Bertrand Huneau

Isaure Masquelier

Yann Marco

Vincent Le Saux

Simon Noizet

Clémentine Schiel

Pierre Charrier

Émentine Cl

Schiel

Fatigue crack initiation in a carbon black-filled natural rubber

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The total life of industrial rubber parts subjected to fatigue can be decomposed in two phases: crack initiation and crack propagation. Consequently, two approaches are generally considered to analyze and to predict the fatigue life of rubber: the crack initiation (nucleation) approach and the crack propagation approach, as proposed by Mars and Fatemi. 1 The knowledge of the size of initial crack/flaws (generally denoted a 0 or c 0 ) is particularly important for the crack propagation approach because it explicitly considers preexisting cracks or flaws. 1 Moreover, the integration of the propagation law requires this value and allows for the prediction of fatigue life, as originally shown by Gent et al. 2 But what really represents this initial flaw size at a microstructural scale? From the early studies on rubber fatigue, the initial flaws responsible for initiation of fatigue cracks are generally considered inhomogeneities, as called by Beatty, 3 reminiscent of the seminal concept of ''flaws'' proposed by Griffith to explain the mechanical weakness of structural materials. 4 For filled natural rubber (NR), which exhibits excellent fatigue properties, 3,[5][6][7] these inhomogeneities can be filler agglomerates (silica or carbon black [CB]) or oxides, such as zinc oxide. [8][START_REF] Bennani | Title[END_REF][START_REF] Jago | Fatigue Life of Rubber Components: 3D Damage Evolution from X-ray Computed Microtomography[END_REF][START_REF] Le Cam | [END_REF] The size of these inclusions ranges from a few microns to a few hundreds of microns. This was confirmed in a previous study in which the damage of a sample was followed throughout its fatigue life. [START_REF] Huneau | Fatigue Damage in Carbon Black Filled Natural Rubber Investigated by X-ray Microtomography and Scanning Electron Microscopy[END_REF] However, the precise damage mechanisms leading to the initiation have not been well understood. A recent study has allowed us to clarify this point. [START_REF] Masquelier | Title[END_REF] The damage induced by fatigue loading in rubber can be investigated by different experimental techniques. One can cite relatively recent techniques in the field of materials characterization such as X-ray computed microtomography [START_REF] Huneau | Fatigue Damage in Carbon Black Filled Natural Rubber Investigated by X-ray Microtomography and Scanning Electron Microscopy[END_REF][START_REF] Jago | [END_REF]15 but also a more classical observation technique: scanning electron microscope (SEM). SEM remains indeed a powerful tool to observe either fatigue fracture surfaces [START_REF] Le Cam | [END_REF]16 or interrupted fatigued samples to acquire more precise information about the mechanisms of crack initiation and crack propagation. 8,[17][18][19][20] Nevertheless, these studies rarely focus only on crack initiation, probably because this phenomenon is not easily predictable in terms of location (where the crack is initiated) or time (after how many cycles).

The first objective of the present study is to obtain some statistical information about fatigue crack initiation sites: location, distribution, nature, and evolution, by SEM observations of the whole damaged surface of the samples at different moments in the fatigue life. The second objective is to describe the fatigue crack initiation mechanisms for the two main types of inclusions responsible for it.

EXPERIMENTAL PROCEDURE MATERIALS AND SAMPLES

The investigated elastomer is an NR filled with CB and vulcanized with sulfur. The chemical composition is given in Table I. Most of the results presented here are obtained on an NR filled with CB N550 type, but some results presented in the section ''Fatigue Crack Initiation Mechanism for CB Agglomerates'' are obtained with other CB types: N220, N326, N375, and N772. For the latter type, the CB amount is raised to 50 phr in order to have a stiffness comparable with one of the other materials.

In the following, the material's designation refers to its CB content and CB type. For example, the main studied compound is referred to as NR43-N550.

FATIGUE TESTS

Fatigue tests are performed at Trelleborg Vibracoustic on a specific device on hourglassshaped specimens (referred to as AE2 in the following). AE2 geometry is shown in Figure 1a; the median zone of those specimens is 10 mm in diameter. Because of this particular geometry, the strain field is very heterogeneous. The maximum strains are reached at the surface of the sample and are calculated by finite element analysis (FEA). Figure 1b is an example of an FEA result showing the maximum principal strain field for a local maximum value of 100%. Fatigue results are plotted with the local maximum strain in the median zone (slice 1, Figure 2). The lower strains corresponding to slices 2/2 0 and 3/3 0 will be considered to investigate the influence of strain level on the number of initiation sites (Figure 2).

Fatigue tests are monitored with prescribed displacement and for a minimum displacement equal to 0 mm. No mechanical accommodation is performed on samples before fatigue tests. The 2 frequency of the tests varies from 5 Hz to 1 Hz in order to limit heat buildup. By using 25 specimens, a Wöhler curve, shown in Figure 3a, is plotted with the maximum strain value (at the surface of slice 1) as a function of the number of cycles to the fatigue end of life of the specimens. This number is called Ni and is detected by a loss of specimen stiffness (see Ostoja-Koczynski et al. [START_REF] Ostoja-Kuczynski | Crack Initiation in Filled Natural Rubber: Experimental Database and Macroscopic Observations[END_REF] for details).

The Wöhler curve is used as a reference to define the strain level and the number of cycles for each sequence of the interrupted fatigue tests. Figure 3b shows a typical evolution of the maximum force during the fatigue tests with the number of cycles. Ni, as well as the three percentages of fatigue life (20% Ni, 40% Ni, and 60% Ni ) considered in the present study, are plotted on this curve.

In a previous study, [START_REF] Huneau | Fatigue Damage in Carbon Black Filled Natural Rubber Investigated by X-ray Microtomography and Scanning Electron Microscopy[END_REF] we investigated the damage evolution along the fatigue life by following the same sample. Here, the following protocol is chosen for the interrupted fatigue tests: each sample is subjected to one of the three strain levels (60%, 100%, and 200%) and to one of the three percentages of fatigue life (20% Ni, 40% Ni, and 60% Ni ). Consequently, nine samples were observed for NR43-N550 and 12 additional samples for the complementary tests on the influence of CB type. This particular protocol is chosen to have statistical information and to avoid the cumulative damage that can be caused by the electron beam of the SEM, when one follows the same sample. Finally, the very low scattering of fatigue lives for the considered NR (see Figure 3a) justifies this protocol.

SEM OBSERVATIONS AND ENERGY DISPERSIVE SPECTROMETER ANALYSIS

The SEM investigations are performed with a JEOL 6060-LA. The acceleration voltage is 20 kV, and the images are taken by using the secondary electrons signal. The acceleration voltage was chosen to be relatively high when one considers observation of elastomers. However, thanks to the relatively high CB content, the material is conductive enough to support his voltage. The previously described procedure consisting in observing each sample only once also justified the use of such a high voltage, which could be involved in fatigue life reduction in the case of multiple observations on the same sample. The chemical composition of inclusions is determined thanks to an energy dispersive spectrometer (EDS) of X-rays. Thanks to the use of the high acceleration voltage, the detection of the high atomic number element is facilitated.

Observations are performed on fatigued samples stretched by 3 mm by using four polymethyl methacrylate beams. The observed zone, which is 3.75 mm in height and 6.9 mm in width, is analyzed throughout the five slices defined in Figure 2. To observe the whole circumference of the samples, this operation is repeated four times, after successive rotations of 908. For one sample, 140 SEM images are taken (seven per slices3five slices3four sides), which makes 1260 SEM images for NR43-N550 (140 images 3 nine samples). The magnification is 3130 for each image, which enables the detection of inclusions larger than 5 lm. Figure 4 gives an example of one of these images, which were used for the statistical analysis. Higher magnification SEM images are performed when necessary.

RESULTS AND DISCUSSION

DAMAGE MECHANISMS IN FATIGUE

For the considered experimental conditions and elastomers, the fatigue loading always induces a multicracking phenomenon. The numerous cracks observed are initiated either at the parting line or out of the parting line, as illustrated in Figure 5. This was first observed on fracture surfaces and then confirmed by interrupted fatigue tests. These interrupted tests also reveal that initiation of cracks starts very early in fatigue life and is followed by a relatively slow propagation. The multiinitiation and the low fatigue crack growth rates can partly explain the very low scattering obtained for fatigue lives (Figure 3a).

INFLUENCE OF THE NUMBER OF CYCLES AND OF THE MAXIMUM STRAIN

The density of surface cracks (only cracks longer than 5 lm are considered here) is plotted in Figure 6 for the three percentages of the fatigue life Ni (20%, 40%, 60%) and for the three maximum strain values: 60%, 100%, 200%. These maximum values are related to the observations made for slice 1 (Figure 2). In addition, the results obtained in slices 2/2 0 and in slices 3/3 0 are plotted with their corresponding strain calculated by FEA. Figure 6 clearly illustrates that the density of cracks increases with the percentage of fatigue life, which means that new crack initiation sites are activated all along the fatigue life. As expected, the density of cracks also increases with the strain level.

NATURE OF THE INCLUSIONS AND CRITICALITY

As mentioned previously, the cracks are initiated either at the parting line, which can be considered as a geometrical defect acting as a stress concentrator, or around inclusions. In the following, the attention is focused only on cracks initiated on inclusions.

Thanks to EDS, it was possible to analyze the chemical nature of those inclusions, which are sorted here in five categories. The first two categories correspond to CB agglomerates (denoted as ''Carbon Black'') and to large ZnO particles (between 2 and 5 lm), denoted as ''ZnO.'' The third category is related to ''Talc,'' and the fourth concerns all the ''other oxides,'' such as calcium/silicon oxides (''SiO 2, '' ''CaO'') or oxides containing aluminum and silicon (probably kaolinite, referred to as ''Al-Si-O'' in the following). All those ''other oxides'' can be introduced into the materials by the NR gum that contains some impurities, the covering powder of the sulfur, and CBS pellets or the process (minor pollution due to a previous mixing/injection, for example). Sometimes, it was not possible to clearly see the inclusions related to the origin of a crack. This could be due to the low magnification used in this study or simply to the absence of inclusions. In the latter case, it should be noted that inclusions could be lost during the sample manipulation. Finally, a fatigue crack initiation due to a microgeometrical defect related to the roughness of the mold is also likely. The fifth category is thus called ''nonvisible inclusions.'' Figure 7a-c corresponds to the distribution, in number, of these five categories for the three considered maximum strain levels: 60%, 100%, and 200%. For the highest strain level, CB clearly appears to be the most common inclusion, but for the two lowest strain levels, no type of inclusion seems to be majoritarian: the number of inclusions identified as being CB, ZnO, talc, or other oxides are in the same order of magnitude. Thus, it appears necessary to investigate the relationship between the chemical nature of the inclusions and their criticality in terms of long cracks contributing to the fatigue failure.

To perform such an analysis, the size of the cracks is compared with the size of the inclusions for the four kinds of inclusions listed before (''nonvisible inclusions'' are not considered in the following) and for the three strain levels. The results are in Figure 8a-c graphs, the line (having a slope of 1) corresponds to sites that do not give rise to the propagation of a crack, as the crack length is almost equal to the size of the inclusion. In that case, the sites are considered ''activated'': there is some visible damage around the inclusion (cavity), but the size of the cavity is not bigger than the size of the inclusion. Critical cracks are therefore those whose length is much higher than the size of the inclusion responsible for its initiation. From Figure 8, it can be seen that for the three strain levels, CB agglomerates are the inclusions leading to the longer cracks; those that are initiated by the other types of inclusions propagate only a little or not at all, except for Talc and Other oxides, which very rarely lead to the formation of long cracks. Moreover, it can be noted that the relatively high number of activated sites is due to ZnO inclusions, but they never lead to a crack longer than a few tens of micrometers. This kind of inclusion appears to be the less deleterious, which could be due to its relatively small size, most being smaller than 10 lm. But this is probably not the only reason, because some CB agglomerates smaller than 20 mm lead to long cracks, especially at 200% strain.

The initiation mechanisms of fatigue cracks due to the most harmful inclusions (i.e., CB) and to the less harmful ones (i.e., ZnO) are more deeply investigated in the next section, in order to better understand the cause of this difference. We will also try to find out what drives the initiation: a critical mechanical quantity and/or energy accumulation?

FATIGUE CRACK INITIATION MECHANISM FOR CB AGGLOMERATES

From the analysis of the SEM images, the fatigue crack initiation mechanism around CB agglomerates can be divided into three stages represented in Figure 9:

Stage 1: debonding at the pole Stage 2: opening on the sides Stage 3: growth at the surface and in the volume, which can be also considered as the early stages of crack propagation Note that these three SEM pictures are related to three different inclusions. Stages 1 and 2 are associated with the initiation of the crack and are driven by the inclusion and its interface with the matrix, whereas the matrix controls stage 3, with no influence of the inclusion. Cracks in stage 3 show some ligaments at the crack tip, which is in agreement with the literature. 17,19 Figure 10 illustrates schematically the initiation mechanism from the initial situation, which can be called stage 0, to the propagation (stage 3).

CB agglomerates observed in the present study seem to have high internal cohesion (no broken agglomerates were observed) and good interface properties (i.e., a good adhesion to the matrix). The initiation mechanism occurs here by debonding, showing that the internal cohesion of the agglomerates is stronger than the filler-matrix interface. We remark also that all CB agglomerates exhibit a spherical shape. It suggests that, during the mixing, the CB pellets could be first broken and then eroded, producing the spherical shape. Those two mechanisms are classically involved in the dispersion of CB. [START_REF] Collin | [END_REF] To see the internal structure and composition of such agglomerates, one of them has been collected and fractured between two small glass slides used for optical microscopy. Figure 11a shows a CB agglomerate before fracture, and Figure 11b shows the same agglomerate after fracture. Besides the fact that this fracture is very brittle (suggesting no or little occluded rubber), it appears to be homogeneous in terms of appearance and carbon content (see the EDS map of carbon in Figure 11c). Consequently, CB agglomerates do not seem to be covered by a layer of rubber. This would confirm a debonding mechanism rather than cavitation in the matrix.

