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URL:http://www.emath.fr/cocv/

ON A MODEL OF ROTATING SUPERFLUIDS

Sylvia Serfaty

1

Abstract. We consider an energy-functional describing rotating superfluids at a rotating velocityω, and prove similar results as for the Ginzburg-Landau functional of superconductivity: mainly the existence of branches of solutions with vortices, the existence of a critical ω above which energy- minimizers have vortices, evaluations of the minimal energy as a function ofω, and the derivation of a limiting free-boundary problem.

Mathematics Subject Classification. 35Q99, 35J60, 35J50, 35B40, 35B25.

Received February 1, 2000. Revised November 28, 2000.

1. Introduction

1.1. The energy functional

The aim of this paper is to study a question that was asked by Pomeau, concerning a model of rotating superfluids. The evolution of a superfluid, such as superfluid helium II at zero temperature, is generally modelled (after some rescaling) by the following nonlinear Schr¨odinger equation, called the Gross-Pitaevskii equation:

−i~∂u

∂t =~2∆u+u(1− |u|2). (1.1)

The Gross-Pitaevskii equation is also used to model the evolution of Bose-Einstein condensates. Here u is a complex-valued function characterizing the local state of the superfluid (it is a pseudo wave-function and 0≤ |u| ≤1). If the superfluid is in a cylindrical bucket of two-dimensional section Ω, smooth, bounded and simply connected, and rotating around a vertical axis at the angular velocityω; then, its energy, written in the rotating frame, taking into account the Coriolis force, is

Z

~2|∇u+iuω×x|2+1

2(1− |u|2)2,

supplemented with the boundary condition u= 0 on ∂Ω. Herex = (x1, x2) Ω with the origin set at the rotation axis, and×is the vectorial product inR3. Byiuω×xwe mean the complex-valued vectoriuω(x2, x1), then consideringω as a positive real number.

Keywords and phrases:Vortices, Gross-Pitaevskii equations, superfluids.

1CMLA, ´Ecole Normale Sup´erieure de Cachan, 61 avenue du Pr´esident Wilson, 94235 Cachan Cedex, France;

e-mail:serfaty@cmla.ens-cachan.fr

c EDP Sciences, SMAI 2001

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This model could also serve to describe Bose-Einstein condensates, whose evolution is given by the Gross- Pitaevskii equation. For a rotating Bose-Einstein condensate trapped in a harmonic potential, a more realistic model includes a termR

(a(x)−|u|2)2wherea(x) is a quadratic function vanishing on∂Ω instead ofR

(1−|u|2)2, (see [9] and [2]), but would also lead to the same kind of analysis.

We replace the study of (1.1) by the study of J(u) = 1

2 Z

|∇u+iuω×x|2+ 1

2(1− |u|2)2, (1.2)

over H01(Ω,C), whereεis a small parameter. If we expand the first term, we obtain J(u) =1

2 Z

|∇u|2+ 1

2(1− |u|2)2+ω2 2

Z

|u|2|x|2+ω Z

(iu, x2ux1−x1ux2). (1.3) Here (., .) denotes the scalar product in R2, where complex numbers are seen as belonging to R2. Another minimization problem which can be considered to derive this is the following: minimize a Hamiltonian of the form

H =1 2 Z

|∇u|2+ 1

2(1− |u|2)2, whereu∈H01(Ω,C), with fixed angular momentum

M = Z

(iu, x× ∇u) = Z

(iu, x1ux2−x2ux1).

(The HamiltonianH and the momentumM are quantities that are conserved in time for the evolution of the type (1.1).) Using a Lagrange multiplier λ, this is equivalent to minimizing

H−λM =1 2 Z

|∇u|2+ 1

2(1− |u|2)2−λ Z

(iu, x1ux2−x2ux1).

Up to the term 12ω2R

|u|2|x|2, this is the same expression as (1.3) for ω =λ. Thus, the rotation velocity ω can be seen as the Lagrange multiplier in the previous problem. On the other hand, we shall see that if ω is sufficiently small compared to 1ε, the term 12ω2R

|u|2|x|2is of lower order in the energy, hence can be neglected, since, up to slight adjustments in our proofs, it would lead to the same qualitative results.

