Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow

Transcription

1 Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow Part I: Study of the perturbed Ginzburg-Landau equation Sylvia Serfaty Courant Institute of Mathematical Sciences 5 Mercer st, New York, NY 00, USA. serfaty@cims.nyu.edu July 005, this revised version March 006 Abstract We study vortices for solutions of the perturbed Ginzburg-Landau equations u + ε u u = f ε where f ε is estimated in L. We prove upper bounds for the Ginzburg- Landau energy in terms of f ε L, and obtain lower bounds for f ε L in term of the vortices when these form unbalanced clusters where i d i i d i. These results will serve in Part II of this paper [S] to provide estimates on the energydissipation rates for solutions of the Ginzburg-Landau heat-flow, which allow to study various phenomena occurring in this flow, among which vortex-collisions; allowing in particular to extend the dynamical law of vortices passed collisions. Introduction and statement of the main results. Presentation of the problem In this paper, we study the forced Ginzburg-Landau equation. { u u + ε u = f ε u = g resp. ν = 0 in Ω on Ω. where f ε is a forcing right-hand side which is given in L Ω. Here Ω is a two-dimensional domain, assumed to be smooth, bounded and simply connected, and u is a complex-valued function, assumed to satisfy either one of the boundary conditions. u = g on Ω with g a fixed regular map from Ω to S, in which case we also assume that Ω is strictly starshaped with respect to a point; or.3 ν = 0 on Ω Supported by NSF Career award DMS039 and a Sloan Foundation fellowship

2 in which case no further assumption is made. This equation with f ε = 0 is the standard Ginzburg-Landau equation, which has been intensively studied, in the asymptotic limit ε 0, in particular since the work of Bethuel-Brezis-Hélein [BBH]. Our motivation for studying the L perturbed equation, which we will develop in Part II of this paper [S], is to study the two-dimensional parabolic Ginzburg-Landau equation:.4 t u log ε = u + ε u u in Ω R + u., 0 = u 0 ε in Ω, with the same boundary conditions as above. However, the results we present here have an interest of their own and can be read independently of Part II. The Ginzburg-Landau heat flow is an L gradient-flow or steepest descent for the Ginzburg-Landau functional.5 E ε u = u + u ε. Ω This energy functional is a simplified version without magnetic field of the Ginzburg- Landau model of superconductivity. Such functionals also appear in other models from physics: for superfluidity, nonlinear optics, Bose-Einstein condensates; and the complexvalued function u, called order parameter, plays the role of a condensed wave-function. In this model, the interesting objects are the vortices, or zero-set of the complex-valued function u carrying a topological degree: since u is complex-valued, it can have a nonzero integer degree around each of its zeroes. Vortices can also be seen as having a core, where u is small, of characteristic lengthscale ε; and a tail where u is close to, but the phase of u still carries a lot of energy; they can be clearly extracted in the asymptotic limit ε 0. Vortices in the Ginzburg-Landau model have been the object of intense studies, generally in the asymptotic limit ε 0 where they become point singularities, in particular since the work of [BBH] on.5, under the assumption E ε u C log ε bounding the possible number of vortices; refer also to [SS] for the analysis of the full model with magnetic field. In both cases, some Γ-convergence type results were obtained. A very precise description of the vortices and of the energy of nonminimizing solutions of the Ginzburg-Landau equation, i.e.. with f ε 0, was given by Comte and Mironescu in [CM, CM]. We are interested here in generalizing these results, and in studying how much the situation can differ from the f ε 0 case. Since we are interested in studying vortexcollisions for solutions of.4, we focus on understanding static situations where vortices are very close to each other. We will characterize pathological vortex situations for. as those for which we have a group of vortices which are far from the others, and degrees d i and i d i i d i in the group, which we call an unbalanced cluster of vortices. We study the equation. with an L -perturbation term because.4 is precisely an L gradient flow for.5 and thus for u ε solving.4, we have d dt E εu ε x, t = log ε Ω u ε + ε u ε u ε Thus, if we write that.4 holds with f ε = tuε log ε, we precisely have that log ε f ε L Ω is the energy-dissipation rate for solutions of.4. This will be crucially used in Part II [S].

3 and this motivates our need for estimates, in particular lower bounds, on f ε L. If f ε L is large, then the energy dissipates fast in the flow.4, thus decreasing to a point which allows to rule out certain configurations for example if E ε decreases so much that E ε C then there can be no more vortices. On the other hand, if f ε is small, then. can be seen as a small perturbation of the Ginzburg-Landau equation { u = u u in Ω.6 ε u = g or ν = 0 on Ω, for which a number of qualitative facts about vortices is known. The idea is thus to use this alternative in a quantitative way, in order to deduce from the static study information on vortex-collisions or other pathological situations in the dynamics. More precisely, it is known that if u is a solution of Ginzburg-Landau in the plane, with vortices a i, d i then we must have.7 d i = i i d i equivalent to the fact that i j d id j = 0, or to the fact that the forces exerted by the vortices balance each other. This follows from suitable applications of the Pohozaev identity, as in [BMR]. Similarly, as seen in [BBH, CM], if u ε, a solution of.6 in a bounded domain, has some vortices a i of degree d i accumulating as ε 0 around a single point p, then the same rule i d i = i d i holds. Now, if u ε is a configuration with say, two vortices, one of degree, one of degree, at a distance o as ε 0 which is what happens during a vortex-collision of a + with a then this rule is obviously violated and it s the same for any situation with i d i i d i, so we can trace how much it is violated in the Pohozaev identity for., and get a lower bound for f ε L. The technique thus relies on some adaptations of the Pohozaev identities with error term f ε. Observe that Pohozaev identities have already been widely used in the context of Ginzburg-Landau statics and dynamics [BMR, BBH, BCPS, RuS, SS]. Some similar results and the balanced clusters condition.7 also appear in the recent preprint of Bethuel-Orlandi-Smets [BOS] see Theorem 5 on the parabolic Ginzburg-Landau equation.. Main results on. Before stating the results, let us make a few assumptions. Since we are going to consider nice initial data u 0 ε for.4 with a fixed number of vortices as ε 0, and since the energy decreases during the flow, it is natural to restrict to.8 E ε u ε M log ε and.9 u ε u ε M ε. It is well-known that.4 is well-posed and that if these estimates are true for u 0 ε, they remain satisfied at all times for solutions of.4. 3