A complementary analysis is then performed to go deeper into the understanding of the proposed mechanism: it consists of classifying the flaws due to fatigue into the three previous stages and for the three percentages of the fatigue life. A flaw can be an activated site or a crack. As shown in Figure 12a, at 100% strain, the percentage of flaws in stage 1 decreases during the fatigue life, whereas the percentages of flaws in stages 2 and 3 increase. These evolutions, also observed at 200% strain (see Figure 12b), tend to validate the proposed mechanism, in so far as it is progressive, and it confirms that the CB agglomerates are responsible for the critical cracks (stage 3).

In addition, to determine whether initiation is driven by an accumulation of energy and/or requires a critical mechanical quantity, the minimal size of CB agglomerates generating initiation is followed for each stage and reported in Table II for the different strain levels and fatigue life percentages. In this table, it can be seen, on one hand, that for stages 2 and 3 and for a given strain level, the minimum size of the flaws generally decreases when the percentage of fatigue life increases. This suggests that the transitions from stage 1 to 2 and from stage 2 to 3 require an accumulation of energy: the inclusions involved in the different damage stages are progressively smaller along the fatigue life. The cracks associated with these inclusions need a given number of cycles, that is, a certain amount of energy (cumulative damage?), to be created. On the other hand, for stages 2 and 3, for the same percentage of fatigue life, the minimum size always decreases when the strain level increases, suggesting that a critical mechanical value has to be reached to create cracks and to start propagating them.

Consequently, it seems that the different stages of the initiation mechanism involving CB agglomerates require both energy accumulation and a critical mechanical value.

Finally, the question arises regarding the influence of the average size of CB agglomerates (on which cracks are initiated) on the fatigue lifetime. To consider this question, an additional study has been conducted by using NR filled with four other CB types, as mentioned in the section ''Materials and Samples.'' Figure 13 reports the fatigue end of life at 100% strain with respect to the average size of the CB agglomerates for the five considered materials, including NR43-N550. As expected, this figure proves that bigger CB agglomerates shorten the fatigue life of CB-filled NR. Even if this result is logical and known for a long time for other materials (see for example, ref. 23, which studied this effect on steels), it is here clearly demonstrated in the case of CB-filled NR on the basis of a large number of observations. It definitively proves that fatigue life is more driven by the initiation phase rather than by the propagation one.

FATIGUE CRACK INITIATION MECHANISM FOR ZNO INCLUSIONS

As shown previously, ZnO inclusions are responsible for many initiations of cracks that in most cases do not propagate. Indeed, considering the definition of the three stages previously established, we find very few flaws in stage 3 and a majority in stage 1, regardless of the strain level.

The analysis of the SEM images shows that there are two mechanisms for stage 1: fracture of the inclusion, in the case in which the internal cohesive energy of the inclusion is smaller than the bonding to the matrix (Fig. 14a) and debonding at one pole or at both poles, in the opposite situation (Fig. 14b,c).

ZnO are inclusions with low internal cohesion and low adhesion to the matrix. This explains that initiation can be easily activated (stage 1) but rarely followed by the growth of long cracks (stage 3). Figure 15 shows schematically the initiation mechanisms for stage 1 around a ZnO inclusion.

If one considers the average size of ZnO inclusions activated (stage 1) with respect to the percentage of fatigue life, it depends neither on the number of cycles nor on the strain level, as illustrated in Figure 16. This suggests that initiation does not require energy accumulation in this case. If a critical mechanical value is necessary for initiation, it is reached from 60% strain, as the size does not change with the strain level.

The size of ZnO inclusions may explain why, in some particular cases, they lead to crack propagation. The size of ZnO associated with the rare stage 3 cracks is between 14.6 lm and 28.7 lm, which is above the average size of ZnO (7 lm) and close to or even higher than the average size of CB agglomerates (varying from 9 lm and 17lm). This suggests that the size of the inclusion is crucial for the propagation. However, it is surprising that some big inclusions of ZnO (greater than 20 lm) do not generate crack propagation. The shape of the inclusion, which contributes to the properties of the interface, also probably plays an important role.

SUMMARY

In the case of CB-filled NR, fatigue damage is due to multicracking at the surface of the samples initiated on geometric accidents (parting line) and on inclusions.

The number of cracks increases with the number of cycles and with the strain level. Analysis of the initiation sites around inclusions shows that they are of various types, including two types of inclusions having a different effect on fatigue crack initiation: CB agglomerates and ZnO inclusions.

CB agglomerates have a high cohesive energy and a good adhesion to the matrix. The crack initiation mechanism involves three steps: debonding at the interface (meaning that the internal cohesion is stronger than the bonding between the inclusion and the matrix), opening on the sides, and then cracking at the surface and in the volume. Finally, at 100% strain for NR filled with different CB types, fatigue life appears to be related to the average size of the CB agglomerates: the number of cycles leading to the end of life increases when the average agglomerate size decreases.

The internal cohesion of ZnO inclusions as well as their interface properties with the matrix are weak. Consequently, fatigue damage occurs by debonding or fracture of the inclusion and does not seem to be progressive. In most cases, there is no crack propagation from this kind of inclusions.

FIG. 1 .

 1 FIG. 1. -(a) Geometry of the AE2 sample. (b) FEA giving the maximum principal strain field for a local maximum strain of 100%.

FIG. 3

 3 FIG. 3. -(a) Wöhler curve obtained on NR43-N550 for the three local maximum strain levels and the three percentages of fatigue life (% Ni ). (b) Evolution of the maximum force versus the number of cycles with the three percentages of fatigue life.

FIG. 4 .

 4 FIG. 4. -Typical SEM image at 3130 magnification with one fatigue crack.

FIG. 6 .

 6 FIG.6. -Density of surface cracks with respect to the local maximum strain for the three percentages of fatigue life (% Ni ).

FIG. 5 .

 5 FIG. 5. -SEM image of the typical locations of fatigue crack initiation sites: at the parting line or out of the parting line.

  FIG. 7. -Distribution of the different categories of inclusions at the origin of fatigue damage: (a) at 60% strain, (b) at 100% strain, (c) 200% strain.

FIG. 9 .

 9 FIG. 9. -The three stages of fatigue crack initiation around a CB agglomerate: debonding at one pole (stage 1), opening on the sides (stage 2), growth at the surface and in the volume (stage 3).

FIG. 10 .

 10 FIG. 10. -Schematic representation of fatigue crack initiation mechanism around a CB agglomerate.

FIG. 12 .

 12 FIG. 12. -Evolution of the three stages for flaws associated with CB agglomerates in NR43-N550 with respect to the percentage of fatigue life (% Ni ): (a) at 100% strain and (b) at 200% strain.

FIG. 14 .

 14 FIG. 14. -The different kinds of fatigue crack initiation mechanisms around ZnO inclusion (stage 1): (a) fracture of the ZnO, (b) debonding at one pole, (c) debonding at both poles.

TABLE II MINIMUM

 II SIZE (S MIN ) OF THE CB AGGLOMERATES, FOR EACH (S1, S2, S3) AS A FUNCTION OF THE STRAIN AND THE PERCENTAGE OF FATIGUE LIFE (% NI)

	Strain, %	% Ni	s min S1, lm	s min S2, lm	s min S3, lm
	60	20	5.2	16.5	None
		40	6.2	10.2	None
		60	3.0	3.6	None
	100	20	5.1	7.4	41
		40	4.8	3.5	12.4
		60	5.2	2.7	14.4
	200	20	5.0	3.3	17.3
		40	5.1	3.2	3.4
		60	2.8	1.5	2.0

FIG. 13. -Size of the CB agglomerates on which cracks are initiated with respect to the fatigue end of life at 100% strain, for different types of CB.

ACKNOWLEDGEMENTS

The authors would like to thank the French National Research Agency (ANR) for its financial support (ANR-2010-RMNP-010-01) and all the partners of the PROFEM project: TrelleborgVibracoustic, LMBS, GeM, UBS, and LRCCP.


K L Mitchell
email: karl.l.mitchell@jpl.nasa.gov

Philippe Paillou

B W Stiles

H Zebker

G Mitri

J I Lunine

S D Wall

R D Lorenz

R M C Lopes

S Hensley

E R Stofan

R L Kirk

S J Ostro

F Pagan- Elli

Are Titan's Lakes Liquid-filled?

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction: SAR imagery obtained during Cassini's T16 Titan fly-by revealed numerous radar-dark features at > ~70° N, interpreted to be lakes [1] on the basis of their low radar reflectivity, morphology and consistency with predictions [START_REF] Mitri | Icarus[END_REF]. Later fly-bys revealed more lakes, and also overlapped with previous scenes, facilitating multi-angle, multi-temporal studies, with several more such opportunities over the coming months. Here we introduce our efforts to understand the nature of the lakes using such studies, focusing on one anomalous lake in particular, and address the issue of whether the observed lakes are liquid-filled or dry.

Observations: To date we have catalogued 156 radar-dark lakes, based on T16, T18 and T19 SAR imagery, observing a trend from broader lakes with more complex margins at higher latitudes, to smaller, more circular lakes with apparently steeper rims at lower latitudes. We also see many morphologically similar but radar-bright features, interpreted to be drained lakes [1] or cryovolcanic calderas [START_REF] Wood | [END_REF]. These are primarily, though not exclusively, at the lower lakeland latitudes, and certainly significantly outnumber radar-dark lakes in these locations.

Overlapping T16 and T19 SAR imagery permits multi-look analysis of ~20 lakes. Within this region they have opposite look directions and a large contrast in incidence angles: 25-31° for T16 and 13-19° for T19. A visual comparison reveals no signs of change that cannot be accounted for by parallax effects due to topography or radar artifacts, with one exception.

"Lake 20" (Fig. 1) exhibits an exceptionally strong brightening of the left-most portion of the lake from T16 to T19, but within this region retains some of the same apparent internal structure. Also, the contrast at the lake boundary is different: for T16 there is a sharp contrast (roughly 6 to 8 dB) all around the lake, whereas for T19 there is only a sharp contrast on the top side of the lake, with a much more gradual change on the bottom side, giving it an appearance of being much less differentiated from the surrounding region.

Analysis: 4 large lakes in the overlap region were compared using thermal/quantization noise-subtracted pixel-averaged backscatter data. The contrast was also stretched to correct for a contrast reduction effect inherent in the compression. Pixels were averaged used a 5 5 pixel box filter to reduce speckle noise. We explore various hypotheses about the nature of the lakes, including their incidence-angle-dependent behavior, by fitting the backscatter data to a multilayer scattering model [4] using several candidate materials (liquid methane, water-ice, water-ice-ammonia, tholins) in both 1-and 2-layer configurations, including: (i) a solid surface; (ii) a pure liquid reservoir; and (iii) a liquid reservoir over a solid surface.

First, it is necessary to explain the extremely low radar backscatter. Liquid methane and ethane have been predicted to be abundant at these latitudes, and their interaction with 2.2-cm microwave is known to be weak. Hence they should be largely transparent to Cassini SAR, although a deep enough reservoir should appear radar dark, similar to the Earth's oceans when viewed from space in visible light. Unfortunately, the loss tangent is not well constrained, and so the correlation of signal with depth is diffcult to estimate. If a dry lake is considered, a dielectric constant of < 3 is necessary, inconsistent with most solid materials proposed on the surface of Titan. However, tholins -radar dark materials (~2.2-0.01j dielectric constant [5]) that are widely predicted to have rained out over much of the planet -may be consistent with observations. Second, the observations should be consistent with multiple incidence angle data. Using the current lim-ited dataset, all configurations that involve a liquid methane or ethane layer and/or a solid tholin layer are consistent with the observations. Data to be acquired over the coming months, including some 3-look opportunities, should help us to constrain this further.

"Lake 20" analysis: As noted previously, Lake 20 presents us with an anomaly. We explore two hypotheses, both sub-sets of previous ones: (a) the lake is static between scenes, with a stronger dependence on incidence angle for the bright portion of the lake than for the surrounding terrain; and (b) the lake changes between scenes in its extent and/or depth.

Figure 2. Two-layer modelling of Lake 20 at 31° (T16) and 16° (T19) incidence angles, assuming liquid methane covering a tholin or ice substrate. (a) The difference between T16 and T19 curves is due only to incidence angle change (b) As above, but with a higher loss tangent for liquid methane, and a lake level decrease from T16-T19 of 3 m, starting at ~3 m at the left part of the lake, and >25 m at the right part.

Static explanations. The backscatter observed in

Lake 20 is only consistent with models that involve either: (1) an extremely low dielectric constant solid, with rather constant roughness (surface height standard deviation, =0.5cm, and surface correlation length, L=2.4cm), but a volume scattering component that decreases from left (albedo = 0.2) to right (albedo = 0.0) due to structural heterogeneity; or (2) a smooth ( =0.35cm and L=2.5cm) liquid layer (here, we assume methane with a dielectric constant taken to 1.8-0.00007j) covering a rougher ( =2.5cm and L=7.0cm) lake bottom -modeled as either tholins (dielectric constant 2.2-0.01j, volume scattering with albedo=0.2) or water-ice (dielectric constant 3.1-0.000006j, volume scattering with albedo=0.1) -with lake depth varying from either very shallow or exposed (left part) to > 200m (right part) (Fig. 2a).

