Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement

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1 Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement Etienne Sandier and Sylvia Serfaty November 8, 00 Abstract We prove some improved estimates for the Ginzburg-Landau energy with or without magnetic field in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localisation of the ball construction method combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy displaced from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite order renormalized energy of vortex interaction, up to the best possible precision i.e. with only a o error per vortex, and is complemented by local compactness results on the vortices. This is used crucially in our forthcoming paper [SS5]. It can also serve to provide lower bounds for weighted Ginzburg-Landau energies. keywords: Ginzburg-Landau, vortices, vortex balls construction, renormalized energy, mass displacement. MSC classification: 35B5, 8D55, 35Q99, 35J0. Introduction We are interested in proving lower bounds and compactness results for Ginzburg-Landau type energies of the form Ωε A u + curl A + u G ε u, A = ε where ε is a small parameter, u is a complex-valued function called order parameter, A is R valued and is the vector potential of the magnetic field h := curl A and A = ia. Here the domain of integration Ω ε is a smooth bounded domain in R, which can depend

2 on ε. We are interested in particular in the case where Ω ε gets large as ε 0. Note that one may set A 0 to recover the simpler Ginzburg-Landau energy Ωε u + u E ε u = ε without magnetic field. Our results apply to this energy functional by making this trivial choice of A. The Ginzburg-Landau energy is a famous model for superconductivity. In this model the order-parameter u often has quantized vortices, which are the zeroes of u with nonzero topological degree. Obtaining ansatz-free lower bounds for G ε in terms of the vortices of u has proven to be crucial in studying the asymptotics of minimizers of G ε, in particular via Γ-convergence methods. The first study establishing lower bounds for Ginzburg-Landau was the work of Bethuel- Brezis-Hélein [BBH] for solutions to the Ginzburg-Landau equations without magnetic field with energy E ε bounded by C log ε. Such an energy bound ensures that the total number of vortices remains bounded as ε 0. This was later improved and extended in two different directions by Han-Shafrir [HS] and Almeida-Bethuel [AB] for arbitrary configurations, still with a number of vortices that remains bounded. The main limitation of such estimates is that the error terms blow up as the number of vortices gets large. Then, Jerrard [Je] and Sandier [Sa] introduced the ball construction method, which provides lower bounds in terms of vortices for arbitrary configurations, allowing unbounded numbers of vortices and much larger energies. This is crucial for many applications, since energy minimizers of the functional with applied magnetic field do not always satisfy a C log ε bound on their energy. Subsequent refinements of the ball construction method were given see for example [SS4] Chap. 4 for a recent result. The lower bound provided by the ball construction method also provides a crucial compactness result on the vorticity roughly the sum of Dirac masses at the vortex centers, weighted by their degrees, these are the so-called Jacobian estimates, see Jerrard-Soner [JS] and [SS4] Chap. 6 and references therein. They say roughly that the vorticity is controlled by times the log ε energy. For other subsequent works refining those results in a slightly different direction, see also [SS3, JSp, ST]. In a way our objective here can be seen as obtaining next order terms order as opposed to order log ε in such estimates, both energy estimates and compactness results. For a given u, A, let us define the energy density e ε u, A = A u + curl A + u. ε If u, A is clear from the context and defined on a set E, we will often use the abbreviation e ε E for E e εu, A, and e ε for the density e ε u, A. We then introduce the measure f ε := e ε π log ε B d B δ ab

3 where the a B s are the centers of the vortex balls constructed via Jerrard s and Sandier s ball construction, the d B s are the degrees of the balls and δ is the Dirac mass. Calculating f ε corresponds to subtracting off the cost of all vortices from the total energy: what remains should then correspond to the interaction energy between the vortices, which we can call renormalized energy by analogy with [BBH]. In order to obtain next order estimates of the energy G ε, we show here lower bounds on the energy f ε, as well as coerciveness properties of f ε, which say, roughly, that f ε, or in other words, the renormalized energy, suffices to control the vorticity. This is again to be compared with the previous ball construction and Jacobian estimate, where the vorticity is controlled by e ε / log ε. The motivation for this is our joint paper [SS5] where we establish a next order Γ-convergence result for the Ginzburg-Landau energy with applied magnetic field, and derive a limiting interaction energy between points in the plane, thus making the link to the question of the famous Abrikosov lattice the Abrikosov lattice is a hexagonal lattice of vortices in superconductors observed in experiments and predicted by Abrikosov. More precisely, we show in [SS5] an asymptotic expansion for the minimal energy of the form min G ε = G N ε + N min W + on where N is the optimal number of vortices determined by the intensity of the applied field, G N ε is a constant of order N the leading order estimate and W is a renormalized energy governing the pattern formed by the vortices after blow-up at the scale N. Moreover, we show that the patterns formed by the vortices of minimizers after this blow-up minimize W almost surely, in some sense. We prove in addition that among lattice configurations of fixed volume, W is uniquely minimized by the hexagonal lattice. The natural conjecture is that this lattice is also a minimizer among all point configurations, and if this were proved, it would completely justify the emergence of the Abrikosov hexagonal lattice. To achieve this, with an error only on, we needed lower bounds on the cost of vortices with a precision o per vortex with still a possibly infinite number of vortices, which is finer than was available in the literature. We also needed to control the local number of vortices by the renormalized energy. In fact the energy density we end up having to analyze in [SS5] is exactly f ε, and we need to be able to control the vortices through it. The other problem we need to overcome for [SS5] is that f ε is obviously not positive or even bounded below, and this prevents from applying standard lower semi-continuity ideas, and the abstract scheme for Γ-convergence of -scale energies which we introduce in [SS5]. This reflects the fact that the energy e ε is not exactly where the vortices are, as we will explain below. The remedy which we implement here, is that we can deform f ε into an energy density g ε which is bounded below and enjoys nice coerciveness properties. To accomplish this we show that we can transport the positive mass in f ε into the support of the negative mass in f ε, with mass travelling at most at fixed finite distances say distance, and so that the result of the operation, g ε, is bounded below. This is done by using the following rather elementary transport lemma: 3