4 We sometimes assume in addition that.0 f ε L Ω ε β for some β <. If this assumption is not true, then clearly we have a large lower bound on f ε L. If.0 holds, then after blow-up at the scale ε, solutions of. converge to solutions of Ginzburg- Landau in the plane U = U U which enables to define what we shall call a good collection of vortices a i with degrees d i depending on ε for u ε. Without going into full details of what it means and how they are found, these are points such that the balls B i := Ba i, R ε ε with some R ε log ε, are disjoint and cover all the zeroes of u ε, and d i = deg u ε, Ba i, R ε ε 0. We can then give a more precise definition although we will mostly use a slightly weaker condition, see Theorem Definition. The a i s and d i s being as above, we say that u ε has a cluster of vortices at the scale l at x 0 if.. Bx 0, l {a i } = dist{a i /a i / Bx 0, l}, Bx 0, l l, as ε 0. We say u ε has an unbalanced cluster of vortices at the scale l at x 0 if the previous conditions hold and if d i. i/a i Bx 0,l i/a i Bx 0,l Once these vortices are found, it allows to define a canonical harmonic phase θ in Ω ε := Ω\ n i= Ba i, R ε ε as the harmonic conjuguate of Φ solution to Φ = π i d iδ ai with suitable boundary conditions. Once this is done, denoting ϕ the phase of u ε, i.e. u = ρe iϕ in Ω ε, we may consider the phase-excess ψ = ϕ θ. The first main result consists in evaluating the energy-excess due to both the phase-excess and the modulus of u, in terms of only one natural quantity: the L norm of f ε, the natural norm to consider for the study of the parabolic flow. The method is inspired by that of Comte-Mironescu in [CM, CM], however their result was for the case f ε 0, and used some precise L and decay estimates for solutions of Ginzburg-Landau away from the vortices. Here we retrieve the result with the only control on f ε L and no a-priori bounds, other than.8 and.9. We obtain in addition a scaled version of the estimate, localised in any small ball. The result is Theorem. Let u ε satisfy.,.8,.9 and.0. The a i, d i, B i being as above, we have.3 ψ ε o + C f ε L Ω. Ω ε d i.4 Ωε u ε + u ε ε 4 o + f ε L Ω.

5 For any x Ω, and any l ε log ε, we have.5 ψ ε min Ω Bx,l\ i B i and.6 Moreover, we have, Ω Bx,l\ ibi u ε + u ε.7 α <, απ and.8 π n i= n d i log ε + W da,, a n + i= π C + Cl log l f ε L Ω, o + l log 4 l f ε L Ω ε o + o l log l f ε L Ω. d i E εu ε log ε + C log ε 7/ ε α f ε L Ω + o, n γv i + o E ε u ε i= n d i log ε + W da,, a n + i=, n γv i + C f ε L Ω + o, where the V i s are the limiting blown-up profiles of u ε around a i at scale ε, and the γv i are constants equal when d i = ± to a universal constant γ introduced in [BBH]. Moreover, all the constants C and o above depend only on β, M, Ω and g if applicable. This result allows to bound the phase-excess with scaled versions of it, cf..5.6, and in turn to bound the energy-excess in terms of f ε L and of the vortices of u ε only, in.8. This way, it provides a lower bound for f ε L and it allows, for solutions of.4, to bound from below the energy-dissipation rate, and to bound from above the number of vortices through.7. Let us mention that the energy-quantization result for solutions of.4 shown in [BOS] Theorem 6 and appendix is equivalent at leading order to.8. From this first theorem, we may implement the Pohozaev strategy described above and obtain the following. Theorem. There exist constants l 0 > 0 and K 0 > 0 such that, assuming that u ε is as in Theorem, and that there exists a nonempty subcollection {B i } k i= of the balls {B i} which are included in Bx 0, l/, ε log ε l < l 0 as ε 0, and such that for some K > K 0, either. Bx 0, Kl Ω and Bx 0, Kl intersects no other ball in the collection {B i }, and we have k k.9 d i d i.. x 0 Ω and Bx 0, Kl intersects no other ball in the collection {B i }. Then C.0 f ε L Ω min l log ε, C l log. l i= i= All the constants above depend only on β, M, Ω and g. 5 i=

6 This is exactly the desired lower bound on f ε L : it shows it blows up like /l log ε in most cases, as the scale of the unbalanced cluster of vortices l gets small. As a byproduct, we retrieve in the case f ε = 0 Corollary.. Let u ε be solutions of the Ginzburg-Landau equation.6 such that E ε u ε C log ε. Then, there exists a constant l 0 > 0, such that for ε small enough, u ε has no unbalanced cluster of vortices at any scale l < l 0 ; and has no vortex at distance < l 0 from the boundary. Some sharper but of same order lower bounds for f ε L will be given in Proposition 5. in [S], by blowing up at the scale l in the case where l is not too small log 4 l C log ε. Observe that all these results in particular.0 can be viewed as obtaining lower bounds for the higher-order energy-functional F ε u = Ω u + u u under the ε assumption E ε u ε C log ε. It was proved in [Li, SS] that denoting here and in the rest of the paper by.,. the scalar product in C identified with R if curl iu ε, u ε π n i= D iδ pi as ε 0 i.e. the limiting vortices of u ε as ε 0 are the p i s with degrees D i, then lim inf ε 0 log ε F ε u ε π n i W D p,, p n i= This is the lower bound part of a Γ-convergence result the upper bound should not be hard to prove. The lower bounds we obtain here and in Proposition 5. of Part II are in agreement with this, but in general sharper since they involve the locations and degrees of the vortices at the ε level, and blow up when these get very close. Let us point out that such a study of forced equations, with its dual Γ-convergence point of view, was performed for the Allen-Cahn equation the same equation as. but with real-valued functions an important model for phase-transitions with a lower bound by the Wilmore functional, see [To, RS] and the references therein. We are not aware of any other singularly perturbed equation for which this has been done. In this first paper, we start by performing a Pohozaev ball-construction which is an adaptation of that done in [SS] but with nonzero error term f ε. This allows to bound the number of vortices and define a good collection of vortices. Then, we prove Theorem and Theorem. In the second part [S], we will present the applications of both of these theorems to the dynamics and collisions of vortices under.4. Acknowledgments: We wish to thank warmly Petru Mironescu and Etienne Sandier for helpful discussions. A Pohozaev ball-construction for. and applications This construction, which is a combination of the Pohozaev identity with the ball-growth method of Jerrard/Sandier, consists in an adjustment of the one presented in [SS], taking into account the nonzero right-hand side in.. The main result is 6

7 Proposition.. Let u ε satisfy.,.8,.9 and.0. Then,. {x Ω, ux where C depends only on β in.0, M, Ω and g.. Pohozaev identities for. u log ε } ε C, The Pohozaev identity consists in multiplying. by x u and integrating by parts. However, because of the boundary conditions, we will need a more general version of it, as in [SS], Chapter 4. Introducing the stress-energy tensor associated to the equation. T ij = u + ε u δ ij i u, j u, an easy computation yields that.3 div T ij = j u, u + u ε u = j u, f ε, where div T ij denotes i= it ij. Multiplying the relation.3 by a vector field X, we find Lemma.. Let u satisfy.. For any U open subset of Ω and any smooth vector-field X, we have.4 X j ν i T ij = i X j T ij f ε, X u U U U i,j where ν denotes the outer unit normal to U and the indices i, j run over,. i,j The most standard Pohozaev identity follows by applying this in U = Ω Bx 0, s to X = x x 0, it yields.5 Bx 0,s x x 0 ν Bx 0,s Ω ν + Bx 0,s τ u ε + x x 0 τ τ, ν u ε = f ε, x x 0 u. Bx 0,s Ω Bx 0,s Bx 0,s Ω In particular, if Bx 0, s does not intersect Ω, one obtains.6 r + u s ε = τ + u ε + s We deduce the following lemma. number to be checked Bx 0,s f ε, x x 0 u. 7

