)
1. Let us prove that the matching upperbound also holds for ε (cid:28)
1. Indeed, for any β ∈ (0 , ) let B ε − β (¯ x i ) ∈ ¯Ω bea disk of radius ε − β centered at ¯ x i defined in Proposition 3.13, and consider¯Ω β = ¯Ω \ ∪ Ni =1 B ε − β (¯ x i ). Note that by Proposition 3.16 the disks B ε − β (¯ x i ) donot intersect for ε (cid:28)
1. In fact, by Proposition 3.16, for any α ∈ ( β, ) we havedist ( B ε − β (¯ x i ) , B ε − β (¯ x j )) > ε − α for ε (cid:28) v defined in (3.4) is attained in ¯Ω β for ε (cid:28)
1. Let ¯ x be such that v (¯ x ) = min and let ¯ x i be the center of a dropletwhich is closest to ¯ x . Recalling the definition in (3.41) and Proposition 3.13, wecan write v (¯ x ) = v i (¯ x ) − ¯ r i | ln ε | ln( ε / (1 + | ¯ x − ¯ x i | )) + o (1) , (3.64)where we used (2.4). In particular, for any δ > v (¯ x ) > v i (¯ x i ) + ¯ r i (1 − β ) − δ , if | ¯ x − ¯ x i | ≤ ε − β and ε (cid:28)
1, in view of (3.42), where accordingto Proposition 3.16, we can use α defined above, whenever ε (cid:28)
1. On theother hand, choosing γ ∈ ( β, α ) and picking any ¯ x (cid:48) such that | ¯ x (cid:48) − ¯ x i | = ε − γ ,we see that for any δ > v (¯ x (cid:48) ) < v i (¯ x i ) + ¯ r i (1 − γ ) + δ for ε (cid:28) δ sufficiently small this implies that v (¯ x (cid:48) ) < v (¯ x ) for small enough ε , contradicting minimality of v at ¯ x .Now, we demonstrate that v (¯ x ) > ¯ δκ − − √ − δ , for any δ >
0, providedthat ε (cid:28)
1. Indeed, suppose the opposite inequality holds for some δ > ε →
0. Then, inserting a new droplet in the form of a disk of radius¯ r = O (1) centered at ¯ x results in the change ∆ ¯ E of energy | ln ε | ∆ ¯ E = V v (¯ x ) (¯ r ) + o (1) , (3.65)where V is given by (3.18), and we used (2.4) and (3.42). Since by assumption2 v (¯ x ) − ¯ δκ − < √
9, it is easy to verify that V v (¯ x ) attains a minimum at some¯ r = ¯ r > √
3, with V v (¯ x ) (¯ r ) <
0. Therefore, inserting a droplet with radius ¯ r and center at ¯ x would reduce energy for some ε (cid:28)
1, contradicting minimalityof ¯ E .This, in turn, implies that v i (¯ x i ) > ¯ δκ − − √ − δ for all i . Indeed, since¯ x ∈ ¯Ω β , from (3.64) we have v i (¯ x (cid:48) ) > ¯ δκ − − √ − π (1 − β ) + o (1), for any¯ x (cid:48) ∈ ∂B ε − β (¯ x i ), for ε (cid:28)
1. On the other hand, by (3.42) the same inequalityholds for ¯ x (cid:48) = ¯ x i . The estimate then follows in view of arbitrariness of β < .Finally, since ¯ v i (¯ x i ) > ¯ δκ − − √ − δ when ε (cid:28)
1, and by Proposition3.13(ii) the values of ¯ r i are close to the minimizers of V v i (¯ x i ) (¯ r ) for ¯ r > √ − δ ,by direct inspection we have ¯ r i < √ δ as well, for any δ > ε (cid:28) v i (¯ x i ) to a space-independent constant as ε → v i (¯ x i ) → κ (cid:32) ¯ δ − √ κ (cid:33) . (3.66)In fact, from the proof of Proposition 3.18 we can conclude that v stays closeto the constant in (3.66) in ¯Ω β for β arbitrarily close to , provided ε (cid:28) ε →
0. Below we prove this fact, which also gives theleading order behavior of energy in the limit.Let us rewrite the energy ¯ E N for the system of interacting droplets, using(3.18): ¯ E N = 1 | ln ε | N (cid:88) i =1 (cid:0) V v i (¯ x i ) (¯ r i ) − πv i (¯ x i )¯ r i (cid:1) + 4 π | ln ε | N − (cid:88) i =1 N (cid:88) j = i +1 G ( ε / | ln ε | − / (¯ x i − ¯ x j ))¯ r i ¯ r j + o (1) . (3.67)To proceed, let us go back to the original scaling in x and introduce x i = ε / | ln ε | − / ¯ x i . Also, for any 0 < σ (cid:28) G σ ( x ) = 14 π ε σ (cid:88) n ∈ Z (cid:90) R e − | x − y | ε σ K ( κ | y − n | ) dy. (3.68)26ere G σ is a mollified version of G , with Fourier transformˆ G σ ( q ) = (cid:90) Ω e iq · x G σ ( x ) dx = e − ε σ | q | κ + | q | , (3.69)and which can, e.g., be estimated as G σ ( x ) = G ( x ) + o ( ε σ/ ) , | x | > ε σ/ , (3.70)and G σ ( x ) = O ( σ | ln ε | ) , | x | < ε σ . (3.71)Therefore, in view of Lemmas 3.4 and 3.6 we can write¯ E N = 1 | ln ε | N (cid:88) i =1 (cid:0) V v i (¯ x i ) (¯ r i ) − πv i (¯ x i )¯ r i (cid:1) + 2 π | ln ε | N (cid:88) i =1 N (cid:88) j =1 G σ ( x i − x j )¯ r i ¯ r j + O ( σ ) . (3.72)Now, let us introduce the quantity ρ ( x ) = 1 | ln ε | N (cid:88) i =1 δ ( x − x i ) . (3.73)Note that by Lemma 3.6 we have (cid:82) Ω ρ ( x ) dx = O (1). In view of Proposition3.18 and (3.66), we can further rewrite ¯ E N as¯ E N = 1 | ln ε | N (cid:88) i =1 V v i (¯ x i ) (¯ r i ) − π √ κ (cid:32) ¯ δ − √ κ (cid:33) (cid:90) Ω ρ ( x ) dx +6 π √ (cid:90) Ω (cid:90) Ω ρ ( x ) G σ ( x − y ) ρ ( y ) dx dy + O ( σ ) . (3.74)In fact, by Proposition 3.18 and (3.66), the first term in (3.74) goes to zero.Therefore, in terms of the Fourier coefficientsˆ ρ q = (cid:90) Ω e iq · x ρ ( x ) dx, (3.75)we can write ¯ E N = − π √ κ (cid:32) ¯ δ − √ κ (cid:33) ˆ ρ + 6 π √ κ ˆ ρ +6 π √ (cid:88) q ∈ π Z \{ } e − ε σ | q | | ˆ ρ q | κ + | q | + O ( σ ) . (3.76)27inimizing this with respect to ˆ ρ , we obtain¯ E N ≥ − κ (cid:32) ¯ δ − √ κ (cid:33) + 6 π √ (cid:88) q ∈ π Z \{ } e − ε σ | q | | ˆ ρ q | κ + | q | + O ( σ ) . (3.77)Finally, from Lemma A.2 one can see that min ¯ E N ≤ − κ (cid:0) ¯ δ − √ κ (cid:1) + o (1) for ε (cid:28)
1. Hence, in view of arbitrariness of σ the constant in (3.77) isthe limit of ¯ E N and, by Proposition 3.15, also of ¯ E , as ε →