4 Lemma 0.. Assume f is a finite Radon measure on a compact set A, that Ω is open and that for any positive Lipschitz function ξ in Lip Ω A, i.e. vanishing on Ω \ A, ξ df C 0 ξ L A. Then there exists a Radon measure g on A such that 0 g f + and such that f g LipΩ A C 0. Thus what is needed is a control on the negative part of f ε, which will be provided by the ball construction lower bounds and additional improvements of it. The norm f ε g ε LipΩ Ω will measure how far mass has been displaced in the process. This control appears in Theorem below and more particularly Corollary.. Since g ε will be close to f ε, it also can be seen as a renormalized energy. Since g ε is bounded below, we can then hope that it enjoys nice coerciveness properties, we can in fact obtain the desired compactness results which allow to control the vorticity locally by g ε. This will be the object of Theorem below. Finally, let us point out that our results can in principle serve to obtain lower bounds for weighted Ginzburg-Landau energies, see Remark.4. Let us now describe a little bit the method that we use, which will allow to control the negative part of f ε as needed. The best vortex ball construction lower bound on e ε available such as that in [SS4] Chap. 4 is of the following type: given u ε, A ε and any small number r, there exists a family of disjoint closed balls B covering all the zeros of u ε, the sum of the radii of the balls being bounded above by r, and such that 0. e ε u ε, A ε πd log r εd C, B B B where D = B B d B with d B = degu ε, B if B Ω and 0 otherwise. We shall reprove here in Proposition. a version of this result using Jerrard s ball construction. This above estimate says that a vortex of degree d costs an energy at least π d log ε, but this is only really true when the vortex is well isolated from other vortices and from the boundary, and if there are not too many of them locally, as the factor r/d in the logarithm above somewhat reflects: an ideal lower bound would be e ε B π d B log r ε C, and compared to this, the lower bound above contains a negative error πd log D which tends to if the total number of vortices becomes large when ε 0. In truth, this ideal lower bound cannot hold in general as can be seen in the case of n vortices of degree all positioned regularly near the boundary of the domain, a case where 0. is optimal. Moreover the energy density e ε is not localized exactly where the vortices are: vortices can be viewed as points, while their energy is spread over annular regions around these 4

5 points. The ball construction lower bounds such as 0. capture well the energy which lies very near the vortices, but some energy is missing from it, in particular when vortices accumulate locally around a point. The missing energy in that case can be recovered by the method of lower bounds on annuli which we introduced in [SS] and re-used in [SS4], Chap. 9. It is based on the following: let Bx 0, r \Bx 0, r 0 be an annulus which contains no zeros of u, roughly speaking we have e ε Bx 0, r \Bx 0, r 0 πd log r r 0 where D = degu, Bx 0, r = degu, Bx 0, r 0. In other words, if a fixed size ball in the domain contains some large degree D of vorticity, then there is an energy of order D lying not in that ball, but in a thick enough annulus around that ball. This energy of order D should suffice to neutralize the error term πd log D found above through the ball construction. However, it lies at a certain finite distance from the center of the vortices. The main technique is then to combine in a systematic way the ball construction lower bounds and the lower bounds on annuli, in order to recover enough energy. Let us finally emphasize a technical difficulty. Since we want a local control on the vortices, the lower bound 0. provided by the ball construction is not quite sufficient because it cannot be localized in general, i.e. we cannot deduce a bound for B e ε for each B B. It is only possible when a matching upper bound on the total in 0. is known, see Proposition. for more details. The idea to remedy this difficulty is to localize the construction, i.e. split the domain into pieces on which one expects to have a bounded vorticity, then apply the ball construction on each piece, and paste together the constructions and lower bounds obtained this way, whose error terms will now be bounded below by a constant. However, this is not completely easy: one needs to localize the construction and still get a global covering of the vortices by balls while preserving the disjointness of the balls. In applications, trying to split the domain into pieces where the vorticity is expected to be bounded leads us to splitting the domain into very small as ε 0 pieces. Equivalently after rescaling one can consider very large domains cut into bounded size pieces. In other words, in order to be able to treat the case where the vortex density becomes large, we need to be able to treat the case of unbounded domains as ε 0. This is precisely what we do in this paper: we consider possibly large domains. This way we may in practice rescale our domains as much as needed until the local density of vortices remains bounded as ε 0. We consider vortex ball constructions obtained over coverings of Ω ε by domains of fixed size, and we work at pasting together these lower bounds while combining them with the method of lower bounds on annuli, as explained above, and finally retrieving finite numbers of vortices estimates of [BBH] type which bound from below the energy f ε or g ε by the exact renormalized energy of [BBH] type up to only o errors. 5

6 Statement of the main results In this paper we will deal with families u ε, A ε ε defined on domains {Ω ε } ε in R which become large as ε 0. The example we have in mind is Ω ε = λ ε Ω where Ω is a fixed bounded smooth domain and λ ε + as ε 0, but we don t need to make any particular hypothesis on {Ω ε } ε, which could even be a fixed bounded domain. Next we introduce some notation. For E R we let Ê = {x Ω ε, distx, E }. We also define, for any real-valued or vector-valued function f in Ω ε, fx = sup{ fy, y Bx, Ω ε }. Note that both f and Ê depend on ε, but the value of ε will be clear from the context. The choice of in the definitions is arbitrary but constrains the choice of other constants below. In all the paper, f + and f will denote the positive and negative parts of a function or measure, both being positive functions or measures, and f is the total variation of f. If f and g are two measures then f g means that g f is a positive measure. Given a family {u ε, A ε } ε, where u ε : Ω ε C and A ε : Ω ε R we define the currents and vorticities to be j ε = iu ε, Aε u ε, µ ε = curl j ε + h ε, where a, b = a b + āb and h ε = curl A ε is the induced magnetic field. We denote by Lip Ω A the set of Lipschitz functions on A which are 0 on Ω \ A, and let f LipΩ A = sup ξ df, the supremum being taken over functions ξ Lip Ω A such that ξ L A. We say a family {f α } α is subordinate to a cover {A α } α if Suppf α A α for every α. Despite the slightly confusing notation, the covering A α will have nothing to do with the magnetic gauge A ε. Also, the densities f α and g α, as well as n α and ν α will implicitly depend on ε, and should be really f ε,α and g ε,α, etc, but for simplicity we do not indicate this dependence. Theorem. Let {Ω ε } ε>0 be a family of bounded open sets in R. Assume that {u ε, A ε } ε, where u ε, A ε is defined over Ω ε, satisfies for some 0 < β < small enough. G ε u ε, A ε ε β. Then the following holds, for ε small enough:. Vortices There exists a measure ν ε, depending only on u ε and not on A ε of the form π i d iδ ai for some points a i Ω ε and some integers d i such that, C denoting a generic constant independent of ε,. µ ε ν ε C 0, 0 Ωε C εg ε u ε, A ε, 6