8 Lemma.. Let u satisfy.. Then, if R is such that Bx 0, R Ω and 0 < r < R, we have R u.7 r s Bx 0,s ε ds + Bx 0,R\Bx 0,r r = Bx 0,R\Bx 0,r τ + u R ε + x x 0 u, f ε ds, r s Bx 0,s with.8 R r f ε, x x 0 u ds s Bx 0,s Bx 0,R\Bx 0,r r + R f ε + r log R 4 r f ε L Bx 0,r u L Bx 0,r. Bx 0,R\Bx 0,r Proof..7 follows from integrating the relation.6 for s [r, R]. For.8, we write R.9 f ε, x x 0 u r s Bx 0,s R x x 0 r s Bx 0,r ν f ε + x x 0 Bx 0,s\Bx 0,r ν f ε ds r log R r ν f ε + R ν f ε. Inserting the fact that for every λ > 0,.0 ν f ε λ Bx 0,R\Bx 0,r applied to λ = R/, we are led to.8. Bx 0,r Bx 0,R\Bx 0,r ν + λ Bx 0,R\Bx 0,r Bx 0,R\Bx 0,r Another standard relation consists in writing in the Dirichlet case, as in [BBH], a global Pohozaev identity using.4 on the whole Ω. Using the fact that Ω is strictly starshaped, one obtains Lemma.3. Let Ω be strictly starshaped and let u satisfy. with u = g on Ω. Then. u ν + Ω ε C + u L Ω f ε L Ω Ω where the constant C depends only on Ω and g. Proof. Assume Ω is strictly starshaped with respect to the point x 0 hence x x 0 ν β > 0 on Ω, and apply.4 to U = Ω and X = x x 0, this yields. Ω β ν + Ω u ε from which the result follows easily. C Ω g τ f ε + ν, g + x x 0 f ε u τ Ω 8

9 . Proof of Proposition. - interior case For simplicity, we will start by presenting the proof of Proposition. assuming no balls intersect Ω. Proof of Proposition.: The proof is a ball-construction that is very similar to that presented in [SS], Chapter 4. Following [Sa], since.8 holds, by the coarea-formula, one may cover the set {x, ux log ε } by a finite union of disjoint closed balls B i 0 of radii r i such that i r i Cε log ε 3. We grow all these balls in parallel according to the method of Jerrard and Sandier, presented for example in [SS], which yields: Lemma.4. For every t 0 there exists a finite collection of disjoint closed balls Bt such that. B0 = {B i 0} i. rbt = e t rb0 for every t 0, where rbt denotes the sum of the radii of the balls in the collection 3. For every t s B Bs B B Bt There exists a finite set T R + such that if [t, t ] R + \ T, then Bt = e t t Bt, where λb denotes the collection of balls obtained from B by keeping the same centers and multiplying all the radii by λ. The times t T correspond to merging times when some of the balls have intersecting closures. Assuming first the balls remain disjoint through the growing, we may apply.7 to r = r i and R = e t r i to find.3 e t r i u r i s B i log s ε ds B r i t i B. u + u ε +e t r i f ε L B i t u L B i t, where we wrote R r s Bx 0,s x x 0 u, f ε ds R Bx 0,R f ε u R f ε L B i t u L B i t. We easily deduce u.4 t B i 0 ε E ε u, B i t + rb i t f ε L B i t u L B i t where we denote for any set U, E ε u, U = U u + ε u. Now these relations add up nicely over all the balls in the collection Bt, including through possible merging of balls, and we have for every t, and every B k t Bt, u.5 t i/bi 0 B k tb i 0 ε E ε u, B k t + rb k t f ε L B k t u L B k t. 9

10 Summing this over k, using.8, and applying this relation to t = α log ε for some 0 < α <, we find u.6 α log ε ε E ε u, k B k t + Cε α f ε L k B k t log ε 7/, i B i 0 where we observe, rbt Cε α log ε 3. Since.0 is satisfied, we may choose α > 0 such that α β > 0, and find u.7 ε E εu + o C. α log ε i B i 0 u ε We conclude that i B i 0 C and since the B i 0 were constructed to cover the set {x Ω, ux } we deduce.. This proof is valid if none of the balls B log ε k t intersect Ω..3 Proof of Proposition. - boundary issues The method follows that of [SS], Chapter 4, with the only modifications due to the f ε term. We sketch the main steps..3. Dirichlet case In the Dirichlet case, instead of using.6 and.7, we use.5. Decomposing Bx 0, s Ω into Bx 0, s Ω and Ω Bx 0, s, we find.8 Bx 0,s Ω + s ν Ω Bx 0,s Using Lemma.3, we deduce + u s Bx 0,s Ω ε Bx 0,s Ω τ + u ε g x x 0 ν τ g ν x x 0 τ τ, ν + x x 0 u, f ε s Bx 0,s Ω.9 Bx 0,s Ω ν + u s ε Bx 0,s Ω τ + u ε + C + u L Ω f ε L Ω Bx 0,s Ω and integrating.0 R r s Bx 0,s Ω u ε ds + τ + u ε Bx 0,R\Bx 0,r Ω Bx 0,R\Bx 0,r Ω ν and we may reproduce the proof above with this relation instead of.7. + CR + u L Ω f ε L Ω 0

11 .3. Neumann case In the Neumann case, we extend u by performing a reflection with respect to Ω. Let Ω denote a large enough tubular neighborhood of Ω, i.e. Ω Ω. Let ψ be a smooth mapping of Ω onto the unit disc. It can be extended to a mapping from Ω to a domain strictly containing the unit disc. Let then R denote the reflection with respect to the unit circle defined in complex coordinates by Rz = z. The mapping ϕ = ψ R ψ then maps Ω\Ω to Ω. z One can check that it is the identity on Ω, that it is C in Ω\Ω, and that Dϕx converges to the orthogonal symmetry relative to the tangent to Ω at x 0 as x x 0 Ω, at a rate bounded by Cdist x, Ω. We can then extend u with ν = 0 on Ω, by u = u in Ω and ux = uϕx if x Ω\Ω Since Dϕ converges to a reflection with respect to the boundary as x Ω and ν = 0 on Ω, we find that u is C in Ω. We also define f ε = f ε ϕx in Ω\Ω and f ε = f ε in Ω. We will use the same proof as above through ball-growth in Ω for u. The relation.7 still applies inside Ω. For the balls that intersect Ω, we need to replace it with a variant for u. Let Bx 0, s be a ball intersecting Ω and let D = Bx 0, s Ω and D = Bx 0, s\ω. From.5, we have. u D ε = x x 0 ν Ω Bx 0,s τ + s τ ν + u ε Bx 0,s Ω + u ε + x x 0 u, f ε. D In order to get the analogue in D, we apply.4 in D = ϕd with Xϕx = Dϕxx x 0. Arguing as in [SS], this leads to u. D ε + Os = x x 0 ν Bx 0,s Ω τ + u ε + s τ ν + u ε + O s E ε u, D + f ε, X u. D Ω Adding this to the relation., the contributions on Ω cancel and we find u D D ε + Os = s D D τ ν + u ε + O s E ε u, D + O s D D D D u f ε After a change of variables, and since ϕ approaches a reflection, we find as in [SS] u Bx 0,s ε = s Bx 0,s τ ν + u ε u + O s + O s E ε u, Bx 0, s + O s u f ε Bx 0,s ε Bx 0,s.