Dynamic explanations. If Titan's lakes are liquidfilled, then we should expect the lake levels to change over time due to evaporative cooling/heating, advective heating/cooling and solar insolation and reradiation, as previously modelled [START_REF] Mitri | Icarus[END_REF] for methaneethane mixtures, as well as possible precipitation and drainage. Brightening of Lake 20 between T16 and T19 is slightly inconsistent with our model for incidence-angle dependence of methane, implying that if the lake is liquid methane then some sort of change has taken place. A plausibly high loss tangent of 0.0005 is consistent with a change as small as 3 m, with lower loss tangents corresponding to higher lake level decreases. Such a change can be accommodated within the model of [START_REF] Mitri | Icarus[END_REF]. The evaporation model is sensitive to local topography and, over the 79-day interval between T16 and T19, would necessitate very shallow local elevations, consistent with shallow lake basins. If the lake was richer in methane (35% methane, 65% ethane was considered by [START_REF] Mitri | Icarus[END_REF]), more rapid evaporation would be possible. A significantly lower loss tangent would imply greater lake elevation change, in which case some sort of drainage would be needed.

The localized brightening can also modeled by such a change in lake level (Fig. 2b), but compared with the static case we require both a higher volume scattering component in the bottom layer (albedo of 0.4) and a higher (still acceptable) loss tangent for the covering liquid layer (dielectric constant 1.8-0.001j).

Discussion: Our preferred interpretation, on the basis that it involves the least assumptions, is that the lakes are liquid hydrocarbons and that the brightening in Lake 20 can be explained by incidence angle effects given a variable lake depth (static hypothesis 2). We cannot completely rule out the possibility that we are looking at very low dielectric constant solid, which, for Lake 20, would require a spatially varying degree of volume scattering, consistent with an evaporitic or sedimentary tholin (or similar) residue of varying porosity. However, we find this unlikely, on the grounds that the few very dark spots in topographic lows at lower latitudes are both less radar-dark and less copious, despite predictions that tholins should rain out there also. In addition, if we accept the interpretations that radar-bright depressions are drained lakes [1], the implication being that the depressions are largely shaped by the interaction of the lake liquids and the substrate (consistent with karst or thermokarst lakes [6]) then, without the presence of liquid it is difficult to explain the dichotomy of both radar dark and radar bright lakes. Alternatively, if the lake depressions are cryovolcanic calderas [START_REF] Wood | [END_REF], then the tendency of increasing apparent abundances at lower latitudes within the lake region (>~65° N), relative lack of similar forms elsewhere on the planet, and dynamics of caldera formation of Titan, must be addressed.

Figure 1 .

 1 Figure 1. Noise-substracted SAR images of Lake 20 (at 78.3° N 113° W). (a) T16 fly-by, illuminated from the bottom at 31.0° incidence angle. (b) T19 fly-by, illuminated from the top at 16.4°incidence angle. (c) As (a), pixel-averaged and shown in pseudocolour, with contour lines plotted at 2 dB backscatter intervals. (d) As (b), processed as (c).

Lunar and Planetary Science XXXVIII (2007)2081.pdf

Acknowledgements: We acknowledge support from NASA and the Cassini project. KLM was supported by a NASA Postdoctoral Fellowship administered by ORAU and carried out at JPL/Caltech.


Petru Mironescu
email: mironescu@math.univ-lyon1.fr

Itai Shafrir
email: shafrir@math.technion.ac.il

Asymptotic behavior of critical points of an energy involving a loop-well potential

Asymptotic behavior of critical points of an energy involving a loop-well potential

Petru Mironescu, Itai Shafrir

Statement of the problem

Let Ω ⊂ R 2 be a smooth bounded star-shaped domain. Let Γ ⊂ R 2 be a smooth simple curve and let g : ∂Ω → Γ be a smooth boundary datum of degree d. Consider, for every ε > 0, a critical point u ε ∈ H 1 g (Ω; R 2 ) of the energy

E ε (u) = ˆΩ 1 2 |∇u| 2 + W(u) ε 2 . ( 1.1)
Here, W : R 2 → [0, ∞) is a smooth potential vanishing precisely on Γ; for the exact assumptions on W, see (1.5)-(1.10) below.

In the Ginzburg-Landau (GL) case, i.e., when W(u) = (1-|u| 2 ) 2 /4, the asymptotic behavior of {u ε } when ε → 0 was studied by Bethuel, Brezis and Hélein, first for minimizers when the boundary condition has zero degree in [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF], and later for minimizers and, more generally, for critical points for arbitrary boundary datum in the seminal work [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF].

The analysis in [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF] for minimizers of the GL energy can be adapted with no significant difficulty to the case of general W, at least when W is non-degenerate, see (1.9). Using more involved arguments, it is even possible to describe the asymptotic behavior of minimizers in the case of a general boundary condition g that does not necessarily take values into Γ; see André and Shafrir [START_REF] André | On a singular perturbation problem involving a "circular-well" potential[END_REF].

We address here the question of the asymptotic behavior of critical points of the energy (1.1), i.e., of solutions of

   ∆u ε = 1 ε 2 ∇W(u ε ) in Ω u ε = g on ∂Ω (1.2)
that need not be energy minimizing with respect to their own boundary condition. As we will see below, the answer to this question requires new ideas and ingredients. We emphasize that the starshapeness condition on Ω is crucial to our analysis, as it was in [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]Chapter X]. As far as we know the problem about critical points in a general simply connected domain is still open even in the case of the usual Ginzburg-Landau potential.

The method of proof in [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]Chapter X] for critical points of the GL energy is based on a clever decomposition of the gradient ∇u ε . Its starting point is the identity

∂ ∂x 1 u ε × ∂ ∂x 1 u ε + ∂ ∂x 2 u ε × ∂ ∂x 2 u ε = 0, (1.3)
which is a direct consequence of the fact that W(u) = W(|u|) in the GL case. We could not find an analogous identity to (1.3) for general W. Our method is different and relies on two main tools:

1. Selection of "good rays" (see Subsection 5.2).

2. A maximum principle for the phase (see Proposition 2.1).

Combined, they allow us to prove a crucial estimate, namely

E ε (u ε ) ≤ C(| log ε| + 1). (1.4)
The first ingredient is new even for the GL energy (and leads to a simplification of the original arguments in [5, Chapter X]), and the second one is much more subtle in the case of a general potential W than in the GL case.

For the analysis of solutions to (1.2) we will need, in the spirit of [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF], the additional assumption that Ω is strictly star-shaped. This assumption enables us to prove that the second term in the energy (1.1) remains bounded when ε → 0, and then to perform the "bad discs" construction à la Bethuel-Brezis-Hélein [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF], which is the starting point of the study of the location of the vortices.

The remaining part of the analysis is similar to the one in [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF] (with some technical complications), and leads to our main result, Theorem 1.1 below. In order to state it, we first present all the assumptions on Γ and W.

W : R 2 → [0, ∞) is a smooth function satisfying W -1 ({0}) = Γ (1.5)
and Γ is a simple closed smooth curve in R 2 .

(1.6)

We assume without loss of generality that

|Γ| = 2π (1.7)
and consider τ : S 1 → Γ an arc length parametrization of Γ.

(1.8)

We also suppose that W is non-degenerate in the following sense:

W(ζ) ≥ µ dist 2 (ζ, Γ) if dist(ζ, Γ) < δ, ( 1.9)
for some µ, δ > 0 (and then it follows from (1.5) that (1.9) holds on any compact subset of R 2 ).

In addition, we impose the following coercivity assumption on the behavior of W at infinity:

∂W ∂r (z) ≥ 0 for |z| = r > R 0 , (1.10)
for some R 0 > max{|z|; z ∈ Γ}.

Theorem.

Let Ω be a smooth, bounded, strictly star-shaped domain in R 2 . Let W, Γ and τ satisfy (1.5)-(1.10). Let g : ∂Ω → Γ be a smooth boundary condition of degree d. For each ε > 0, let u ε denote a solution of (1.2). Then up to a subsequence we have

u ε n → u * = τ e ıη(z) z -a 1 |z -a 1 | D 1 • • • z -a N |z -a N | D N in C 1,α Ω \ {a 1 , . . . , a N } , (1.11)
where 1. a 1 , . . . , a N ∈ Ω are mutually distinct points.

2. D 1 , . . . , D N ∈ Z \ {0} satisfy the compatibility condition N j=1 D j = d.

3. η is a harmonic function in Ω.

4. α ∈ (0, 1).

In the spirit of [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF], we may also prove that the configuration (a 1 , . . . , a N ) is a critical point of a suitable renormalized energy associated with the degrees (D j ) N j=1 and the boundary condition; see Remark 5.17 in Section 5.

Let us mention that non minimizing solutions do exist. For the GL energy, their existence was established in different situations. In the special case where Ω is the unit disc and g(z) = z d , with |d| ≥ 2, the GL energy has critical points of the form u ε (re ıθ ) = f ε (r) e ıdθ , and these solutions are not minimizing for sufficiently small ε [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]. Non minimizing critical points also exist when d = 0: F.H. Lin [START_REF] Lin | Mixed vortex-antivortex solutions of Ginzburg-Landau equations[END_REF] constructed examples of "mixed vortex-antivortex solutions". More specifically, for all N ≥ 1 there exists g N : ∂Ω → S 1 of degree 0 and non minimizing corresponding critical points u ε n such that

u ε n → u * = e ıη N (z) 2N j=1 z -a j,N |z -a j,N | (-1) j-1 .
Other existence results concerning non minimizing solutions for the the GL energy were proved by Almeida and Bethuel [START_REF] Almeida | Topological methods for the Ginzburg-Landau equations[END_REF] and by F. Zhou and Q. Zhou [START_REF] Zhou | A remark on multiplicity of solutions for the Ginzburg-Landau equation[END_REF], using variational and topological methods. We believe that at least some of these methods lead to the existence of non minimizing critical points of (1.2) for a general W, but we did not investigate this issue.

Except for the upper bound (1.4), we did not establish a more precise estimate for the energy E ε (u ε ). In the case of the GL-energy, Comte and Mironescu [START_REF] Comte | Remarks on nonminimizing solutions of a Ginzburg-Landau type equation[END_REF] proved that the following is true:

E ε (u ε ) = π N j=1 D 2 j | log ε| + O(1). (1.
12)

It would be interesting to generalize the validity of (1.12) to our setting.

The paper is organized as follows. In Section 2 we introduce some notation and prove a maximum principle for the phase, that plays an important role in the remaining part of the paper. In Section 3 we study the case of boundary data of zero degree (d = 0) under the additional assumption that the solutions stay close to Γ, i.e., no vortices appear. The techniques of this section are used in Section 4 to treat the more general case of a boundary data depending on ε (again, for vortex-less solutions). This latter case is very useful in the proof of convergence away from the vortices in Theorem 1.1. Section 5 is devoted to the proof of the main result, Theorem 1.1.

Assume in what follows that

u : ω → R 2 is a smooth map such that u(x) ∈ Γ δ Γ , ∀ x ∈ ω.
(2.1) (Here, ω ⊂ R 2 is some open set.) Locally in ω, we can associate to u two smooth coordinates, t and ϕ, such that Π • u = τ (e ıϕ ) and t is the signed distance of u to Γ (taken with the plus sign inside Γ). Analytically, this means that the functions t and ϕ satisfy (t(x), ϕ(x)) ∈ (-δ Γ , δ Γ ) × R and u(x) = τ e ıϕ(x) + t(x) n τ e ıϕ(x) .

(2.2)

Here, n(z) denotes the inward unit normal to Γ at the point z ∈ Γ. Equivalently, we have

Π(u(x)) = τ e ıϕ(x) and t(x) = (u(x) -Π(u(x))) • n (Π(u(x))). (2.3)
Note that t is globally defined, but ϕ is only locally defined in ω, and that ϕ is (locally) unique mod 2π. It is useful to note that ϕ is globally defined when ω is simply connected.

A simple calculation (see [START_REF] André | On a singular perturbation problem involving the distance to a curve[END_REF]Lemma 4.1]) shows that for u satisfying (2.1) we have (denoting by κ(z) the curvature of Γ at the point z ∈ Γ)

|∇u| 2 = 1 -t κ τ(e ıϕ ) 2 |∇ϕ| 2 + |∇t| 2 = (1 -t κ (Π • u)) 2 |∇ϕ| 2 + |∇t| 2 . (2.4)
Moreover, for such u we have (using (1.9)) that

W(u) = α(ϕ, t) t 2 (2.5)
where α(ϕ, t) is a smooth positive function, 2π-periodic in the ϕ-variable.

Assume next that u = u ε is a solution of (1.2) in Ω and that ω ⊂ Ω is such that (2.1) holds. Then locally in ω we may use (2.5) to write the Euler-Lagrange equations (1.2) for the function u in the new coordinates t and ϕ as follows.

div(a∇ϕ) = b|∇ϕ| 2 -α ϕ t 2 ε 2 , (2.6a)

-∆t + (2α + α t t) t ε 2 = c|∇ϕ| 2 .
(2.6b)

In (2.6), the coefficients a = a(ϕ, t), b = b(ϕ, t) and c = c(ϕ, t) are given by

             a = 1 -tκ τ(e ıϕ ) 2 = 1 + O(t) b = - 1 2 a ϕ = O(t) c = - 1 2 a t = O(1)
.

(2.7)

A maximum principle for the phase

By (2.5)-(2.7), for sufficiently small δ 0 ∈ (0, δ Γ ) there exist positive constants c 0 , . . . , c 5 such that for |t| ≤ δ 0 there holds:

|1 -a| ≤ c 0 |t|, (2.8a) 2α -|α t t| ≥ c 1 , (2.8b) |c| ≤ c 2 , (2.8c) b a ≤ c 3 |t|, (2.8d) a t a ≤ 2c 4 , (2.8e
)

α ϕ a ≤ c 5 .
(2.8f)

Note that δ 0 depends only on Γ.

Next we prove a maximum principle for the phase ϕ, that will be useful throughout the paper. For this purpose, we introduce two numbers, 0 < δ 1 < δ 0 and m > 0, satisfying

c 5 c 1 ≤ m (2.9) and 2c 4 δ 1 + m(mc 2 + c 3 )δ 3 1 < 1.
(2.10)

Note that δ 1 and m depend only on Γ and W.