7 and for any measurable set E ν ε E C e εê log ε.. Covering There exists a cover {A α } α of Ω ε by open sets with diameter and overlap number bounded by a universal constant, and measures {f α } α, {ν α } α subordinate to this cover such that, letting f ε := e ε log ε ν ε, f ε α f α, ν ε = α ν α, ν α ν α for α α. 3. Energy transport Letting n α := ν α /π, for each α the following holds: If dista α, Ω ε c > ε there exists a measure g α C such that either.3 f α g α LipΩ A α Cn α + β log ε and g α A α cn α log ε, or.4 f α g α LipΩ A α Cn α + log n α and g α A α cn α Cn α, where and c, C > 0 are positive universal constants. If dista α, Ω c ε ε there exists g α 0 such that for any function ξ.5 ξ df α g α Cn α ξ L A α + β log ε ξ L A α. 4. Properties of g ε Letting g ε = f ε + α g α f α it holds that.6 C g ε e ε + log ε ν ε, and for any measurable set E Ω ε,.7 g ε E C e εê log ε, g ε + E Ce ε Ê. Moreover, assuming u ε in Ω ε and that E + B0, C Ω ε, for some C > 0 large enough, then for every p <,.8 j ε p C p gε + E + B0, C + E. E The third item admits, or rather implies the following form, from which the covering {A α } α is hidden. 7

8 Corollary.. Under the hypothesis above and using the same notation, for every 0 < η we have if ε > 0 is small enough: First, for every Lipschitz function ξ vanishing on Ω ε.9 ξ dg ε f ε C Ω ε Second, if de, Ω ε > C then Ω ε.0 ν ε E C ξ [d ν ε + β + η dg ε + + log η η ηg ε + Ê + η Ê + e εê Ω ε log ε ] dx + Cβ ξeε. d Ω ε The point in introducting the extra parameter η is that we want to be able to use only a small η-fraction of the remaining energy g ε to control the error f ε g ε between the original energy and the displaced one. This corollary is obtained by simply summing the relations.5 and controlling n α and n α log n α by a small fraction of n α through the elementary relations. x log x ηx + C log η η x ηx + η and then controlling n α by g α A α via.3 or.4. Remark.. If we let η = and choose E to be at distance at least from Ω then.9 and.0 reduce to [. ξ df ε g ε C ξ dgε + + d ν ε ] Ω ε and ν ε E C g ε + Ê + Ê. If one takes ξ = χ R to be a positive cut-off function supported in B0, R and in B0, R then the right-hand side in. scales like a boundary term i.e. like R as R gets large, while the left-hand side scales like an interior term. Remark.. Assume we have proved the above Theorem and Corollary. Then, given {u ε, A ε } ε and {Ω ε } ε satisfying the hypothesis, we may consider for some fixed σ > 0 the rescaled quantities ε = ε/σ, x = x/σ and let ũ ε x = u ε x, Ã ε x = σa ε x, Ωε = Ω ε /σ. Then, letting h = curl A and h = curl Ã, we have e σ ε u, A := σ Au + σ h + 4ε u 8 = e A ũ + h + 4 ε ũ.

9 We may then apply the Theorem to the tilded quantities, yielding a measure g ε. Then if we let g ε x = g ε x, the measure g ε will satisfy the properties stated in Theorem and Corollary., with e ε replaced by e σ ε and with a different C provided we modify the definition of Ê to Ê = {x dist x, Ẽ < } = {x distx, E < σ}, note that we can keep the original definition provided σ. Then we may add to both e ε and g ε the quantity σ h ε and obtain in this manner a new g ε satisfying the listed properties and for the particular choice σ = the lower bound. g ε h ε 4 C. We will then usually assume when applying Theorem that this lower bound holds as well as the other conclusions of the theorem. The next result shows how g ε has the desired coerciveness properties, and behaves like the renormalized energy. Indeed, under the assumption that the family {g ε } ε is bounded on compact sets recall that the domains become increasingly large as ε 0 we have compactness results for the vorticities and currents, and lower bounds on g ε hence f ε via.9 in terms of the renormalized energy W. Before stating that result, we introduce some additional notation. We denote by {U R } R>0 a family of sets in R such that for some constant C > 0 independent of R.3 U R + B0, U R+C and U R+ U R + B0, C. For example {U R } R>0 can be the family {B R } R>0 of balls centered at 0 of radius R. Then we use the notation χ UR for cutoff functions satisfying, for some C independent of R,.4 χ UR C Suppχ UR U R χ UR x = if distx, U R c. Finally, given a vector field j : R R such that curl j = π p Λ δ p + h with Λ, where h is in L loc and Λ a discrete set, we define the renormalized energy of j by where for any χ.5 W j, χ = lim inf η 0 W j = lim sup R W j, χ BR, B R χ j + π log η χp. R \ p Λ Bp,η p Λ Various results on W, in particular on its minimizers, are proved in [SS5]. Note in particular that if we assume div j = 0, then the lim inf in.5 is in fact a limit, because in this case j = H with H = πδ p + h in a neighbourhood of p, and thus H = log p + f with f H in this neighbourhood. 9

10 Theorem. Under the hypothesis of Theorem, and assuming u ε in Ω ε we have the following.. Assume that dist0, Ω ε + as ε 0 and that, for any R > 0,.6 lim sup g ε U R dx < +, ε 0 where {U R } R satisfies.3. Then, up to extraction of a subsequence, the vorticities {µ ε } ε converge in W,p loc R to a measure ν of the form π p Λ δ p, where Λ is a discrete subset of R, the currents {j ε } ε converge weakly in L p loc R, R for any p < to j, and the induced fields {h ε } ε converge weakly in L loc R to h which are such that curl j = ν h in R.. If we replace the assumption.6 by the stronger assumption.7 lim sup g ε U R < CR, ε 0 where C is independent of R, then the limit j of the currents satisfies, for any p <,.8 lim sup j p dx < +. R + U R Moreover for every family χ UR satisfying.4 we have χ UR W j,.9 lim inf ε 0 R U R dg χur ε + U R h + γ U R π U R h + o R, where γ is a constant defined below and o R is function tending to 0 as R +. Remark.3. The constant γ in.9 was introduced in [BBH] and may be defined by γ = lim u 0 R BR + u 0 π log R, where u 0 r, θ = fre iθ is the unique up to translation and rotation radially symmetric degree-one vortex see [BBH, Mi]. Remark.4. Lower bounds immediately follow from this theorem. Indeed f ε is the energy density minus the energetic cost of a vortex, and f ε g ε is controlled by Theorem, see also Remark.. This, combined with the lower bound.9 shows that in good cases the averages over large balls of f ε are bounded below by W plus explicit constants, which proves a sharp lower bound for the energy with a o order error, à la Bethuel-Brezis-Hélein [BBH]. 0