Another question that physicists consider is to minimize an energy of the typeJ or H−λM with a fixed

“number of particles” N =R

|u|2. Again, this can be taken into account through a Lagrange multiplier. It adds a term which is also negligible whenεis small and ω not too large. Thus, we reduce to the study of J given by the expressions (1.2) or (1.3).

As already mentioned,εis a small parameter, we will actually make it tend to zero. This corresponds to the case where the characteristic scale of the phenomenonε, is small compared to the scale of the domain, which is relevant for usual sizes of domains, and is a limit often considered by physicists (see for example [10]). In the physics of Bose-Einstein condensates,εsmall corresponds to the “Thomas-Fermi” approximation (see [2, 9]).

The question is, of course, to find steady states (or critical points) for this energy in the rotating frame, and to describe them. The main feature of rotating superfluids is that, for certain velocities, they exhibitvortices: u has some isolated zeros in Ω, and |uu| has a nonvanishing (topological) degree around these zeros. More precisely, consider a a point where u vanishes and r > 0 small such that udoes not vanish on ∂B(a, r), then |uu| is a mapping from ∂B(a, r) to S1, hence it has a topological degree, or winding number (which is the number of turns of the phase ofu). This is what is called the degree of the isolated zero. The characteristic scale of the phenomenon is thusε, the scale of a vortex. In experiments, there can be up to thousands of vortices in the domain. For more details on the physical aspects, one can refer to the physical litterature ([10, 24] for example).

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This behaviour of superfluids is very similar to the behaviour of superconductors in an external magnetic field. Actually, we prove here that there is a total analogy between this model and the Ginzburg-Landau model of superconductivity, and that we can adjust our results on the Ginzburg-Landau energy to this functional. The Ginzburg-Landau functional for superconductors is

G(u, A) = Z

1

2|∇u−iAu|2+1

2|curl A−hex|2+ 1

2(1− |u|2)2, (1.4) wherehexis the intensity of the applied magnetic field,A= (A1, A2)R2is the vector potential of the magnetic field, andh= curlAthe induced magnetic field in the material. The first termR

|∇u+iuω×x|2is very similar to the termR

|∇u−iAu|2 in the Ginzburg-Landau functional. Actually,J is even simpler, it only depends on one function, and, as we shall see, the role of the external fieldhexis replaced by the angular velocityω.

In [16–18,20–22], we studied in details the functional (1.4) and its minimizers, and proved that they exhibited a vortex-structure whenHc1 ≤hex Hc2, where Hc1 and Hc2 are critical values depending on ε. Here, we adjust these results and obtain very similar ones.

Let us emphasize that the main difference between the two problems is theboundary condition: here u= 0 on ∂Ω whereas, for Ginzburg-Landau, all functions in H1 were admissible, so no boundary conditions were imposed. This conditionu= 0 induces a cost of Cε at least in the energy, becauseuhas to be small on a layer of size of the order of εnear ∂Ω. This cost is very large compared to the Ginzburg-Landau energyG. Hence, if we make comparisons with test maps, all the fine information on the behaviour of uin Ω will be hidden by the energetic cost of the boundary layer. The method for solving this problem was suggested to us by Shafrir, and is one that has been introduced by Lassoued and Mironescu [13] and also used by Andr´e and Shafrir in [4].

It consists in dividingubyρ, the real-valued function which vanishes at∂Ω and minimizes J over the space of real-valued functions. Then, we can prove thatJ splits as

J(u) =J(ρ) + Z

ρ2

2|∇v|2+ ρ4

2(1− |v|2)2+ω Z

ρ2(iv, x2vx1−x1vx2), (1.5) where v = uρ. J(ρ) contains the boundary layer contribution whileJ(u)−J(ρ)J(ρ) can be studied as the Ginzburg-Landau functional. Let us emphasize again that the ideas of the results are not new, but borrowed from those of [4, 13, 16–18, 20, 21], and that this paper consists in showing that these ideas remain valid and can be adjusted to this new problem.

1.2. Notations

We studyJ onH01(Ω,C). Critical points ofJ are solutions of the following associated Euler equation:

(G.P.)