0. In addition, thisimplies that ˆ ρ q → ε → q (cid:54) = 0. Thus, we just proved Proposition 3.19.
Let ¯ δ > √ κ , and let ρ be defined in (3.73), with x i = ε / | ln ε | − / ¯ x i , where ¯ x i are as in Proposition 3.13. Then min ¯ E → − κ (cid:32) ¯ δ − √ κ (cid:33) , (3.78) and ρ → π √ (cid:32) ¯ δ − √ κ (cid:33) (3.79) weakly in the sense of measures, as ε → . We end by noting that the homogenization approach to multi-droplet pat-terns in a related context was first discussed in [57]. Also, let us mention that ina related class of problems existence of limiting density for the ground states ofparticle systems interacting via potentials like our G as the number of particlesgoes to infinity was proved in [58]. The difference with our result, however, isthat in [58] the limit is taken at fixed positive temperature, while in our case thesystem’s temperature (in the usual thermodynamic sense) is strictly zero. Yet,as was pointed out in [58], the “effective” temperature of the system consideredactually goes to zero as the number of particles goes to infinity, making theseresults closely related to ours. Finally, we briefly discuss the appearance of non-trivial minimizers in the vicin-ity of the point ¯ δ m = √ κ (this transition point was identified in [13, 14]).First, we note that by the results just obtained the transition from trivial tonon-trivial minimizers appears to be quite abrupt in the limit ε →
0. In fact,in this limit one goes immediately from no droplets to infinitely many dropletsupon crossing the point ¯ δ = ¯ δ m from below.Observe that the energy ¯ E [ u ] (and, equivalently, E [ u ] − E [ − δ . Therefore, a passage through the neighborhoodof ¯ δ = ¯ δ m at small but finite ε will result in a monotonic increase of the numberof droplets in a minimizer. This number will quickly get large as one moves28way from the transition point. Therefore, in order to analyze droplet creationat the transition, we need to further zoom in on the parameter region around¯ δ = ¯ δ m . Let us introduce the renormalized distance to the transition (with thetransition point shifted appropriately): τ = | ln ε | κ (cid:32) ¯ δ − √ κ − ln | ln ε | √ | ln ε | κ (cid:33) , (3.80)and consider the behavior of energy ¯ E in the limit ε → τ = O (1). As canbe easily seen, all the estimates obtained previously remain valid in this case,and minimizers are close to a collection of N disks separated by large distances,whose energy is given by ¯ E N to O ( ε α ). We can also write the energy ¯ E N in theform ¯ E N = N (cid:88) i =1 ¯ E (¯ r i ) + 4 π | ln ε | − N − (cid:88) i =1 N (cid:88) j = i +1 G ( x i − x j )¯ r i ¯ r j , (3.81)where | ln ε | ¯ E (¯ r ) = 2 π (cid:16) ¯ r − √ r + ¯ r (cid:17) + π | ln ε | − ln | ln ε | (¯ r − √ r − π | ln ε | − (¯ r (ln ¯ κ ¯ r − ) + 2 τ )¯ r (3.82)is the energy of one disk-shaped droplet of radius ¯ r .It is easy to see that in the limit ε → E (¯ r ) ≥ r > E (¯ r ) = 0 if and only of ¯ r = √
3. Therefore, ¯ r i → √ ε → τ fixed. In fact, by convexity of ¯ E near ¯ r = √ r i − √ O ( | ln ε | − ln | ln ε | ) in the limit ε →
0. Therefore, we obtain(the summation is absent in the formula, if N = 1)¯ E N = 12 √ π | ln ε | − (cid:40) N − (cid:88) i =1 N (cid:88) j = i +1 G ( x i − x j ) − N π (cid:32) ln ¯ κ + 13 ln 3 −
14 + 2 τ √ (cid:33)(cid:41) + o ( | ln ε | − ) . (3.83)From this expression it is easy to see that N = O (1) quantity. Thus, in thiscase the problem reduces to minimizing the pair interaction potential given bythe sum in (3.83). We summarize the above discussion by stating the followingresult. Proposition 3.20.