11 The bound.9 may also be interpreted as a lower bound for the Ginzburg-Landau energy with weight. Assuming a fixed domain Ω and G ε u ε, A ε < C log ε for instance, and that µ ε π n i= δ a i, where a i Ω, then by blowing up by a factor independent of ε we may assume the points are at distance, say, from the boundary and then if ξ is a fixed positive weight we may multiply it by a cutoff 0 χ equal to zero on Ω and equal to at each a i. Then.9 becomes Ω ξe ε π log ε n i= ξa i + χξ dg ε C χξ [d ν ε + β + η dg ε + + log η η ] dx. Typically, there will be an upper bound for the energy which implies that g ε + Ω < C and since also g ε C, the integrals on the right-hand side may be bounded below by a constant independent of ε. The paper is organized as follows: In Section we state without proof the result on lower bounds via Jerrard s ball construction the proof is postponed to Section 5 which we adapt for our purposes, and explain how we use it on a covering of Ω ε by a collection U α of balls of finite size. In Section 3, we present the tool used to transport the negative part of f ε to absorb it into the positive part, and deduce Theorem. In Section 4, we prove Theorem. Finally in Section 5, we prove the ball-construction lower bound. Ackowledgements : Etienne Sandier was supported by the Institut Universitaire de France, Sylvia Serfaty by an NSF CAREER award and a EURYI award. Use of the ball construction and coverings of the domain The first step consists in performing a ball construction in Ω ε in order to obtain lower bounds. This follows essentially the method of Jerrard [Je], the difficulty being that we are not allowed more than an error of order one per vortex. This is hopeless if the total number of vortices diverges when ε 0, hence we need to localize the construction in pieces of Ω ε small enough for the number of vortices in each piece to remain bounded as ε 0.. The ball construction lower bound We start by stating the result of Jerrard s ball construction in a version adapted to our situation, in particular including the magnetic field. The proof is postponed to Section 5. In all what follows, if B is a collection of balls, rb denotes the sum of the radii of the balls in the collection. In all the sequel we will sometimes abuse notation by writing B for B B B, i.e. identify the collection of balls and the set it covers.

12 Proposition.. There exists ε 0, C > 0 such that if U R, ε 0, ε 0, and u ε, A ε defined on U is such that G ε u ε, A ε ε β, where β 0,, the following holds. For every r Cε β,, there exists a collection of disjoint closed balls B depending only on u ε and not on A ε such that, letting U ε = {x dx, U c > ε},. { x U ε u ε x < } B.. rb r. 3. For any C r/ε it holds that either e ε B U C log r ε or B B such that B U ε, e ε B π d B log r εc C, where d B = degu ε, B. A natural choice of C above is πd, where D = B B d B and we have let d B = 0 if B U ε. With this choice we find in all cases e ε B U πd log r εd C i.e. we recover the same lower bound as in [SS4], Theorem 4., mentioned in the introduction as 0.. The reason why we don t simply use that theorem directly is that we need to keep the dichotomy above, and thus a lower bound localized in each ball.. Localizing the ball construction For any ε > 0 we construct an open cover {U α } α of Ω ε as follows: We consider the collection B of balls of radius l 0 where l 0 0, is to be chosen below, small enough 8 but independent of ε centered at the points of l 0 Z. The cover consists of the open sets Ω ε B, for B B. This cover depends on ε, but the maximal number of neighbours of a given α defined as the indices β such that U α U β is bounded independently of ε by an integer we denote by m in fact m = 9. Note that m also bounds the overlap number of the cover, i.e. the maximal number of U α s to which a given x can belong. There is also l > 0 independent of ε which is a Lebesgue number of the cover, i.e. such that for every x Ω ε, there exists α such that Bx, l Ω ε U α or, equivalently, distx, Ω ε Uα c l. Assuming β < /4, and applying Proposition. to u ε, A ε in U α for every α we obtain, since ε > Cε β if ε is small enough, a collection Bε α,r for every ε r /. If ρ is chosen small enough depending on l and m only, thus less than a universal constant, we may extract from α Bε α,ρ a subcollection B ε such that any two balls B, B in B ε satisfy Ω ε B B =. We will say B ε is disjoint in Ω ε :

13 Proposition.. Assume ρ l/8m. Then, writing in short Bε α instead of Bε α,ρ, there exists a subcollection of α Bε α call it B ε which is disjoint in Ω ε and such that. { u ε /} {x distx, Ω ε c > ε} B Bε B. Moreover, for every B B ε B α ε we have B Ω ε = B U α and distb, Ω ε c > ε distb, U α c > ε. Proof. Assume C = Ω ε B B k is a connected component of Ω ε α B α ε. Reordering if necessary, we may assume that B i B B i for every i k. There exists x Ω ε B and α such that distx, Ω ε U c α l. Then distb, Ω ε U c α > 3l/4. Assume distb B i, Ω ε U c α 3l 4. Then distb i, Ω ε U c α > l/ hence for every j i the ball B j belongs to B β ε, where β is a neighbour of α. It follows that r + + r i mρ l/8, where r i is the radius of B i, and we deduce that B B i Bx, l/4 and then distb B i, Ω ε U c α 3l 4. We have thus proved by induction that C U α and even that dist C, Ω ε U c α 3l/4 for every i. We delete from {B,..., B k } the balls which do not belong to B α ε and call C the union of the remaining balls. If y belongs to C { u ε /} {x distx, Ω ε c > ε} then, since dist C, Ω ε Uα c 3l/4 and disty, Ω c ε > ε, provided ε < 3l/4 we have that disty, U c α > ε hence y belongs to some ball B Bε α since Bε α covers the set { u ε } {distx, U c α > ε}, thus y C. The balls in C are disjoint in Ω ε since they belong to the collection Bε α which is itself disjoint in Ω ε. Performing this operation on each connected component of Ω ε α Bε α we thus obtain a collection B ε which covers { u ε /} {x distx, Ω c ε > ε} and is disjoint in Ω ε. Moreover, if B B ε Bε α then dist B, Ω ε Uα c 3l/4 hence B Ω ε = B U α and distb, Ω ε c > ε distb, U α c > ε. The value ρ will be fixed to some value smaller than l/8m and independent of ε, to be specified below. The above proposition provides us for any ε > 0 small enough with collections of balls B ε and B α ε. We will also need the following 3

