Alain Audouard

Jean-Yves Fortin

David Vignolles

Rustem B Lyubovskii

Loïc Drigo

Fabienne Duc

Gena V Shilov

Elena I Zhilayeva

Rimma N Lyubovskaya

Enric Canadell

J.-Y Fortin

G Ballon

E I Zhilyaeva

Quantum oscillations in the

Keywords: 71, 18, +y -Fermi surface: calculations and measurements; effective mass, g factor, 71, 20, Rv -Polymers and organic compounds, 72, 15, Gd -Galvanomagnetic and other magnetotransport effects

come

Introduction. -Quasi-two-dimensional (q-2D) multiband organic compounds (ET) 2 X (where ET stands for the bis-ethylenedithio-tetrathiafulvalene molecule and X is a monovalent anion) exhibit very rich phase diagrams with ground states ranging from antiferromagnetic insulator to Fermi liquid. In particular, superconducting and magnetic ordered phases are observed close to each other which is reminiscent of cuprates and iron-based superconductors (for a recent review, see [1] and references therein). These compounds share the same Fermi surface (FS) topology which is an illustration of the linear chain of coupled orbits introduced by Pippard in the early sixties [2]. Such FS is liable to give rise to a network of orbits coupled by magnetic breakdown (MB) in large magnetic fields. Quantum oscillations in such systems have been extensively studied during the two last decades (for a review, see e.g. [3] and references therein). The main feature of the oscillation spectra of these com-pounds is the presence of frequency combinations that are forbidden in the framework of the semiclassical model of Falicov-Stachowiak [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF]. Theoretical studies have demonstrated that these Fourier components result from both the formation of Landau bands due to coherent MB, instead of discrete levels, and the oscillation of the chemical potential enabled by the 2D character of the FS. However, easy to handle analytic tools necessary to quantitatively account for the data are still lacking. As a matter of fact, comprehensive theoretical calculations taking into account coherent MB and oscillation of the chemical potential have been reported in Ref. [START_REF] Gvozdikov | [END_REF] and implemented in de Haas-van Alphen (dHvA) data of the above mentioned compound. Nevertheless, as pointed out in Ref. [START_REF] Gvozdikov | [END_REF], equations are complex and data could be analyzed only numerically.

The aim of this paper is therefore to provide analytical formulae accounting for quantum oscillation spectra of these systems. In the first step, analytical calculation of the Fourier amplitude of the various dHvA oscillations predicted for the FS illustrating the Pippard's model is reported. In the second step, the FS of the recently synthesized strongly 2D charge transfer salt θ-(ET) 4 CoBr 4 (C 6 H 4 Cl 2 ) [6] is presented and, in the third step, analysis of the observed field-and temperature-dependent quantum oscillation spectra are reported. An excellent agreement with the calculations is obtained.

Experimental. -The studied crystals were synthesized by electrocrystallization technique as reported in Ref. [6]. They are elongated hexagonal platelets with approximate dimensions 1.5 × 0.5 × 0.04 mm 3 . X-ray diffraction data were collected both at room temperature in Chernogolovka with a KM-4 single-crystal diffractometer (Kuma Diffraction) and at 100 K with an Xcalibur diffractometer (Oxford Diffraction) at the Laboratoire de Chimie de Coordination in Toulouse. Tight-binding band structure calculations were based upon the effective one-electron Hamiltonian of the extended Hückel method [7], as reported e.g. in Ref. [8]. Magnetic torque and magnetoresistance were measured in pulsed magnetic fields of up to 55 T with a pulse decay duration of 0.32 s. Magnetic torque measurements were performed with a commercial piezoresistive microcantilever [9] in the temperature range from 1.4 K to 4.2 K. The size of the sample studied was approximately 0.12 × 0.1 × 0.04 mm 3 . Variations of the cantilever piezoresistance was measured with a Wheatstone bridge with an ac excitation at a frequency of 63 kHz. The angle between the normal to the conducting plane and the magnetic field direction was θ = 7 • . Magnetoresistance was measured thanks to a contactless radio frequency measurement technique based on tunnel diode oscillator as reported in Ref. [10]. A pair of counter-wound coils made with copper wire of 50 µm in diameter was used. As pointed out in [11], in the case where the interlayer resistivity is much larger than the in-plane resistivity, the contribution of the former parameter to the measured signal is negligible. The time evolution of the oscillator frequency during the field pulse is deduced from successive short-term fast Fourier transforms of the raw signal. In contrast, analysis of the oscillatory magnetoresistance and magnetic torque is based on discrete Fourier transforms of the field-dependent data, calculated with a Blackman window after removal of a non-oscillating background. Zero-field interlayer and in-plane 4-point resistance was measured with 1 µA ac current at 77 Hz. Electrical contacts were made using annealed platinum wires of 20 µm in diameter glued with graphite paste.

Calculation of de Haas-van Alphen amplitudes. -We consider a two-band system with band extrema ∆ 0(1) (see Fig. 1(c)) and effective masses m 0(1) . In the low temperature limit, the grand potential Ω of a two-dimensional slab with area A is given by a series of harmonics p of classical frequencies F η : where Φ 0 = h/e is the quantum flux and u 0 = 2π 2 m e k B /eh = 14.694 T/K. Effective masses are expressed in m e units and chemical potential µ in Tesla. The index η stands for all the possible classical orbits α, β, β + α, 2β -α, etc., excluding their harmonics, with frequencies F η = m η (µ -∆ η ) and effective masses m η . Frequencies F η depend on the chemical potential µ since they represent a filling factor corresponding to classical orbits. Coefficient C η is the symmetry factor of the orbit η. For example, C α = C β = C 2β-α = 1 and C β+α = 2. ϕ η is the phase of the orbit η, i.e. π/2 times the number of extrema of the orbit in the k x or k y direction, for example, ϕ α = ϕ β = ϕ 2β-α = π and ϕ β+α = 2π. These phases are important for determining the sign of amplitudes. In particular, we define two fundamental orbits with frequencies, F α = m 1 (µ -∆ 1 ) (i.e. m α = m 1 ) and F β = m 0 (µ -∆ 0 ) + m 1 (µ -∆ 1 ) which can be written as F β = m β (µ -∆ β ) (see Fig. 1(d)). Therefore, m 0 + m 1 is identified to the effective mass m β of the MB orbit β. The damping factors can be expressed as

Φ 0 A u 0 k B Ω(T, µ) = - m 0 2 (µ -∆ 0 ) 2 - m 1 2 (µ -∆ 1 ) 2 + (B cos θ) 2 2 p≥1 η C η π 2 p 2 m η R η,p (T, B) cos(2πp F η B cos θ + pϕ η ), (1)
R η,p (T, B) = R T η,p R D η,p R MB η,p R s η,p
where the thermal, Dingle, MB and spin damping factors are given by:

R T η,p = u 0 T pm η B cos θ sinh( u0T pmη B cos θ ) (2)
R D η,p = exp(-

u 0 T D pm η B cos θ ) (3) R MB η,p = (ip 0 ) n t η (q 0 ) n r η (4) R s η,p = cos(πp g * m η 2 cos θ ), (5)
respectively. T D is the Dingle temperature (T D = h/2πk B τ where τ is the relaxation time). Integers n t η and n r η are the (even) number of MB tunnelings and reflections, respectively, encountered by the quasiparticle along its closed trajectory. The MB tunneling and reflection probabilities are given by p 0 = exp(-B 0 /2B) and q 2 0 = 1 -p 2 0 , respectively, where B 0 is the MB field. g * is the effective Landé factor1 . The electron density per surface area n e is given by dΩ/dµ = -n e . In zero-field, Eq. 1 yields

n e = (m 0 + m 1 )µ 0 -m 0 ∆ 0 -m 1 ∆ 1 .
In presence of a magnetic field, µ satisfies the following implicit equation:

µ = µ 0 - B cos θ m β p≥1 η 1 πp C η R η,p (T, B) sin(2πp F η B cos θ + pϕ η ) (6)
The oscillating part of the magnetization can be computed systematically by inserting Eq. 6 in Eq. 1 and deriving the resulting expression with respect to the magnetic field. Direct numerical resolution of this implicit equation could be done. Besides, a more user-friendly controlled expansion in term of damping factors R η,p can be derived systematically at any possible order. Keeping successive terms in powers of R η,p up to the second order, yields analytically the following expression for the oscillating part of the magnetization:

m osc = - η p≥1 F η C η πpm η R η,p (T, B) sin 2πp F η B cos θ + pϕ η + η,η ′ p,p ′ ≥1 F η C η C η ′ πp ′ m β R η,p (T, B)R η ′ ,p ′ (T, B) sin 2π pF η + p ′ F η ′ B cos θ + pϕ η + p ′ ϕ η ′ -sin 2π pF η -p ′ F η ′ B cos θ + pϕ η -p ′ ϕ η ′ + • • • (7)
where frequencies F η are evaluated at µ = µ 0 , namely, F η = m η (µ 0 -∆ η ). Eq. 7 can be written as a sum of periodic functions with frequencies F i and Fourier amplitudes A i where the index i stands for either classical orbits η or 'forbidden' orbits such as β -α. Relevant examples for experimental analysis are given below:

A α = F α πm α R α,1 + F α 2πm β R α,1 R α,2 + F α 6πm β R α,2 R α,3 + 2F α πm β R β,1 R α+β,1 , (8)
A 2α = - F α 2πm α R α,2 + F α πm β R 2 α,1 - 2F α 3πm β R α,1 R α,3 , (9)
A β = F β πm β R β,1 + F β 2πm β R β,1 R β,2 + F β 6πm β R β,2 R β,3 + 2F β πm β R β,1 R α+β,1 , (10)
A 2β = - F β 2πm β R β,2 + F β πm β R 2 β,1 - 2F β 3πm β R β,1 R β,3 , (11)
A β-α = - F β-α πm β R α,1 R β,1 - F β-α πm β R α,2 R α+β,1 - F β-α πm β R β,2 R α+β,1 , (12)
A β+α = - 2F β+α πm β+α R β+α,1 + F β+α πm β R α,1 R β,1 , (13)
A 2β-α = F 2β-α πm 2β-α R 2β-α,1 + F 2β-α 2πm β R α,1 R β,2 + F 2β-α 6πm β R α,3 R α+β,2 . (14)
In the framework of the LK model, the amplitude of a Fourier component linked to a basic orbit involves only one damping factor (e.g. A α ∝ R α ). This is also the case for a MB orbit, the effective mass of which is given by the Falicov-Stachowiak model (e.g. m 2β-α = 2m βm α ) [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF]. According to this statement, the leading term of Eqs. 8 and 10 which accounts for the α and β components, respectively, reproduces the LK equation. Furthermore, deviations from the LK behaviour due to the high order terms involving products of damping factors are only significant in the low temperature and high field ranges. This statement also stands for 2β-α since R 2β-α is significantly higher than the product R α,1 R β,2 (see Eq. 14). As a consequence, the Falicov-Stachowiak model should apply for this component as well. In contrast, algebraic sum of damping terms in Eqs. 9 and 13, where minus signs accounts for π dephasings, may cancel at field and temperature values dependent on the effective masses, Dingle temperature, MB field, etc. as displayed in Fig. 2 relevant to 2α. This point has already been reported for the second harmonic of the basic orbits both for compensated [13] and un-compensated [14] metals. In that respect, since algebraic sums are involved, care must be taken of the spin damping factor sign2 . Indeed, the value of the spin damping factor, which is changed through the value of g * in Fig. 2, can have a drastic effect on the field and temperature dependence of the amplitude in this case. Finally, the Fourier amplitude of the 'forbidden orbit' β -α is accounted for by Eq. 12. Although a non-LK behaviour is evidenced, a monotonic field and temperature dependence is expected owing to the negligibly small value of the high order terms. The consistency of Eqs. 8 to 14 with experimental data is examined in the next section.

Results and discussion. -Even though the organic metal θ-(ET) 4 CoBr 4 (C 6 H 4 Cl 2 ) undergoes a phase transition at 343 K [6], no additional phase transition is revealed by X-ray diffraction below room temperature, at which crystals are synthesized, down to 100 K. As displayed in Fig. 1, the central feature of the crystalline structure at room temperature is that the unit cell has two different donors planes labeled A and B. According to band structure calculations, layer A is insulating while layer B is metallic. FS of layer B is composed of one closed tube centered at Y with an area of 18 % of the first Brillouin zone (FBZ) area (yielding α orbit in magnetic field) and two q-1D sheets with rather small gap in between. Therefore, it is expected that the MB orbit β, which corresponds to the hole tube centered at Γ with an area equal to that of the FBZ is observed in moderate fields, giving rise to the textbook where R T η,1 is given by Eq. 2, assuming the LK formalism accounts for the data, i.e. the high orders terms of Eq. 8 can be neglected. As reported in Fig. 5, all the data relevant to the α and β components lies, respectively, on the same curve within the experimental scattering which suggests that the above assumption holds. Dingle temperature and MB field can be derived for components which do not involve tunneling (n t η = 0 in Eq. 5). Although this is the case of α, a very broad maximum is observed in Fig. 5 of Fig. 5(b) displays the field dependence of A τ α /BR T α,1 calculated for the highest field range covered by the experiments, assuming further that g * = 23 . Although the contribution of these factors rises as the field increases, it remains smaller than the experimental scattering for both α and β.

The field and temperature dependence of the Fourier amplitude relevant to frequency combinations is displayed in Fig. 6. Since their amplitude is much smaller than that of the orbits α and β, only mass plots at high field and Dingle analysis at the lowest explored temperature are considered. As already discussed for the second harmonics 2α (see Fig. 2), amplitudes are strongly dependent on the spin damping factors value. Solid lines in Fig. 6 are the best fits of Eqs. 12 to 14, obtained with the same values of m α , m β , B 0 , T D and g * as above, with only a prefactor as a free parameter for each of the data: a very good agreement is obtained, in particular for the 'forbidden orbit' β -α. It can be remarked that the behaviour of both 2β -α and β + α are close to the predictions of the Falicov-Stachowiak model. Whereas this feature is mainly due to the small value of R α,1 R β,2 compared to R 2β-α,1 in the former case, it can be attributed to the peculiar set of values of the spin damping factors (R s α,1 R s β,2 /R s β+α,1 = -0.22) in the latter case. Summary and conclusion. -Analytical formulae for dHvA oscillations relevant to a class of FS illustrating the Pippard's model and observed in many q-2D organic conductors have been derived systematically taking into account the chemical potential oscillations in magnetic field (Eqs. 8 to 14). Although high orders terms are observed, basic (α) and MB-induced (β and 2β -α) orbits can be accounted for by the LK and Falicov-Stachowiak semiclassical models, at least at moderate fields and temperatures. In contrast, the 'forbidden orbit' β -α is accounted for by a non-LK equation, involving products of classical amplitudes relevant to α and β. Furthermore, strongly non-monotonic field and temperature dependence may be observed for the second harmonics of basic frequencies such as 2α and the MB orbit β + α, depending on the value of the spin damping factor R s η . These formulae are in agreement with the dHvA oscillation spectra of the strongly 2D organic metal θ-(ET) 4 CoBr 4 (C 6 H 4 Cl 2 ), assuming g * = 2. However, SdH oscillations spectra exhibit a tremendous number of Fourier components suggesting, besides quantum interference contributions, more severe deviations from the LK model that remains to be explained within a SdH theory including quantum interference contribution. * * *

Fig. 1 :

 1 Fig. 1: (Colour on-line) (a) Crystal structure of θ-(ET)4CoBr4(C6H4Cl2) projected along the a direction. Two different cation layers labeled A and B, respectively, are evidenced. Atomic arrangement of the ET molecules plane and band structure calculations relevant to (b) insulating layer A and (c) metallic layer B. (d) Fermi surface relevant to layer B. Dashed line marks the top of the highest occupied band and the Fermi level in (b) and (c), respectively. Labels ∆0 and ∆1 stand for the band extrema. Γ, X, Y, M and S refer to (0, 0), (a * /2, 0), (0, b * /2), (a * /2, b * /2) and (a * /2, -b * /2), respectively.

Fig. 3 :

 3 Fig. 3: (Colour on-line) Temperature dependence of zero-field in-plane (blue lower curve) and interlayer (black upper curve) resistance normalized to the values at 290 K.

Focusing

  on dHvA oscillations, Dingle plots of α and β can be obtained plotting A τ η /BR T η,1

Fig. 4 :

 4 Fig. 4: (Colour on-line) Fourier analysis in the field range 35-55.3 T of (a) the magnetic torque data (dHvA oscillations) and (b) the tunnel diode oscillator data (SdH oscillations) reported in the respective insets. Solid triangles are marks calculated with Fα(θ = 0) = 0.944 kT and F β (θ = 0) = 4.60 kT. Data have been shifted down from each other by a constant amount.

Fig. 5 :

 5 Fig. 5: (Colour on-line) (a) Effective masses vs. magnetic field for the α (circles) and β (squares) oscillations deduced from best fits of Eq. 8 to the torque (solid symbols) and tunnel diode oscillator (open symbols) data. Horizontal lines stand for the mean values of the effective masses deduced from torque data (mα = 1.81, m β = 3.52). (b) Dingle plot of the α and β torque oscillations. R T η,1, where η = α, β, is calculated with mα and m β deduced from (a). Solid lines are best fits of Eq. 8 and 10 to the data relevant to α and β, respectively, obtained with B0 = 35 T and TD = 0.79 K (see Eqs. 2 to 4). Inset displays an enlarged view of the fits for α.

Fig. 6 :

 6 Fig. 6: (Colour on-line) (a) Mass plots and (b) Dingle plots of βα (circles), β + α (squares) and 2βα (diamonds). Solid lines are bests fits of Eqs. 12, 13 and 14, calculated with mα = 1.81, m β = 3.52, B0 = 35 T, TD = 0.79 K and g * = 2.

FS warping can be accounted for by the additional damping factor J 0 (4πpmη τ ⊥ /ehB), where τ ⊥ is the interlayer transfer integral[12]. In most q-2D organic metals, this factor is close to 1.

Since the LK equation involves a single damping term, the absolute value of the spin damping factor is considered in the case where data are analyzed on this basis. However, π dephasing is observed on either side of spin-zero angles as reported in[15].

Effective Landé factor as low as g * = 1.5 ÷ 1.6 are reported for the strongly correlated compound κ-(ET) 2 Cu(NCS) 2[START_REF] Gvozdikov | [END_REF], nevertheless g * = 2 is consistent with data of other ET salts[17,18].

This work has been supported by EuroMagNET II under the EU contract number 228043, DGES-Spain (Projects FIS2009-12721-C04-03 and CSD2007-00041) and by the CNRS-RFBR cooperation under the PICS contract number 5708. We acknowledge the technical help of Laure Vendier at the X-ray facility of the LCC-Toulouse.

linear chain of coupled orbits. In addition, since layers A are insulating, conducting planes are strongly separated from each other which suggests a strongly 2D behaviour. In line with this statement, the room temperature resistivity anisotropy ρ(interlayer)/ρ (in plane) is as large as 10 4 [6]. Furthermore, according to the data in Fig. 3, this anisotropy further increases up to ρ(interlayer)/ρ(in plane) = 10 6 as the temperature decreases down to liquid helium temperatures.