Let ¯ δ be given by (3.80) with τ fixed. Then, there existsa strictly monotonically increasing sequence of numbers ( τ n ) , with τ n → ∞ as n → ∞ , such that, provided that ε (cid:28) :(i) If τ < τ = √ − −
12 ln ¯ κ ) , then there are no non-trivialminimizers of E . ii) If τ < τ < τ , with τ = τ + 2 π √ G , the minimizer of E is a singledroplet.(iii) If τ n < τ < τ n +1 , all minimizers of E consist of precisely n droplets. Thedroplet centers { x i } nearly minimize V = n − (cid:88) i =1 n (cid:88) j = i +1 G ( x i − x j ) . Let us mention that local minimizers of E without screening (i.e. with κ →
0) which are close to disks of the same radius centered at the minimizersof V were constructed perturbatively in a recent work of Ren and Wei [32, 33].We note that when τ = O (1), existence of these solutions easily follows fromour analysis, if one notices that in the considered regime the excess energy ofa minimizing sequence controls the isoperimetric deficit of each droplet andenforces O (1) distance between them. Therefore, solutions with a prescribednumber of droplets may be obtained by minimizing over all u ∈ BV (Ω; {− , } ),such that the support of 1 + u has a fixed number of disjoint components. Inturn, by Proposition 3.20 the global minimizers of E belong to this class. We now turn to the study of the relationship between the sharp interface energy E and the diffuse interface energy E . Since most of our analysis here does notrely on any particular assumptions on the dimensionality of space, we will treatthe general case of Ω being a d -dimensional torus: Ω = [0 , d . We assume that W is a symmetric double-well potential with non-degenerate minima at u = ± W ∈ C ( R ), W ( u ) = W ( − u ), and W ≥ W (+1) = W ( −
1) = 0 and W (cid:48)(cid:48) (+1) = W (cid:48)(cid:48) ( − > W (cid:48)(cid:48) ( | u | ) is monotonically increasing for | u | ≥
1, lim | u |→∞ W (cid:48)(cid:48) ( u ) = + ∞ ,and | W (cid:48) ( u ) | ≤ C (1 + | u | q ), for some C > q >
1, with q < d +2 d − if d > ε , we need to additionally normalize W as follows:(iv) We have (cid:90) − (cid:112) W ( u ) du = 1 . (4.1)Note that these assumptions are satisfied for, e.g., the rescaled version of theclassical Ginzburg-Landau energy: W ( u ) = (1 − u ) for d ≤
3. Also notethat this assumption is not restrictive, since it is always possible to make (4.1)hold by an appropriate rescaling. 30et us begin our analysis with a few general observations. First of all, it isclear from standard arguments (see e.g. [59]) that for any ε > u ∈ H (Ω) of E satisfying (cid:82) Ω u dx = ¯ u . Note that any critical point u of E , including minimizers, is a weak solution of the Euler-Lagrange equation(here and below G solves (1.2) with periodic boundary conditions and has zeromean) ε ∆ u − W (cid:48) ( u ) − v + µ = 0 , v = (cid:90) Ω G ( x − y )( u ( y ) − ¯ u ) dy, (4.2)where µ = (cid:90) Ω W (cid:48) ( u ) dx (4.3)is the Lagrange multiplier. Furthermore, from the Sobolev imbedding theoremwe have u ∈ L p (Ω) for p = dd − , and hence v ∈ W ,p (Ω) ⊂ C ,α (Ω), for some α ∈ (0 , d <
6. Applying Moser iteration [59], we then find that u ∈ L p (Ω),for any p < ∞ . Therefore, by standard elliptic regularity theory [50], we alsohave u ∈ W ,p (Ω), so u ∈ L ∞ (Ω) and is, in fact, a classical solution of (4.2).We now show that u is uniformly bounded and that | u | cannot much exceed1 whenever E [ u ] is sufficiently small, at least for d not too high. Proposition 4.1.
Let d < and let u be a critical point of E . Then, for every δ > we have | u | < δ and | v | < δ in Ω , whenever E [ u ] is sufficiently small.Proof. Observe first that for every δ > E [ u ] small enough we have |{| u | > δ }| < . Now, suppose that the maximum value u m of | u | is greater than1 + δ . Without a loss of generality, we may assume that u m = max u . By thepreceding observation, we have µ ≤ C + W (cid:48) ( u m ), for some C > u m . Therefore, in view of the monotonic increase to infinity of W (cid:48) ( u ) due tohypothesis (iii) on W , we have µ ≤ W (cid:48) ( u m ) for u m sufficiently large. Now,taking into account that v ≥ − C (cid:48) u m for some C (cid:48) > u m , from(4.2) we find that ε ∆ u ≥ W (cid:48) ( u m ) − C (cid:48) u m > u = u m ,in view of assumption (iii) on W , contradicting the maximality of u . Finally,to see that | u | < δ with any δ >
0, when E [ u ] (cid:28)
1, note that u → ± E [ u ] →
0. Hence, in view of the uniform bound on u obtained earlier, wehave µ →
0. Furthermore, since the non-local term in the energy can be writtenas (cid:82) Ω |∇ v | dx , and v is uniformly bounded in W ,p (Ω) for any p < ∞ , wealso have v → W (cid:48) (1 + δ ), we can apply the same argument as above to complete the proof ofthe statement.We note that while the arguments above hold for every critical point E withsmall energy, it is generally possible for a local minimizer of E to strongly deviatefrom ± O (1) [4, 60, 61]. Of course, these critical points will have O (1)energy when ε →
0, as opposed to minimizers of E whose energy vanishes in31his limit. Let us also mention that numerical evidence shows that generallymax | u | >
1, even for minimizers and ε (cid:28) E from below by theminimal energy of E . For u ∈ H (Ω) with (cid:82) u dx = ¯ u , let us separate thedomain Ω into three pairwise-disjoint subdomains:Ω = Ω δ + ∪ Ω δ − ∪ Ω δ . (4.4)where Ω δ + = { x ∈ Ω : u ( x ) ≥ − δ } , (4.5)Ω δ − = { x ∈ Ω : u ( x ) ≤ − δ } , (4.6)Ω δ = { x ∈ Ω : − δ < u ( x ) < − δ } , (4.7)Next, let us introduce the following three auxiliary functionals (for simplicity ofnotation, we will suppress the index δ in the definition of each functional): E [ u ] = (cid:90) Ω δ (cid:18) ε |∇ u | + W ( u ) (cid:19) dx, (4.8) E [ u ] = 12 κ (cid:90) Ω δ + ∪ Ω δ − ( u − u ) dx + 12 (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dxdy, (4.9)where u ( x ) = ± x ∈ Ω δ ± , respectively, with κ = 1 (cid:112) W (cid:48)(cid:48) (1) , (4.10)and E [ u ] = (cid:90) Ω δ + ∪ Ω δ − (cid:18) W ( u ) − κ ( u − u ) (cid:19) dx. (4.11)It is clear that the energy E can be estimated from below as E ≥ E + E + E . (4.12)Hence, we are going to establish a lower bound for E by considering the lowerbounds for each term in the sum above.We start with the part of energy that is associated with the interfaces: Lemma 4.2.