14 Definition. For any ε r ρ, and any B Bε α, we let Bε B,r in Bε α,r which are included in B. Then we let B r ε = B Bε B B,r ε. be the collection of balls It is disjoint in Ω ε and covers the set { u ε /} {x distx, Ω c ε > ε} and of course if B Bε r Bε α,r, then B Ω ε = B U α and distb, Ω ε c > ε distb, U α c > ε. In other words, the disjoint collection B ε permits us to construct disjoint collections of smaller radius by discarding from Bε α,r those balls which are inside a ball discarded from Bε α,ρ. The collection B ε ε should be seen as the collection of small balls and B ε obtained from Bε α,ρ as the collection of large balls. We will sometimes also use the collection of the intermediate size balls Bε r with ε r ρ. Finally we let. ν ε = πd B δ ab, ν ε = π d B δ ab, ε B Bε distb,ω c ε >ε ε B Bε distb,ω c ε >ε where a B is the center of B, and d B denotes the winding number of u ε / u ε restricted to B. This is the ν ε given by the conclusion of the theorem. Note that since the balls only depend on u ε and not on A ε, ν ε satisfies the same. If B is any ball which does not cross the boundary of balls in B ε ε and distb, Ω c ε > ε then ν ε B = πd B. From the Jacobian estimate see [JS] or the version in [SS4], Theorem 6. we have that. is satisfied. We also have recall that f denotes the total variation of a measure Lemma.. There exists ε 0 > 0 such that if β < /4 in. and ε < ε 0 then ν ε E 6 e εω ε Ê log ε for any measurable set E, so that choosing E = Ω ε and taking logarithms,.3 log ν ε β log ε + C. Proof. We use the properties of B α, ε ε. Letting C = ε/ε = ε 4, it is impossible when ε is small enough that e ε Ω ε B α, ε ε C log ε/ε since we assumed that e ε Ω ε ε β. Thus Proposition. implies that, for every B B α, ε ε such that distb, U c α > ε, e ε B π d B log ε 4 C π 8 d B log ε, if ε is small enough. If, moreover, B B ε ε, then from Definition we have distb, U α c > ε distb, Ω ε c > ε 4

15 Hence for any set E, using. and the fact that balls in B ε ε / if ε is small enough, ν ε E ε B Bε distb,ω c ε >ε B E ν ε B 6 e εω ε Ê. log ε have radius smaller than Definition. For any α we let ν α denote the restriction of ν ε to the balls in B ε Bε α n α = να, so that π ν ε = α ν α, n α = B B ε B α ε ν ε B π, ν ε = π α n α. and We also define 3e α max Mn α, if n α 0,.4 C α = log ε otherwise, where M is a large universal constant to be chosen later and e α = B B α ε e ε B U α. Note that n α is the sum of the absolute values of the degrees of the small balls included in the large balls of B α ε. We have the following Proposition.3. There exists ε 0, C 0 > 0 such that if β < /4 in. and ε < ε 0, ε < r < ρ then C α r/ε and for any B Bε r Bε α,r such that distb, Ω c ε > ε we have.5 e ε B π d B Λ α,r ε, where Λ α,r ε = log r C 0. εc α Moreover, 0 Λ α,r ε log ε and.6 0 log ε Λα,r ε β log ε + log r + C 0. Proof. From the definition.4, from. and Lemma. we have for ε small enough that C α ε β. It follows that if ε < r < then C α r/ε, since β < /4. Also, from the definition of C α it is impossible that e ε Bε α,r U α C α logr/ε since for ε r ρ we have Cα 3e ε Bε α,r / log ε. 5

16 Then from Proposition., letting C = C α, we deduce.5 for any B Bε α,r such that distb, U c α > ε, which is equivalent to distb, Ω c ε > ε if B Bε r Bε α,r. Finally, r/εc α ε 4 using C α r/ε and r ε, which easily implies that Λ α,r ε > 0 if ε is small enough, and Λ α,r ε log ε is clear from the definition. Inequality.6 follows from log ε Λα,r ε = log C α r + C 0 since C α ε β. 3 Mass Transport We proceed to the displacement of the negative part of f ε = e ε log ε ν ε. 3. Mass transport abstract lemmas For the displacements we will use the following two lemmas. The first, more sophisticated one, was already stated in the introduction and uses optimal transportation for the - Wasserstein distance or minimal connection cost. Lemma 3.. Assume f is a finite Radon measure on a compact set A, that Ω is open and that for any positive Lipschitz function ξ in Lip Ω A, i.e. vanishing on Ω \ A, ξ df C 0 ξ L A. Then there exists a Radon measure g on A such that 0 g f + and such that f g LipΩ A C 0. Proof. The proof uses convex analysis. Let X = CA denotes the space of continuous functions and for ξ X let { + if ξ L ϕξ = ξ + df + and ψξ = A > or ξ / Lip Ω A. ξdf otherwise Then ψ is lower semicontinuous because {ξ Lip Ω A ξ L } is closed under uniform convergence, and ϕ is continuous. Moreover both functions are convex, and finite for ξ = 0. Then the theorem of Fenchel-Rockafellar see for instance [ET] yields inf X ϕ + ψ = max µ X ϕ µ ψ µ, 6

17 where X is the dual of X, i.e. the Radon measures on A and We deduce that ϕ µ = sup ξ X ψ µ = inf ξ Lip Ω ξ L ξ dµ sup ξ Lip Ω ξ ξ + df + ξ + df + = ξ dµ + ξ df = { 0 if 0 µ f +, + otherwise ξ df = µ + f LipΩ. max µ + f LipΩ 0 µ f + and then the existence of a Radon measure g such that g maximizes the right-hand side, ie such that 0 g f + and f g LipΩ = inf ξ Lip Ω ξ L ξ + df + ξ df. But inf ξ Lip Ω ξ L ξ + df + ξ df = sup ξ Lip Ω ξ L = sup ξ Lip Ω ξ L = sup ξ Lip Ω ξ L ξ df ξ + df + ξ + df f + ξ df ξ df = inf ξ Lip Ω ξ L ξ df. The assumption of the lemma implies that this last right-hand side is C 0 therefore f g LipΩ A C 0. The second, less sophisticated, displacement result is Lemma 3.. Assume f is a finite Radon measure supported in Ω and such that fω 0. Then there exists 0 g f + such that for any Lipschitz function ξ ξ df g diamω ξ L Ωf Ω. Ω 7