Magnetic torque (dHvA oscillations) and tunnel diode oscillator [Shubnikov-de Haas (SdH) oscillations] data are displayed in Fig. 4. Besides Fourier components linked to the closed orbit α and to the MB orbit β, few frequency combinations typical of the FS displayed in Fig. 1 are observed in the high field range of the torque spectra. In contrast, numerous combinations are observed in SdH spectra. As an example, the Fourier component 2β -3α is clearly detected in Fig. 4(b). This frequency, which to our knowledge is observed for the first time, could correspond to a quantum interferometer path. However, its amplitude is unexpectedly large. The frequencies linked to the β and α orbits are F β (θ = 0) = 4600 ± 10 T, in agreement with the FBZ area at 100 K, and F α (θ = 0) = 944 ± 4 T, respectively. The latter value yields a cross section area of 20.6 % of the FBZ area, in satisfactory agreement with the band structure calculations of Fig. 1. Field-and temperature-dependent Fourier amplitudes can be studied at the light of the results given in the preceding section, keeping in mind that dHvA amplitudes A η are related to torque oscillation amplitudes A τ η as A η ∝ A τ η /Btan(θ). Effective masses, derived from Eq. 8 at various magnetic field values are field-independent: m(α) = 1.81 ± 0.05, m(β) = 3.52 ± 0.19 (see Fig. 5(a)). In agreement with results of Ref. [16], m(β) is twice m(α) within the error bars. At odds with torque results, effective masses deduced from SdH data, in particular those relevant to the β component, definitely increase as the magnetic field increases. This behaviour indicates that the above calculations, relevant to dHvA oscillations, cannot account for the SdH data at high field.


Andres Contreras
email: contreras@ann.jussieu.fr

Sylvia Serfaty
email: serfaty@ann.jussieu.fr

LARGE VORTICITY STABLE SOLUTIONS TO THE GINZBURG-LANDAU EQUATIONS

.

Introduction

Let Ω ⊆ R 2 be a bounded and smooth domain. We are interested in finding local minimizers of the 2D Ginzburg-Landau functional of superconductivity

(1.1) G ε (u, A) := 1 2 Ω |(∇ -iA)u| 2 + |∇ × A -h ex | 2 + (1 -|u| 2 ) 2 2ε 2 .
Here, the complex-valued function u is a pseudo wave function, the "order parameter" in physics, that indicates the local state of the superconductor. The material is in the superconducting state in the regions where |u| ≈ 1, and in the normal state where |u| ≈ 0. The zeros of the order parameter carrying a non-zero degree are called vortices. The parameter h ex > 0 is the intensity of the applied magnetic field, A is the induced magnetic potential, h := ∇ × A is the induced magnetic field, and finally ε is the inverse of the "Ginzburg-Landau parameter." For more background, we refer to standard physics textbooks, such as [START_REF] Tinkham | Introduction to superconductivity[END_REF][START_REF] Tilley | Superfluidity and superconductivity[END_REF], or to [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF] and references therein. We are interested in the asymptotic regime ε → 0. This corresponds to the characteristic lengthscale of the vortices going to 0.

The vortices of a critical point of (1.1) are conveniently studied via the vorticity measure µ(u, A) := ∇ × j(u, A) + ∇ × A, where j(u, A) := (iu, ∇ A u) denotes the superconducting current associated to a configuration and ∇ A = ∇ -iA. We also write j ε := j(u ε , A ε ), for (u ε , A ε ) a pair depending on ε. In the limit ε → 0, the vorticity measure is well approximated by a sum of Dirac masses at the vortex centers, for a precise statement see the Jacobian estimate (1.7) below.

In what follows we write, for quantities A ε and B ε depending on ε, A ε B ε when there is a constant C independent of ε such that A ε ≤ CB ε . We also write A ε ≪ B ε when lim ε→0

The asymptotic study of critical points of (1.1), as ε → 0 for external fields h ex ≪ 1 ε 2 , has been carried out in a series of papers by Sandier and Serfaty, starting from [START_REF] Serfaty | Local minimizers for The Ginzburg-Landau Energy near critical magnetic field; part I[END_REF][START_REF] Serfaty | Local minimizers for The Ginzburg-Landau Energy near critical magnetic field; part II[END_REF]; continuing in [START_REF] Sandier | Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field[END_REF][START_REF] Sandier | Ginzburg-Landau minimizers near the first critical field have bounded vorticity[END_REF][START_REF] Sandier | Limiting Vorticities for the Ginzburg-Landau equations[END_REF]; the monograph [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF], and more recently [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]. It was shown in [START_REF] Sandier | Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field[END_REF] that for h ex = o (|log ε|) there is a critical value λ * of λ = lim ε→0 hex |log ε| below which minimizers have no vortices. It was further shown in [START_REF] Sandier | Ginzburg-Landau minimizers near the first critical field have bounded vorticity[END_REF] that if h ex -λ * |log ε| log |log ε| then the number of vortices of a minimizer remains bounded independent of ε. For log |log ε| ≪ h ex -λ * |log ε| ≪ |log ε|, minimizers have N vortices with 1 ≪ N ≪ h ex as shown in [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF], Chap. 9, while for λ > λ * possibly = +∞, minimizers exhibit a number N proportional to h ex of vortices with uniform density in a subregion determined by an obstacle problem (see [START_REF] Sandier | Limiting Vorticities for the Ginzburg-Landau equations[END_REF], [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF] Chap. 7, and [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]).

The behavior of critical points of (1.1) that are not necessarily minimizers of the energy, was the object of study of [START_REF] Sandier | Limiting Vorticities for the Ginzburg-Landau equations[END_REF]. In [START_REF] Sandier | Limiting Vorticities for the Ginzburg-Landau equations[END_REF] the vorticity measures µ(u ε , A ε ) associated to critical points (u ε , A ε ) of (1.1), normalized by the number of vortices, are shown to converge weakly to a measure µ. The measure µ has the form µ := -∆h + h, where h is a "stationary function" for a free boundary problem (see (1.2) below), and satisfies in some weak sense the relation µ∇h = 0. This expresses that the average force acting on the vortices (∇h) vanishes where the vortices are, i.e. on the support of µ, thus allowing them to be at equilibrium. This result was later proved under less restrictive assumptions in [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF], Chapter 13, and included cases not previously covered.

It is then natural to ask the converse, that is, given a measure µ associated to a stationary function h as above, can one find (u ε , A ε ) critical points of (1.1) such that µ(u ε , A ε ), suitably normalized, converges weakly to µ? We will answer this question partially, by building solutions to the Ginzburg-Landau equations which are local (but not global) minimizers of the energy, and have a prescribed and divergent (as ε → 0) number of vortices N . The same thing has already been accomplished for a bounded number of vortices in [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF], and also in [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF], Chap. 11 for N slowly diverging as ε → 0.

Note that the question of inverse problems, i.e. whether given a critical point of a limiting energy, one can find corresponding critical points of the original problem, has been addressed in a variety of contexts, and for Ginzburg-Landau in particular : see [START_REF] Lin | Minimax solutions of the Ginzburg-Landau equations[END_REF][START_REF] Du | Ginzburg-Landau vortices: dynamics, pinning, and hysteresis[END_REF] for the situation of a finite number of vortices in 2D, and [START_REF] Alama | On the Ginzburg-Landau model of a superconducting ball in a uniform field[END_REF][START_REF] Montero | Local minimizers with vortices in the Ginzburg-Landau system in three dimensions[END_REF][START_REF] Jerrard | Local minimizers of the Ginzburg-Landau energy with magnetic field in three dimensions[END_REF] for lines of vortices in 3D. Our approach, which is not local inversion but rather local minimization, is more in the spirit of [START_REF] Montero | Local minimizers with vortices in the Ginzburg-Landau system in three dimensions[END_REF][START_REF] Jerrard | Local minimizers of the Ginzburg-Landau energy with magnetic field in three dimensions[END_REF] for vortex lines in 3D. All previous results however, concerned only finite numbers of vortices. Another related context where such a question is addressed is that of the Allen-Cahn equation, whose limiting energy is the perimeter functional, for results there see [START_REF] Del Pino | Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature[END_REF][START_REF] Kohn | Local minimizers and singular perturbations[END_REF][START_REF] Pacard | From constant mean curvature hypersurfaces to the gradient theory of phase transitions[END_REF].

The stable solutions found in [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF] and [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF] Chap. 11, are shown to exist for a wide range of h ex (they are in fact globally minimizing for only one value of h ex ): they exist for fields of intensity up to ∼ 1 ε α 0 , where α 0 < 1 2 , and also for fields almost as small as constant. They thus form branches of solutions, each corresponding to a number N of vortices, with N bounded with ε, or even unbounded, but subject to the restrictions

N 2 ≤ ηh ex , h ex ≤ N ε -2η N 2 +η
, where η > 0 is a small but fixed parameter. These restrictions imply in particular (cf. [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF]) that in any case N ≪ (|log ε|)

N

h ex vortices. We address this question and prove for the first time that branches of local minimizers with N vortices continue to exist up to N = O(h ex ), with restrictions on N and h ex specified below, see (1.5)- (1.6).

The analysis in [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF] consisted, roughly, in minimizing the energy G ε "among configurations with N vortices" and exploiting the quantization of N through a careful expansion of the minimal energy of a configuration in terms of its vortices. This expansion came with an error o(N 2 ). However, the method followed there only tolerates an error of o (|log ε|) (hence at least the condition N 2 |log ε|) which thus limited the range of applicability of the approach. What we do here is to follow the same general idea of local minimization but this time over a class which is defined so as to exploit the results and methods of [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] to get a more precise energy expansion with an error of only o(N ). In this way we can cover the range of N even comparable to |log ε|, and fields as large as a power of 1 ε .

1.1. Statement of main result. In order to state the main result of the article in more accurate terms, we first proceed to introducing some auxiliary function.