∆u= εu2(1− |u|2) + 2i∇u.ω×x−ω2r2u in Ω u= 0 on∂Ω

that we call the Gross-Pitaevskii equation. By the maximum principle, a solutionuof (G.P.) satisfies|u| ≤1.

We write x = (x1, x2) and r = |x|, × is the vector product in R3 while (,) is the scalar product on R2. ω denotes the rotation-vector perpendicular to Ω inR3 in the expressionω×x, otherwise its norm. We write

f = (−fx2, fx1). F will denote the functional studied in [5],i.e.

F(u) = 1 2 Z

|∇u|2+ 1

2(1− |u|2)2.

For any subset V Ω, JV or FV will denote the energy-functionals restricted to V. The domain DM over which we perform a local minimization of J, corresponds roughly to the u∈H01 for which F(u)≤ M|log ε|. Rwill denote the space of Radon measures on Ω.

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1.3. Statement of the results of existence of branches of solutions

We prove the following results, where the notion of “vortex” will be specified later. In all the paper, ω is considered as a function ofε such thatωε→0 as ε→0, andC denotes some positive constant, independent ofε.

Theorem 1. SupposeΩ =BR=B(0, R). Defining a rotation velocity ω1by ω1= |log ε|

R2 , (1.6)

there existk(ε) =O(1),k0(ε) =O(|log |logε||), andε0(M)such that forε < ε0, the following holds:

- if and only if ω ≤ω1−k(ε), the minimum ofJ is J(ρ)−o(1) and if ω ≤ω1−k0(ε)any minimizer is vortex-less;

- if ω1+k(ε)≤ω≤ω1+O(1), there exists a minimizer of J overDM which is a solution of (G.P.). In addition, it has exactly one vortexaof degree one, and|a| →0asε→0.

This theorem which is similar to Theorem 1 of [16], shows that there exists a critical value ofω above which vortices become energetically favourable. The expression ofω1, equivalent to that of Hc1 in [16], is an explicit function of the size of the domain, and corresponds to the expressions found in physics literature (see [10]).

Theorem 2. Suppose Ω =BR, and ω is any function of ε such thatω + as ε→0, and ω ≤Cεα for some small α > 0; then ∀n N such that n < Mπ, and ∀ε < ε0, there exists a branch of stable solutions of (G.P.) such that:

1) uhas exactly nvortices of degree 1, located at aεi. 2) |aεi| →0asε→0, and if we seta˜i=ai

ω, the a˜i’s tend to minimize w(x1,· · ·, xn) =−πX

i6=j

log|xi−xj|+πX

i

|xi|2

so that|ai| ≤ Cω, and|ai−aj| ≥ Cω. 3)

J(u) =J(ρ) +πn |logε| −R2ω +π

2(n2−n)log ω+w( ˜a1,· · ·,a˜n) +Qn+o(1).

The solution withnvortices minimizesJ inDM exactly forωn≤ω≤ωn+1, whereωn has an expression of the form

ωn=|logε|

R2 +n−1

R2 |log|logε||+O(1). (1.7) The result can also be reformulated as follows: ∀n∈N, there existsε0(n) such that∀ε < ε0(n), there exists a branch of stable solutions of (G.P.) satisfying 1), 2) and 3).

This theorem is the analogue of Theorem 2 of [18]. It proves, in the case of a disc, the existence ofbranches of stable solutions withnvortices of degree 1. These solutions coexist for a wide range ofω, their energy follows a simple explicit formula. In addition, they are globally minimizing,i.e. they achieve the minimum over allH01, forωn≤ω≤ωn+1; this has been proved for the Ginzburg-Landau energy in a forthcoming paper [23].