2.1 Proposition. Let u = u ε be a critical point of E ε in a bounded simply connected domain ω, continuous on ω and satisfying dist(u(x), Γ)

≤ δ 1 , ∀ x ∈ ω. Consider t = t ε , ϕ = ϕ ε associated to u via (2.2). Then min ω ϕ - mt 2 2 = min ∂ω ϕ - mt 2 2 , (2.11a) max ω ϕ + mt 2 2 = max ∂ω ϕ + mt 2 2 . ( 2
min ∂ω ϕ ≤ ϕ(x) - mt 2 (x) 2 ≤ ϕ(x) + mt 2 (x) 2 ≤ max ∂ω ϕ, ∀ x ∈ ω.
(2.12)

In particular, We also have

min ∂ω ϕ ≤ ϕ(x) ≤ max ∂ω ϕ, ∀ x ∈ ω. ( 2
∇ mt 2 2 -ϕ 2 =m 2 t 2 |∇t| 2 -2mt∇ϕ • ∇t + |∇ϕ| 2 ≥ m 2 t 2 |∇t| 2 -2m|t||∇ϕ||∇t| + |∇ϕ| 2 .
(2.17) By (2.16)-(2.17) we obtain, for any k > 0,

-∆ mt 2 2 -ϕ -k ∇ mt 2 2 -ϕ 2 ≤ (mc 2 |t| + c 3 |t| -k) |∇ϕ| 2 -m + km 2 t 2 |∇t| 2 + (2c 4 + 2km|t|) |∇ϕ||∇t| + (c 5 -mc 1 ) t 2 ε 2 .
(2.18)

Next we are looking for conditions that will insure that the right-hand side of (2.18) is nonpositive. First, by our assumption (2.9) the last term is indeed nonpositive. The sum of the first three terms on the right-hand side of (2.18) is a quadratic form in the two variables |∇ϕ|, |∇t| whose discriminant ∆ is given by

∆/4 = (c 4 + km|t|) 2 -(k -mc 2 |t| -c 3 |t|) m 1 + kmt 2 =c 2 4 + m(mc 2 + c 3 )|t| -km 1 -2c 4 |t| -m(mc 2 + c 3 )|t| 3 .
(2.19) By (2.10) and (2.19) it follows that for sufficiently large k we have ∆ ≤ 0, implying that the right-hand side of (2.18) is nonpositive. For such k it follows that the function v := mt 2 /2 -ϕ satisfies ∆(e kv ) = ke kv (∆v + k|∇v| 2 ) ≥ 0 in ω.

By the maximum principle, max ω v = max ∂ω v, which is equivalent to (2.11b).

By similar calculations, the function w := mt 2 /2 + ϕ satisfies ∆(e kw ) ≥ 0, implying (2.11a).

Asymptotic behavior of solutions without vortices

In this section we shall study the asymptotic behavior of solutions u ε of (1.2) in a smooth bounded simply connected domain Ω in R 2 . We assume a priori that the solutions are vortex-less. Actually, we shall assume a stronger condition, namely that the solutions are "sufficiently close" to Γ, in a sense to be precised below (see (3.1)). We are given a smooth boundary condition g : ∂Ω → Γ of degree zero and a family of solutions {u ε } of (1.2). Since g is of degree zero, we may globally write it as g = τ(e ıϕ 0 ) for some smooth ϕ 0 : ∂Ω → R.

We next assume that

dist(u ε (x), Γ) ≤ δ 1 , ∀ x ∈ Ω, (3.1)
where δ 1 is chosen to satisfy the hypotheses of Proposition 2.1.

Then we may write, globally in Ω and with smooth t ε and ϕ ε ,

u ε (x) = τ e ıϕ ε (x) + t ε (x) n τ e ıϕ ε (x) , ∀ x ∈ Ω. (3.2)
Let ζ denote the harmonic extension of ϕ 0 to Ω and define the Γ-valued map u 0 by

u 0 := τ e ıζ . ( 3.3)
The main result of this section establishes, in the spirit of [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF], a convergence result of u ε to the limit u 0 .

3.1 Theorem. Let, for 0 < ε < ε 0 , u ε denote a solution of (1.2) satisfying (3.1). Then we have

u ε → u 0 in C 1,α (Ω) as ε → 0, ∀ α < 1, (3.4
)

∆u ε ∞ ≤ C, (3.5)
u ε -u 0 ∞ ≤ Cε 2 , ( 3.6)
∇(u ε -u 0 ) ∞ ≤ Cε. (3.7)
Theorem 3.1 is an immediate consequence of several intermediate estimates (Lemma 3.2 to Proposition 3.9) that we now state and prove.

We start with two simple estimates satisfied by the solutions. These estimates are valid in any bounded domain Ω provided |u ε | ≤ R 0 on ∂Ω.

Lemma. We have

u ε L ∞ (Ω) ≤ R 0 , (3.8)
where R 0 is given by (1.10).

Proof. We claim that the set E := {x ∈ Ω; |u ε (x)| > R 0 } is empty. Indeed, this follows from the maximum principle for subharmonic functions since, on the one hand, we have |u ε | = R 0 on ∂E and, on the other hand, u ε satisfies in E

∆(|u ε | 2 ) = 2 |∇u ε | 2 + ∆u ε • u ε ) ≥ 2 ε 2 ∇W(u ε ) • u ε ≥ 0
(the latter inequality following from (1.10)).

From Lemma 3.2 we deduce the following gradient bound.

Lemma.

We have for some constant C,

∇u ε L ∞ (Ω) ≤ C ε . (3.9)
The proof of Lemma 3.3 uses the same rescaling argument as in [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF] and is therefore omitted.

Next we prove:

3.4 Lemma. We have lim ε→0 t ε = 0 uniformly on Ω.

Proof. Arguing by contradiction, assume that for a subsequence ε n → 0 and a sequence of points {x n } ⊂ Ω we have lim n→∞ t ε n (x n ) = T with T = 0. We distinguish two cases:

1. lim n→∞ dist(x n , ∂Ω) ε n = ∞. 2. lim inf n→∞ dist(x n , ∂Ω) ε n < ∞.
In Case 1 we define a rescaled sequence on B R n (0), with

R n := dist(x n , ∂Ω) ε n , by u ε n (x) := u ε n (x n + ε n x). (3.10)
By our assumptions, R n → ∞ and, by standard elliptic estimates, a further subsequence, still denoted by { u ε n }, converges in C 1,β loc (R 2 ) to a limit u, solution of ∆ u = ∇W( u) on all of R 2 and such that dist( u(x), Γ) ≤ δ 1 , ∀ x ∈ R 2 . The associated t, ϕ then solve the system (2.6), with ε = 1 on R 2 and t(0) = T = 0. But then the proof of Proposition 2.1 shows that the two functions e k v and e k w , where

v := m t 2 2 -ϕ and w := m t 2 2 + ϕ,
are subharmonic and bounded on R 2 . It follows that both v and w are identically constant in R 2 , and therefore the same holds for t and ϕ. In particular t ≡ T = 0 and ∇ ϕ ≡ 0. But then, in view of (2.8b), equation (2.6b) is violated. Contradiction.

Consider next Case 2. We may assume that L = lim n→∞ dist(x n , ∂Ω) ε n exists. By Lemma 3.3, we have

L > 0.
Arguing similarly to Case 1 we define the rescaled sequence { u ε n } by (3.10). Again, a subsequence converges to a solution of ∆ u = ∇W( u), this time on a half-plane H, with a constant boundary condition u = γ on ∂H, for some point γ ∈ Γ.

With no loss of generality, we may assume that H = R × (0, ∞). We know that for some point (x 0 , y 0 ) ∈ H with y 0 = L we have t(x 0 , L) = T = 0. In addition, the boundary condition u = γ implies that the corresponding coordinates t and ϕ satisfy t = 0 on ∂H and ϕ = Φ =const. on ∂H.

As above, the functions e k v and e k w are subharmonic. Since they are also bounded, the maximum principle applies on H and we obtain that both functions attain their maximum on ∂H. We obtain that

Φ ≤ ϕ(x) - m t 2 (x) 2 ≤ ϕ(x) + m t 2 (x) 2 ≤ Φ, ∀ x ∈ H.
It follows that t ≡ 0, contradicting t(x 0 ) = T = 0.

Next we prove strong convergence of {u ε } to u in H 1 .

Proposition.

As ε → 0, we have

u ε → u 0 in H 1 (Ω) and E ε (u ε ) → 1 2 ˆΩ |∇u 0 | 2 .
(3.11)

Proof. Write ϕ ε = ψ ε + ζ (see (3.3)).
The phase ϕ ε is determined up to an integer multiple of 2π. We fix ϕ ε by imposing

ψ ε = 0 on ∂Ω. (3.12)
Note that by Corollary 2.2 we have

ψ ε ∞ ≤ 2M := 2 ϕ 0 ∞ . (3.13)
We rewrite (2.6a) (dropping the subscript ε) as

-div(a∇ψ) = div((a -1)∇ζ) + b |∇ψ| 2 + 2∇ψ • ∇ζ + |∇ζ| 2 - α ϕ t 2 ε 2 . (3.14)
Multiplying (3.14) by ψ ∈ H 1 0 (Ω) and integrating yields

ˆΩ a|∇ψ| 2 = ˆΩ (1 -a)∇ζ • ∇ψ + b |∇ψ| 2 + 2∇ψ • ∇ζ + |∇ζ| 2 ψ - α ϕ t 2 ε 2 ψ .
Using Cauchy-Schwarz inequality, (3.13), (2.8a), (2.8d), (2.8f), Lemma 3.4 and Poincaré inequality, it follows that for some constant C = C(g) and for sufficiently small ε we have

ˆΩ |∇ψ| 2 ≤ C ˆΩ t 2 ε 2 . (3.15)
Similarly, we rewrite (2.6b) as

-∆t + (2α + α t t) t ε 2 = c |∇ψ| 2 + 2∇ψ • ∇ζ + |∇ζ| 2 . ( 3
+ α t t) t 2 ε 2 = ˆΩ ct |∇ψ| 2 + 2∇ψ • ∇ζ + |∇ζ| 2 .
(3.17) Using (2.8b) and (2.8c) in (3.17) gives

ˆΩ |∇t| 2 + c 1 t 2 ε 2 ≤ C t ∞ 1 + ˆΩ |∇ψ| 2 .
(3.18) Plugging (3.15) into (3.18) yields (using Lemma 3.4) 3.6 Remark. Note that Proposition 3.5 implies a uniform bound for E ε (u ε ) for all ε > 0. Indeed, it suffices to consider only small values of ε, e.g., ε < ε 0 , since for all ε ≥ ε 0 we deduce from the Euler-Lagrange equation (1.2), Lemma 3.2 and standard elliptic estimates that

ˆΩ |∇t| 2 + t 2 ε 2 = o(1). ( 3
u ε W 2,p ≤ C(p, ε 0 ), ∀ p < ∞, and u ε C 1,α ≤ C(α, ε 0 ), ∀ α < 1. (3.21)
We shall use this observation below for other estimates as well.

Lemma. {ϕ

ε } is bounded in W 1,4 (Ω).
Proof. We use the same notation as in the proof of Proposition 3.5 and write

ϕ ε = ψ ε + ζ = ψ + ζ.
We will actually show that

ˆΩ |∇ψ ε | 4 = o(1), (3.22)
that clearly implies the result for small ε (and then the result for any ε > 0 follows from Remark 3.6). Rewrite (2.6a) as

                     -∆ψ = b|∇ψ| 2 + 2b∇ψ • ∇ζ + b|∇ζ| 2 + div (a -1)∇ζ + (a -1)∇ψ R=R ε - α ϕ t 2 ε 2 S=S ε = b|∇ψ| 2 + R + S in Ω, ψ = 0 on ∂Ω.
(3.23)

We split ψ = ψ 1 + ψ 2 + ψ 3 where

-∆ψ 1 = b|∇ψ| 2 , -∆ψ 2 = R, -∆ψ 3 = S in Ω, ψ 1 = ψ 2 = ψ 3 = 0 on ∂Ω. (3.24)
Fix any p > 2. By standard elliptic estimates, using (2.8a) and (2.8d),

∇ψ 2 p ≤C 1 2b∇ψ • ∇ζ p + b|∇ζ| 2 p + (a -1)∇ζ p + (a -1)∇ψ p ≤C 2 t ∞ ∇ψ p + 1 . (3.25)
Next we estimate ψ 1 . Let p > 1 and set q := 2p p + 2 . Then, by Sobolev embedding (in two dimensions),

W 2,q (Ω) → W 1,p (Ω). Note also that 1 2q = 1/2 2 + 1/p 2 , hence f 2 2q ≤ f 2 f p , ∀ f ∈ L p (Ω). (3.26)
By elliptic estimates, (2.8d) and (3.26) we obtain

∇ψ 1 p ≤C 1 ψ 1 W 2,q ≤ C 2 b|∇ψ| 2 q ≤ C 3 t ∞ ∇ψ 2 2q ≤C 4 t ∞ ∇ψ 2 ∇ψ p ≤ o(1) • t ∞ ∇ψ p , (3.27)
where we used (3.20) in the last inequality. Finally, we turn to ψ 3 . Multiplying (2.6b) by t, integrating and using (2.8b) and the Cauchy-Schwarz inequality yields

c 1 ε 2 t 2 2 ≤ ˆΩ ct|∇ϕ| 2 ≤ C t 2 ∇ϕ 2 4 , implying that (for small ε), ˆΩ t 4 ≤ ˆΩ t 2 ≤ Cε 4 ∇ϕ 4 4 .
(3.28)

Recall also that by (3.19),

t 2 = o(ε). (3.29)
Again by elliptic estimates and (3.26) we get 3.9 Proposition. We have