12 Dividing by s and integrating, we find.3 R u s Bx 0,s ε ds + CRE ε u, Bx 0, R\Bx 0, r + R r Bx 0,R\Bx 0,r Bx 0,R u + u ε u + CR u L Ω f ε L Ω Replacing.7 by this relation, and growing the balls the same way, we are led to the same result for u, and Proposition. is proved..4 Application: construction of the vortex collection We now show how to define a good collection of vortex-balls for solutions of.. If ux 0 <, the assumption.9 implies standardly that u 3 4 in some ball Bx 0, λε and thus that there exists a constant µ > 0 such that.4 Bx 0,λε u ε µ. Using this, the result of Proposition. suffices to conclude as in [BBH], that the set { ux 3 4 } can be covered by a bounded independently of ε number of disjoint balls of centers a i and radii Rε where R is fixed, and changing R if necessary, we may always assume that a i a j ε for i j. We may also assume that each ball contains a point x 0 where ux 0 < otherwise the ball can simply be removed from the collection. The next step is to perform a blow-up analysis. If.0 is verified, then the perturbation term f ε in. disappears after blow-up at the scale ε, and we can use the known results on.6. Lemma.5. Let u ε satisfy.,.8,.9 and.0. If a ε is a sequence of points such that dista ε, Ω ε and degu, Ba ε, Rε = d, then up to extraction, v ε x = u ε a ε + εx converges uniformly over compact subsets of R as ε 0 to a solution U of.5 U = U U in R with.6 R U = πd. If a ε is such that dista ε, Ω Cε then up to extraction, v ε x = u ε a ε + εx converges locally uniformly to a constant of modulus. Proof. Setting v ε x = u ε a ε + εx we have.7 v ε + v ε v ε = ε f ε a ε + εx, and we also know that v ε x C and v ε. Thus, v ε is compact in L by Ascoli s theorem. But we have ε f ε a ε + εx L B R ε f ε L Ω o by.0; thus u ε is strongly compact in L B R for every R. In the first case, up to extraction, we thus find that v ε converges locally uniformly and in Hloc to U solution of.5. It was proved in [BMR] ε

13 that under our assumptions,.6 holds. In the case dista ε, Ω Cε, up to translation and extraction, we find that v ε converges to a solution of U = U U on the halfplane R +, with either U = boundary condition or U ν = 0. In the Dirichlet case, a result of Sandier [Sa] allows to conclude that U is a constant; in the Neumann case, a simple reflection yields a solution to.5 of degree zero, hence a constant of modulus from [BMR]. For U a solution of.5, following [BMR], we have.8 U + U = πd log R + γu as R. B0,R where d is the degree of U and γu is a constant depending on the solution. When d = ±, it has been proved by Mironescu [M] that there exists a unique solution to.5 up to multiplication by a constant of modulus, which is the radial solution of.5 and then γu = γ, a universal constant first defined in [BBH]. Proposition.. Let u ε satisfy.,.8,.9 and.0. Then, after extraction of a sequence ε 0, we can find R ε + with R ε C log ε and a family of balls n i= B i = n i= Ba i, R ε ε, with a i depending on ε and n bounded independently of ε, such that. u ε L Ω\ i Ba i,r εε 0 as ε 0.. a i a j R ε ε for i j and dista i, Ω R ε ε for every i. 3. The d i = degu, Ba i, R ε ε are all nonzero lim. ε 0 u ai ε V i ε L Ba i,r ε ε = 0 where V i is some solution of degree d i of.5, lim ε 0 Ba i,r εε u ε ν = 0 Ba i,r εε ε ν C. u ε.3 lim ε 0 Ba i,r ε ε ε = πd i.3 lim E ε u ε, Ba i, R ε ε V i + V i ε 0 B0,R ε Moreover, for every α <, and every subset I of [, n], we have.33 απ d i E ε uε, i I Ba i, R ε ε α + C log ε 7/ ε α f ε log ε L Ω + o i I = 0. 3

14 Proof. We have already found the points a i. In view of Lemma.5, we may blow up around them to find that v ε x = u ε a i + εx converges in H B0, R for every R > 0 to some V i solution of.5. Now, following [CM], we may find by an abstract argument R ε, with R ε C log ε and, such that { u ε 3 4 } k i= Ba i, R ε ε and lim v ε V i H ε 0 B0,R ε = 0. We deduce.9,.30,.3 and.3: H convergence implies H / convergence of the derivatives on the boundary by trace. 3 comes from the fact that if d i = 0 then.3 contradicts.4. But the choice of 3/4 was arbitrary, the same can be done to cover { u ε m} for any m <. By a diagonal argument, one can then obtain. There remains to prove.33. In the previous subsection, we may apply the method of Proposition. with initial balls B i 0 equal to the Ba i, R ε ε, and obtain exactly as in.6 that for every α <, u α log ε ε E ε u, B k + Cε α f ε L B k log ε 7/, i/bi 0 B k B i 0 where the B k are disjoint balls of sum of radii e α log ε R ε ε = R ε ε α. Combining this with.3 leads to.33 and thus to.7. 3 Canonical phase and energy lower-bounds We introduce the Green kernel Gx, y solution to { x Gx, y = δ y in Ω 3. G g ν = ig, τ resp. G = 0 for Neumann boundary condition on Ω. and Sx, y defined by 3. Sx, y = πgx, y + log x y. It is standard that G is symmetric, and S is a C function in Ω Ω. Also the renormalized energy W, as introduced in [BBH], can be written with these notations 3.3 W d a,, a n = π d i d j log a i a j + π d i d j Sa i, a j i j i,j + d i log x a i + Ω i i d i Sx, a i ig, τ g where the a i s are distinct points in Ω, the d j s are integers, and the last integral is taken to be 0 in the Neumann case. If there are no vortices then we consider instead 3.4 W 0 = Φ where Φ = 0 in the Neumann case, and Φ is a harmonic function with Φ g ν = ig, τ on Ω in the Dirichlet case. 4 Ω

15 3. Estimates for the canonical phase The balls B i = Ba i, R ε ε being given by Proposition., we denote by Ω ε = Ω\ i Ba i, R ε ε. We consider { Φ = π i 3.5 d iδ ai in Ω Φ g ν = ig, τ resp. Φ = 0 for Neumann on Ω, with Ω Φ = 0 in the Dirichlet case. Then, we consider θ the canonical phase associated to the a i, d i, the harmonic conjugate of Φ in Ω ε. It is not univalued, however e iθ is well defined. Observe that θ depends implicitely on ε since the points a i do. We will use the estimate: 3.6 θx C r where r = distx, {a i } Ω, and the following result: Lemma 3.. Let Bb j, ρ j be any finite collection of disjoint balls in number bounded with ε 0, with ρ j R ε ε depending on ε, such that. i Ba i, R ε ε j Bb j, ρ j,. ρ j b i b j for every i j and ρ j dist i {b i }, Ω. 3. a i Bb j, ρ j, a i b j ρ j, these hypotheses allow in particular to take b i = a i and ρ i = R ε ε. We have 3.7 θ = π Di log + W D b,, b n + o Ω\ j Bb j,ρ j ρ i j where D j = dege iθ, Bb j, ρ j = i/a i Bb j,ρ j d i. Proof. The proof is quite standard, similar to results in [BBH] except that here the a i depend on ε or to [SS], Chap From 3., we have 3.8 Φx = n d i log x a i + i= n d i Sx, a i. Using the fact that a i b j ρ j b j b k for j k, and a i Bb j, ρ j, and computing explicitly, we find Φx ν Φx i= = D j + o on Bb j, ρ j, ρ j C min j x b j, in Ω\ j Bb j, ρ j. For assertion 3.7, integrating by parts using 3.5, we first have 3. θ = Φ = Ω\ j Bb j,ρ j 3 number to be confirmed Ω\ j Bb j,ρ j j Bb j,ρ j Φ Φ ν + Φ Φ Ω ν, 5