What seems most interesting to us is the minimization ofw: this says that we can replace the minimization ofJoverH01by the minimization of the explicit functionwover Ωn. After rescaling, the positions of the vortices of our branches of solutions tend to minimize w. Then, the natural question is to ask what minimizers ofw look like. This is not so easy to calculate. Shafrir and Gueron have worked on this problem (see [11]). They prove that forn≤6, the regular polygons centered at the origin are local (and very likely) global minimizers, for 4 n≤6 there are other stable critical shapes: the regular “stars” which are regular polygons centered

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at the origin plus the origin. For 7 n 11, they are again local minimizers (and probably global). For highern, numerically, the minimizers look like series of concentric polygons and then triangular lattices, first concentrated around the origin, then scattered all over Ω, asnincreases. Observations have been made (since the 70’s) on the vortices in rotating superfluid helium II, which show pictures of vortices which are exactly the ones described for the minimizers ofw: i.e. regular polygons, stars, lattices. One can refer to [24] for pictures.

Thus, our results agree with the physical observations and theoretical predictions (see [10] for superfluids), and particularly with those found in [8, 9] on Bose-Einstein condensates. Moreover, they state precise values of theωn for which the n-th vortex becomes energetically favourable, which seems to be a new result, they say that the vortices are concentrated around 0 at a scale Cω and prove the multiplicity of stable solutions for a givenωaroundω1.

1.4. Methods of the proofs

As the proofs are borrowed from other papers, we only explain their main step s. For Theorems 1 and 2, let us just say that the method consists in splittingJ as (1.3) and then splittingJ−J(ρ) similarly as in [16]. The

term Z

BR

ρ2

2|∇v|2+ ρ4

2(1− |v|2)2,

can be replaced by Z

BR0

1

2|∇v|2+ 1

2(1− |v|2)2=FBR0(v)

whereR0=R−εβ(0< β <1) andFis the functional studied in [5]. Then, we prove that ifuis a configuration with a bounded number of vorticesai of degreedi, then the angular momentumM can be expressed as:

M= Z

BR

(iu, x1ux2−x2ux1)' −Z

BR

(iv, dv∧dX)' −2πX

i

diX(ai)X

i

di(R2− |ai|2),

where X = |x|22R2. Here X plays the same role as ξ in [16–18], hence we can perform the same analysis to evaluate the cost and gain of each vortex, and see that vortices will tend to the point of minimum ofX (which is the origin). To find our branches of n-vortices solutions, we perform a local minimization exactly as in [18], over domains of the type

Un=

u∈H01(BR,C)/n|logε|< F u

ρ

<

n+1

2

|logε|

and prove that it yields asolutionof (G.P.) which hasnvortices.

1.5. Statement of the results on global minimizers

The following results are the analogues of those of [21] and [22] on the Ginzburg-Landau functional.

We assume thatω(ε) is such thatω ε12 and that λ= lim

ε0

|logε|

ω (1.8)

exists and is finite. Then, for anyλ, we define the limiting functionalE as:

E(f) =λ 2 Z

|∆f+ 2|+1 2 Z

|∇f|2, (1.9)

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over

{f ∈H01(Ω)/∆f+ 2∈ R}, where Ris the space of bounded Radon measures onω.

We study any family (uε) of global minimizers of J overH01(Ω). Such a uεis solution of (G.P.), therefore one can check that it satisfies

div((iu,∇u)−ω∇X) = 0 whereX =|x2|2. We will see that we can find a uniqueUε∈H01such that

U = (iu,∇u)−ω∇X. (1.10)

This equation is the analogue of the second Ginzburg-Landau equation. It yields a relevant quantityU which plays the same role as the induced magnetic fieldhfor Ginzburg-Landau. We shall see howU is related to the total vorticity ofu.

Theorem 3. 1) Assume λexists and is finite, ω ε12, uε minimizes J and Uε is associated to uε by (1.10).

Then, as ε→0,

Uε

ω * U weakly inH01(Ω),

whereU is the unique minimizer ofE, and solution of the following obstacle problem:













U= 0 on∂Ω

U≤λ

2 in Ω

(∆U+ 2)

U−λ 2

= 0 in Ω

∆U+ 20.

(1.11)

In additionU∈C1,α(Ω),∀α <1. Moreover,

minJε0F(ρ) +ω2E(h). (1.12)

2) Ifλ= 0, then U = 0, and the convergence is strong in H01. Ifλ >0, for ε < ε0, we can find a family of balls (Bi)iIε= (B(ai, ri))iIε such that

x,dist(x, ∂Ω)≥εβ/||v|(x)1| ≥ 1

|logε|

⊂ ∪iIεBi, (1.13) X

iIε

ri 1

|logε|6, (1.14)

∀i∈Iε, 1 2 Z

Bi|∇U|2≥π|di||log ε|(1−o(1)), (1.15) wheredi=deg(u, ∂Bi).