An Obstacle Problem. Let N be a positive number (typically an integer representing the number of vortices). Following [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], we denote by h ε,N the minimizer of

(1.2) 1 2 Ω |∇h| 2 + |h -h ex | 2 ,
over the set of functions h that are equal to h ex on the boundary and such that

Ω |-∆h + h| = 2πN.
As long as 2πN ≤ h ex |Ω| (an assumption we will need to make), this minimizer h ε,N exists and it is the solution to an obstacle problem (see Lemma 5.1 and Appendix A in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]). There exists a coincidence set ω ε,N ⊂ Ω and a number 0 < m ε,N ≤ 1 (for us the normalized vortex density) such that

(1.3) µ ε,N := -∆h ε,N + h ε,N = h ex m ε,N 1 ω ε,N and h ex m ε,N |ω ε,N | = 2πN,
where 1 denotes the characteristic function of a set and | • | its area. If Ω is convex, then ω ε,N is also convex, by a result in [START_REF] Caffarelli | Convexity of solutions of semilinear elliptic equations[END_REF]. To avoid technical difficulties, we will assume in the sequel that Ω is convex. It is also checked in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] that m ε,N is bounded universally from below by a positive constant m 0 and that m ε,N → |ω ε,N | is increasing. Without loss of generality we can thus assume that

(1.4) lim ε→0 m ε,N =: m ≥ m 0 > 0,
where N may also depend on ε.

As in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] (cf. Lemma 5.2), the function h ε,N will serve to "split" the energy G ε in a convenient way. The vorticity of the solutions we construct will ressemble µ ε,N , which is a constant over the subregion ω ε,N , hence we will have a uniform density of vortices there, and a vanishing one outside.

Range of fields and vorticity studied. We consider fields and vortex numbers satisfying the following conditions

(1.5) 1 ≪ N ≤ min c h ex |Ω| 2π , K log 1 ε √ h ex ,
for constants 0 < c < 1 and K > 0 independent of ε, and

(1.6) h ex ε -s 0 for some s 0 < 1 7 .

A few comments are in order. The first condition in (1.5) N ≤ c hex|Ω| 2π with c < 1, seems like a natural one. It corresponds to the maximum number of vortices that a certain domain can fit according the the applied field, at least it is the maximum permitted at the level of the limiting obstacle problem. On the other hand, we do not believe the second condition N ≤ K log 1 ε √ hex and (1.6) to be optimal, but rather to be technical assumptions. Optimal conditions should rather be of the type N ≤ hex|Ω| 2π and h ex less than some superheating field at which vortices lose their stability, probably of order 1/ε (but this is all a bit speculative). In any case, we expect our condition (1.5) on N to be optimal when h ex = |log ε| .

We also note that N ≤ K log 1 ε √ hex could equivalently be written N ≤ K|log ε| (changing K if necessary). Let us also point out that (1.5) implies h ex ≫ 1 and 0 < m 0 ≤ m < 1 (see (1.3)), and thus 1 -m ε,N is always bounded away from 0 for ε small enough, and h ex (1 -m ε,N ) ≫ 1. We will make use of these facts repeatedly during the proofs.

Our main result is:

Theorem 1.1.
Let Ω ⊆ R 2 be a convex smooth bounded set. Assume that N = N (ε) a family of positive integers and h ex = h ex (ε) satisfy conditions (1.5)- (1.6). Then, for ε small enough (depending on the constants above), there exists a local minimizer (u ε , A ε ) of (1.1) such that the following holds: (i) There exists a measure ν ε of the form 2π i d i δ a i , with d i ∈ Z and a i ∈ Ω, such that

(1.7) µ(u ε , A ε ) -ν ε (C 0,1 0 (Ω))
* ≤ ε α for some α > 0, and we have

(1.8) ν ε (Ω) = 2πN,
i.e. the total vorticity of u ε is N. (ii) The asymptotics

(1.9) G ε (u ε , A ε ) = G N ε + πN log 1 ε √ h ex + N W m -2πN h ex (1 -m ε,N ) + o(N ),
hold as ε → 0, where W m and G N ε are constants explicited in (1.11) and (1.12) respectively, and m is as in (1.4). (iii) For any 1 < p < 2,

(1.10) µ(u ε , A ε ) -µ ε,N W -1,p (Ω) ≤ C p √ N ,
where C p depends only on p.

(iv) For 1 < p < 2, denoting by P ε the probability measure on L p loc (R 2 , R 2 ) which is the pushforward of the normalized uniform measure on ω ε,N by the map x → √ h ex j ε ( √ h ex (x + •)), 1 we have that, up to subsequence, P ε converges weakly to a translation invariant probability measure P on L p loc (R 2 , R 2 ) such that P -a.e. j ∈ A m and P -a.e. j minimizes W over A m , where W and A m are defined just below.

As announced, this theorem proves the existence of locally minimizing solutions of Ginzburg-Landau with N vortices. Moreover, their vortices tend to follow the average distribution µ ε,N , and their superconducting currents, after blow-up at scale √ h ex , tend to minimize the "Coulombian renormalized energy" W , introduced in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], whose definition we recall below.

1.2. The renormalized energy W . For each m > 0, consider the class of currents j such that div j = 0 and ∇ × j = p∈P 2πδ p -m for a discrete set P, and #(P

∩ B R )/R 2 is bounded for R > 1. We call this class A m .
For any family of sets {U R } R>0 in R 2 , χ U R will be a family of associated positive cut-off functions satisfying

|∇χ U R | ≤ C, Supp(χ U R ) ⊂ U R , χ U R (x) = 1 if d(x, (U R ) c ) ≥ 1,
where the constant C is independent of R. We specialize this to the family of balls B R of radius R and center the origin.

Definition 1.2. W is defined, for j ∈ A m , by W (j) = lim sup R→∞ W (j, χ B R ) |B R | ,
where for any positive function χ, we denote

W (j, χ) = lim η→0 1 2 R 2 \∪ p∈P B(p,η) χ |j| 2 + π log η p∈P χ(p) .
For properties of W , we refer to [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]. It is proven there in particular that min j∈Am W is achieved and finite. We also set γ

γ := lim R→∞ 1 2 rf ′2 + (1 -f 2 ) 2 r 2 + f 2 r dr -π ln R , constant introduced in [2],
where f is such that f (r)e iθ is the unique (cf. [START_REF] Mironescu | Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale[END_REF]), up to a phase shift, degree one radial solution of

-∆u + (1 -|u| 2 )u = 0 in R 2 . We finally let (1.11) W m := 2π m min j∈Am W + γ.
In [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], it is conjectured that the minimum of W over A m is achieved for currents corresponding to a configuration of points in a perfect hexagonal lattice (of suitable volume). A small hint to this is provided there by the fact that W is uniquely minimized, among perfect lattices of fixed volume, by the hexagonal lattice. In superconductivity, the hexagonal lattice also corresponds to the famous "Abrikosov lattice", predicted from (1.1) and observed in experiments. It was proven in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] that global minimizers of the Ginzburg-Landau functional (1.1) have currents which minimize W , after a suitable blow inside the free-boundary region that contains the vorticity. If the conjecture about the minimum of W is true, then one expects the vortices of global minimizers to arrange themselves in such an Abrikosov hexagonal lattice. From item (iv) of Theorem 1.1 the exact same thing is expected of the lo cally minimizing solutions found here. The only qualitative difference with the global minimizers is in the total number of vortices N , which manifests itself in a different µ ε,N hence a different local density of vortices and a different location for the free-boundary that encloses them.

1.3. The local minimization method. The local minimization first relies on the same energy splitting introduced in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], which we now recall. For any A, set

A 1 := A -∇ ⊥ h ε,N
where h ε,N is the solution of the obstacle problem (1.2) above. Let us also define accordingly

J ε,N (u, A) := 1 2 Ω |(∇ -iA)u| 2 + |∇ × A -µ ε,N | 2 + (1 -|u| 2 ) 2 2ε 2 dx,
with µ ε,N as in (1.3), the configuration independent quantity (1.12)

G N ε := h ε,N -h ex 2 H 1 (Ω) + 2πN (m ε,N -1)
h ex , and finally the function ξ := h ex -h ε,N (implicitly depending on ε and N , note that it is nonnegative in Ω and vanishes on ∂Ω). We denote (1.13) ξ max = max

Ω ξ = h ex (1 -m ε,N ).
Lemma 1.3. For any (u, A), G ε (u, A) can be decomposed as:

(1.14) G ε (u, A) = G N ε + J ε,N (u, A 1 ) - Ω ξ µ(u, A 1 ) + 1 2 Ω (|u| 2 -1) |∇h ε,N | 2 .
Proof. This splitting formula was written with slightly different notation in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]. Decomposition (1.14) follows easily from the relations:

|(∇ -iA)u| 2 = |(∇ -iA 1 )u| 2 + |u| 2 ∇ ⊥ h ε,N 2 -2j(u, A 1 ) • ∇ ⊥ h ε,N ,
and

|∇ × A -h ex | 2 = |∇ × A 1 + ∆h ε,N -h ε,N + h ε,N -h ex | 2 = |∇ × A 1 -µ ε,N | 2 + |h ε,N -h ex | 2 + 2(∇ × A 1 -µ ε,N )(h ε,N -h ex ).
The last term in the right-hand side of (1.14) will be proven later to be o(1) for the configurations of interest to us.