∇ψ 3 p ≤ C ε 2 t 2 q = C ε 2 t 2 2q ≤ C ε 2 t 2 t p . ( 3
∇ψ 3 4 ≤ C ε 2 • o(ε) • ε ∇ϕ 4 ≤ o(1) • ∇ψ 4 + 1 . ( 3
t ε ∞ ≤ Cε 2 , (3.33) ∇t ε ∞ ≤ Cε, (3.34) ψ ε ∞ ≤ Cε 2 , (3.35) ∇ψ ε ∞ ≤ Cε. (3.36)
Proof. We use an argument from [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF]Step B.4]. Fix q > 2. Multiplying (2.6b) by |t| q-2 t/(ε 2 ) q-1 and integrating gives

c 1 ˆΩ |t| ε 2 q ≤ ˆΩ (q -1) ε 2(q-1) |t| q-2 |∇t| 2 + (2α + α t t) |t| ε 2 q = ˆΩ c|∇ϕ| 2 |t| ε 2 q-2 t ε 2 . (3.37)
We conclude, using Hölder inequality and (2.8c), that the function

f ε = f := t ε 2 satisfies c 1 f q q ≤ ˆΩ c|∇ϕ| 2 | f | q-1 ≤ c 2 ∇ϕ 2 2q f q-1 q , i.e., f q ≤ c 2 c 1 ∇ϕ 2 2q . (3.38)
By Lemma 3.8 and Sobolev embedding, {∇u ε } is uniformly bounded in L r (Ω) for every r ∈ [1, ∞), and we obtain from (3.38) that f q ≤ C q . It follows that for each q > 2 the right-hand side of the equation in (1.2) is bounded in L q (Ω). Hence {∇u ε } is uniformly bounded in L ∞ (Ω), and therefore

∇ϕ ∞ ≤ C, (3.39)
for some constant C. Going back to (3.38) we obtain that

f q ≤ c 2 c 1 C 2 |Ω| 1/q .
(3.40)

Passing to the limit q → ∞ in (3.40) yields

f ∞ ≤ c 2 c 1 C 2 ,
a -1 = O(t) = O(ε 2 ) and b = O(t) = O(ε 2 ),
which allow us to rewrite (3.23) in the form

∆ψ = F + divG, with F ∞ = O(ε 2 ) and G ∞ = O(ε 2 ).
The estimate (3.35) follows by elliptic estimates and finally (3.36) is deduced via interpolation as above.

The proof of the main result of this section is an easy consequence of our previous estimates.

Proof of Theorem 3.1. Since, by (2.5), |∇W(u ε )| = O(t ε ), (3.5) follows from (1.2) and (3.33). By standard elliptic estimates we obtain that {u ε } is uniformly bounded in C 1,β (Ω) for all β < 1, and (3.4) follows by the Arzelà-Ascoli theorem (the identification of the limit as u 0 follows from Proposition 3.5). Finally, (3.6) is a consequence of (3.33) and (3.35), while (3.7) follows from (3.34) and (3.36).

Boundary condition depending on ε

In the next sections we shall also need a version of Theorem 3.1 in the case where the boundary condition depends on ε, and does not necessarily take values into Γ (analogously to [4, Theorem 2] which deals with minimizers for the GL energy). For Ω as in Section 3, assume that the family {g ε } of maps g ε : ∂Ω → R 2 , ε > 0, satisfies:

g ε H 1 (∂Ω) ≤ C, (4.1)
ˆ∂Ω

W(g ε ) ≤ Cε 2 . ( 4.2)
From (4.1)-(4.2) it follows in particular that, possibly up to a subsequence,

g ε → g in H s (∂Ω), ∀ 0 < s < 1, and thus in C α (∂Ω), ∀ α ∈ (0, 1/2), (4.3)
for some g ∈ H 1 (∂Ω; Γ).

For each ε > 0 (or ε ∈ (0, ε 0 )), let u ε denote a solution of    ∆u ε = 1 ε 2 ∇W(u ε ) in Ω u ε = g ε on ∂Ω . ( 4.4)
We now make the crucial assumption that u ε satisfies (3.1) (at least for small ε). Then we have

deg Π • g ε = 0 and thus deg g = 0. ( 4.5)
(Recall that Π is the Euclidean projection on Γ.) As before, we write g(x) = τ(e ıϕ 0 (x) ), with ϕ 0 ∈ H 1 (∂Ω; R). Define, in Ω, the Γ-valued map u 0 by (3.3), i.e., u 0 = τ(e ıζ ), where ζ is the harmonic extension of ϕ 0 to Ω. Our main result establishes the convergence of {u ε } towards u 0 when ε goes to zero:

u ε → u 0 strongly in H 1 (Ω) and in C 0 (Ω), (4.6) ∆u ε L ∞ (K) ≤ C K , (4.7)
u ε → u 0 strongly in C 1,α (K), ∀α < 1, (4.8)
u ε -u 0 L ∞ (K) ≤ C K ε 2 and ∇(u ε -u 0 ) L ∞ (K) ≤ C K ε, ( 4.9)
for every compact K ⊂⊂ Ω.

The proof follows similar steps to those of Section 3 and part of the analysis carries over with slight modifications to the current situation. This is the case for the analogous results to Lemma 3.2 and Lemma 3.4 that we state in the next proposition.

Proposition.

We have u ε ∞ ≤ R 0 and lim ε→0 t ε = 0 uniformly on Ω.

Next we turn to an H 1 -convergence result, generalizing Proposition 3.5.

Proposition. We have

u ε → u 0 in H 1 (Ω) and E ε (u ε ) → 1 2 ˆΩ |∇u 0 | 2 .
(4.10)

Proof. We define the pair of functions t ε and ϕ ε associated with u ε via (3.2). We let ζ ε denote the harmonic extension of ϕ ε|∂Ω to Ω. Analogously to the proof of Proposition 3.5, we then write

ϕ ε = ψ ε + ζ ε , with ψ ε = 0 on ∂Ω.
Clearly, (4.3) implies that, possibly after subtracting suitable integer multiples of 2π from the ϕ ε 's, we have ϕ ε|∂Ω → ϕ 0 in H 1/2 (∂Ω), and thus

lim ε→0 ˆΩ |∇ζ ε | 2 = ˆΩ |∇ζ| 2 . (4.11)
Repeating the calculations at the beginning of the proof of Proposition 3.5, with ζ ε playing the role of ζ, yields, analogously to (3.15),

ˆΩ |∇ψ ε | 2 ≤ C ˆΩ t 2 ε ε 2 . ( 4.12)
Now, since in the current setting t ε is not identically zero on ∂Ω, multiplying (3.16) by t ε , integrating and using (2.8b) yields

ˆΩ |∇t ε | 2 + (2α + α t t ε ) t 2 ε ε 2 = ˆ∂Ω t ε ∂t ε ∂n + ˆΩ c t ε |∇ψ ε | 2 + 2∇ψ ε • ∇ζ ε + |∇ζ ε | 2 , (4.13)
where n stands for the outward normal on ∂Ω. In order to deal with the boundary term in (4.13), we use a Pohozaev identity type argument, as in [4, Proposition 3]. So let V = (V 1 , V 2 ) be a smooth vector field on Ω satisfying V = n on ∂Ω. We consider the vector field

V • ∇u ε = (V • ∇(u ε ) 1 , V • ∇(u ε ) 2 )
. We take the scalar product of both sides of the equation in (4.4) and V •∇u ε and integrate. A direct computation (see [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF]) gives

ˆΩ(∆u ε ) • (V • ∇u ε ) = ˆΩ 1 2 div V |∇u ε | 2 -(V 1 ) x 1 |(u ε ) x 1 | 2 -(V 2 ) x 2 |(u ε ) x 2 | 2 - ˆΩ (V 1 ) x 2 + (V 2 ) x 1 (u ε ) x 1 • (u ε ) x 2 + 1 2 ˆ∂Ω ∂u ε ∂n 2 - ∂g ε ∂σ 2 . ( 4.14)
(Here, ∂σ stands for the tangential derivative on ∂Ω.) On the other hand, we have

1 ε 2 ˆΩ ∇W(u ε ) • (V • ∇u ε ) = 1 ε 2 ˆΩ V • ∇ W(u ε ) = 1 ε 2 -ˆΩ(divV )W(u ε ) + ˆ∂Ω W(u ε ) . ( 4
≤ C 2 1 + ˆΩ |∇u ε | 2 + W(u ε ) ε 2 ≤ C 3 1 + ˆΩ |∇t ε | 2 + |∇ψ ε | 2 + t 2 ε ε 2 . ( 4
≤ C ε 1 + ˆΩ |∇t ε | 2 + |∇ψ ε | 2 + t 2 ε ε 2 1/2 . ( 4
.17) Substituting (4.17) in (4.13) leads to Analogously to Lemma 3.7, and in particular to (3.22), we have:

ˆΩ |∇t ε | 2 + c 1 t 2 ε ε 2 ≤Cε 1 + ˆΩ |∇t ε | 2 + |∇ψ ε | 2 + t 2 ε ε 2 1/2 + C t ε ∞ 1 + ˆΩ |∇ψ ε | 2 . ( 4
4.4 Lemma. ψ ε → 0 in W 1,4 (Ω).
Proof. We first notice that since {ϕ ε |∂Ω } is bounded in H 1 (∂Ω) by (4.1), the family {ζ ε } is bounded in H 3/2 (Ω). Since H 3/2 (Ω) → W 1,4 (Ω), we get:

{ζ ε } is bounded in W 1,4 (Ω). (4.21)
Arguing as in the proof of Lemma 3.7 we use (3.23) to split

ψ ε = ψ 1,ε + ψ 2,ε + ψ 3,ε .
The same arguments that led to (3.25) and (3.27) (with p = 4) yield

∇ψ 2,ε 4 ≤ C t ε ∞ ∇ψ ε 4 + 1 (4.22)
and

∇ψ 1,ε 4 ≤ C ψ 1,ε W 2,4/3 ≤ o(1) • t ε ∞ ∇ψ ε 4 . (4.23)
The only difference with respect to the case where g ε ≡ g stands in the estimate of ψ 3,ε . Multiplying (2.6b) by t ε and integrating gives

∇t ε 2 2 + c 1 ε 2 t ε 2 2 ≤ ˆΩ c t ε |∇ϕ ε | 2 + ˆ∂Ω t ε ∂t ε ∂n ≤ C t ε 2 ∇ϕ ε 2 4 + Cε, (4.24)
where in the last inequality we used the Cauchy-Schwarz inequality and (4.17) combined with (4.19)-(4.20).

Next we claim that

t ε 2 ∇ϕ ε 2 4 ≤ ε, ( 4.25)
for sufficiently small ε. Indeed, arguing by contradiction, assume that (4.25) does not hold, i.e., for a sequence ε n → 0 we have

t ε n 2 ∇ϕ ε n 2 4 > ε n . (4.26)
Then, from (4.24) we get that

c 1 ε 2 n t ε n 2 2 ≤ C t ε n 2 ∇ϕ ε n 2 4 ,
and the argument of the proof of Lemma 3.7 applies, so thanks to (4.21) we get, as in (3.31), that

∇ψ 3,ε n 4 ≤ C ε 2 n • o(ε n ) • ε n ∇ϕ ε n 4 = o(1) • ∇ψ ε n 4 + 1 . ( 4
∇t ε 2 2 + c 1 ε 2 t ε 2 2 ≤ Cε, (4.29)
which implies, in particular, that

ˆΩ |t ε | q = o(1) ˆΩ t 2 ε = o(ε 3 ), (4.30)
for any q > 2. By (4.30), Sobolev embedding and elliptic estimates we obtain

∇ψ 3,ε 4 ≤ C ∆ψ 3,ε 4/3 ≤ C ˆΩ t 8/3 ε ε 8/3 3/4 ≤ C ε 2 • o(ε 9/4 ) = o(ε 1/4
∇ψ ε 4 ≤ o(1) • ∇ψ ε 4 + 1 ,
implying that ∇ψ ε 4 = o(1), as claimed.

We next prove local estimates in Ω. It suffices to consider a sequence ε n → 0, but for simplicity we will drop the subscript n.

Fix some small r 0 > 0, depending on Ω, such that the nearest point projection onto ∂Ω is smooth in the set {x ∈ Ω; dist(x, ∂Ω) < r 0 }. Set, for 0 < r < r 0 , Ω r := {x ∈ Ω; dist(x, ∂Ω) > r}, which is a smooth domain. Using (4.29) and the Fubini theorem we can find some r = r ε such that r 0 /2 < r < r 0 and

ˆ∂Ω r |∇t ε | 2 + t 2 ε ε 2 ≤ Cε. (4.32)
For such r, we claim the following.

4.5 Lemma. We have

ˆΩr |∇t ε | 2 + t 2 ε ε 2 ≤ Cε 2 . ( 4
u ε H 2 (Ω s ) ≤ C,
and then, by Sobolev embeddings,

∇u ε L p (Ω s ) ≤ C p , ∀ p ∈ [1, ∞). (4.38)
Next we argue similarly to the proof of Proposition 3.9. For any q > 2, multiplying (2.6b) by |t ε | q-2 t ε /(ε 2 ) q-1 and integrating over Ω s gives

c 1 ˆΩs |t ε | ε 2 q ≤ ˆΩs (q -1) ε 2(q-1) |t ε | q-2 |∇t ε | 2 + (2α + α t t ε ) |t ε | ε 2 q = ˆΩs c|∇ϕ ε | 2 |t ε | ε 2 q-2 t ε ε 2 + ˆ∂Ω s |t ε | ε 2 q-2 t ε ε 2 ∂t ε ∂n . (4.39)
We apply the above with q = 5/2. Using (4.37) and Cauchy-Schwarz inequality we estimate the boundary integral by

ˆ∂Ω s |t ε | ε 2 1/2 t ε ε 2 ∂t ε ∂n ≤ 1 ε 3 ˆ∂Ω s |t ε | 3 1/2 ˆ∂Ω s |∇t ε | 2 1/2 = 1 ε 3 • o(ε 2 ) • O(ε) = o(1). (4.40)
From (4.39)-(4.40) and Hölder inequality we deduce that the function

f ε := t ε ε 2 satisfies c 1 f ε 5/2 L 5/2 (Ω s ) ≤ ˆΩs c|∇ϕ ε | 2 | f ε | 3/2 + o(1) ≤ c 2 ∇ϕ ε 2 L 5 (Ω s ) f ε 3/2 L 5/2 (Ω s ) + o(1).
Applying (4.38) to the above yields

f ε 5/2 L 5/2 (Ω s ) ≤ C f ε 3/2 L 5/2 (Ω s ) + o(1), implying that f ε L 5/2 (Ω s ) = O(1) and therefore ∆u ε L 5/2 (Ω s ) = O(1)
. By elliptic interior estimates we obtain that u ε W 2,5/2 (Ω r ) = O(1), and (4.36) follows by Sobolev embedding.