Let δ > be sufficiently small, let u ∈ H (Ω) and suppose that | Ω δ − ∪ Ω δ + | > . Then there exists u ∈ BV (Ω; {− , } ) such that u ( x ) = ± whenever x ∈ Ω δ ± , and E [ u ] ≥ ε − a δ ) (cid:90) Ω |∇ u | dx (4.13) for some a > independent of δ and ε . roof. First of all, if either | Ω δ + | or | Ω δ − | is zero, we can simply choose u to beconstant (e.g. u = − | Ω δ + | = 0). So, let us assume that both | Ω δ ± | > u in H (Ω) by a piecewise linear function ˜ u with ∇ ˜ u (cid:54) = 0almost everywhere in Ω. Then, using the Modica-Mortola trick [62, 63] and theco-area formula [64], we find E [˜ u ] ≥ ε (cid:90) | ˜ u | < − δ (cid:112) W (˜ u ) |∇ ˜ u | dx = ε (cid:90) − δ − δ (cid:112) W ( t ) |{ ˜ u = t }| dt. (4.14)Since the function |{ ˜ u = t }| is continuous for all t ∈ [ − δ, − δ ], there existsa constant c ∈ [ − δ, − δ ] such that the right-hand side of (4.14) equals |{ ˜ u = c }| (cid:82) − δ − δ (cid:112) W ( t ) dt ≥ (1 − a δ ) |{ ˜ u = c }| , for some a > δ small enough.Now, define ˜ u ∈ BV (Ω; {− , } ) as˜ u ( x ) = (cid:40) +1 , ˜ u ( x ) > c, − , ˜ u ( x ) ≤ c. (4.15)The preceding arguments imply the desired inequality for ˜ u . Passing to thelimit in the approximation, we obtain the result, with u = lim ˜ u in L (Ω)upon extraction of a subsequence. Lemma 4.3.
Let u and u be as in Lemma 4.2, let u satisfy (cid:82) Ω u dx = ¯ u , andlet | u | ≤ δ and (cid:12)(cid:12)(cid:82) Ω G ( x − y )( u ( y ) − ¯ u ) dy (cid:12)(cid:12) ≤ δ in Ω , for δ > sufficientlysmall. Then E [ u ] ≥ (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dxdy − a δ E [ u ] , (4.16) for some a ≥ independent of δ and ε , whenever E [ u ] ≤ δ ( d +6) .Proof. Let us write u as follows u = u + u + u , u ( x ) = − κ (cid:90) Ω G ( x − y )( u ( y ) − ¯ u ) dy. (4.17)Note that by assumption, || u || L ∞ (Ω) ≤ Cδ and || u || L ∞ (Ω) ≤ C , for some C >
0. Now, observe that u solves − ∆ u + κ u = − κ ( u + u − ¯ u ) , (4.18)and, therefore, we also have u ( x ) = − κ (cid:90) Ω G ( x − y )( u ( y ) + u ( y ) − ¯ u ) dy. (4.19)33ubstituting u in the form (4.17) into (4.9), we obtain E ( u + u + u ) = 12 κ (cid:90) Ω δ + ∪ Ω δ − ( u + u ) dx + 12 (cid:90) Ω (cid:90) Ω ( u ( x ) + u ( x ) + u ( x ) − ¯ u ) G ( x − y )( u ( y ) + u ( y ) − ¯ u ) dydx = − κ (cid:90) Ω δ u dx − κ (cid:90) Ω δ u u dx − (cid:90) Ω (cid:90) Ω u ( x ) G ( x − y ) u ( y ) dydx + 12 (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dydx + 12 κ (cid:90) Ω δ + ∪ Ω δ − u dx ≥ (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dydx − κ (cid:90) Ω δ u dx + (cid:90) Ω δ (cid:90) Ω u ( x ) G ( x − y )( u ( y ) − ¯ u ) dydx + 12 κ (cid:90) Ω δ + ∪ Ω δ − (cid:32) u ( x ) − κ u ( x ) (cid:90) Ω δ + ∪ Ω δ − G ( x − y ) u ( y ) dy (cid:33) dx. (4.20)In fact, the last line in (4.20) is non-negative. Indeed, writing the integral in thelast line of (4.20) with the help of the Fourier Transform ˆ a q of ˜ u = u χ Ω δ + ∪ Ω δ − ,where χ Ω δ + ∪ Ω δ − is the characteristic function of Ω δ + ∪ Ω δ − :ˆ a q = (cid:90) Ω δ + ∪ Ω δ − e iq · x u ( x ) dx, (4.21)we obtain (cid:90) Ω δ + ∪ Ω δ − (cid:32) u ( x ) − κ u ( x ) (cid:90) Ω δ + ∪ Ω δ − G ( x − y ) u ( y ) dy (cid:33) dx = (cid:88) q ∈ π Z d | q | | ˆ a q | κ + | q | ≥ . (4.22)To estimate the remaining terms in (4.20), we note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ (cid:90) Ω δ u u dx + (cid:90) Ω δ (cid:90) Ω u ( x ) G ( x − y )( u ( y ) − ¯ u ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω δ (cid:90) Ω u ( x ) G ( x − y ) u ( y ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) Ω δ (cid:90) {| u |≥ − δ } | u ( x ) | G ( x − y ) | u ( y ) | dydx + (cid:90) Ω δ (cid:90) {| u | < − δ } | u ( x ) | G ( x − y ) | u ( y ) | dydx. (4.23)34ince G ∈ L p (Ω) for all p < dd − (any p < ∞ in d = 2), by H¨older inequality wecan see that for any ˜Ω ⊆ Ω (cid:90) ˜Ω G ( x − y ) | u ( y ) | dy ≤ C (cid:18)(cid:90) ˜Ω | u | q dx (cid:19) /q ≤ C || u || L ∞ (˜Ω) | ˜Ω | /q , (4.24)for any q > d . Therefore, continuing the estimates in (4.23), we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω δ (cid:90) Ω u ( x ) G ( x − y ) u ( y ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) δ + δ − /q E /q [ u ] (cid:17) | Ω δ | ≤ Cδ | Ω δ | , (4.25)whenever E [ u ] ≤ δ q +2) , where we took into account that by the assumptionsof the lemma | u | ≤ | u − u | + | u | ≤ Cδ in {| u | > − δ } , and that E [ u ] ≥ (cid:82) {| u | < − δ } W ( u ) dx ≥ cδ |{| u | < − δ }| for some c > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω δ u dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω δ u u dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ | Ω δ | . (4.26)The statement of the lemma then follows from the fact that E [ u ] ≥ (cid:82) Ω δ W ( u ) dx ≥ cδ | Ω δ | , for some c >
0, by choosing q = ( d + 2). Lemma 4.4.