As explained above for the case of smaller values of N , we still seek our local minimizers as configurations that minimize G ε among configurations with N vortices. The question is to define (following [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF]) a suitable set corresponding to this heuristic idea. This is achieved here by minimizing G ε over (1.15)

S β ε,N := (u, A) ∈ H 1 (Ω, C) × H 1 (Ω, R 2 ) s.t. J ε,N (u, A 1 ) ≤ πN log 1 ε √ h ex + N W m + βN ; Ω ξµ ≥ 2πN h ex (1 -m ε,N ) -βN ,
where β is chosen to be a small enough positive constant. The idea of this class is that it allows for at most N vortices when β is small, it leaves enough room for a particular test function to be admissible (the precision with which we can compute its energy is o(N ) while the error allowed is N), and it can easily be related to the Ginzburg-Landau energy G ε via (1.14). As in [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF], the task is then, exploiting the quantization of N , to show that min S β ε,N G ε is achieved in the interior of S β ε,N , thus providing a local minimizer of G ε , and to show it has the other desired properties. The class S β ε,N differs from those in [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF]. Indeed, following these works it would be tempting to try minimizing over the set of configurations (u, A) such that:

J ε,N (u, A 1 ) -πN log 1 ε √ h ex + N W m ≤ βN. However, if J ε,N (u, A 1 ) = πN log 1 ε √ hex + N W m -βN
we cannot conclude that u has strictly less than N vortices, which is crucial to assure that such a configuration on the boundary of the set would have an energy that exceeds that of a local minimizer. If one replaces βN by β log 1 ε √ hex a similar problem occurs; in this scenario one is able to prove that if (u, A) is in the corresponding lower boundary then it has at most N -1 vortices, but the -β log 1 ε √ hex in the value of J(u, A 1 ) makes the energy of the hypothetical configuration possibly drop below the one in Proposition 3.1, rendering that approach doomed. To overcome this difficulty (and this is a novelty of the paper) we proceed by bounding the total vorticity from below indirectly by requiring the elements of S β ε,N to satisfy

Ω ξµ(u, A 1 ) ≥ 2πh ex (1 -m ε,N )N -βN,
this implies there are at least N vortices as long as there are no vortices with negative degree outside the free boundary ω ε,N . This is proven to be the case in Proposition 4.1.

Finally, the question of treating the case of N larger than the order |log ε| remains open, we believe it would require much more precise estimates on the energy than those provided by [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] that we use here, and this seems out of reach for the moment.

The plan of the paper is as follows. In Section 2 we introduce notation and recall the Jacobian estimate. In Section 3 we construct an explicit configuration with N vortices uniformly distributed according to µ ε,N and a superconducting current almost minimizing W , we check the test configuration belongs to the admissible class, and by computing its energy G ε , we obtain an almost optimal upper bound. In Section 4 we analyze general properties of the elements of S β ε,N , in particular characterize their total vorticity as well as give lower bounds for the different components of the splitting of G ε . We conclude with the proof of Theorem 1.1 in Section 5, by showing minimizers do not lie on the boundary of the admissible class.

Acknowledgment: Both authors are supported by S.S's EURYI award.

Preliminaries

We will mostly work in blown-up coordinates at the characteristic lengthscale 1/ √ h ex . Let

(2.1)

ε ′ := ε h ex and x ′ := h ex x.
With this notation (1.5) becomes

N ≤ K |log ε ′ | in the regime where |log ε ′ | ≪ h ex .
The domains Ω ε and (ω ε,N ) ′ correspond to Ω and ω ε,N in the blown-up coordinates x ′ . We transform the rest of the variables in a natural fashion, that is

(2.2) u ′ (x ′ ) = u(x), ξ ′ (x ′ ) = ξ(x) and A ′ (x ′ ) = 1 √ h ex A(x),
for any magnetic potential A.

The blown up currents and measures are defined in terms of these quantities in a clear way, also denoted with a prime.

We define, for U ⊂ Ω ε , the sets: We now recall the basic concentration estimate relating the Jacobian and the total vorticity, à la Jerrard-Soner [START_REF] Jerrard | The Jacobian and the Ginzburg-Landau energy[END_REF].

(2.3) U := {x ∈ Ω ε /d(x, U ) ≤ 1} (2.
Theorem 2.1. (cf. [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], Theorem 5) Assume G ε (u ε , A ε ) (ε ′ ) -s , where s < 1. Then there exists a measure ν ′ ε that depends only on u ε (not on A ε ) of the form

ν ε = 2π i d i δ a i , for d i ∈ Z and a i ∈ Ω ε , with (2.6) µ ′ -ν ′ ε (C 0,1 0 (Ωε)) * (ε h ex ) 1 2 G ε (u ε , A ε ),
and there exists a constant C 1 > 0 such that for any measurable set

E ⊆ Ω ε (2.7) |ν ′ ε | (E) ≤ C 1 Ê e ε |log ε ′ | , where (2.8) e ε (u, A) := |∇ A u| 2 + |∇ × A| 2 + (1 -|u| 2 ) 2 2(ε ′ ) 2 .

Local minimization class and energy upper bound

As mentioned, we follow the idea of [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF], and Chapter 11 of [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF], and seek for local minimizers of G ε among configurations with "N vortices" by minimizing G ε over the class S β ε,N , cf. (1.15), with β to be determined later. From now on we write (u ε , A ε ) an element in the Argmin of G ε in S β ε,N . The fact that such a (u ε , A ε ) actually exists follows from classical arguments (a proof can be adapted from [START_REF] Serfaty | Stable configurations in superconductivity: Uniqueness, multiplicity and vortex-nucleation[END_REF] or Chapter 11 in [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF]). The task is then to show a minimizer cannot lie on the boundary.

We first show that the classes S β ε,N are non-empty and provide a priori estimates on the energy of a minimizer in those classes.

Proposition 3.1. The classes S β ε,N are non-empty. Furthermore, if (u ε , A ε ) is a minimizer of G ε over S β ε,N , then, as ε → 0, (3.1) G ε (u ε , A ε ) ≤ G N ε + πN |log ε ′ | + N W m -2πN h ex (1 -m ε,N ) + o(N ).
Proof. The proof relies on computing the energy of a carefully built test configuration ( u ε , A ε ).

The construction is carried out in Theorem 6 in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] where the argument makes use of the fact that in their case N corresponds to the number of vortices of a global minimizer. We show here that that the construction still works for N satisfying (1.5) and also make explicit why the test configuration belongs to S β ε,N , an issue not present in the global minimization setting of [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF].

In Theorem 6 of [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], a test configuration ( u ε , A ε ) is built in the following way.

Step 1:

For M > 0, consider a discrete set of points P ε , such that ∪ p∈Pε B(p, M ε) ⊂ ω ε,N and such that

|P ε | = N. A current  ε is built that satisfies (3.2) lim sup η→0 1 m ε,N h ex |ω ε,N | 1 2 Ω\ S p∈Pε B " p,η(m ε,N hex) -1 2 " |  ε | 2 + πN log η ≤ min A 1 W + o ε (1).
The current  ε is constructed under the assumption that Ω is convex (Proposition 4.2 and Corollary 7.4 in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]). In addition it is necessary that the coincidence set ω ε,N be well contained inside Ω. More precisely one must guarantee

(3.3) dist (ω ε,N , Ω c ) (m ε,N h ex ) -1 2
In [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] an explicit lower bound for dist (ω ε,N , Ω c ) was proven in the case where N is the number of vortices corresponding to global minimizers of G ε . In our case we recall that (1.5) implies m < 1 and therefore 1 -m ε,N is bounded from below by a positive number. From Appendix A in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], we know that

(3.4) ∇h ε,N ∞ h ex 1 -m ε,N .
Then since h ε,N is equal to m ε,N h ex on ω ε,N and to h ex on the boundary, ξ = h ex -h ε,N and (1.13), one has:

dist (ω ε,N , Ω c ) ξ max ∇h ε,N L ∞ 1 -m ε,N .
On the other hand, m ε,N ≥ m 0 > 0, 1 -m ε,N is bounded below, and

h ex ≫ 1, therefore 1 -m ε,N 1 m ε,N h ex ,
which proves (3.3) in our case.

Step 2:

The functions u ε and A ε are then built satisfying

(3.5) | u ε | = 1 on Ω \ ∪ p∈Pε B(p, M ε). (3.6) A ε := A 1,ε + ∇ ⊥ h ε,N , where A 1,ε is chosen to satisfy ∇ × A 1,ε = µ ε,N and (3.7) 1 2 B(p,M ε) ∇ d A 1,ε u ε 2 + (1 -| u ε | 2 ) 2 2ε 2 = π log M + γ + o M (1) + o ε (1),
for all p ∈ P ε , where o M (1) is a quantity that goes to zero as M → ∞, uniformly with respect to ε.

(3.8) 1 2 B " p,η(m ε,N hex) -1 2 " \B(p,M ε) ∇ d A 1,ε u ε 2 + (1 -| u ε | 2 ) 2 2ε 2 ≤ π log η m ε,N h ex M ε + O(η).
(3.9)

Ω µ( u ε , A 1,ε ) = Ω µ ε,N = 2πN, and µ( u ε , A 1,ε ) ≡ 0 on (ω ε,N ) c .
From this it easily follows that (

u ε , A ε ) ∈ S β ε,N . Indeed J ε,N ( u ε , A 1,ε ) = 1 2 Ω ∇ d A 1,ε u ε 2 + ∇ × A 1,ε -µ ε,N 2 + (1 -| u ε | 2 ) 2 2ε 2 = 1 2 Ω ∇ d A 1,ε u ε 2 + (1 -| u ε | 2 ) 2 2ε 2 .
Above, the second term disappears because of (3.6). The identity |∇ A u| 2 = |∇ |u|| 2 + |u| 2 |j| 2 implies, together with (3.5) that

∇ d A 1,ε u ε 2 = |  ε | 2 on Ω \ ∪ p∈Pε B(p, M ε).
Therefore, combining (3.2), (3.7) and (3.8), one gets

J ε,N ( u ε , A 1,ε ) ≤ πN |log ε| + 2πN min A 1 W + πN log 1 m ε,N h ex + N (γ + o ε (1) + o M (1) + O(η)) .
Next, passing to a subsequence, one can let M → ∞ as ε → 0, so that the above becomes, after taking η → 0 :

(3.10) J ε,N ( u ε , A 1,ε ) ≤ πN |log ε| + 2πN min A 1 W + πN log 1 m ε,N h ex + N γ + o(N ).
By assumption m ε,N → m, therefore N |log m ε,N | = N |log m| + o(N ). According to (1.17) and (1.18) of [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], by scaling, one has min