We are now ready to complete the proof of the main result of this section.

Proof of Theorem 4.1. The strong convergence u ε → u 0 in H 1 (Ω) was established in Proposition 4.3. To complete the proof of (4.6) we need to prove the uniform convergence. This follows from the two uniform convergences on Ω: t ε → 0 (see Proposition 4.2) and ψ ε → 0 (which results, by Morrey's theorem, from the W 1,4 -convergence that was established in Lemma 4.4). For the proof of (4.7) we only need to verify the following estimate:

t ε L ∞ (K) ≤ C K ε 2 , (4.41)
for every compact K ⊂⊂ Ω. We shall prove (4.41) using an argument from [START_REF] Bethuel | Asymptotics for the minimization of a Ginzburg-Landau functional[END_REF]. We first use Kato's inequality in (2.6b) to get

∆|t ε | ≥ sgn(t ε ) ∆t ε = (2α + α t t ε ) |t ε | ε 2 -c|∇ϕ ε | 2 sgn(t ε ).
Hence, by (2.8b) and (4.36),

-∆|t ε | + c 1 |t ε | ε 2 ≤ C r in Ω r . (4.42)
Now recall [4, Lemma 2] that states that the radial solution ω = ω(r) of

-ε 2 ∆ω + ω = 0 in B R (0) ω = 1 on ∂B R (0) (4.43) satisfies, for ε < 3 4 R, ω(r) ≤ e (r 2 -R 2 )/(4εR) in B R (0). ( 4

.44)

Let d := dist(K, ∂Ω), so that (4.42) is satisfied with r := d/2. Let x 0 be an arbitrary point in K. With no loss of generality we may assume x 0 = 0. From (4.42)-(4.44) and the maximum principle we obtain that

|t ε | ≤ Cε 2 + exp c 1 (|x| 2 -d 2 /4)/(2εd) in B d/2 (0).
In particular,

|t ε (0)| ε 2 ≤ C + 1 ε 2 exp -d c 1 /(8ε) . (4.45)
Since the right-hand side of (4.45) remains bounded as ε → 0, (4.41) follows, completing the proof of (4.7).

From (4.7) and elliptic estimates we obtain that u ε is bounded in W 2,p loc (Ω) for every p < ∞, and (4.8) follows from Morrey's theorem. Finally, (4.9) follows from the previous estimates by the same arguments as in the proof of Theorem 3.1.

We will need in the next section also the following variant of Theorem 4.1 and Theorem 3.1. The proof is very similar to the proofs of these theorems, and is therefore omitted.

Theorem.

Let Ω be a smooth bounded and simply connected domain in R 2 . Let x 0 ∈ ∂Ω and suppose that R > 0 is sufficiently small such that ∂B R (x 0 ) ∩ ∂Ω consists of exactly two points.

Suppose that g : ∂(B R (x 0 )∩Ω) → Γ is a continuous map of degree zero such that the restriction g |∂Ω∩B R (x 0 ) is smooth. Let ϕ 0 be a continuous function such that g = τ(e ıϕ 0 ). Let ζ be the harmonic extension of ϕ 0 to Ω ∩ B R (x 0 ) and set u 0 := τ(e ıζ ).

For each ε > 0 let g ε : ∂(Ω ∩ B R (x 0 )) → R 2 satisfy:

g ε = g on ∂Ω ∩ B R (x 0 ) (4.46) g ε H 1 (∂B R (x 0 )∩Ω) ≤ C, (4.47) ˆ∂B R (x 0 )∩Ω W(g ε ) ≤ Cε 2 , (4.48) g ε → g in H s (∂B R (x 0 ) ∩ Ω), 0 < s < 1. ( 4

.49)

Let u ε be a solution of (4.4) on Ω ∩ B R (x 0 ) (instead of Ω) satisfying (3.1). Then for every R 1 ∈ (0, R) we have:

∆u ε L ∞ (Ω∩B R 1 (x 0 )) ≤ C R 1 , (4.50) u ε → u 0 in C 1,α (Ω ∩ B R 1 (x 0 )), (4.51) u ε -u 0 L ∞ (Ω∩B R 1 (x 0 )) ≤ C R 1 ε 2 , (4.52) ∇(u ε -u 0 ) L ∞ (Ω∩B R 1 (x 0 )) ≤ C R 1 ε. (4.53)
Note that (possibly after passing to a subsequence) the condition (4.49) actually follows from conditions (4.46)-(4.48) via the compact embedding

H 1 (∂B R (x 0 ) ∩ Ω) → H s (∂B R (x 0 ) ∩ Ω), 0 < s < 1.

General solutions

Preliminary estimates

Assume that Ω is a smooth bounded domain in R 2 , strictly star-shaped with respect to a point z ∈ Ω. With no loss of generality, we may assume that z = 0, and thus

x • n = x • n(x) ≥ c > 0, ∀ x ∈ ∂Ω (5.1)
(with n = n(x) the outward normal to ∂Ω at x ∈ ∂Ω).

Let g : ∂Ω → Γ be a smooth boundary datum of degree d. For each ε > 0, let u ε denote a solution of (1.2). As in the previous sections, we denote by t(x) = t ε (x) the signed distance of u ε (x) to Γ. In contrast with the previous sections, we do not impose a condition like (3.1), and thus we allow solutions with vortices.

We start with some basic estimates satisfied by the solutions u ε . We first notice that the results of Lemma 3.2 and Lemma 3.3 hold true since their proofs do not rely on the degree of g.

Next we prove a Pohozaev identity that does rely heavily on the star-shapeness assumption.

5.1 Lemma. We have

ˆΩ W(u ε ) ε 2 + ˆ∂Ω ∂u ε ∂n 2 ≤ C, (5.2)
for some C independent of ε.

Proof. The proof is standard and requires only a simple adaptation of the proof in [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]. We argue as in the proof of Proposition 4.3 multiplying both side of the equation in (1.2) by V • ∇u ε , but this time with V = (x 1 , x 2 ). For this choice of V , (4.14) reads

ˆΩ ∆u ε • (V • ∇u ε ) = ˆ∂Ω ∂u ε ∂n • (x • ∇u ε ) - 1 2 (x • n)|∇u ε | 2 , (5.3) while (4.15) becomes 1 ε 2 ˆΩ ∇W(u ε ) • (V • ∇u ε ) = 1 ε 2 ˆΩ V • ∇(W(u ε )) = - 2 ε 2 ˆΩ W(u ε ).
(5.4)

Combining (5.3) with (5.4) yields 2

ε 2 ˆΩ W(u ε ) + 1 2 ˆ∂Ω (x • n) ∂u ε ∂n 2 = ˆ∂Ω 1 2 (x • n) ∂g ∂σ 2 -(x • σ) ∂u ε ∂n • ∂g ∂σ ,
which, in view of (5.1), clearly implies (5.2).

Since by (1.9) there exists 0

< µ 0 < µ such that W(ζ) ≥ µ 0 dist 2 (ζ, Γ) for ζ ∈ B R 0 , it follows from (5.2) that ˆΩ dist 2 (u ε (x), Γ) ε 2 ≤ C. (5.5)
Using the two estimates (5.2) and (3.9), we can show, using the argument of [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]Chapter 4], that for any small δ 2 > 0 (we will always take δ 2 < δ 1 , see Proposition 2.1) the set

S ε,δ 2 := {x ∈ Ω; dist(u ε (x), Γ) > δ 2 } (5.6)
can be covered by a finite number of "bad discs" {B λε (x ε j )}

k ε j=1 with {x ε j } k ε j=1 ⊂ S ε,δ 2 , (5.7)
where k ε is bounded uniformly in ε. Indeed, we first use (3.9) to choose λ > 0 such that dist(u ε (x), Γ) > δ 2 =⇒ B λε/4 (x) ⊂ Ω and dist(u ε (y), Γ) > δ 2 /2, ∀y ∈ B λε/4 (x) .

(5.8)

Then, we take a collection of mutually disjoint discs {B λε/4 (x ε j )} k ε j=1 which is maximal with respect to the property that (5.7) holds true. Note that by (1.5) there exists η = η(δ 2 ) such that

W(z) > η, ∀z ∈ B R 0 \ Γ δ 2 /2 ,
(5.9)

where Γ δ 2 /2 = {z ∈ R 2 ; dist(z, Γ) < δ 2 /2}. Taking into account (3.8) we get from (5.8)-(5.9) that

1 ε 2 ˆBλε/4 (x ε j ) W(u ε ) ≥ πλ 2 η/16, j = 1, . . . , k ε .
(5.10)

The uniform bound for k ε follows by combining (5.10) with (5.2). By construction S ε,δ 2 ⊂ k ε j=1 B λε (x ε j ). Next, by increasing λ if necessary, we may also assume that the bad discs are well-separated, in the sense that B 4λε (x ε j ) ∩ B 4λε (x ε ) = if j = (this may results in decreasing the value of k ε ). Passing to a subsequence ε n → 0, but continuing to denote ε n by ε, for simplicity, we may assume k ε = k is independent of ε. Note that outside the bad discs the function t(x) is well-defined and that we have

|t(x)| ≤ δ 2 , ∀ x ∈ Ω ε := Ω \ k j=1 B λε (x ε j ).
(5.11)

The definitive value of δ 2 satisfying δ 2 ≤ δ 1 will be chosen in Section 5.3; see the proof of Proposition 5.12.

We next prove that the x ε j 's are relatively far away from ∂Ω.

Lemma.

We have

lim ε→0 dist(x ε j , ∂Ω) ε = ∞, j = 1, . . . , k.
(5.12)

Proof. We argue by contradiction and assume that (5.12) does not hold for some j along some sequence ε n → 0. For notational simplicity, we drop the subscript n. We will obtain a contradiction via a blow up analysis. Let, for small ε, y ε denote the projection of x ε j onto ∂Ω, and let R ε denote the rotation of R 2 such that R ε (0, -1) = n(y ε ). Consider

v ε (x) := u ε y ε + ε R ε x , x ∈ 1 ε R ε -1 Ω -y ε .
Using (3.8) and (5.2), together with the boundary condition in (1.2), we find that, up to a subsequence and uniformly on compacts of

H := {x ∈ R 2 ; x 2 > 0}, v ε converges to a solution v of          ∆w = ∇W(w) in H w = w 0 ∈ Γ on ∂H ∂w ∂x 2 = 0 on ∂H ; (5.13)
here, w 0 is a constant. Let us note that w is not a constant. Indeed, we assumed by contradiction that (5.12) does not hold, and then the fact that w is not constant follows from (5.7). Consider now the map

w = w, in H w 0 , in R 2 \ H . ( 5.14)
In view of (5.13), the map w satisfies ∆ w = ∇W( w) in R 2 , first in the distributions sense, then, by elliptic regularity, in the classical sense. By unique continuation, we have w = w 0 . (The unique continuation property follows from [START_REF] Müller | On the behavior of the solutions of the differential equation ∆U = F(x,U) in the neighborhood of a point[END_REF]; there, w is a scalar function, but this is not relevant for the proof. For an explicit result relevant for vector-valued functions, see e.g. [START_REF] Lopes | Radial symmetry of minimizers for some translation and rotation invariant functionals[END_REF]Appendix].) This contradicts the fact that w is not a constant, and achieves the proof of the proposition. Now that we know that the "bad discs" B λε (x ε j ) are well-inside Ω, we may define the integer d ε j as the degrees of u ε on ∂B λε (x ε j ). By (3.9), these integers are uniformly bounded, so we may assume that their values are independent of ε as well, and thus

deg(u ε , ∂B λε (x ε j )) = d j , ∀ ε, j = 1, . . . , k. (5.15)
In the sequel, in case there is no risk of confusion, we shall often drop the subscript ε.

Our next estimate yields in particular a simple answer to Open Problem 19 in the book [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF] (previously solved in [START_REF] Comte | Remarks on nonminimizing solutions of a Ginzburg-Landau type equation[END_REF] using a different method); see Corollary 5.5 below.

Proposition. We have ´Ωε

[|∇t| 2 + t 2 /ε 2 ] ≤ C.
Proof. The proof uses the following pointwise inequality:

|∇W(z)| 2 ≤ MW(z), ∀z ∈ B R 0 , (5.16)
for some M > 0. The validity of (5.16) for z in a neighborhood of Γ follows from (2.5); the extension to arbitrary z ∈ B R 0 is clear (see also Remark 5.4 below for a simple alternative argument valid also for degenerate W). Arguing as in [5, Ch. V], we obtain using the Galgardo-Nirenberg inequality, (5.16) and (5.2) that

∇u L 4 (Ω) ≤C 1 u 1/2 H 2 u 1/2 ∞ ≤ C 2 u 1/2 H 2 ≤ C 3 1 ε 2 ∇W(u) 2 + 1 1/2 ≤C 4 1 ε 2 W(u) 1/2 1 + 1 1/2 ≤ C 5 ε 1/2 .
(5.17)

Next we multiply (2.6b) by t and integrate over Ω ε . Using (5.17), (2.8b) (recall that δ 2 ≤ δ 1 ≤ δ 0 ) and (5.5) we get ˆΩε

|∇t| 2 + c 1 t 2 ε 2 ≤ C 1 ˆΩε |∇ϕ| 2 |t| + k j=1 ˆ∂B λε (x ε j ) t ∂t ∂n ≤ C 1 ˆΩε |∇u| 4 1/2 ˆΩε t 2 1/2 + C 2 ≤ C 1 C 2 5 ˆΩε t 2 ε 2 1/2 + C 2 ≤ C 3 .
(5.18)

For the bound of the boundary integrals we used the estimate ∂t ∂n ≤ C ε on ∂B λε (x ε j ) (by (3.9)). The conclusion of the proposition is a direct consequence of (5.18).

5.4

Remark. The inequality (5.16) was proved by Dieudonné in [START_REF] Dieudonné | Sur un théorème de Glaeser[END_REF], in connection to his simplified proof to a result of Glaeser [START_REF] Glaeser | Racine carrée d'une fonction différentiable[END_REF] about the square root of a nonnegative C 2 -function. A variant of Dieudonné's argument, valid for any

W ∈ C 2 (R 2 ), goes as follows. Fix a function ξ ∈ C ∞ c (R 2 ; [0, 1]) such that ξ ≡ 1 on B R 0 and set, in R 2 , F(z) := ξ(z)W(z) + (1 -ξ(z))|z| 2 . Note that F is a smooth nonnegative function on R 2 . Let K := 1 2 max z∈R 2 D 2 F(z) ,
where A stands for the spectral norm of the matrix A. By Taylor formula

0 ≤ F(z + h) ≤ F(z) + ∇F(z) • h + K|h| 2 , (5.19)
for every z, h ∈ R 2 . Applying (5.19) for h := -∇F(z) 2K yields |∇F(z)| 2 ≤ 4K F(z), whence (5.16).

Corollary. Let u

ε satisfy (1.2). Then ˆΩ |∇ (dist(u ε , Γ))| 2 ≤ C, ∀ ε > 0.
(5.20)

In particular, in the GL case, i.e., W(u) = (1 -|u| 2 ) 2 /4, we have

ˆΩ ∇|u ε | 2 ≤ C, ∀ ε > 0. (5.21) Proof. Since ∇(dist(u ε , Γ)) ∞ ≤ ∇u ε ∞ ≤ C ε
(by (3.9)), we have

ˆ k j=1 B λε (x ε j ) |∇ (dist(u ε , Γ))| 2 ≤ C. (5.22)
The result of the corollary readily follows from Proposition 5.3 and (5.22). (Recall that, in Ω ε , we have dist(u ε , Γ) = |t ε |.)

In the GL case, it suffices to note that dist(u ε , Γ) = 1 -|u ε |.

A O(|log ε|) bound for the energy

The main result of this section is the following.

Proposition. We have E

ε (u ε ) ≤ C(| log ε| + 1), ∀ ε > 0.
In view of Proposition 5.3, of (3.9) and (5.2), it suffices to obtain the following bound for the energy of the phase ϕ:

ˆΩε |∇ϕ| 2 ≤ C(| log ε| + 1), ∀ ε > 0.
(5.23)

Since ϕ is defined only locally in Ω ε (only its gradient ∇ϕ is defined globally), it will be convenient to introduce a new function, which is globally defined in Ω ε .

Definition.

Let Π denote the nearest point projection on Γ in a δ 2 -tubular neighborhood of Γ. The S 1 -valued map

Ω ε z → τ -1 (Π(u)) • k j=1 z -x j |z -x j | -d j
(with d j as in (5.15)) has zero degree around each of the holes B λε (x j ), j = 1, . . . , k. Hence, there exists a unique (up to addition of an integer multiple of 2π) scalar function η = η ε such that

τ -1 (Π(u)) = e ıη k j=1 z -x j |z -x j | d j in Ω ε .
(5.24)

By adding an appropriate multiple of 2π we may assume that min ∂Ω η ε ∈ [0, 2π).

(5.25)

Since g is smooth, we deduce from (5.25) that

η ε L ∞ (∂Ω) ≤ C(g). (5.26)
Our first step consists of proving an L ∞ bound for η. In order to be able to apply the maximum principle of Proposition 2.1 we will remove from Ω ε a collection of rays, connecting the boundaries of the holes B λε (x j ), j = 1, . . . , k, to the boundary of Ω. The choice of these "good rays" will depend on energy considerations. For any j = 1, . . . , k and α ∈ [0, 2π), we let D j (α) be the half-line D j (α) := {x ε j + r e ıα ; r ∈ [λε, ∞)}, and then set R j (α) := D j ∩ Ω ε .

Lemma.

For each j = 1, . . . , k and 0 < ε < 1/2, there exists α j = α j (ε) ∈ [0, 2π) such that R j := R j (α j ) satisfies

ˆR j ∂η ∂r ≤ C| log ε| 1/2 ∇η L 2 (Ω ε ) .
(5.27)

Here, ∂/∂r stands for the tangential derivative along R j .

Proof. Since ˆΩε

|∇η| 2 ≥ ˆ2π 0 ˆR j (α)
|∇η| 2 rdr dα, there exists α j ∈ [0, 2π) such that

ˆR j (α j ) |∇η| 2 rdr ≤ 1 2π ∇η 2 L 2 (Ω ε ) . ( 5
≤ C| log ε| 1/2 ∇η L 2 (Ω ε ) .
Next, we denote

ω ε := Ω ε \ k j=1 R j .
For each j, let θ j denote the polar coordinate around the point x j , taking values in [α j , α j + 2π). Then the function

Θ = Θ ε := k j=1 d j θ j , (5.29)
is smooth in ω ε and satisfies

Θ L ∞ (ω ε ) ≤ 4π k j=1 |d j |.
(5.30)

We define ϕ = ϕ ε := η + Θ in ω ε . Note that

τ -1 (Π(u)) = e ıη k j=1 z -x j |z -x j | d j
= e ı(Θ+η) = e ıϕ in ω ε , so that ϕ is a well-defined phase of u in ω ε .

5.9 Lemma. We have

η L ∞ (∂ω ε ) ≤ C | log ε| 1/2 ∇η L 2 (Ω ε ) + 1 (5.31) and lim sup δ→0 sup{|ϕ(x)|; x ∈ ω ε , dist(x, ∂ω ε ) ≤ δ} ≤ C | log ε| 1/2 ∇η L 2 (Ω ε ) + 1 .
(5.32)

Proof. We may assume that 0 < ε < 1/2. Let r j (α) be the smallest r > λε such that x ε j + r e ıα ∈ ∂Ω. By Lemma 5.8 and (3.9), for each x ∈ [x ε j + λ e ıα , x ε j + r j (α) e ıα ] we have η(x) -η x ε j + r j (α j ) e ıα j ≤ C | log ε| 1/2 ∇η L 2 (Ω ε ) + 1 .

(5.33)

Note that (3.9) is needed in case R j intersects some of the other discs {B λε(x ε l ) } l = j before reaching ∂Ω for the first time, at x ε j + r j (α j ). In particular, the following holds:

η(x ε j + λε e ıα j ) -η x ε j + r j (α j ) e ıα j ≤ C | log ε| 1/2 ∇η L 2 (Ω ε ) + 1 .

(5.34)

On the other hand, by (3.9) we have Finally, (5.32) follows from (5.31) and (5.30).

|η(x) -η( y)| ≤ C, j = 1, . . . , k, ∀ x, y ∈ ∂B λε (x ε j ). ( 5

Lemma. We have

η L ∞ (Ω ε ) ≤ C | log ε| 1/2 ∇η L 2 (Ω ε ) + 1 .
Proof. We apply the maximum principle in Proposition 2.1 to ϕ on each component of the open set {x ∈ ω ε ; dist(x, ∂ω ε ) > δ}, then we let δ → 0 (with fixed ε). Using (5.32), we find that sup

ω ε |ϕ| ≤ C | log ε| 1/2 ∇η L 2 (Ω ε ) + 1 .
(5.36)

The bound for η is a consequence of (5.30) and (5.36).

Proof of Proposition 5.6. By (2.6), η satisfies in Ω ε div(a∇η) = -div(a∇ϕ) + div(a∇Θ) = b|∇ϕ| 2 -α ϕ t 2 ε 2 + div(a∇Θ) = f + div(a∇Θ),

(5.37) with

f = f ε := b |∇ϕ| 2 - α ϕ t 2 ε 2 .
(5.38)

Above we denoted by ∇Θ the vector field

∇Θ = k j=1 d j ∇θ j = k j=1 d j (x -x ε j ) ⊥ |x -x ε j | 2 , which is smooth in R 2 \ {x ε 1 , . . . , x ε k }.
Here we used the notation

V ⊥ = (-v 2 , v 1 ) for a vector V = (v 1 , v 2 ) ∈ R 2 . We claim that f ε L 1 (Ω ε ) ≤ C.
(5.39)

Indeed, the second term on the right-hand side of (5.38) is bounded in L 1 (Ω ε ) by Proposition 5.3. The L 1 boundedness of the first term b |∇ϕ| 2 follows from the calculation (5.18) and the inequality (2.8d). Multiplying (5.37) by η and integrating yields ˆΩε

a|∇η| 2 = ˆΩε f η - ˆΩε a∇Θ • ∇η + ˆ∂Ω ε a ∂ϕ ∂n η ≤ f 1 η ∞ + C| log ε| 1/2 ∇η L 2 (Ω ε ) + C η ∞ ≤ C | log ε| 1/2 ∇η L 2 (Ω ε ) + 1 .
(5.40)

The first inequality in (5.40) uses (5.2) on ∂Ω and (3.9) on ∂B λε (x ε j ). The second inequality follows from Lemma 5.10.

From (5.40) we get

∇η 2 L 2 (Ω ε ) ≤ C(| log ε| + 1), (5.41)
and therefore

∇ϕ 2 L 2 (Ω ε ) ≤ 2 ∇Θ 2 L 2 (Ω ε ) + ∇η 2 L 2 (Ω ε ) ≤ C(| log ε| + 1).
(5.42)

As explained above, estimate (5.42) implies Proposition 5.6.

Combining Lemma 5.10 with (5.41) we obtain the following.

Corollary.

We have η L ∞ (Ω ε ) ≤ C(| log ε| + 1).

An L p -bound for the gradient, p ∈ [1, 2)

The main result of this section is 5.12 Proposition. We have

∇u ε L p (Ω) ≤ C p , 1 ≤ p < 2.
Proof. Fix any p ∈ (1, 2). We will apply the bad discs construction from Subsection 5.1 with a δ 2 = δ 2 (p) ≤ δ 1 , that we define below. By standard elliptic estimates, there exists a constant A p = A p (Ω) such that the solution w of the problem where c 0 is defined in (2.8a). We choose δ 2 = δ 2 (p) accordingly such that (5.11) holds. In the sequel, Ω ε denotes the set given in (5.11) for this choice of δ 2 . Note that the number of discs and the value of λ may change with δ 2 , but we shall use the same notation as before.

-∆w = div g in Ω w = 0 on ∂Ω (5.43) with g ∈ (L p (Ω)) 2 satisfies ∇w L p (Ω) ≤ A p g L p (Ω) . ( 5
Let H = H ε denote the harmonic function in Ω satisfying H = η ε on ∂Ω. By (5.26) and the maximum principle,

H ε L ∞ (Ω) = η ε L ∞ (∂Ω) ≤ C(g).
(5.46) Note that

η ε W 1-1/p,p (∂Ω) ≤ C, since k j=1 z -x j |z -x j | d j W 1,p (Ω)
≤ C, see (5.24). Therefore, we also have

H W 1,p (Ω) ≤ C. (5.47) Consider a function ξ = ξ ε ∈ C ∞ (Ω) satisfying 0 ≤ ξ ≤ 1, ξ ≡ 1 on Ω \ k j=1 B 2λε (x ε j ), ξ ≡ 0 on k j=1 B 3λε/2 (x ε j ), ∇ξ ∞ ≤ 4 λε . (5.48)
Note that, by (5.12), for small ε the discs {B 2λε (x j )} do not intersect the boundary, and thus ξ = 1 on ∂Ω. From the properties of ξ we obtain, in particular, that

∇ξ p = ∇ξ L p ( k j=1 B 2λε (x ε j )) ≤ C ε 2/p-1 = o(1).
(5.49)

In Ω, we set η := η ε = ξ 2 η and H = H ε := ξ 2 H. From (5.46)-(5.49) we conclude that

H W 1,p (Ω) ≤ C.
(5.50) Note that, although η is defined only in Ω ε , the function η is globally defined (and smooth), since η = 0 on a neighborhood of B λε (x ε j ). The function η satisfies (5.53)

-div(a∇ η) = -div(aξ 2 ∇η) -div(aη∇(ξ 2 )) = -ξ 2 div(a∇ϕ) F 1 -a∇(ξ 2 ) • ∇ϕ F 2 + div(aξ 2 ∇Θ G 1 ) + div(-2aηξ∇ξ G 2 ) := F 1 + F 2 + divG 1 + divG 2 . Therefore, -∆( η -H) = F 1 + F 2 + divG 1 + divG 2 + div(a∇ H) + div((a -1)∇( η -H)) in Ω η -H = 0 on ∂Ω . ( 5
Similarly, we can find for each j ∈ {1, . . . , N} a number β j ∈ [0, 2π) such that the set R j = R j (β j ) := {a j + s e ıβ j ; s ≥ r} ∩ Ω r satisfies ˆ R j ∂η ∂s ds ≤ C 2 (r).