18 Proof. This follows from the previous Lemma but can be proved directly by letting assuming f 0, otherwise g = 0 is the answer, g = f + f Ω. f + Ω Then g is positive because fω 0 implies f Ω f + Ω and f Ω ξ df g = ξ d f + f + Ω f = ξ ξ d f + f Ω f + Ω f where ξ is the average of ξ over Ω, and the right-hand side is clearly bounded above by diamω ξ f Ω. 3. Mass displacement in the balls Definition 3. For B B ε B α ε. We let where Λ α,r ε f B ε = e ε Λ α ε ν ε B Ωε. is defined in.5 and we have set Λ α ε = Λ α,ρ ε. This corresponds to the excess energy in the balls i.e. the energy remaining after subtracting off the expected value from the ball construction. There is a difference of order ν ε B log C α between f ε B and fε B B which will be dealt with later. Proposition 3.. There exists ε 0, C > 0 such that for any ε < ε 0, and any B B ε Bε α, there exists a positive measure gε B defined in B Ω ε and such that 3. gε B e ε + Λ α ε ν ε and ξ dfε B gε B C ξ L B Ω ε ν ε B, B Ω ε for any Lipschitz function ξ vanishing on Ω ε \ B. Proof. To prove the existence of gε B, in view of Lemma 3. and since fε B + = e ε + Λ α ε ν ε on B it suffices to prove that for any positive function ξ defined on B and vanishing on B \ Ω ε we have 3. ξ dfε B C ξ L B ν ε B. We turn to the proof of 3.. Let B B ε Bε α and ξ be as above. Then ξ dfε B = E t B dt, where we have set E t = {x B ξx t} and f B ε A = A f B ε. 0 8 f B ε,

19 We will divide the integral 3.3 into t ε t ε, with t ε = ε ξ L. The first integral is straightforward to bound from below. Indeed fε B B C log ε ν ε B hence 3.4 tε fε B 0 E t dt Cε log ε ξ L ν ε B C ξ L ν ε B. On the other hand, if t > t ε, and this motivated our choice of t ε, then since ξ = 0 in B \ Ω ε we have diste t, Ω ε c > ε. Let then t > t ε, and a E t be a point in the support of ν ε. Then for any r [ ε, ρ], there exists a ball B a,r B r ε containing a. Since {B r ε} is monotonic with respect to r, B a,r B. We call ra, t = sup{r [ ε, ρ, B a,r E t } if this set is nonempty, and 0 otherwise. We then let B t a = B a,ra,t. If 0 < ra, t < ρ then ra, t bounds from above the distance of a to the complement of E t. In particular 3.5 ξa t ra, t ξ L. Indeed for any ra, t < s < ρ we have B a,s B and B a,s E t c hence there exists b B a,s E t. Then ξa ξb s ξ L and since E t {ξ = t} we deduce ξa t s ξ L, proving 3.5 by making s tend to ra, t from above. A second fact is that if ra, t = 0, then B a, ε intersects B \ E t and as above we deduce 3.6 ξa t ε ξ L B. The third fact is that the collection {Ba} t a, where a ranges over E t and the a s for which ra, t = 0 have been excluded, is disjoint. Indeed take a, b E t and assume that ra, t rb, t. Then, since B ra,t is disjoint, the balls B a,ra,t and B b,ra,t are either equal or disjoint. If they are disjoint we note that ra, t rb, t implies that B b,rb,t B b,ra,t and therefore Bb t = B b,rb,t and Ba t = B a,ra,t are disjoint. If they are equal, then B b,ra,t E t and therefore rb, t ra, t, which implies rb, t = ra, t and then Bb t = Bt a. Now, for any B {Ba} t a we have B E t and diste t, Ω c ε > ε hence distb, Ω c ε > ε and from Proposition.3, we have, since Λ α,r ε e ε B ν ε B = Λ α,ρ ε log ρ, r Λ α ε log ρ r where r is the common value of ra, t for a s in B which are in the support of ν ε. We may rewrite the above as e ε B ν ε a Λ α ε log ρ ra, t, a B Supp ν ε + 9, +

20 and summing over B {Ba} t a we deduce e ε E t B a P t Λ α ε log ρ ra, t + ν ε a, where P t is the set of points in E t Supp ν ε such that ra, t > 0. We will let Q t be the set of points in E t Supp ν ε such that ra, t = 0. Since ν ε E t = ν ε P t + ν ε Q t, subtracting from the above Λ α ε ν ε E t we find fε B E t ν ε aλ α ε ρ ν ε a log ra, t. a Q t a P t From 3.6, a given a Supp ν ε B can belong to Q t only if t ξa ε ξ L. Therefore integrating the above with respect to t yields, using the fact that t ξa if a E t, that t ε hence f B ε E t dt a Supp ν ε B t ε f B ε E t dt Λ α ε ε ξ L ν ε B ξa+ ε ξ L ν ε a Λ α ε dt + ξa ε ξ L a Supp ν ε B ν ε a ξa 0 ξa 0 ρ log dt ra, t + ρ log dt. ra, t + We now note that since Λ α ε log ε ελ α ε is bounded independently of ε and, using the inequality 3.5, we get ξa 0 ρ log dt ra, t + ξa 0 log ρ ξ ξa L dt = log ρ ξ L ξa t + ξa ρ ξ L ξa t dt, and the rightmost integral is equal, by change of variables u = ξa t, to ρ ξ ρ ξ L L. Therefore + fε B E t dt C ν ε B ξ L. t ε In view of 3.3, adding 3.4 yields the result. Remark 3.. Note that in the proof of 3., the final radius ρ may be replaced by any r ε, ρ. This yields the following result: Assume that r ε, ρ and that B Bε r is included in some ball in B ε Bε α. Then, for any positive function ξ vanishing on B \ Ω ε, 3.7 e ε Λ α,r ε ν ε ξ C ξ L B ν ε B. B 0

21 We record the following lower bounds: Proposition 3.. For ε small enough and B B ε Bε α : 3.8 e ε Ω ε B log ε C ν ε B. 8 For ε small enough and B B ε Bε α such that distb, Ω c ε > ε, we have 3.9 gε B Ω ε B log ε C ν ε B 8 log ε ν εb. If in addition d B < 0, then 3.0 g B ε Ω ε B log ε Λα ε ν ε B log ε C ν ε B. 8 The meaning of this lower bound is that e ε B is not only bounded below by Λ α ε ν ε B, which to leading order is log ε ν εb this is the positivity of gε B in the above proposition but also by some constant times log ε ν ε B, even though the constant is no longer guaranteed to be the optimal value /. This information is valuable in the case where ν ε B is much smaller than ν ε B. The precise value of the constants is unimportant. Proof. As we noticed, C α < ε/ε implies ε/εc α ε 4 thus, using Proposition.3, e ε B Ω ε B ε Bε B B distb,ω c ε >ε e ε B B ε Bε B B distb,ω c ε >ε π d B log ε 4 C = ν ε B 8 log ε C, which proves the first assertion. Secondly, note that from 3., if distb, Ω c ε > ε, choosing ξ compactly supported in Ω ε such that ξ = in B, we have f B ε B Ω ε = g B ε B Ω ε. Since Λ α ε log ε we deduce 3.9 in view of gε B B Ω ε = fε B B Ω ε ν ε B log ε C 8 log ε ν εb. For the last assertion, since ν ε B = πd B < 0, we write gε B B Ω ε log ε Λα ε ν ε B = e ε B Ω ε log ε ν εb e ε B Ω ε, and this is bounded below using 3.8.