A 1 W = 1 m min Am W + 1 4 log m,
Thus, by definition of W m and ε ′ , (3.10) is equal to

(3.11) J ε,N ( u ε , A 1,ε ) ≤ πN |log ε ′ | + N W m + o(N ).
This shows that ( u ε , A 1,ε ) satisfies the upper bound in the definition of S β ε,N . To see it also satisfies the lower bound, we see by (3.9) that

Ω ξµ( u ε , A 1,ε ) = ω ε,N ξµ( u ε , A 1,ε ) = ξ max ω ε,N µ( u ε , A 1,ε ) = ξ max Ω µ( u ε , A 1,ε ) = ξ max Ω µ ε,N = 2πN h ex (1 -m ε,N ).
(3.12) By (3.11) and (3.12) we obtain that ( u ε , A ε ) ∈ S β ε,N . To conclude, from the splitting formula (1.14), we see that

G ε ( u ε , A ε ) = G N ε + J ε,N ( u ε , A 1,ε ) - Ω ξµ( u ε , A 1,ε ) + 1 2 Ω (| u ε | 2 -1) |∇h ε,N | 2 .
The last term is identically zero because |∇h ε,N | = 0 on ω ε,N , while | u ε | = 1 on (ω ε,N ) c by (3.5). Hence, by (3.11) and (3.12) again, we see that

G ε ( ûε , Âε ) ≤ G N ε + πN |log ε ′ | + N W m -2πN h ex (1 -m ε,N ) + o(N )
, and the proposition is proved.

In the next section, the lower bound component of Theorem 1.1 requires a very precise approximation of the vorticity measure by ν ε . We also need a better control on the rightmost term in the splitting formula (1.14). Lemma 3.2. Let (u ε , A ε ) be a minimizing pair on S β ε,N . Assume N and h ex satisfy (1.5) and (1.6). Then there exists an α > 0 such that

(3.13) µ ′ -ν ′ ε (C 0,1 0 (Ωε)) * = o(ε α ) and (3.14) Ω (1 -|u ε | 2 ) |∇h ε,N | 2 = o(1
µ ′ -ν ′ ε (C 0,1 0 (Ωε)) * ε 1 2 h 1 4 ex • h 2 ex ε 1 2 -9 4 s 0 ,
The fact that this is o(ε α ) for some α > 0 follows from (1.6). Thus, we have proved (3.13), and (1.7) also follows. For the second assertion, Cauchy-Schwarz yields

Ω (|u ε | 2 -1) |∇h ε,N | 2 ≤ Ω (1 -|u ε | 2 ) 2 1 2 ∇h ε,N 2 ∞ |Ω| 1 2 ε 2 G ε (u ε , A ε ) ∇h ε,N 2 ∞
Using and the bound h

2 ex G ε (u ε , A ε ), the above leads to (1 -|u ε | 2 ) |∇h ε,N | 2 εh 3 ex ≤ ε (1-3s 0 ) = o(1),

by (1.6).

The case h ex |log ε| can be dealt with similarly.

Further Properties of minimizers and lower bound for

G ε (u ε , A ε )
The next proposition, which is the crucial part of the proof, shows that the elements of S β ε,N

have N vortices and also that the upper bound in (3.1) is sharp. We recall that Ω ε and ω ′ ε,N correspond to Ω and ω ε,N in the blown up coordinates. At this point, it is convenient to introduce the notion of "vorticity adapted" family of functions. To that end, for a function ψ ε , let

ω ψε := x ∈ Ω ε s.t. ψ ε (x) = 1 2 |log ε ′ | . We will restrict to considering families {ψ ε } ε>0 such that (4.1) ∇ψ ε ∞ = o(|log ε ′ |).
Now, define the class:

VA := {ψ ε } ε>0 such that ψ ε : Ω ε → R, ψ ε satisfies (4.1), ω ψε ⊃ ω ′ ε,N ; 0 ≤ ψ ε (x) < 1 2 |log ε ′ | if x ∈ Ω ε \ ω ψε ; ψ ε = 0 on ∂Ω ε
Abusing notation, we sometimes call ψ ε a vorticity adapted function when {ψ ε } ε>0 is a vorticity adapted family. This notion will help us in making use of a family of estimates available in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] in a transparent way. The proof in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] of such estimates, is presented for a particular vorticity adapted function, while here we use the estimates in their full generality. Ultimately, these estimates, together with the quantization of the vorticity measure, will prove that S β ε,N consists only of functions with N essential vortices.

In what follows, a + or -superscript denotes the positive resp. negative part of a function or measure. Proposition 4.1. Let N and h ex be sequences satisfying (1.5) and (1.6). Assume (u ε , A ε ) ∈ S β ε,N where β > 0 is small enough but fixed independent of ε. Then, for ε small, we have: (i) There are no vortices with negative associated degree outside the free boundary, i.e.:

ν ε = ν + ε on (ω ε,N ) c . (ii)
The total vorticity is N, in other words:

ν ε (Ω) = 2πN. (iii) Ω ξµ(u ε , A ε 1 ) ≤ 2πN h ex (1 -m ε,N ) + o(1). (iv) J ε,N (u ε , A ε 1 ) ≥ πN |log ε ′ | + N W m + o(N ). Proof. Recall that (u ε , A ε ) ∈ S β ε,N means that (4.2) J ε,N (u ε , A ε 1 ) ≤ πN |log ε ′ | + N W m + βN, and
(4.3) Ω ξµ(u ε , A ε 1 ) ≥ 2πN h ex (1 -m ε,N ) -βN.
The proof relies on lower bounds on quantities of the form

(4.4) J ε,N (u ε , A ε 1 ) - Ω ψ ε ( h ex •)µ(u ε , A ε 1 ),
where {ψ ε } ε>0 ∈ VA. More precisely, in Steps 3-6 below we derive assertions (i)-(iv) with the aid of sufficiently good lower bounds on (4.4) that are available for pairs (u ε , A ε ) such that their superconducting current j is divergence free. We first show that we can reduce to this case by proving (4.6) below for a A ε 0 such that j(u ε , A ε 0 ) is divergence free.

Step 1: First we claim we may assume that (u ε , A ε ) satisfies

(4.5) div j(u ε , A ε ) = 0.
This assumption is justified by the following: we claim there is a A ε 0 such that div j(u ε , A ε 0 ) = 0 and

J ε,N (u ε , A ε 1 ) - Ω ψ ε ( h ex •) µ(u ε , A ε 1 ) ≥ J ε,N (u ε , A ε 0,1 ) - Ω ψ ε ( h ex •) µ(u ε , A ε 0,1 ) + o(1), (4.6)
holds for any {ψ ε } ε>0 ∈ VA. Indeed, letting

A ε 0 be such that G ε (u ε , A ε 0 ) = min A∈H 1 (Ω,R 2 ) G ε (u ε , A) (such a minimizer is easily seen to exist), by minimality (u ε , A ε 0 ) satisfies the second Ginzburg- Landau equations, that is -∇ ⊥ (∇ × A ε 0 ) = (iu ε , ∇ A ε 0 u ε ), whence j(u ε , A ε 0 ) is divergence free. By definition of A ε 0 , we have (4.7) G ε (u ε , A ε 0 ) ≤ G ε (u ε , A ε ) max{h
Ω (1 -|u ε | 2 ) |∇h ε,N | 2 = o(1).
We can use the splitting formula (1.14) to conclude:

G N ε + J ε,N (u ε , A ε 1 ) - Ω ξµ(u ε , A ε 1 ) ≥ G N ε + J ε,N (u ε , A ε 0,1 ) - Ω ξµ(u ε , A ε 0,1 ) + o(1).
From this, we can assert that

J ε,N (u ε , A ε 1 ) - Ω ψ( h ex •)µ(u ε , A ε 1 ) ≥ J ε,N (u ε , A ε 0,1 ) - Ω ψ ε ( h ex •)µ(u ε , A ε 0,1 ) - Ω ψ ε ( h ex •) -ξ µ(u ε , A ε 1 ) -µ(u ε , A ε 0,1 ) + o(1).
If the field h ex is not too large, the Jacobian estimate (2.6) implies that the last term is o(1).

To see this, recall that the measure ν ε in (2.6) does not depend on A ε and therefore:

Ω ψ( h ex •) -ξ µ(u ε , A ε 0,1 ) -µ(u ε , A ε 1 ) ≤ Ω ψ( h ex •) -ξ µ(u ε , A ε 0,1 ) -dν ε + Ω ψ( h ex •) -ξ (dν ε -µ(u ε , A ε 1 )) ψ( h ex •) -ξ ∞ • max ν ε -µ(u ε , A ε 0,1 ) (C 0,1 0 (Ω)) * , ν ε -µ(u ε , A ε 1 ) (C 0,1 0 (Ω)) * ( ξ ∞ + ∇ψ ∞ ) • √ ε ′ G ε (u ε , A ε ) (h ex + h 1 2 ex |log ε|) ε 1 2 h 1 4 ex h 2 ex = o(1),
thanks to (1.6) and (4.7). This proves the claim.

Step 2: Lower bound with vorticity adapted functions. We switch to blown-up coordinates in order to appeal to the proof of Proposition 6.1 in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]. Recalling that the Jacobian estimate allows us to approximate µ ′ by ν ′ ε , we can replace bounding below (4.4) by bounding below

J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) - Ωε ψ ε dν ′ ε .
But it was proved in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] that there exists a measure g ′ ε depending only on ((u ε ) ′ , (A ε ) ′ ) such that for any vorticity adapted function ψ ε , and for some constant C 1 > 0,

J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) - Ωε ψ ε dν ′ ε ≥ 1 6 Ωε 1 - 2ψ ε |log ε ′ | e ε + 1 C 1 (g ′ ε ) + ( ω ′ ε,N ) c +g ′ ε ( ω ′ ε,N ) c + o( ω ′ ε,N ), (4.8)
where e ε is as in (2.8).