(5.59)

Repeating the proof of Lemma 5.10 and using (5.58) and (5.59), we find that

η L ∞ ( Ω r ) ≤ C 3 (r).
(5.60)

For ε sufficiently small we have Applying the Cauchy-Schwarz inequality to I 2 and the above estimates in (5.62) leads to

|x ε -a j | < r/2, ∀ ∈ J j , j = 1, . . . , N. ( 5
ˆ Ω r a|∇η| 2 ≤ C 6 (r) + ˆ Ω r a 2 |∇η| 2 + ˆ Ω r a 2 |∇Θ| 2 .
(5.63)

Since

´ Ω r (a/2) |∇Θ| 2 ≤ C 7 (r)(| log r| + 1), we get from (5.63) that

´ Ω r |∇η| 2 ≤ C 8 (r). It follows that also

´ Ω r |∇ϕ| 2 ≤ C 9 (r), which clearly implies (5.57).

Convergence of {u ε n }

The bound of Proposition 5.12 implies that for a subsequence {u ε n } we have

u ε n u * weakly in W 1,p (Ω), ∀ p ∈ [1, 2), (5.64)
for some u * ∈ p∈[1,2) W 1,p (Ω; Γ). The fact that u * is Γ-valued follows from (5.64) and the estimate (5.2) that implies the convergence t ε n → 0 in L 2 (Ω).

We can now further state 5.14 Proposition. We have

u ε n → u * in C 1,α (Ω \ {a 1 , . . . , a N }), ∀ α ∈ (0, 1).
(5.65)

The limit u * is a Γ-valued harmonic map in Ω \ {a 1 , . . . , a N }.

Proof. We argue as in [5, Proof of Theorem VI.1]. For notational simplicity, we drop in what follows the subscript n. It suffices to show that for every x 0 ∈ Ω \ {a 1 , . . . , a N } there exists R > 0 such that u ε → u * in C 1,α (Ω ∩ B R (x 0 )). Consider first the case x 0 ∈ Ω. We choose R > 0 such that B 2R (x 0 ) Ω \ {a 1 , . . . , a N }. Since, by (5.57),

E ε (u ε ; B 2R (x 0 )) ≤ C,
we can use Fubini's theorem to find R ∈ (R, 2R) such that (after passing to a further subsequence),

ˆ∂B R (x 0 ) 1 2 |∇u ε | 2 + W(u ε ) ε 2 ≤ C.
Then, applying Theorem 4.1 we obtain that, up to a further subsequence, u ε → u 0 in C 1,α (B R (x 0 )), and that u 0 is a harmonic map in B R (x 0 ) (since it can be written as u 0 = τ(e ıζ ) where ζ is a harmonic function in B R (x 0 )). Using the uniqueness of the limit, we find that u 0 = u * , and that the original subsequence {u ε n } converges to u * in C 1,α (B R (x 0 )). It remains to consider the case x 0 ∈ ∂Ω\{a 1 , . . . , a N } (at this stage we do not exclude the possibility that some of the a j 's belong to ∂Ω). We choose a small R > 0 such ∂B R (x 0 ) ∩ ∂Ω consists of exactly two points and R < min Again by (5.57), we have

E ε (u ε ; Ω ∩ B 2R (x 0 )) ≤ C,
and by Fubini's theorem there exists R ∈ (R, 2R) such that (after passing to a further subsequence),

ˆ∂B R (x 0 )∩Ω 1 2 |∇u ε | 2 + W(u ε ) ε 2 ≤ C.
Applying Theorem 4.7 we obtain that u ε → u * in C 1,α (B R (x 0 ) ∩ Ω).

Next we deduce further properties of the map u * that will enable us to conclude the proof of Theorem 1.1.

Proposition.

We have {a 1 , . . . , a N } ⊂ Ω.

Proof. The proof is the same as that of [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]Theorem X.4], so we just mention the main idea. By Pohozaev identity (5.2) and Proposition 5.14 it follows that

ˆ∂Ω ∂u * ∂n 2 < ∞.
(5.66)

The map v * := τ -1 • u * is an S 1 -valued smooth harmonic map on Ω \ {a 1 , . . . , a N }, and satisfies: v * ∈ W 1,p (Ω; S 1 ) for all p ∈ [1, 2), v * = τ -1 • g on ∂Ω, and thanks to (5.66), also

ˆ∂Ω ∂v * ∂n 2 < ∞.
Therefore, all the hypotheses of [5, Lemma X.14] are satisfied, and we can conclude that v * is smooth in a neighborhood of ∂Ω. Clearly, the same holds for u * .

To conclude the proof of Theorem 1.1 we need to show that the limit u * has the form given in (1.11). Equivalently, Proposition 5.16 asserts that the S 1 -valued harmonic map τ -1 • u * is the canonical harmonic map associated with τ -1 • g and {(a j , D j )} N j=1 , as defined in [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]Sec I.3].

Proof. We apply the same argument as in [5, Ch. VII], which uses the Hopf differential. Setting

ω = ω ε := (u ε ) x 1 2 -(u ε ) x 2 2 -2ı (u ε ) x 1 • (u ε ) x 2 ,
we find by a direct computation Note that up to a further subsequence we have

∂ω ∂z = 1 2 ω x 1 + ı ω x 2 = ∆u ε • (u ε ) x 1 -ı (u ε ) x 2 = 2∆u ε •
W(u ε ) ε 2 * N j=1 m j δ a j , (5.70)
for some positive m j 's. (Convergence is in the weak star topology of C(Ω).) Indeed, combining (4.9) and (4.52) we obtain, for any sufficiently small R > 0,

W(u ε ) ≤ C R ε 2 in Ω \ N j=1
B R (a j ), which clearly implies (5.70) with m j ≥ 0. The fact that m j > 0 for all j follows from (5.10).

Defining the distribution

α = α ε := ∂ ∂z 1 πz * χ Ω W(u ε ) ε 2 ,
we obtain by a direct calculation that β = β ε := ω -2α is a holomorphic function in Ω (see also [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF]). Since, by (4.7)-(4.8),

1 ε 2 ∇W(u ε ) ∂u ε ∂x j L ∞ (Ω\ N j=1 B R (a j ))
≤ C, j = 1, 2, we obtain that W(u ε ) ε 2 is bounded in C 1 (Ω \ N j=1 B R (a j )), and we deduce by the argument of [START_REF] Bethuel | Ginzburg-Landau Vortices[END_REF] that {β ε } is bounded in C 0 loc (Ω). It follows that, up to a further subsequence, β ε → β * in C k loc (Ω), ∀ k, for some holomorphic function β * in Ω. In addition, using (5.70) we find that (5.72) Fix any j ∈ {1, . . . , N} and assume without loss of generality that a j = 0. Recall that τ -1 • u is a harmonic map in Ω \ {a 1 , . . . , a N } (Proposition 5.14) and belongs to W 1,p (Ω) when 1 ≤ p < 2 (Proposition 5.12). In addition, we have deg(u * , 0) = D j . Arguing as in [5, Remark I.1] we may write, near 0, u * = τ exp(ıD j θ + ıc j log r + ıh) , where h is a harmonic function.

It follows that if we write, locally near 0, u * = τ(e ıϕ ) with ϕ := D j θ + c j log r + h, then we have . The fact that c j = 0 for all j implies that u * has the form (5.67).

ω * = (u * ) x 1 2 -(u * ) x 2 2 -2ı (u * ) x 1 • (u * ) x 2 = ϕ x 1 -ı ϕ x 2 2 = c j -ı D j z + 2 ∂h
Since we know already that m j = 0 for all j, it follows that also D j = 0 for all j.

( 3 .

 3 20) The conclusion (3.11) clearly follows from (3.19)-(3.20).

4. 1

 1 Theorem. Under the assumptions (4.1)-(4.4) and (3.1) we have, as ε → 0,

2 ˆR j (α j ) ∂η ∂r 2 rdr 1 / 2

 2212

  .35) We obtain (5.31) by combining (5.33)-(5.35) with (5.26).

  .51) By elliptic estimates, there exists B p = B p (Ω) > 0 such that the solution w of the problem -∆w = v in Ω w = 0 on ∂Ω (5.52) with v ∈ L 1 (Ω), satisfies ∇w p ≤ B p v 1 .

  1≤ j≤N |x 0 -a j |.

5. 16

 16 Proposition. We have u * (z) = τ e ıη(z) za 1 |z -a 1 |for some smooth harmonic function η in Ω and D 1 , . . . , D N ∈ Z \ {0}.

  a j ) 2 in D (R 2 ). (5.71)Therefore, ω ε = β ε + 2α ε → β * + 2α * in D (Ω). Since Proposition 5.14 implies that ω ε → ω * := (u * ) x 1 2 -(u * ) x 2 2 -2ı (u * ) x 1 • (u * ) x 2 in C 0 loc (Ω \ {a 1 , .. . , a N }), we obtain ω * = β * + 2α * in D (Ω \ {a 1 , . . . , a N }).

From ( 5 .

 5 71)-(5.73) we obtain (c j -ı D j ) 2 = -2 m j π , implying that c j = 0 and also m j = π D 2 j 2

  Proof. Again by Remark 3.6, it suffices to consider small ε. Using the L 4 -bound of Lemma 3.7 for ∇ϕ ε in (3.28) yieldsˆΩ t 2 ≤ Cε 4 .Since |∇W(u ε )| = O(t ε ), we deduce from (3.32) that the right-hand side of the equation in (1.2) is bounded in L 2 (Ω) and the conclusion follows from elliptic estimates.

	.31)
	Combining (3.25),(3.27) and (3.31) and using Lemma 3.4 we obtain
	∇ψ 4 ≤ o(1) • ∇ψ 4 + 1 ,
	and (3.22) follows.
	3.8 Lemma. {u ε } is bounded in H 2 (Ω).
	(3.32)

  Next, using(3.39) and(3.33) in (2.6b) gives the ∆t ∞ ≤ C. Combining this estimate with (3.33) and applying an interpolation inequality (see [4, Lemma A.2]) yields (3.34). To prove (3.35)-(3.36) for ψ, we use (3.39) and the estimates

and (3.33) follows.

  Choose r ∈ (r 0 /6, r 0 /5) satisfying (4.32) on ∂Ω r . Then the above arguments apply for Ω r . In particular, (4.33) holds on Ω r , and using Fubini theorem we can find s ∈ ( r 0 /4, r 0 /3) such that ˆ∂Ω s

		|∇t ε | 2 +	t 2 ε ε 2 ≤ Cε 2 .		(4.37)
	Since |∇W(u ε )| = O(t ε ), the estimate (4.33) on Ω r implies that ∆u ε L 2 (Ω r ) = O(1). By standard interior
	elliptic estimates, it follows that
									.33)
	Proof. By (4.32) and the Cauchy-Schwarz inequality we have
	ˆ∂Ω r	t ε	∂t ε ∂n	≤ Cε 1/2 • ε 3/2 = Cε 2 .	(4.34)
	Similarly to the proof of (4.24), we multiply (2.6b) by t ε and integrate by parts on Ω r . For the boundary
	integral we use the improved bound (4.34) and to bound ϕ ε 4 we use Lemma 4.4. This yields
	ˆΩr	|∇t ε | 2 + c 1	t 2 ε ε 2 ≤ C	ˆΩr	t 2 ε	1/2	+ Cε 2 ,	(4.35)
	which clearly implies (4.33).

4.6 Lemma. We have

∇u ε L ∞ (Ω r ) ≤ C. (4.36)

Proof.

  .61) Next, we multiply the equation (5.37) satisfied by η and integrate over Ω r . This yields as in (5.40) ˆ By (5.39) and (5.60) we have |I 1 | ≤ C 4 (r). We claim that also |I 3 | ≤ C 5 (r). Indeed, we use (5.2) and (5.60) for the integral on ∂ Ω r ∩ ∂Ω and for the integral on ∂B r (a j ) ∩ Ω we use (5.58) and the fact that thanks to (5.61) we have

	Ω r	a|∇η| 2 =	ˆ Ω r	f η -	ˆ Ω r	a∇Θ • ∇η +	ˆ∂	Ω r	a	∂ϕ ∂n	η := I 1 + I 2 + I 3 .	(5.62)
							∂Θ ∂n	≤	C r	on ∂B

r (a j ).

Acknowledgments.

The first author (PM) was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The second author (IS) was supported by the Israel Science Foundation (Grant No. 999/13). Part of this work was done while IS was visiting the University Claude Bernard Lyon 1. He thanks the Mathematics Department for its hospitality.

Note that F 1 = ξ 2 f is bounded in L 1 (Ω); here, f is defined in (5.38). The same holds for F 2 since, by (3.9),

Using the inequality

by Corollary 5.11 and (5.48). Finally, a∇ H is bounded in L p (Ω) by (5.50).

We also note that

Using the above in (5.51) we get by (5.44) and (5.53) that

(5.54)

Combining (5.45) and (5.54), we find that ∇( η -H) L p ≤ C, which in conjunction with (5.50

(5.55)

The conclusion of Proposition 5.12 follows from (5.55) and the fact that, by (3.9), {∇u ε } is bounded in L p (B 2λε (x ε j )).

A bound for the energy away from the singularities

We denote by a 1 , . . . , a N ∈ Ω the different limits of the families {x ε j }, j = 1, . . . , k (possibly along a subsequence). Since two different families can converge to the same limit, we have N ≤ k. At this point we do not exclude the possibility that some of the a i 's belong to ∂Ω. Consider some r > 0 such that r < min{|a i -a j |; i = j} and r < dist(a j , ∂Ω), ∀ j such that a j ∈ Ω.

(5.56)

We denote

and by D j the degree of u ε on ∂(B ρ (a j ) ∩ Ω) for small ε and (small but fixed) ρ. The following equality is clear: if J j := { ; x ε → a j }, then D j = ∈J j d .

Theorem.

For each r as in (5.56) we have E ε (u ε ; Ω r ) ≤ C(r).

(5.57)

Proof. By the boundedness of {∇η} in L 1 (Ω ε ) (see Proposition 5.12), it follows that there exists r = r(ε) ∈ (r/2, r) such that N j=1 ˆ∂B r (a j )∩Ω |∇η| dσ ≤ C 1 (r).

(5.58)