16 where ν denotes the outer unit normal to Bb j, ρ j. Inserting 3.8, we find 3. = Bb j,ρ j Bb j,ρ j Φ Φ ν i/a i Bb j,ρ j = Bb j,ρ j d i log x a i d i log x a i + d i Sx, a i Φ ν i/a i / Bb j,ρ j i D j log D k log b j b k + D k Sx, b k + o Φ ρ j ν, k j k where we have used the continuity of S, and the facts that on Bb j, ρ j, if a i Bb j, ρ j, log x a i = log x b j + log + b j a i x b j = log x b j + o = log ρ j + o, because x b j = ρ j a i b j ; and similarly if a i Bb k, ρ k, log x a i = log b j b k +o on Bb j, ρ j. On the other hand, Φ ν = Φ = πd j. Bb j,ρ j Bb j,ρ j Inserting this into 3. and using the regularity of S, we get 3.3 Φ Φ Bb j,ρ j ν = πd j log π D j D k log b j b k ρ j k j + π k D j D k Sb j, b k + o. Combining 3. and 3.3, we conclude that 3.4 Φ = π Dj log π D j D k log b j b k Ω\ j Bb j,ρ j ρ j j k j + π D j D k Sb j, b k + log x b j + D j Sx, b j ig, τ g + o j,k Ω j j thus 3.7 holds. With the same kind of techniques, we can get the following result, which will be useful in the sequel. Lemma 3.. Let B i be a family of balls as in Proposition., and let θ be the canonical phase associated to the a i, d i s. If B R and B l are two concentric balls such that B R \B l/ is included in Ω and does not intersect any of the balls B i, then 3.5 π log R θ π log R l B R \B l l + O. i/b i B l d i 6 i/b i B l d i

17 3.6 B R \B l θ r O. If B R and B l are two concentric balls centered on Ω such that B R \B l/ does not intersect any of the B i s, then 3.7 θ O. B R \B l Here the O depends only on Ω, on the number of points a i and on some upper bound on i d i. Proof. Let us first deal with the interior case, and the left-hand side inequality in 3.5. Since B R \B l/ does not intersect any ball, θ is well defined in B R \B l and the degree is constant equal to i,b i B l d i ; that is, for every l r R, we have B r θ τ = π Thus, using the Cauchy-Schwarz inequality, we have 3.8 B R \B l θ R l R l B r θ τ πr B r dr θ τ i/b i B l d i. dr = π i,b i B l d i log R l. This proves the left-hand inequality in 3.5. For the other inequality, for both the interior and boundary case, let us evaluate θ = Φ. B R \B l B R \B l Clearly, since B R \B l does not contain any a i, integrating by parts yields 3.9 Φ = Φ Φ B R \B l B R \B l ν, where ν is the outer unit normal to each disc. Let us now study Φ closer. For each point a i, let us denote by a i its symmetric with respect to Ω there might be several choices, but it does not matter. Let then 3.0 Ψx = i d i log x a i + i d i log x a i. One can check that Ψ and Ψ ν remain bounded on Ω by some constant independent of the a i s. On the other hand, Φ Ψ = 0 in Ω, so in view of the boundary conditions for Φ see 3.5 we find, by the maximum principle, that Φ Ψ is bounded in Ω in both Dirichlet and Neumann cases. Thus, we may write 7

18 3. Φx = i d i log x a i + i d i log x a i + O. Let us now first focus on the interior case. Denoting by x 0 the center of the balls B l and B R, let x B l and a i B l/, we have log x a i = log x x 0 + log a i x 0 x x 0 But, since a i B l/, we have a i x 0 x x 0, thus log a i x 0 x x 0 remains uniformly bounded and we can write log x a i = log l + O. Assume now that x B l and a i / B l/, that means that a i is outside of B R, then log x a i = log a i x 0 + log x x 0 a i x 0 and since x x 0 R and a i x 0 R we have x x 0 a i x 0 and thus log x x 0 a i x 0 remains uniformly bounded. The same holds for a i which is always in Ω\B R. We can thus write 3. Φx = d i log l d i log a i x 0 log a i x 0 + O for x B l. i/b i B l/ i/b i Ω\B R Similarly, for x B R, we have Φx = d i log R i/b i B l/ i/b i Ω\B R d i log a i x 0 log a i x 0 + O. Thus, B l Φ Φ ν 3.3 = = i/b i B l/ d i log l i/b i B l/ d i log l d i log a i x 0 a i/b i R \B i x 0 R d i log a i x 0 a i/b i R \B i x 0 R B l Φ π ν + O Φ ν B l j/b j B l/ d j + O, where we have used 3.5 and the estimate 3.6 or in other words Φ C l on B l. Similarly, Φ Φ B R ν = d i log R a i/b i B i x Φ 0 B l ν + O. l/ i/b i R \B R d i log a i x 0 Substracting those two relations and returning to 3.9, we find Φ = π B R \B l i/b i B l/ d i 8 log R l + O.

19 This finishes the proof of 3.5. Comparing it to 3.8, we conclude that 3.6 must hold. For the boundary case, one may check with similar ideas that 3. implies that Φ is bounded independently of the location of the points and R and l in B R \B l. Therefore B R \B l Φ Φ ν = Ω B R \B l Φ Φ ν + O B R Ω Φ ν + O B l Ω Φ ν The contribution on Ω is zero in the Neumann case and is bounded in the Dirichlet case in view of the bound on Φ and the boundary condition on Φ ν. The contributions on B R and B l are bounded by the same argument as above using Φ C R or C l. We conclude that 3.7 holds. 3. Lower bounds on the energy Returning to u ε, we introduce ρ ε = u ε, and ϕ ε such that 3.4 u ε = ρ ε e iϕ ε in Ω ε. We also introduce the phase-excess ψ ε = ϕ ε θ in Ω ε, observe it is a single-valued function. Afterwards, we most often drop the subscripts ε. We claim that from.9, for each i, there exists a constant c i such that 3.5 ψ = ϕ θ = c i + o on Ba i, R ε ε. Also, ψ = cst on Ω in the case of the Dirichlet boundary condition, and ψ ν = 0 on Ω in the case of the Neumann boundary condition. In the next section, we will work alternatively in Ω ε or in Bx, l\ i B i where Bx, l is some ball of radius l possibly depending on ε included in Ω such that Bx, l Ω ε. In the sequel denotes either Ω ε or any subset of Ω ε of the form Ω Bx, l\ i B i, and D denotes Ω or Bx, l Ω respectively. Lemma 3.3. Assume u ε satisfies.,.8,.9 and.0, and hence the results of Proposition.. Then, E ε u ε 3.6 θ + π d i log R ε + γv i + o Ω ε i i = π i d i log ε + W da,, a n + i γv i + o where V i is given by.9. Proof. This follows arguments of [BMR, CM]. Let = Ω ε or = B l \ i B i. We claim that 3.7 E ε u ε, E ε e iθ, + ρ ψ + ρ 5 Dε ε + ρ θ ψ + o. Indeed, 3.8 ρ ϕ = = ρ θ + ψ D ε ρ θ + ρ ψ + ρ θ ψ. 9