Let u and u be as in Lemma 4.2. Then E [ u ] ≥ − a δ E [ u ] . (4.27) for some a ≥ independent of δ and ε , for sufficiently small δ > .Proof. By assumption (iii) on W , we have W ( u ) ≥ κ ( u − u ) whenever | u | >
1. Hence E [ u ] ≥ (cid:90) { − δ ≤| u |≤ } (cid:18) W ( u ) − κ ( u − u ) (cid:19) dx ≥ − Cδ (cid:90) { − δ ≤| u |≤ } ( u − u ) dx ≥ − a δ E [ u ] , (4.28)for some a ≥ | Ω δ + ∪ Ω δ − | > E [ u ] small enough, we obtain Proposition 4.5.
Let δ > be sufficiently small, let u ∈ H (Ω) satisfy (cid:82) Ω u dx =¯ u , let | u | ≤ δ and (cid:12)(cid:12)(cid:82) Ω G ( x − y )( u ( y ) − ¯ u ) dy (cid:12)(cid:12) ≤ δ in Ω , and let E ≤ δ ( d +6) . Then there exists a function u ∈ BV (Ω; {− , } ) such that E [ u ] ≥ (1 − δ / ) E [ u ] , with κ given by (4.10). ε → u ∈ BV (Ω; {− , } ) obeying suitable bounds (satisfied byminimizers of E in d = 2): Proposition 4.6.
Let u ∈ BV (Ω; {− , } ) , with the jump set of class C ,let the principal curvatures of the jump set of u be bounded by ε − α for some α ∈ [0 , , let the distance between different connected portions of the jump setbe bounded by ε α , and let (cid:12)(cid:12)(cid:82) Ω G ( x − y )( u ( y ) − ¯ u ) dy (cid:12)(cid:12) ≤ δ for some δ > smallenough. Then there exists a function u ∈ H (Ω) with (cid:82) Ω u dx = ¯ u , such that E [ u ] ≤ (1 + δ / ) E [ u ] , with κ given by (4.10), whenever E [ u ] ≤ δ ( d +3) and ε (cid:28) .Proof. For simplicity of presentation, we only give the proof in the case d = 2.With minor modifications, the proof remains valid for all d .Here we adapt the standard construction of a trial function for the localpart of the Ginzburg-Landau energy. Let U ( ρ ) be the solution of the ordinarydifferential equation d Udρ − W (cid:48) ( U ) = 0 , U ( −∞ ) = 1 , U (+ ∞ ) = − , U (0) = 0 , (4.29)where the last condition fixes translations. As is well-known (see e.g. [65]),this solution exists, is unique and is a strictly monotonically decreasing oddfunction, approaching the equilibria at ρ = ±∞ exponentially fast. Therefore,for any δ > | U ( ρ ) | ≤ − δ , if and only if | ρ | ≤ l , with some positive l = O ( | ln δ | ). Also note that by hypothesis (iv) on W (cid:90) l − l (cid:40) (cid:12)(cid:12)(cid:12)(cid:12) dUdρ (cid:12)(cid:12)(cid:12)(cid:12) + W ( U ) (cid:41) dρ = (cid:90) − δ − δ (cid:112) W ( s ) ds = 1 + O ( δ ) . (4.30)Now, introduce the signed distance function r ( x ) = ± dist( x, Ω ± ), whereΩ ± = { u = ± } , whenever x ∈ Ω ∓ , and define a regularized version u ε of u : u ε ( x ) = U ( ε − r ( x )) , | r ( x ) | ≤ εl, (1 − δ + ε − δ ( | r ( x ) | − εl )) u ( x ) , εl ≤ | r ( x ) | ≤ ε ( l + 1) ,u ( x ) , | r ( x ) | ≥ ε ( l + 1) . (4.31)Then, it is easy to see that the function u ( x ) = u ε ( x ) − κ (cid:90) Ω G ( x − y )( u ε ( y ) − ¯ u ) dy. (4.32)is in H (Ω), with (cid:82) Ω u dx = ¯ u . Moreover, we have for any q > | u ( x ) − u ε ( x ) | ≤ κ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω G ( x − y )( u ( y ) − ¯ u ) dy (cid:12)(cid:12)(cid:12)(cid:12) + κ (cid:90) Ω l G ( x − y ) | u ε ( y ) − u ( y ) | dy ≤ C ( δ + | Ω l | /q ) ≤ C (cid:48) ( δ + | ln δ | /q E /q [ u ]) ≤ C (cid:48)(cid:48) δ, (4.33)36here we defined Ω l = {| r | ≤ ε ( l + 1) } , estimated (cid:82) Ω l G ( x − y )( u ε ( y ) − u ( y )) dy as in Lemma 4.3, and used the curvature bound and the assumption on E with d = 2 and q = 2. On the other hand, after a few integrations by parts, from(4.32) we also obtain (cid:90) Ω |∇ ( u − u ε ) | dx = − κ (cid:90) Ω ( u − u ε )( u − ¯ u ) dx = κ (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dydx ≤ κ E [ u ] . (4.34)To estimate E [ u ], let us introduce a system of curvilinear coordinates ( ρ, ξ )consisting of the signed distance ρ to the jump set of u and the projection ξ onto the jump set. By our assumptions this is possible whenever | r ( x ) | < ε α .Therefore, for ε (cid:28) E [ u ] = (cid:90) ε ( l +1) − ε ( l +1) (cid:90) ∂ Ω + (cid:32) ε (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂ρ (cid:12)(cid:12)(cid:12)(cid:12) + ε ρK ) − (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂ξ (cid:12)(cid:12)(cid:12)(cid:12) + W ( u ) (cid:33) × (1 + ρK ) d H ( ξ ) dρ + (cid:90) Ω \ Ω l (cid:32) ε |∇ u | + W ( u ) (cid:33) dx + 12 (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dydx, (4.35)where K = K ( ξ ) is the curvature at point ξ on the jump set of u . Substitutingthe ansatz of (4.32) into (4.35), taking into account that |∇ ( u − u ε ) | ≤ C inΩ l for some C > ε (we have u − u ε uniformly bounded in W ,p (Ω), for any p < ∞ ) and that ∇ u = ∇ ( u − u ) = ∇ ( u − u ε ) in Ω \ Ω l , andusing (4.33) and (4.34), we obtain (estimating each line in (4.35) separately) E [ u ] = ε (cid:0) O ( ε − α | ln δ | ) + O ( δ | ln δ | ) (cid:1) (cid:90) Ω |∇ u | dx + 12 κ (cid:90) Ω \ Ω l ( u − u ) dx + O ( ε E [ u ]) + O ( δ E [ u ])+ 12 (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dydx. (4.36)Now, using the identity (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dydx + κ − (cid:90) Ω ( u − u ε ) dx = (cid:90) Ω (cid:90) Ω ( u ε ( x ) − ¯ u ) G ( x − y )( u ε ( y ) − ¯ u ) dydx, (4.37)37e can further write (4.36) as E [ u ] = E [ u ] + O ( δ | ln δ | E [ u ]) + O ( δ E [ u ]) − κ (cid:90) Ω l ( u − u ε ) dx + (cid:90) Ω l (cid:90) Ω ( u ε ( x ) − u ( x )) G ( x − y )( u ε ( y ) − ¯ u ) dydx + 12 (cid:90) Ω l (cid:90) Ω l ( u ε ( x ) − u ( x )) G ( x − y )( u ε ( y ) − u ( y )) dydx, (4.38)for ε (cid:28)
1. Finally, using the same estimates as in (4.33), we obtain(1 + O ( δ )) E [ u ] = (1 + O ( δ | ln δ | )) E [ u ] + O ( δ | Ω l | ) + O ( | Ω l | / )= (1 + O ( δ | ln δ | )) E [ u ] , (4.39)from which the result follows immediately.The last two propositions show asymptotic equivalence of the diffuse inter-face energy E with the sharp interface energy E for sufficiently well-behavedcritical points and ε (cid:28)
1. In particular, the energies of minimizers of both E and E are asymptotically the same in the limit ε →
0. It would also be naturalto think that the minimizers (even local, with low energy) of E are, in somesense, close to minimizers of E when ε (cid:28) Here we complete the proofs of Theorems 2.1–2.3.
Proof of Theorem 2.1
The main point of the proof is the lower bound in(2.1), since the upper bound is easily obtained by constructing a suitable trialfunction (as in Lemma A.1). The basic tool for the lower bound is a kind ofinterpolation inequality obtained in Lemma B.1. Note that the proof for E works in any space dimension.To prove the lower bound, let us denote by u a minimizer of E . Introducingˆ a q = (cid:90) Ω e iq · x ( u ( x ) − ¯ u ) dx, (5.1)where q ∈ π Z d , we can estimate the energy of the minimizer as followsmin E ≥ (cid:90) Ω (cid:90) Ω ( u ( x ) − ¯ u ) G ( x − y )( u ( y ) − ¯ u ) dxdy = 12 (cid:88) q | ˆ a q | κ + | q | ≥ | ˆ a | κ = 12 κ (cid:18)(cid:90) Ω ( u − ¯ u ) dx (cid:19) = 2 κ (cid:18) | Ω + | − u (cid:19) , (5.2)38here we introduced the set Ω + = { u = +1 } .In view of the upper bound in (2.1), it follows from (5.2) that | Ω + | = (1 +¯ u ) + O ( ε / ), implying that | Ω + | is bounded away from 0 or 1 for ε (cid:28)
1. Hence,by isoperimetric inequality there exists p > P = (cid:90) Ω |∇ u | dx ≥ p, (5.3)whenever ε (cid:28)
1. Applying Lemma B.1 to u − ¯ u , we conclude thatmin E ≥ εP + CP , (5.4)for some C > ε , for ε (cid:28)
1. The result then follows from anapplication of Young inequality and Propositions 4.1 and 4.5.
Proof of Theorem 2.2
This theorem combines a number of results provedin Sec. 3 in the original, unscaled variables. Part (i) of the theorem is thestatement of Proposition 3.5. Part (ii) of the theorem is the collection of resultsfrom Lemma A.1 (taking into account that E [ u k ] < E [ −
1] = ε / | ln ε | / κ − ¯ δ for ¯ δ > √ κ ), Corollary 3.2, Lemma 3.6, and Propositions 3.13, 3.15, and3.16 with α = − σ . Part (iii) of the theorem is contained in the statements ofPropositions 3.18 and 3.19. Proof of Theorem 2.3
First of all, we have min
E (cid:28) ε (cid:28) u = − O ( ε / | ln ε | / ), since min E ≤ E (¯ u ) = O ( ε / | ln ε | / ) in that case.Then, from Proposition 4.1 and Lemma 3.8 we conclude that the assumptionsof Propositions 4.5 and 4.6 are satisfied for the minimizers of E . Therefore,the energies E and E are asymptotically the same in the considered limit, andthe conclusion follows from Theorem 2.2 (the case ¯ δ = √ κ is included bymonotone decrease of ¯ E with ¯ δ ). Acknowledgments
The author would like to acknowledge valuable discussions with M. Kiessling,H. Kn¨upfer, V. Moroz, M. Novaga and G. Orlandi. This work was supported,in part, by NSF via grant DMS-0718027.