The argument can be found in (6.28)-(6.35) of [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] where ζ ′ ε can be replaced by ψ ε ; indeed it can be readily checked that the only fact about ζ ′ ε that was used in the proof was that it was vorticity adapted. Note that here we also use the fact that div j(u ε , A ε ) = 0, obtained in Step 1.

We also record the followings facts from [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF][START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] 1. There exists a family of disjoint balls {B i } i∈I of bounded total radius such that the support of

ν ′ ε is contained in ∪ i∈I B i , a universal constant C 2 and functions {g B i ε } such that if ν ′ ε (B i ) < 0 (4.9) B i g B i ε ≥ 1 8 (|log ε ′ | -C 2 ) |ν ε (B i )| ≥ 1 16 |log ε ′ | |ν ε (B i )|
(The family {B i } i∈I is built in [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF], where (4.9) is proved in Proposition 3.2)

2. There exists a universal constant C 3 such that

(4.10) (g ′ ε ) + ≥ 3 4   {i, ν ′ ε (B i )<0} g B i ε -C 3 1 B i  
where both sides are understood as measures. (This is a consequence of the definition of g ′ ε in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], combined with (3.22) in [START_REF] Sandier | Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement[END_REF] and (4.9)).

3. Let P ε be the probability measure on L p loc (R 2 , R 2 ) which is the push-forward of the normalized uniform measure on ω ε,N by the map

x → √ h ex j ε ( √ h ex (x + •)). Let x ′ ε ∈ ω ′ ε,N where d(x ′ ε , (ω ε,N ) c ) → +∞.
By the definition of g ′ ε , one can see as in the proof of Proposition 6.1 in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] that if

∀R > 0, lim sup ε→0 - B(x ′ ε ,R) dg ′ ε (x ′ ε + •) < +∞ then assumption (4.5) together with m ε,N → m implies that j((u ε ) ′ , (A ε 1 ) ′ )(x ′ ε + •) converges in L p loc (R 2 , R 2 )
for any p < 2 to some j ∈ A m . Thus, since the field h ex is not too large (cf. (1.6)), Proposition 6.3 in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] tells us that P ε converges weakly to a translation invariant probability measure P on L p loc (R 2 , R 2 ) such that P -a.e. j ∈ A m and furthermore (recalling the definitions (1.11) and (2.5))

(4.11) g ′ ε ( ω ′ ε,N ) ≥ N 2π m W (j) dP (j) + γ + o(1) + o( ω ′ ε,N ) ≥ N W m + o(N ),
since, in view of (1.3)

ω ′ ε,N = |ω ε,N | h ex = 2πN m ε,N ≤ 2πN m 0 = O(N ). (4.12)
Using (4.8) together with (4.11)-(4.12), we have thus found that for any ψ ε ∈ VA, (4.13)

J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) - Ωε ψ ε dν ′ ε ≥ 1 6 Ωε 1 - 2ψ ε |log ε ′ | e ε + N W m + o(N ).
Step 3: All vortices of u ε are contained in a neighborhood of ω ′ ε,N .

We will quite often need to use the fact that for any measurable set A,

(4.14) ν ′ ε (A) ∈ 2πZ, while (ν ′ ε ) + (A) , (ν ′ ε ) -(A) , |ν ′ ε | (A) ∈ 2πN.
In what remains, α is set to be the constant

α := |log ε ′ | 2h ex (1 -m ε,N )
.

Note that α depends on ε and N but we omit to indicate this dependence. Also, from here on, all the primed quantities denote the blown up versions of the unprimed ones, according to the transformations (2.1) -(2.2). We begin by noticing that αξ ′ ∈ VA. Indeed, since h ex ≫ 1, and m = lim ε→0 m ε,N < 1 one can readily see, using (3.4) and the fact that h ex (1 -m ε,N ) ≫ 1, that (4.15)

∇(αξ ′ ) ∞ = O |log ε ′ | h ex (1 -m ε,N ) • ∇ξ ∞ • 1 √ h ex = O |log ε ′ | (h ex ) 3 2 (1 -m ε,N ) • ∇h ε,N ∞ O |log ε ′ | (h ex ) 3 2 (1 -m ε,N ) h ex 1 -m ε,N O |log ε ′ | h ex (1 -m ε,N ) = o(|log ε ′ |).
By (4.2), (4.3) and definition of α, we find that

(4.16) J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) -α Ωε ξ ′ µ ′ ((u ε ) ′ , (A ε 1 ) ′ ) ≤ πN |log ε ′ | + N W m + βN -α(2πN h ex (1 -m ε,N ) -βN ) = (1 + α)βN + N W m .
Conversely, the bound (4.13) applied to ψ ε = αξ ′ together with (4.16) will allow us to rule out the presence of vortices with negative degrees where ξ ′ is small. In turn, we shall see that this implies an upper bound on the total vorticity ν ε (Ω ε ).

As announced, we apply (4.13) to ψ ε = αξ ′ which is indeed vorticity adapted, combining with the Jacobian estimate (2.6), (4.16) and (4.13) we obtain that

(1 + α)βN + N W m ≥ J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) -α Ωε ξ ′ µ ′ ((u ε ) ′ , (A ε 1 ) ′ ) ≥ J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) -α Ωε ξ ′ dν ′ ε + O ∇(αξ ′ ) ∞ µ ′ -ν ′ ε (C 0,1 0 (Ωε)) * ≥ 1 6 Ωε 1 - 2αξ ′ |log ε ′ | e ε + N W m + o(N ) + O |log ε ′ | ε 1 2 h 1 4 ex G ε (u ε , A ε ) . (4.17)
The last term is o(1) because G ε (u ε , A ε ) max{h 2 ex , h ex |log ε ′ |} and h ex satisfies (1.6). Thus, we obtain

(4.18) 1 6 Ωε 1 - 2αξ ′ |log ε ′ | e ε ≤ β(1 + α)N + o(N ).
For any Z > 0 small, we introduce the following neighborhood of ∂Ω:

U Z :=    x ′ /ξ ′ (x ′ ) ≤   1 - 6C 1 K + c|Ω|(1-m ε,N ) 2π Z π   h ex (1 -m ε,N )    ,
where C 1 and K are the constants in (2.7) and (1.5) respectively. From the definition of α, it follows that

1 - 2αξ ′ |log ε ′ | ≥ 6C 1 K + c|Ω|(1-m ε,N ) 2π β π 1 U β .
Using this together with (2.7), we obtain (cf. (2.4))

Ωε 1 - 2αξ ′ |log ε ′ | e ε ≥ 6C 1 K + c|Ω|(1-m ε,N ) 2π β π U β e ε ≥ 6 K + c|Ω|(1-m ε,N ) 2π β π |ν ′ ε | ( Ǔ β ) |log ε ′ | . (4.19)
On the other hand, by (1.5)

, N ≤ K |log ε ′ | and αN |log ε ′ | ≤ c|Ω| 2π(1-m ε,N ) ,
|( Ǔ β ) π ≤ 1 + o N |log ε ′ | < 2 (for ε small enough), which implies (4.20) |ν ′ ε | ( Ǔ β ) = 0,
in light of (4.14).

Step 4: There are no vortices with negative degree outside ω ′ ε,N

We turn to the proof of item (i). We invoke (4.8) applied to αξ ′ and (4.11)-(4.12) again to get

J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) - Ωε αξ ′ dν ′ ε ≥ 1 C 1 (g ′ ε ) + (( ω ′ ε,N ) c ) + N W m + o(N ),
which, arguing as in (4.17) yields

(4.21) (1 + α)βN + o(N ) ≥ 1 C 1 (g ′ ε ) + (( ω ′ ε,N ) c
).

Recall that from (1.5) we have αβN

β |log ε ′ | N hex β |log ε ′ | hence (1 + α)βN + o(N ) β|log ε ′ | + o(|log ε ′ |).
On the other hand, from (4.9) and (4.10) one can bound (g ′ ε ) + (( ω ′ ε,N ) c ) from below by (letting C denote a generic positive constant)

(g ′ ε ) + (( ω ′ ε,N ) c ) ≥ C|log ε ′ | ν - ε (( ω ′ ε,N ) c ) - C 3 2π i∈I |B i | |log ε ′ | ν - ε (( ω ′ ε,N ) c
) .

Going back to (4.21), this implies

β + o(1) ≥ C ν - ε ( ω ′ ε,N ) c
, where C is a positive constant independent of β and ε. Since |ν - ε | ∈ 2πN, we may choose β small so that for any ε small enough, it imposes

(4.22) (ν ′ ε ) -( ω ′ ε,N ) c = 0. Since ( ω ′ ε,N ) c contains (ω ′ ε,N ) c
, we have thus proved item (i).

Step 5: The total vorticity of u ε is N

We proceed, once more, applying (4.13) only this time to a vorticity adapted function ψ ε such that ω ψε ⊇ Ω ε \ U β . Such a function exists. Indeed, by (4.15)

, for x ∈ ∂(Ω ε \U β ) we have |αξ ′ (x)| ≤ o(|log ε ′ |)d(x, ∂Ω ε ) while αξ ′ (x) = 1 2 (1 -O(1)β)|log ε ′ |. It follows that d(∂(Ω ε \U β ), ∂Ω ε ) ≫ 1 as ε → 0 (if β is a small enough constant), and thus d(∂( Ω ε \U β ), ∂Ω ε ) ≫ 1.
One can therefore build a family of functions ψ ε which are equal to 1 2 |log ε ′ | in Ω ε \U β = Ω ε \ Ǔ β , which vanish on ∂Ω ε and satisfy ∇ψ ε ∞ = o(|log ε ′ |), i.e. belongs to VA.