22 3.3 Mass displacement of the remainder Proposition 3. will allow to replace fε B by the positive gε B, and we have 3. f ε fε B = e ε Bε c + log ε Λα ε ν α. B B ε α We now proceed to absorb the negative part of f ε fε B, which is log ε Λα ε να +. This will be easy if C α = 3eα and if not, in view of.5, we have log ε 0 log ε Λα ε log n α + C, which allows to bound the mass of the negative part by C α n αlog n α +. Following the method in [SS] see also [SS4], Chap. 9, this will be balanced by a lower bound by c[n α ] for the energy on annuli surrounding U α. Recall that U α = Bx α, l 0 Ω ε. We let A α = Bx α, r, where r = 3l 0. Choosing l 0 small enough, we may require that diama α < and A α Ω εc = A α { x distx, Ω ε < }. We will denote below by m a bound, uniform in ε for the overlap number of the {A α } α. Now we choose ρ such that for any ε > 0 T α ε l 0, where T α ε is the set of t r 0, r such that { x x α = t} B ε =, where r 0 = l 0. Indeed, the number of U β s which intersect Bx α, r is bounded by a certain number N independent of ε and α. Choosing ρ = l 0 /N, the sum of the radii of balls in β B β ε which intersect Bx α, r is bounded above by l 0, hence T α ε r r 0 l 0 = l Lower bounds on annuli For any α let 3. g α ε + = 4m e ε Bε c + B B ε g B ε Aα, g α ε = log ε Λα ε ν ε + Bε B ε α, and g ε α = g ε α + g ε α. We have g ε g ε α 3 e ε Bε c + gε B + 4 α B B ε α log ε Λα ε ν ε Bε B ε α. In particular g ε α + A α g 3m ε β g β ε A α.

23 Proposition 3.3. There exist ε 0, C, c > 0 such that if β < /4 in., then for any ε < ε 0 and any index α 3.3 g α ε A α πn α β log ε + C. If moreover dista α, Ω ε c > ε then at least one of the following is true: 3.4 g α ε A α πn α β log ε + C, g α ε + A α cn α log ε or 3.5 g α ε A α πn α log n α + C, g α ε + A α cn α. Proof. The bound 3.3 follows from 3.,.6. Now assume dista α, Ω c ε > ε. First, if n α = 0 then g ε α = 0, g ε α + 0 hence 3.4 is true. Second, if 3e α / log ε Mn α then, since for B A α, we have gε B B = fε B B = e ε B Λ α ε ν ε B and Λ α ε log ε it follows that g ε α + A α e 4m ε A α 4m Λα ε d B 4m B B ε A α M m n α log ε πn α log ε U α e ε 4m Λα ε M m π d B B B ε A α n α log ε. Together with 3.3, this implies 3.4 if M was chosen strictly greater than m π. The last case is that where C α = Mn α. Then log ε Λα ε = log n α + C and therefore, using.3, 3.6 g ε α A α πn α log n α + C n α πβ log ε + C. We define D + 0 = d B, D = d B, B B ε B Bx α,r 0 d B >0 B B ε B Bx α,r d B <0 and again we distinguish several cases. First from 3.6 we will have proven 3.4 if we prove that 3.7 g α ε + A α cn α log ε, for some c > 0. This inequality holds in the following two cases. First case : D > n α /0. This means there is a significant proportion of balls with negative degrees. For each such negative ball we have from 3.0, and since ν ε B ν ε B, gε B B gε B B log ε Λα ε ν ε B log ε C π d B. 8 3

24 This implies that hence 3.7 is satisfied when D > n α /0. g ε α + A α log ε C πd 4m, 8 Second case : D 0 + n α /0 and D n α /0. Then for each B B ε Bε α, Proposition 3. yields { gε B 8 B log ε C ν ε B log ε ν εb if d B > 0 log ε C ν 8 ε B if d B < 0. Summing with respect to B we find, since B B ε Bε α g ε α + A α log ε C 4m 8 implies B Bx α, r 0, that n α 4m D+ 0 log ε, which again yields 3.7 when D 0 + n α /0. We are left with the third case, when D 0 + nα and 0 D nα. In this case 3.7 and 0 then 3.4 do not necessarily hold. We need to prove 3.5 instead, which in view of 3.6 reduces to proving g ε α + A α cn α. For this we really need to use the lower bounds on annuli of the type first introduced in [SS]. We denote C α ε = Bx α, r \ Bx α, r 0 B ε. For any t Tε α we let B t = Bx α, t and γ t = B t and recall that γ t does not intersect B ε. If t Tε α then u ε / on γ t because of. and the fact that dista α, Ω c ε > ε. It follows see for instance [SS4] Lemma 4.4, or 5.4 below that for some constant c > 0 we have 3.8 Au + u + curl A 4ε Bt c dt ε, t γ t where d t ε is the degree of u ε / u ε on γ t. Integrating 3.8 with respect to t Tε α, which has measure less than, the left-hand side will be bounded above by e ε A α. In view of the lower bound d t ε D 0 + D, which is valid for any t T α ε, since Tε α l 0, and from the assumption on D 0 + and D we deduce that e ε A α \ B ε c D + 0 D cnα. Then, since g α ε + = 4m e ε on B ε c we deduce g α ε + A α cn α and 3.5 is proved. 4

25 3.4 Proof of Theorem and Corollary. Item. The estimate. was already mentioned after the definition. of ν ε, and the bound ν ε E Ce ε Ê/ log ε was proved in Lemma.. Item. We define f α = B B ε B α ε f B ε g B ε + g α ε + g α ε. Then clearly f α is supported in A α. Moreover, using the fact see 3. that f ε fε B = e ε Bε c log ε Λα ε ν α B B ε α and since α A α m we easily obtain 3.9 f ε f α = log ε Λα ε ν α + e ε Bε c + gε B 4m Aα. α α B B ε α Since α A α m we find 3.0 f ε α f α α log ε Λα ε ν α + 3 e ε Bε c + gε B 0. 4 B B ε Item 3. We define g α. In the case dista α, Ω c ε ε we let g α = g ε α +. Then ξ df α g α = ξ dfε B gε B ξ d g ε α. B B ε B α ε This implies.5, summing 3. over B B ε Bε α and using 3.3. In the case dista α, Ω c ε > ε we let g α c α = ε A α A α We deduce easily from 3.4, 3.5 and if β is small enough that c α C and applying Lemma 3. in A α to g α ε +c α we obtain ϕ α defined on A α and such that 0 ϕ α g α ε + +c α and, for any Lipschitz function ξ, A α ξ d g α ε g α C ξ L A α g α ε A α, where g α := ϕ α c α. Moreover C c α g α g α ε +. 5.