20 But, with Cauchy-Schwartz, 3.9 ρ θ ε By definition of θ, we have Ω ε θ ρ θ = o ρ ε Dε C R εε, thus, using R ε +, ρ ε θ 4.. Hence, 3.3 ρ θ θ ρ + o Dε ε + o. We deduce 3.7. Arguing similarly, we also have 3.3 ρ θ ψ ρ ε θ L Dε ε ψ Dε C ρ R ε ε + ψ ρ o Dε ε + ψ where we have used 3.6. Combining this with 3.7, we find 3.33 E ε u ε, E ε e iθ, + θ ψ + o. Particularising to = Ω ε, using the fact that θ is harmonic, we also have θ ψ = ψ θ Ω ε i B i ν. Inserting 3.5, and using θ B i ν = Φ B i τ = 0, we find that θ ψ = o θ Ω ε i B i ν. Using 3.6 again, we conclude that Ω ε θ ψ = o and hence, from 3.33, we get 3.34 E ε u ε, Ω ε E ε e iθ, Ω ε + o. On the other hand, from.3 and.8, we have 3.35 E ε u ε, B i = πd i log R ε + γv i + o. Adding to 3.34 and combining with Lemma 3., we have the result. 0

21 As a corollary, we get the lower bound: Lemma 3.4. Assume that u ε satisfies the results of Proposition. and Bb k, ρ k is a family of balls satisfying the hypotheses,, 3 of Lemma 3.. Let the p j s be the points of accumulation of the a i s with nonzero total degree. Then, with the same notations as in Lemma 3., 3.36 E ε u ε π k D k log ε + W D p j π D k D k log b k b k + D k γ + o, j k k /b k p j,b k p j k where D j = b k p j D k. Moreover, if there is equality in 3.36 then each D k = ± and each Bb k, ρ k contains only one a i. Proof. From the results of Proposition., we have a family of small balls a i, d i. Applying Lemma 3.3, we have 3.37 E ε u ε π i d i log ε + W da,, a n + i γv i + o. On the other hand, one can check that 3.38 W d a,, a n = W D p j π j i i /a i,a i p j d i d i log a i a i + o. For each given j, let us now study π d i log ε π i/a i p j i i /a i,a i p j d i d i log a i a i. The points a i converging to the same p j belong to several of the Bb k, ρ k. It is again easy to check from the properties on the Bb k, ρ k that 3.39 π i i /a i,a i p j d i d i log a i a i = π π k/b k p j So we are led to studying for each k, 3.40 π d i log ε π i/a i Bb k,ρ k D k D k log b k b k k k /b k,b k p j d i d i log a i a i + o. i i,a i /a i Bb k,ρ k i i /a i,a i Bb k,ρ k d i d i log a i a i. We examine the a i s belonging to one Bb k, ρ k. Let l be the smallest distance between two of the a i s. Let us group together all the a i s that are at distance Ol from each other.

22 This makes several clusters of points. Over each cluster C m, since the total number of vortices is bounded, we have π d i d i log a i a i = π d i d i log l + O i i C m i i C m = π d i d i log l + O. i C m i C m Therefore, 3.4 π i C m d i log ε π i i C m d i d i log a i a i = π d i log l ε π i C m i C m d i log l + O. We now need to sum this over all m s and add the interactions between the clusters themselves, which have total degree δm = i C m d i. Since they are all at a distance l from each other, we may consider l l the minimum of their distance. Let us again group the clusters into clusters of size Ol, at a distance l 3 l from the others. The interaction within each cluster of size l can be counted as π δ δmδ m log l = π m δ m log l. Adding up over all clusters of size l, we find an energy π d i log l ε + π δ l m log π δ l m log l + O. i m m Again there remains to add this over all clusters of clusters, and add the interaction between them, which is at the scale l 3 l, etc... Iterating this process which stops after a finite number of steps since the total number of balls is bounded we are left with an energy bounded from below by 3.4 π π i/a i Bb k,ρ k i/a i Bb k,ρ k d i log ε π d i log l ε + π m i i /a i,a i Bb k,ρ k d i d i log a i a i δ l m log + π δ l 3 l m log + + πdk l log + O, l q m where D k is the total degree on Bb k, ρ k and each δ h, the total degree of a cluster at scale l h, is the sum of the degrees over all the clusters at scale l h that it contains. In other words, we have i d i i d i D k and similarly m δ h m m δh m D k. This means we can bound from below 3.4 by 3.43 π d i log ε π d i d i log a i a i i/a i Bb k,ρ k i i /a i,a i Bb k,ρ k π D k log l ε + log l + + log + O = π D k log l l q ε + O.

23 Moreover, this inequality is sharp if and only if i d i = i d i = D k and m δ h m = m δh m for every h. The first relation implies that each d i is equal to ±, the sign being equal to that of D k. The second relation implies that each δm h = ±, which means that there is only one cluster at each scale, so in fact there can be at most one vortex a i of degree ± in Bb k, ρ k, and D k = ± or 0. In that case, then the lower bound above can be replaced simply by π D k log ε. If this is not the case, then we have dropped some term in 3.4 of size π log l h l h which tends to + by construction of the l j s. Thus, in all cases, we may replace 3.43 by 3.44 π i/a i Bb k,ρ k d i log ε π i i /a i,a i Bb k,ρ k d i d i log a i a i π D k log ε + R ε, where R ε +, unless D k = ± or 0, with at most one vortex of degree ± in each Bb k, ρ k, in which case R ε = 0. Combining this to 3.37, 3.38 and 3.40, we find 3.45 E ε u ε π k D k log ε +W D p j π j D k D k log b k b k +R ε + γv i +o. k k /b k p j,b k p j i If R ε + this implies the desired relation If not then all the small vortices are of degree ±, so γv i = γ for each i, which implies again The substitution lemma and Theorem This section is inspired by the analysis of [CM, CM]. It needs to be readjusted to the case where only. is known, and also to be localized in small balls. The main result we obtain by this method is the following we recall we work in which is alternatively Ω ε = Ω\ i B i or Bx, l\ i B i. Proposition 4.. Assume u ε satisfies.,.8,.9 and.0 and the results of Proposition.. Then, with the same notations as above, as ε 0, 4. ψ + ρ + Ω ε ε ρ C f ε L Ω + o, and 4. E ε u ε, Ω ε E ε e iθ, Ω ε + C f ε L Ω + o. Let x be a given point in Ω, and let us denote 4.3 F l = ψ + Bx,l Ω\ i B i ρ + 5 If either ρ ε.. x Ω, l dist x, Ω and dist i {a i }, Bx, l ε log ε is satisfied, or. x Ω and dist i {a i }, Bx, l ε log ε 3