A Upper bound
Here we construct a trial function that achieves the lower bound for the energyof the non-trivial minimizers of E . Lemma A.1.
Let ¯ u = − ε / | ln ε | / ¯ δ , with ¯ δ > √ κ fixed. Then thereexists u ∈ BV (Ω; {− , } ) , such that E [ u ] = ε / | ln ε | / (cid:40) √ (cid:18) ¯ δ − √ κ (cid:19) + O (cid:18) ln | ln ε || ln ε | (cid:19)(cid:41) , (A.1)39 or ε (cid:28) .Proof. First, consider u ( x ) = − χ B r (0) ( x ), where χ B r (0) is the characteristicfunction of a disk of radius r centered at the origin. If v ( x ) = (cid:82) Ω G ( x − y )( u ( y ) − ¯ u ) dy , then by using (2.3) we explicitly have (see (3.40)) v ( x ) = − uκ + 2 κ (1 − κrK ( κr ) I ( κ | x | )) , (A.2)+ 2 κ (cid:88) n ∈ Z \{ } rI ( κr ) K ( κ | x + n | )) , | x | ≤ r, (A.3)where K n and I n are the modified Bessel functions of the first and second kind.Therefore, expanding the Bessel functions for r (cid:28) | x | ≤ rv ( x ) = − uκ − r κr + 2 γ − ln 4 − − | x | r (cid:88) n ∈ Z \{ } K ( κ | x + n | ) + O ( r | ln r | ) , (A.4)where γ ≈ . E , after integration we get E [ u ] = 2 πεr + (1 + ¯ u ) κ − − π (1 + ¯ u ) κ − r − πr (ln κr + γ − ln 2 − ) + πr (cid:88) n ∈ Z \{ } K ( κ | n | ) + O ( r | ln r | ) . (A.5)Now, consider a new test function u k ( x ) = − k (cid:88) k =1 k (cid:88) k =1 χ B r ( e ( k − )+ e ( k − )) ( x ) , (A.6)consisting of k disks of radius r arranged periodically in Ω (here e and e arethe unit vectors along the coordinate axes). We have E [ u k ] = (1 + ¯ u ) κ − + πk (cid:16) εr − u ) κ − r − r (ln κr + γ − ln 2 − ) + r (cid:88) n ∈ Z \{ } K ( κk − | n | ) (cid:17) + O ( k r | ln r | ) . (A.7)Approximating the sum in (A.7) by an integral: k − (cid:88) n ∈ Z \{ } K ( κk − | n | ) = (cid:90) R K ( κ | x | ) dx + O ( k − ln k )= 2 πκ − + O ( k − ln k ) , (A.8)and expanding for r (cid:28)
1, we can further write E [ u k ] = (1 + ¯ u ) κ − + πk (cid:16) εr − u ) κ − r − r ln r + 2 πκ − r k (cid:17) + O ( k r ln k ) . (A.9)40e now substitute r = ε / | ln ε | − / √ E [ u k ] = ε / | ln ε | / (cid:16) κ − ¯ δ − π √ | ln ε | − κ − (cid:16) ¯ δ − √ κ (cid:17) k +6 π √ κ − | ln ε | − k (cid:17) + O ( ε / | ln ε | − / k ln k ) . (A.10)Finally, setting k = | ln ε | π √ (cid:32) ¯ δ − √ κ (cid:33) + O (1) , (A.11)we obtain (A.1) with u = u k .Let us also quote without proof a similar result concerning the upper boundfor the reduced energy E N . Lemma A.2.
Let ¯ u = − ε / | ln ε | / ¯ δ , with ¯ δ > √ κ fixed. Then min E N ≤ − κ ε / | ln ε | / (cid:32) ¯ δ − √ κ (cid:33) + O (cid:18) ε / ln | ln ε || ln ε | / (cid:19) . (A.12) B Interpolation inequality
Here we present the lemma that connects the non-local part of the energy withthe interfacial energy via a kind of an interpolation inequality between BV (Ω), H − (Ω) and L ∞ (Ω), for functions bounded away from zero. Lemma B.1.
Let u ∈ BV (Ω) , where Ω = [0 , d is a torus, and assume that m ≤ | u | ≤ M in Ω for some M ≥ m > . Let also (cid:82) Ω |∇ u | dx ≥ p > , andlet G solve (1.5) in Ω with periodic boundary conditions. Then there exists aconstant C = C ( d, κ/p, m /M ) > such that (cid:90) Ω (cid:90) Ω u ( x ) G ( x − y ) u ( y ) dx dy ≥ C (cid:18)(cid:90) Ω |∇ u | dx (cid:19) − . (B.1) Proof.