26 3. Then ξ df α g α = ξ df α g ε α + ξ d g ε α g α A α A α A α = ξ dfε B gε B + ξ d g ε α g α A α B B ε B α ε C ξ L A α nα + g α ε A α, where we have used 3. to bound the integral involving fε B gε B. Moreover, g α A α = g ε α A α. If 3.4 holds, then.3 follows immediately from 3. when πβ < c/, with c the constant in 3.4. If 3.5 holds we deduce.4 from 3. by noting that cn α Cn α log n α + cn α C n α if C is chosen large enough depending on c, C. Item 4,.8. We adapt an argument in [St]. First, g ε α g α = f ε α f α thus from 3.0 and since α g α C we find 3. g ε 3 4 e ε Bε c + B B ε g B ε C. Then, assuming U α Ω ε, denote by Bε r,α the set of balls in Bε r which are included in some ball belonging to Bε α B ε, so that ν α B ε = ν ε Bε α B ε = ν ε Bε r,α. Applying Remark 3. for some r ε, ρ with ξ = and summing 3.7 over B Bε r,α we find e ε Bε r,α Λ α,r ε ν ε Bε r,α and then e ε B ε Bε α \ Bε r,α e ε B ε Bε α Λ α ε ν α B ε + Λ α ε Λ α,r ε ν α B ε = gε B B + log r ν αb ε, B B ε B α ε where we have used the fact that fε B B = gε B B. It follows using 3. that 3.3 e ε B ε Bε α \ Bε r,α C g ε + U α + n α log r +. Then comes the argument in [St]: For any integer k, let r k = k ρ, and let C k be the intersection of B r k ε \B r k+ ε and Bε α. Then C k C k ρ, since ρ k bounds the total radius of the balls in B r k ε Bε α. Moreover j ε = iu ε, u ε ia ε and thus assuming u ε we have j ε e ε. Then using Hölder s inequality in C k and 3.3 we find for p < j ε p C k p/ e ε C k p/ C k p/ e ε B ε Bε α \ B r k+ ε p/ C k C p pk e ε B ε B α ε \ B r k+ ε p/ C p pk g ε + U α + kn α log + p/. 6

27 Using.0 we find C k j ε p C p pk + k log p/ g ε + U α + p/. Summing these inequalities for k ranging from 0 to the largest integer K such that r K ε so that in particular r K ε we find B ε B α ε \B ε ε j ε p C p gε + U α + p/, where C p is a constant times the sum of the convergent series k pk + k log log ρ p/. To this inequality we add B α ε B ε ε j ε p Cε p/ e ε U α p/, which follows from Hölder s inequality after estimating as above B ε ε since e ε = f ε + log ε ν ε we may write using.9,.0, B α ε by Cε. But 3.4 e ε U α Cg ε + Ûα+C ν ε Ûα+ log ε C log ε g ε + U α + B0, +. Thus B α ε B ε ε j ε p Cε p p log ε g ε + U α + B0, p + C g ε + U α + B0, p +. We also add which follows from 3.. Finally we obtain U α\b ε j ε p C g ε + U α + U α j ε p C p gε + U α + B0, +. Summing with respect to the α s such that E U α, this proves.8 and concludes the proof of Theorem. Proof of Corollary.. Note that ξ df ε g ε = ξ df α g α. α Three types of indices occur. 7

28 First we consider indices α such that dista α, Ω ε c > ε and.3 holds. Since 3.5 g α g ε β α g β g ε + C, we deduce from.3 that if n α and ε is small enough, g ε A α cn α log ε and then using.3 again that 3.6 ξ df α g α C ξ L A α nα + βg ε + A α. If n α = 0 the same inequality holds since from.3 the left-hand side is zero. Second we consider indices α such that dista α, Ω c ε > ε and.4 holds. We note that if C is large enough then x log x ηx + C log η/η holds for every x > 0 and η, for instance by distinguishing the cases η > log x and η log x. We use this and 3.5, x x together with.4 to find that if n α then 3.7 ξ df α g α C ξ L A α n α + ηg ε + A α + log η. η Again the inequality is true if n α = 0 since from.4 the left-hand side is zero in this case. Finally we consider indices α such that dista α, Ω c ε ε. In this case, noting that from Lemma. we have n α log ε Ce ε A α, we rewrite.5 as 3.8 ξ df α g α C ξ L A αn α + β ξ L A αe ε A α. To conclude we sum either 3.6, 3.7 or 3.8 according to the type of index α, noting that since diama α, we have f L A α f on A α for any function f. Since the overlap number of the A α s is bounded by a universal constant, we deduce.9. We prove.0. We start by proving that when dista α, Ω ε c > ε we have 3.9 min n α, n α log ε C g ε + A α +. If n α = 0 this is trivial, if not then it follows from either.3 or.4 using 3.5. Assume α is such that dista α, Ω ε c > ε, then since x ηx + /η and since x ηx log ε is trivially true if / log ε < η, we deduce from 3.9 that 3.30 n α C ηg ε + A α + /η. On the other hand Lemma. implies that for any α 3.3 n α C e εa α Ω ε. log ε Summing 3.30 or 3.3 according to whether dista α, Ω ε c is > ε or ε we deduce.0. 8

29 4 Proof of Theorem 4. Convergence We study the consequences of the hypothesis 4. R > 0, M R := lim sup ε 0 U R g ε x dx < +. and prove that it implies the convergence of the vorticities and currents in the appropriate sense. Note that we assume dist0, Ω ε + so that for every R, U R Ω ε for ε small enough. From.3 there exists C > 0 such that for any R large enough B R/C U R B CR, C U R C. R We now gather several easy consequences of Theorem and 4.. Proposition 4.. Assume 4. holds, and let g ε be as in Theorem. Then for any R and ε small enough depending on R we have 4. minn α, n α log ε CM R+C + R, α A α U R 4.3 ν ε U R CM R+C + R, 4.4 f ε g ε χ UR C α A α U R+C \U R C n α log n α + CM R+C + R, where {χ UR } R are any functions satisfying.4. For any p < there exists C p > 0 such that for any R > 0, and ε small enough 4.5 j ε p C p M R+C + R. U R Moreover, up to extraction of a subsequence, {j ε } ε converges weakly in L p loc R, p < to some j : R R ; {ν ε } ε converges in the weak sense of measures to a measure ν on R of the form π p Λ d pδ p where Λ is a discrete set and d p Z, {µ ε } ε converges to the same ν in W,p loc R for any p < and {h ε } converges weakly in L loc R to h. Moreover it holds that 4.6 curl j = ν h. 9