24 then the function F satisfies a relation of the form 4.4 F l l + Kl F l + Kl f ε L Bx,l + F l + o. where K and the constants and o above is a constant depending only on β, M, Ω and g. The proof requires many steps which we separate into lemmas. Lemma 4. Substitution lemma. Under the hypotheses of Proposition 4., 4.5 E ε e iθ, = E ε u ε, + f ε, e iϕ ρ + ρ ϕ ψ + 4ε ρ + where we recall u = ρe iϕ in Ω ε. ρ D ρ + ψ D ε ρ ρ ρ ν + o, Proof. For any real-valued functions ζ and η 4 3 in, we may consider v = ηe iζ u ε = ηρe iϕ+ζ, and we have 4.6 E ε v, = ρη + ρ η ϕ + ζ + ε η ρ. Expanding all the terms, we find 4.7 E ε v, = E ε u ε, + η ρ + ρ η + ηρ ρ η + ρ η ϕ + ρ η ζ + ρ η ϕ ζ + ρ ε ρ η + ρ 4 η But, taking the scalar product of. with e iϕ yields 4.8 ρ + ρ ϕ = ρ ε ρ + f ε, e iϕ in. Multiplying 4.8 by η ρ and integrating, we find 4.9 η ρ ρ ν + η ρ + ηρ η ρ + ρ η ϕ + ρ ε ρ η = Inserting this into 4.7, and using.30, we find 4.0 E ε v, = E ε u, + f ε, e iϕ ρη + ρ Dε η + ρ η ζ + ρ4 ε η + ρ η ϕ ζ + Choosing specifically ζ = ψ and η = ρ, we find 4.5. D f ε, e iϕ ρη η ρ ρ ν + o. 4

25 Lemma 4.. Under the same hypotheses, 4. ρ ψ + θ ψ + ρ 5 Dε ε + ρ D ε C ρ ρ ν + o. and 4. E ε u ε, E ε e iθ, + C Proof. Adding up the relations 3.7 and 4.5, we find D D ρ ρ ν D + ϕ ψ + o. ε ρ + ψ ρ 4 Dε ε + ρ ρ + ρ θ ϕ ψ ρ ρ ρ ν D C f ε ρ + o. ε Hence, splitting ϕ as θ + ψ, we get 4.4 ρ D θ ψ + ψ + ρ 5 Dε ε + ρ C ρ ρ ν D + C f ε ρ + o, ε where f ε ρ f ε L ρ L Ω Cε log ε f ε L o by.0, hence the result. Similarly 4.5 implies 4.. Lemma 4.3. Under the same hypotheses, 4.5 ρ ϕ ψ C f ε L Ω ψ L Ω ε + o, Ω ε 4.6 θ ρ ϕ ψ C f ε L Ω ψ L Ω ε + o. Ω ε and 4.7 θ ρ ϕ ψ Bx,l Ω\ i B i l + Cl ψ + ρ 5ε Bx,l Ω + Cl f ε L Bx,l ψ L Bx,l Ω\ i B i + o. 5

26 Proof. Taking the inner product of. and iu, we find 4.8 div ρ ϕ = f ε, iu Thus, div ρ ϕ θ = f ε, iu in We now let ψ denote, if D is a ball not intersecting Ω, the average value of ψ on D, if D is a ball intersecting Ω, either in the Dirichlet case, the constant value of ψ on Ω, or in the Neumann case, the mean value of ψ on D Ω; finally if D = Ω, either the value of ψ on Ω in the Dirichlet case, or the average of ψ on Ω in the Neumann case. Let us then multiply 4.8 by ψ ψ, and integrate by parts. We find 4.9 Moreover, θ ρ ϕ ψ = ψ ψ B i ψ ψ θ ν = by 3.6 and 3.5. Also, ψ ψρ ϕ B i ν B i ψ c i θ ν + θ ϕ ρ ν ν + O B i c i ψ Φ τ = o = ψ c i ρ ϕ B i ν + c i ψρ ϕ B i ν = o c i ψ f ε, iu, B i f ε ψ ψ where we have used.30 and 4.8. We may always extend ψ inside D i B i into a function ψ in such a way that Ba i,r εε ψ C Ba i,r εε\ba i,r εε ψ see for example [BMR], so that we have D ψ C ψ. Moreover, we can do it in such a way that ψ c i L B i ψ c i L B i = o. Using this, we find c i ψ f ε, iu B i ψ ψ f ε + ψ c i f ε B i B i ψ ψ f ε + or ε ε f ε L Ω. B i We may thus conclude with 4.4 and.0 combined with R ε log ε that 4.0 θ ρ ϕ ψ = O D f ε ψ θ ϕ ψ + ψ ψ ρ + o. D ν ν In the case D = Ω, in view of the boundary conditions ψ = ψ or ϕ ν = 0, the second term in the right-hand side vanishes identically, so 4. θ ρ ϕ ψ = O f ε ψ ψ + o Ω ε Ω 6

27 Following the exact same steps, we can deduce that 4. ρ ϕ ψ = O f ε ψ ψ + o. Ω ε Ω But, by a Poincaré type inequality, we always have 4.3 f ε ψ ψ C D f ε L D ψ L D where D denotes the half-diameter of D a constant if D = Ω and l if D = Bx, l. Recall that if D intersects Ω, then it is a ball centered at a point of the boundary, essentially a half-disc if l is small, by smoothness of Ω. From 4. and 4., we already deduce that 4.6 and 4.5 hold. There remains to bound the other term of the right-hand side of 4.0. In the case D = Bx, l Ω the only one left to consider we observe that since ϕ = θ + ψ, in view of the boundary conditions and the choice of ψ, we have θ ϕ 4.4 ψ ψ ρ = ν ν D Bx,l Ω ψ ψ ρ θ ν Bx,l Ω ψ ψρ ψ ν. Let us now distinguish between the cases where Bx, l intersects Ω and not. If D = Bx, l Ω, we may use, as in [BMR], a sharp scaled Poincaré inequality on Bx, l: observe that Bx,l ψ ψ l Bx,l ψ τ and Bx,l ψ τ Bx,l ψ ν Bx,l ψ. Inserting this into the above, and using ρ, we are led to ψ ψρ ψ l ψ ψ Bx,l ν Bx,l τ Bx,l ν l 4.5 ψ. Bx,l When D = Bx, l Ω and x Ω, then, we may calculate explicitely L min h h H0 L = π [0,L] 0 h L and 4.7 min R L 0 L 0 h L = π h=0 0 h L Applying this to the curve Bx, l Ω parametrized by arclength, we find, using 4.6 in the Dirichlet case and 4.7 in the Neumann case, in view of the choice of ψ, 4.8 ψ ψ Bx, l Ω ψ π τ, Bx,l Ω Bx,l Ω where Bx, l Ω denotes the length of Bx, l Ω. Using the fact that Ω is smooth, we can write Bx, l Ω πl + Cl 7

28 that is Bx, l Ω tends to a half-circle as l 0. Inserting into 4.8, we find, in place of 4.5, 4.9 Bx,l ψ ψρ ψ ν l + Cl 3 Bx,l Ω l + Cl ψ. Bx,l Ω ψ τ Bx,l Ω ψ ν On the other hand, for both cases boundary and interior, using 3.6, we have θ C log ε ε on Bx, l, hence Bx,l Ω ψ ψ ρ θ ν o ψ ψ log ε Bx,l Ω Bx,l Ω o l ψ log ε Bx,l Bx,l Ω ρ ε ρ ε where we have used a trace inequality. Using the fact that Ω ε ψ Ω u + Ω ε θ C log ε, we deduce ψ ψ ρ θ ν o + l ρ ε. Bx,l Bx,l Combining with 4.0 and 4.3, we conclude that 4.7 holds. Lemma 4.4. Under the same hypotheses, we have the estimates ρ θ ψ ψ C ψ L + o ρ θ + 3 ψ ψ 4.3 Ω ε o Ω ε ε ρ + ψ + o. Ω ε Proof. For the first relation, let us write ρ ψ ρ ψ + ρ ψ { u log ε } { u log ε } C log ε ψ + ρ 4.3 Ω ε { u log ε } ε ψ where we have used the fact that ψ C ε. Now, using the fact that Ω ε ψ C log ε and combining this to the result of Proposition., we conclude that ρ ψ o+c ψ L. A similar reasoning using θ C ε in Ω ε works for ρ θ ψ, and we deduce The other relation is a direct consequence of 3.3 and ρ o in Ω ε. 8