For any such family, if we defineµε=ω P

iIεdiδai, we have µε* µ= ∆U+ 2

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andω

X

iIε

|diai * µ

in the sense of measures.

3) If we set Uλ ={x∈Ω/U(x) = λ2}, we have µ = 21Uλ, where1Uλ denotes the characteristic function ofUλ. Uλ=∅⇔λ≥2 maxξ0 whereξ0 is the solution of

∆ξ0= 2 in Ω

ξ0= 0 on∂Ω. (1.16)

This theorem is mostly relevant in the intermediate caseω=O(|logε|) corresponding toλ >0. We then isolate the zeroes ofuε(which are not too close to ∂Ω) in vortex ballsBi and define a vorticity measureµε, proved to be closely related toUε. µεconverges weakly toµ which is a uniform measure of density 2 on a subsetUλ of Ω. Thus, qualitatively, we expectuε to have vortices of positive degrees, regularly scattered overUλ with a density∼2ω whenε is sufficiently small. Uλ is determined by (1.7) which is a free boundary problem. It is a classical obstacle problem (see [12]). If∂Uλ is smooth (which is not always the case, but is the case at least for almost every value ofλfrom a result of [6]), then (1.7) can be rewritten more simply:







U= 0 on∂Ω

∆U= 2 in Ω\Uλ

U=λ2 on∂Uλ

∂U

∂n = 0 on∂Uλ.

The size of the vortex-regionUλ depends on λ. Ifλ is very large (corresponding to smallω’s), thenUλ =∅. More precisely, ifω≤ω12max|logεξ|0, thenUλ=∅, and following [20], we could have proved rigorously thatuε

has no vortex in this case. Thus, someω12max|logεξ|0 orλ= 2maxξ0 corresponds to a critical value (first critical velocity), and is compatible with the result of Theorem 1. Indeed, if Ω is a ballB(0, R), thenξ0= R2−|2x|2,and thus 2maxξ0=R2. This theorem generalizes the result of Theorem 1 to arbitrary simply connected geometries.

Ifω≥ω1, thenUλis nonempty and minimizers have vortices. Uλincreases asλdecreases (i.e. asωincreases), until, forλ= 0, corresponding toω |logε|,Uλ= Ω, and the vortices fill all Ω. The main difference compared to the result of [22] on the Ginzburg-Landau functional is that the limiting measureµ always has density 2, whereas in Ginzburg-Landau it had a density 1λ2, thus depending onλand on the applied field.

(1.12) provides an asymptotic expansion of the minimal energy, in whichF(ρ) carries the boundary layer cost of any configuration due to the boundary conditionu= 0. Indeed,F(ρ) =12R

|∇ρ|2+12(1−ρ2)2is of the order of 2l(∂Ω) as we shall see in Section 2, andF(ρ) =J(ρ)12

R

ω2r2ρ2,hence, as soon asω2 1ε, F(ρ)∼J(ρ) is the term of highest order in (1.12). In the case ofω |log ε|i.e. whenλ= 0, then this theorem only states that minJF(ρ), and Uωε 0. We are in fact able to get more precise results (adjusted from [21]) in the following theorems:

Theorem 4. Assume|log ε| ω 1ε. Then J(ρ)−

Z

ω2

2 r2+ω||log 1 ε√

ω(1−o(1))≤ min

H01(Ω,C)

J ≤F(ρ) +ω||log 1 ε√

ω +O(ω), where|.| denotes the volume.

If in additionω≤ ε4/5C , then

min

H01(Ω,C)J =F(ρ) +ω||log 1 ε√

ω(1 +o(1)).