First, extend u periodically to the whole of R d . Then, introducing χ δ ( x ) = δ − d | B | − χ ( δ − x ), where χ is the characteristic function of the unitball B centered at the origin, we have (cid:90) Ω (cid:90) R n u ( x ) χ δ ( x − y ) u ( y ) dy dx = 1 | B | (cid:90) Ω (cid:90) B u ( x ) u ( x + δy ) dy dx ≥ m − M δ | B | (cid:90) Ω (cid:90) B (cid:90) |∇ u ( x + δty ) | dt dy dx ≥ m − M δ (cid:90) Ω |∇ u | dx, (B.2)41here the inequality is obtained by approximating u by C functions and passingto the limit. Therefore, choosing δ = (cid:18) Mm (cid:90) Ω |∇ u | dx (cid:19) − , (B.3)we obtain m ≤ (cid:90) Ω (cid:90) R n u ( x ) χ δ ( x − y ) u ( y ) dx dy = (cid:88) q ˆ χ δ ( q ) | ˆ u q | , (B.4)where we introduced Fourier transform ˆ u q of u :ˆ u q = (cid:90) Ω e iq · x u ( x ) dx, (B.5)with q ∈ π Z d . The Fourier transform ˆ χ δ of χ δ is, in turn, explicitly given byˆ χ δ ( q ) = (cid:18) δ | q | (cid:19) d/ Γ (cid:18) d (cid:19) J d/ ( δ | q | ) , (B.6)where J d/ ( x ) is the Bessel function of the first kind and Γ( x ) is the gamma-function. Now, applying Cauchy-Schwarz inequality, we obtain m ≤ (cid:32)(cid:88) q | ˆ u q | κ + | q | (cid:33) (cid:32)(cid:88) q ˆ χ δ ( q )( κ + | q | ) | ˆ u q | (cid:33) ≤ sup q (cid:8) ˆ χ δ ( q )( κ + | q | ) (cid:9) (cid:88) q | ˆ u q | × (cid:90) Ω (cid:90) Ω u ( x ) G ( x − y ) u ( y ) dx dy. (B.7)Taking into account that (cid:80) q | ˆ u q | = || u || L (Ω) ≤ M and that [45] δ ˆ χ δ ( q )( κ + | q | ) ≤ (cid:40) C ( κ m M − p − + 1) , | q | δ ≤ ,C ( κ m M − p − + | q | δ )( | q | δ ) − d − , | q | δ > , (B.8)for some C , > d , we conclude that Cδ ≤ (cid:90) Ω (cid:90) Ω u ( x ) G ( x − y ) u ( y ) dx dy. (B.9)for some C > d , κ/p , and m /M . The result then followsimmediately from (B.3).Let us also make some remarks regarding a few extensions of these argu-ments. First, the same estimate holds true in the case where G is the Green’sfunction of the Laplacian in Ω and u has zero mean. Note that in this case42he constant C in (B.1) becomes independent on p . The proof easily follows bypassing to the limit κ → Proposition B.2.
Let u be as in Lemma B.1. Then (cid:90) Ω u (1 − ∆) − d +12 u dx ≥ C (cid:18)(cid:90) Ω |∇ u | dx (cid:19) − d − , (B.10) for some C = C ( d, p, m, M ) > . C First and second variation
Here we present the derivation of the first and second variation of ¯ E in d = 2,adapted from [14]. Lemma C.1.
Let ¯Ω + ⊂ ¯Ω be a set with boundary of class C and v be given by(3.4). Then, the functional ¯ E is twice continuously Gˆateaux-differentiable withrespect to C -perturbations of ∂ ¯Ω + . Furthermore, the first and second Gˆateauxderivatives of ¯ E are given by (3.6) and (3.7).Proof. Let a >
0, let ρ ∈ C ( ∂ ¯Ω + ), and let ¯Ω + a be the set obtained from ¯Ω + bytransporting each point of ∂ ¯Ω + by aρ in the direction of the outward normal.Note that for sufficiently small a the set ∂ ¯Ω + a is of class C , in view of regularityof ∂ ¯Ω + . Then, if ¯ E a = ¯ E ( ¯Ω + a ) and ¯ E = ¯ E ( ¯Ω + ), from (3.2) we have explicitly | ln ε | ( ¯ E a − ¯ E ) = (cid:90) ∂ ¯Ω + (cid:16)(cid:112) (1 + aK (¯ x ) ρ (¯ x )) + a |∇ ρ (¯ x ) | − (cid:17) d H (¯ x )+ (cid:90) ∂ ¯Ω + (cid:90) aρ (¯ x )0 (4 v (¯ x + rν (¯ x )) − δκ − )(1 + K (¯ x ) r ) dr d H (¯ x )+2 | ln ε | − (cid:90) ∂ ¯Ω + (cid:90) ∂ ¯Ω + (cid:90) aρ (¯ x )0 (cid:90) aρ (¯ y )0 (1 + K (¯ x ) r )(1 + K (¯ y ) r (cid:48) ) × G (cid:0) ε / | ln ε | − / (¯ x + rν (¯ x ) − ¯ y − r (cid:48) ν (¯ y )) (cid:1) dr (cid:48) dr d H (¯ y ) d H (¯ x ) , (C.1)where K (¯ x ) is curvature, ν (¯ x ) is the outward unit normal at ¯ x ∈ ∂ ¯Ω + , and werewrote the integrals in terms of the curvilinear coordinates consisting of theprojection ¯ x of a point x ∈ ¯Ω to ∂ ¯Ω + and signed distance r = ν (¯ x ) · ( x − ¯ x ),which is possible for sufficiently small a . Now, Taylor-expanding the integrandsin the powers of r and integrating over r and r (cid:48) , after some tedious algebra weobtain that for any α ∈ (0 ,
1) it holds¯ E a = ¯ E + a d ¯ E a da (cid:12)(cid:12)(cid:12)(cid:12) a =0 + a d ¯ E a da (cid:12)(cid:12)(cid:12)(cid:12) a =0 + O ( a α ) , (C.2)43here the derivatives are given by (3.6) and (3.7). In estimating the remainderterm in (C.2) we took into account that v ∈ C ,α ( ¯Ω) and the following estimateof the terms involving the convolution integral: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ ¯Ω + (cid:90) aρ (¯ y )0 (cid:16) G ( ε / | ln ε | − / (¯ x + ν (¯ x ) r − ¯ y − ν (¯ y ) r (cid:48) )) − G ( ε / | ln ε | − / (¯ x − ¯ y )) (cid:17) dr (cid:48) d H (¯ y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:90) ∂ ¯Ω + (cid:90) aρ (¯ y )0 (cid:16) a + (cid:12)(cid:12)(cid:12)(cid:12) ln | ¯ x − ¯ y + ν (¯ x ) r − ν (¯ y ) r (cid:48) || ¯ x − ¯ y | (cid:12)(cid:12)(cid:12)(cid:12)(cid:17) dr (cid:48) d H (¯ y ) ≤ C (cid:32) a + (cid:90) ∂ ¯Ω + ∩| ¯ x − ¯ y |≥ Ma (cid:90) aρ (¯ y )0 | ν (¯ x ) r − ν (¯ y ) r (cid:48) || ¯ x − ¯ y | dr (cid:48) d H (¯ y ) (cid:33) ≤ Ca (cid:90) ∂ ¯Ω + ∩| ¯ x − ¯ y |≥ Ma d H (¯ y ) | ¯ x − ¯ y | ≤ C (cid:48) a | ln a | , (C.3)for a (cid:28)
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