Applying (4.13) to that family yields

J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) - Ωε ψ ε dν ′ ε ≥ N W m + o(N ).
We can then appeal to (3.13) again to obtain

J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) - Ωε ψ ε µ ′ ((u ε ) ′ , (A ε 1 ) ′ ) ≥ J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) - Ωε ψ β ε dν ′ ε + O( ∇ψ ε ∞ ε α ) ≥ N W m + o(N ). Since (u ε , A ε ) ∈ S β ε,N , we must have πN |log ε ′ | + βN ≥ J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) -N W m ≥ Ωε ψ ε dν ′ ε + o(N ). (4.
ψ ε dν ′ ε = Ωε\ Ǔ β ψ ε dν ′ ε = Ωε\ Ǔ β 1 2 |log ε ′ | dν ′ ε = 1 2 |log ε ′ | ν ′ ε (Ω ε \ Ǔ β ).
Plugging this into (4.23) leads us to

N + β π N |log ε ′ | ≥ 1 2π ν ′ ε (Ω ε \ Ǔ β ) + o N |log ε ′ | .
Since N is taken satisfying (1.5), if β is chosen small enough, the above implies that for ε small enough,

(4.25) N ≥ ν ′ ε (Ω ε \ Ǔ β ) 2π = ν ′ ε (Ω ε ) 2π ,
because of (4.14) and (4.20). We now show that there is actually equality in (4.25). To that end we first prove

Ωε ξ ′ µ((u ε ) ′ , (A ε 1 ) ′ ) ≤ ν ε (Ω ε )ξ ′ max + o(1).
This is not immediately clear since ξ is not constant and ν ε is not necessarily equal to ν + ε . However, from (1.6) and (2.6), we have

ξ ′ µ((u ε ) ′ , (A ε 1 ) ′ ) = Ωε ξ ′ dν ′ ε + O(ε 1 2 h 1 4 +2 ex |∇ξ ′ | ∞ ) = g ω ′ ε,N ξ ′ dν ′ ε + c g ω ′ ε,N ) ξ ′ dν ′ ε + o(ε α ).
On the other hand, from (4.22),

c ( g ω ′ ε,N ) ξ ′ dν ′ ε = c ( g ω ′ ε,N ) ξ ′ d(ν ′ ε ) + ≤ ξ max (ν ′ ε ) + (( ω ′ ε,N ) c ).
Also,

( g ω ′ ε,N ) ξ ′ dν ′ ε = ξ max ν ′ ε ( ω ε,N ′ ) because ξ ′ = ξ max on ( ω ′ ε,N ).
Collecting these estimates we are led to

(4.26) Ωε ξ ′ µ ′ ((u ε ) ′ , (A ε 1 ) ′ ) ≤ ξ max ν ′ ε ( ω ′ ε,N ) + (ν ′ ε ) + (( ω ′ ε,N ) c ) + o(1) = ξ max ν ′ ε (Ω ε ) + o(1)
where the last equality follows from (4.22). Consequently ν ′ ε (Ω ε ) > 2π(N -1), for otherwise we would have (cf. (1.13))

Ωε ξ ′ µ ′ ((u ε ) ′ , (A ε 1 ) ′ ) ≤ 2π(N -1)ξ max + o(1) = 2π(N -1)h ex (1 -m ε,N ) + o(1)
and the right-hand side is strictly smaller than 2πN h ex (1 -m ε,N ) -βN, if β is small enough, by virtue of (1.5). This contradicts (u ε , A ε 1 ) ∈ S β ε,N . Therefore, ν ′ ε (Ω ε ) ≥ 2πN (recall ν ′ ε (Ω ε ) ∈ 2πZ.) Thus, with (4.25), we have proved item (ii).

Step 6: Proof of (iii) and (iv) Item (iii) follows readily from (4.26) since we already know that ν ′ ε (Ω ε ) = 2πN. For item (iv), note that from (4.20), (4.23) and (4.24) we deduce

J ′ ε,N ((u ε ) ′ , (A ε 1 ) ′ ) ≥ Ωε ψ ε dν ′ ε + N W m + o(N ) = 1 2 |log ε ′ | ν ′ ε (Ω ε \ Ǔ β ) + N W m + o(N ) = πN |log ε ′ | + N W m + o(N ).
The proposition is proved.

Conclusion

In this section we finish the proof of our main result.

Proof of Theorem 1.1

As mentioned earlier, a minimizer in S β ε,N exists, so to show it actually corresponds to a local minimizer, it only remains to prove (u ε , A ε ) / ∈ ∂S β ε,N . We decompose ∂S We prove item (iii). It essentially follows the proof of (1.34) in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]. First, arguing as in item 2 of Proposition 6.1 in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF], replacing F ′ ε with J ′ ε,N -α Ωε ξ ′ µ ′ , we can sandwich the term Ω |∇ × A 1,ε -µ ε,N | 2 in J ε (u ε , A 1,ε ) between the upper and lower bound of (4.17) and hence control it by O(N ), i.e.

(5.3)

∇ × A 1,ε -µ ε,N L 2 √ N
Using the same item of Proposition 6.1 in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF] also to control, for 1 < p < 2,

Ωε |j(u ′ ε , A ′ 1,ε )| p ≤ C p J ′ ε,N (u ′ ε , A ′ 1,ε ) -α Ωε ξ ′ µ ′ + |ω ′ ε,N | ≤ C(N + |ω ′ ε,N |).
Rescaling we find that for any 1 < p < 2,

(5.4) j(u ε , A 1,ε ) L p (Ω) h thanks to (4.12), then (1.5) giving N h ex . On the other hand, by direct calculation we have

µ(u ε , A ε ) -µ(u ε , A 1,ε ) = ∇ × ((1 -|u ε | 2 )∇ ⊥ h ε,N )
from (1.6)and (3.4) 1), from which one obtains (5.5) µ(u ε , A 1,ε ) -µ(u ε , A ε ) W -1,p = o(1).

(1 -|u ε | 2 ) |∇h ε,N | L 2 ε 2 G ε • |∇h ε,N | ∞ ε max{h 2 ex , h ex |log ε ′ |} = o(
We may now write, using the triangle inequality

µ(u ε , A ε ) -µ ε,N W -1,p ≤ µ(u, A ε ) -µ(u ε , A 1,ε ) W -1,p + µ ε,N -∇ × A 1,ε W -1,p + µ(u ε , A 1,ε ) -∇ × A 1,ε W -1,p N
where we have used (5.3) (together with an embedding inequality), (5.5) and the fact that µ(u ε , A 1,ε )-∇×A 1,ε = ∇×j(u ε , A 1,ε ) combined with (5.4). Consequently, item (iii) is proved. We turn to the proof of (iv). Since αξ ′ ∈ VA, (4.8), (4.11) and (1.9) allow us to assert

N W m + o(N ) ≥ N 2π m W (j) dP (j) + γ + o(1)
which in turns translates into W m ≥ 2π m W (j) dP (j) + γ.

To conclude, since P -a.e. j ∈ A m (as given by [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF]) this implies P -a.e. j minimizes W over A m . The result follows.

  4) Ǔ := {x ∈ U/d(x, ∂U ) ≥ 1} and (2.5) U := {x ∈ U/d(x, ∂U ) ≥ 2}.

  23) Now, by (4.20) and definition of ψ ε (4.24) Ωε

  β ε,N = (∂S β ε,N ) + (∂S β ε,N ) -where (∂S β ε,N ) + := (u, A) ∈ H 1 × H 1 such that:J ε,N (u, A 1 ) = πN |log ε ′ | + N W m + βN and (∂S β ε,N ) -:= (u, A) ∈ H 1 × H 1 such that: Ω ξµ(u, A 1 ) = 2πN h ex (1 -m ε,N ) -βN .Since (u ε , A ε ) is a minimizer in S β ε,N , we may apply Lemma 3.2 and Proposition 4.1. By (1.14) and (3.14), we have(5.1) G ε (u ε , A ε ) = G N ε + J ε,N (u ε , A 1,ε ) -Ω ξµ(u ε , A 1,ε ) + o(1).If (u ε , A ε ) ∈ (∂S β ε,N ) + then by (iii) in Proposition 4.1, we have G ε (u ε , A ε ) ≥ G N ε + πN |log ε ′ | + N W m + βN -2πN h ex (1 -m ε,N ) + o(N ). If instead, (u ε , A ε ) ∈ (∂S β ε,N ) -, invoking (iv) in Proposition 4.1 yields G ε (u ε , A ε ) ≥ G N ε + πN |log ε ′ | + N W m -(2πN h ex (1 -m ε,N ) -βN ) + o(N ).In either case there is a contradiction with the upper bound (3.1), so necessarily (uε , A ε ) / ∈ ∂S β ε,N , whence (u ε , A ε ) is a local minimizer.From (5.1), (iii) and (iv) in Proposition 4.1, we must have(5.2) G ε (u ε , A ε ) ≥ G N ε + πN |log ε ′ | + N W m -2πN h ex (1 -m ε,N ) + o(N ). Thus, the asymptotic expansion (1.9) follows from (5.2) and (3.1). (1.7), (1.8) were proved in Lemma 3.2 and Proposition 4.1 respectively.

  Proof. Let us assume first h ex ≫ |log ε| . It is easy to see that in this case G N

	the rest of the positive terms in (3.1) are o(h 2 ex ). Therefore	ε	h 2 ex , while
	G ε (u ε , A ε ) h 2 ex .
	From (2.6) and (1.6) we can deduce
	(3.15)

).

  2 ex , h ex |log ε ′ |}, which follows from the decomposition (1.14) together with G N ε max{h 2 ex , h ex |log ε ′ |} and the definition of S β ε,N . As in the proof of Lemma 3.2 this upper bound implies

  so combining (4.18) and (4.19), we get |ν ′

	ε

Here j ε is extended by 0 outside Ω.