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Theorem 5. Let |log ε| ω≤ ε4/5C , anduε be a corresponding minimizer ofJ. Then, forε < ε0, there exists a family of disjoint disks(Bεi)with radii each less than 1ω and sum less than||√

ω, such that|uε| ≥ 12 on∂Biε and, ifaεi is the center of Biεanddεi = deg(|uuε

ε|, ∂Biε), then µε= 2π

ω X

i

dεiδaεi−→ε02dx

in the weak sense of measures, wheredxis the Lebesgue measure on R2 restricted toΩ.

Moreover,

πX

i

|dεi| 'πX

i

dεi 'ω||, and most of the vortex-energy is concentrated in the balls, i.e.

J\∪iBεi(uε)−F(ρ) =o(J(uε)−F(ρ)).

Of course, for any value ofω, we have the trivial solutionu≡0 which has an energy |2|. We believe that, forω higher than some critical valueω≥ Cε, it becomes minimal.

2. The splitting of the energy

We introduceρε, for a general domain Ω. It is defined to be the minimizer of the following problem:

min

H01(Ω,R)

1 2 Z

|∇ρ|2+ 1

2(1−ρ2)2+ω2r2ρ2. (2.1) We will often drop the subscript and writeρinstead ofρε.

Lemma 2.1. ρε satisfies the following:

ρ∈C 0≤ρε1, |∇ρ| ≤C

ε (2.2)

if Ω =BR, thenρis radial and is a solution of (G.P.), (2.3)

∆ρε+ω2r2ρε= 1

ε2ρε(1−ρ2ε) (2.4)

1−ρε≤Ceδ(x) +O(ε2ω2) where δ(x) = dist(x, ∂Ω) (2.5) J(ρ)

2

l(∂Ω) + ω2 2

Z

|x|2+O(1), (2.6)

wherel(∂Ω)denotes the length of∂Ω.

Z

(1−ρ2)2≤C(ε+ω4ε4). (2.7)

Proof. It is well-known since the work of Brezis and Oswald [7] that, as soon asεis small enough, there exists a positive minimizer for the functional (2.1), and that it is the only positive solution of

∆ρ+ω2r2ρ=ε12ρ(1−ρ2) in Ω

ρ= 0 on∂Ω. (2.8)

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It is also standard thatρ≤1 and|∇ρ| ≤ Cε.If Ω =BR, the fact thatρis radial comes from the uniqueness of the positive solution, and the fact thatρsatifies (G.P.) comes from (2.8).

We then prove (2.5). The proof is similar to that of Proposition 2.1 of [4].

Consider x0 such that δ(x) := dist(x0, ∂Ω)> Kε, for some K to be determined afterwards. Let φ1 be a positive eigenfunction corresponding to the first eigenvalueλ1 of∆ onB(0,1):

∆φ1=λ1φ1 inB(0,1) φ1= 0 on∂B(0,1), and satisfyingφ1 12 onB(0,1).

Let us writeφ(x) =φ1 xx0

, then ∆φ= Kλ21ε2φonB(x0, Kε). If K is chosen large enough (independent from ε), then

λ1

K2φ≤φ(1−φ2)−ω2ε2r2φ inB(x0, Kε), for smallε, sinceωε→0. Hence,

∆φ φ

ε2(1−φ2)−ω2r2φ, and thusφis a subsolution for (2.8), implying

ρ≥φ inB(x0, Kε).

Therefore, there exists 1> a >0, independent fromε, such that ρ≥ φ≥a >0 inB(x0, Kε).

Hence

ρ≥a >0 in ˜Ω :=

x/δ(x)> 2

· (2.9)

Now, as in [4], it is enough to prove the estimate (2.5) on ˜Ω. We prove it by using suitable subsolutions.

Consider againx0 Ω, and let˜ µ= dist(x0, ∂Ω). On˜ B(x0, µ), we considerw1, the subsolution of [4], defined by:

w1(η) =th

th1a+µ2−η2 3µε

, whereη=|x−x0|. As in [4], we havew1≥aand

∆w1 8

2(1−w21)w1+ 4

3µε(1−w21).