29 Lemma 4.5. Under the same hypotheses, 4.33 Bx,l Ω\ ibi ρ + ρ ε C Bx,l Ω ρ ρ +o + ψ, Bx,l Ω\ i B i and 4.34 Ωε ρ + ρ ε o + ψ. Ω ε Proof. Indeed, returning to 4., we find ρ Dε + ρ ε C ρ ρ ν D +o+c ρ θ ψ +C ρ ψ ε Using 3.3 and the fact that ρ = o in from. of Proposition.,.30 to get rid of the terms on B i, we easily find that 4.33 and 4.34 hold. These relations will be used later. We are now in a position to give the full Proof of Proposition 4.. The control of the excess energy comes from 4.. We just observe that 4.35 ρ θ ψ + ψ = ψ + θ ρ ϕ ψ + ρ θ ψ + 3 ψ, and combine 4. with 4.6 and 4.3, to find 4.36 o ψ + o Ω ε Ω ε 5ε ρ + Ω ε 4 ρ C ψ L Ω f ε L Ω +o. The relation 4. follows directly. Similarly, using 4.30, we are led to 4.37 Bx,l Ω ψ + ρ + 5ε ρ l + Cl Bx,l ψ + ρ + 5ε ρ + C ψ L Bx,l\ i B i l f ε L Bx,l + + o. Setting F l = Bx,l Ω\ i B i ψ + ρ + ρ 5, we deduce that F satisfies 4.4. ε Also, returning to 4., and using 4.5 and the same other arguments, and combining to 4., we find 4.. 9

30 4. ODE approach Here, we give estimates for F given that it satisfies the differential inequality 4.4. Lemma 4.6. Let f be a nondecreasing function on an interval [r, R] with R L, satisfying 4.38 x [r, R], fx x + cx f x + gx fx + b for some continuous function g, then 4.39 fr C r R fr + r GR Gr + b r, R where G is an antiderivative of gx x and C = + Lc. Proof. Dividing 4.38 by x fx, we find fx x f x x fx + cf x fx + gx b x + x fx. Setting hx = fx x, we observe that h x = f x fx, and thus x fx x h gx b x + c f x + x + x fx. Integrating between r and R, and using the monotonicity of f for the last term, we find fr fr + c fr c fr + GR Gr + b r R fr r. R Thus fr r fr R + Lc + GR Gr + b R fr r. R We observe that this is of the form λ a λ + a where λ = fr. Using the fact that for such an equation we have λ a + a, we deduce r + Lc fr fr + rgr Gr + b r, R R the relation 4.39 follows directly. 4. Proof of Theorem Proof of.3,.4,.8 and.7.3 follows directly from 4., and.4 from.3 combined with For.8, we start from 4., which, combined with Lemma 3. applied to the Ba i, R ε ε yields E ε u ε, Ω ε π i d i log R ε ε + W da,, a n + C f ε L + o. 30

31 But, from.3 and.8, we find E ε u ε, Ba i, R ε ε = πd i log R ε + γv i. With Lemma 3.3, the result.8 follows. Finally,.7 was proved in Proposition.. Localised estimates We recall that l ε log ε, so we can find a quantity log ε ε Q ε l. Let us first consider the boundary case, i.e. x Ω, and let F l be defined as in 4.3. Since the number of points a i remains bounded by some n 0, the set S = {l R/ Bx, l i Ba i, Q ε } is a finite union of fewer than n 0 intervals, with total length CQ ε. Let us write S [0, R] = [t, t ] [t, t ] [t k, t k ], where t < t < t < t < t k+ = R. Assume now l is given, and l [t i, t i+], 4.4 holds in that interval. We may use Lemma 4.6 with f = F, gl = Kl f ε L Bx,R+ and b the o found in 4.4, then Gl = K log l f ε L Bx,R l, thus we find F l C l t F t i+ + i+ But, F t i+ F t i+ and t i+, so l log t i+ f ε l L Bx,R + + b t i+. l 4.4 F l C l t F t i l log i+ l f ε L Bx,R + b. Similarly, using 4.4 on [t i+, t i+], we have 4.4 F t i+ C t i+ t F t i t i+ log i+ t f ε L Bx,R + b. i+ The same relation holds for any i j k +. Now observe that since t j+ t j+ Q ε, we have l t i+ t k R l R + CQ n0 ε Cl l R t i+ t i+ in view of the assumption l Q ε. Using this and combining all the relations of the type 4.4, we are led, after some calculations, to 4.43 F l C l R F R + Cl log l f ε L Bx,R + C where C is a constant depending on n 0. On the other hand, from 4., we have F R C f ε L Ω + o, thus, taking R =, 4.44 F l Cl f ε L Ω log l + C. If l belongs to some interval [t i, t i ], then we may get 4.44 for t i, and using F l F t i and t i l + Q ε l, we deduce that a relation like 4.44 still holds. 3

32 For the interior case, let x Ω and let R = dist x, Ω. Let us denote F by F x to keep track of the center point. If l R, then there exists x 0 Ω such that Bx, l Bx 0, l and thus F x l F x0 l and the result follows from the boundary case. If l R, then, arguing exactly as above, since Bx, R Ω and 4.4 holds in the interior case, we can get similarly 4.43, that is 4.45 F x l C l R F xr + Cl log l f ε L Bx,R + C. If R, using 4., we are done. If not, we can find x 0 Ω such that Bx, R Bx 0, R, thus, using the result 4.44 for the boundary case Combining with R l and 4.45, we find F x R F x0 R CR f ε L log R + C. F x l Cl f ε L log l + C, that is 4.44 is proved in the interior case as well, and we always have 4.46 Bx,l Ω\ ibi ψ ε + ρ + u ε C + Cl log l f ε L Ω. In order to prove.6, let us use 4.33 on Bx, s: 4.47 Bx,s Ω\ ibi ρ + ρ ε C Bx,s Ω Let us recall that l ε log ε ε. Thus, from 4.47, Bx,s Ω\ ibi ρ + ρ ε Cε Bx,s Ω ρ ρ +o+o Integrating this relation for s [l, l], we easily deduce that l ρ Bx,l Ω\ ibi + ρ ε Cε Bx,l Ω Bx,s Ω\ i B i ψ ρ + ρ ε +o+o ψ Bx,s Ω\ i B i ρ + ρ ε +o+o l ψ Bx,l Ω\ i B i Inserting 4.46 and the fact that B i ρ + ρ = O from [BMR] for example, we ε are led to ρ Bx,l\ ibi + ρ C ε l + o l log l f ε L Ω + C + o ε and since we assumed ε/l 0, we conclude that.6 holds. 3