As previously, we may consider only µ≥24εa , then 3µε4 18εa2 18εw12 and

∆w1 8

9+ 1 18

1

ε2w1(1−w21). (2.10)

We then definew2 byw2=w1−M ω2ε2. From (2.10), asw2≤w1,

∆w2 1

ε2w2(1−w22) +ω2r2w2 17

18ε2(1−w12)w1 1

ε2w2(1−w22) +ω2r2w2

17

18ε2(1−w22)(w2+M ω2ε2) 1

ε2w2(1−w22) +ω2r2w2

≤ − 1

18ε2w2(1−w22) + 17

18ε2(1−w22)M ω2ε2+ω2r2w2.

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But, forεsmall enough,w2a2, hence

∆w2 1

ε2w2(1−w22) +ω2r2w2 ≤ −w2

ε2

1−w22 18 17

18(1−w22)2

aM ε2ω2−ω2ε2r2

≤ −w2

ε2

1−w22

18 (1−o(1))−ω2ε2r2

. (2.11)

On the other hand,

1−w22 1

2(1−w2)≥M 2ω2ε2 forεsmall enough. Then,

1−w22

18 (1−o(1))−ω2ε2r2 M

40ω2ε2−r2ω2ε2>0 ifM is chosen large enough compared to maxr. Therefore, for a suitable choice of M,

∆w2 1

ε2w2(1−w22) +ω2r2w20 onB(x0, µ),

and w2≤a≤ρon ∂B(x0, µ), hence w2 is a subsolution for (2.8) and we deduce thatρ≥w2≥w1−M ω2ε2 onB(x0, µ), and, as in [4],

1−ρ≤Cεµ +O(ω2ε2) onB(x0, µ).

Asµ=δ(x0)12Kε, we obtain that

1−ρ(x0)≤Cεδ(x0 ) +O(ε2ω2),

and finally, changingCif necessary, this estimate is true on all Ω, which proves (2.5).

For (2.6), it is well-known (see [4] or [14]) that min

H01(Ω,R)

1 2 Z

|∇u|2+ 1

2(1−u2)2

2

l(∂Ω) +O(1).

Hence, by definition ofρ,

J(ρ)

2

l(∂Ω) +ω2 2

Z

|x|2+O(1).

This implies that

1 4ε2

Z

(1−ρ2)2+ω2 2

Z

r2ρ2 C ε +ω2

2 Z

r2, thus

1 4ε2

Z

(1−ρ2)2≤C ε +ω2

2 Z

r2(1−ρ2)≤C ε +2

2 Z

(1−ρ2)2 12

,

from which we deduce (2.7).

Thus, in the case of a disc domain,ρεis a vortex-less solution of (G.P.). As explained in the introduction, the fact thatu= 0 on ∂Ω induces a cost of Cε in the energy. That cost can be, as in [AS], removed by considering v= uρ. Then, v 'u except near the boundary, and the boundary cost is “carried” byρ. This can be proved by using the fact that the energy splits very conveniently under the decompositionu=ρv, exactly as in [LM]

or [AS]. More precisely, we have the following lemma, in whichHρ12 denotes the H1 space with respect to the measureρ2dx, and the same for L2ρ2.

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Lemma 2.2. Letu∈H01(Ω,C). ∃ε0,∀ε < ε0,v=uρ is well-defined, belongs toHρ12 and J(u) =J(ρ) +

Z

ρ2

2|∇v|2+ ρ4

2(1− |v|2)2+ω Z

ρ2(iv, x2vx1−x1vx2). (2.12) Proof. Letdx denote the Lebesgue measure onR2.

Step 1: We prove thatv∈Hρ12.

ρonly vanishes on∂Ω, hencev=uρ is well-defined on Ω. Furthermore, Z

|v|2ρ2dx= Z

|u|2dx <∞ hencev ∈L2ρ2. Then, ∇v= ∇u

ρ u

ρ2∇ρ. Asu∈H01(Ω,C), |∇u|2

ρ2 isρ2dx integrable. On the other hand, we can say from (2.5) that

∃λ >0, δ(x)≥λε=⇒ρε(x) 1

2· (2.13)

Hence, we have

Z

{x/δ(x)λε}

u ρ2∇ρ

2ρ2≤Ck∇ρk2L

Z

|u|2<∞, (2.14)

while, with (2.5), Z

{x/δ(x)λε}

u ρ2∇ρ

2ρ2 = Z

{x/δ(x)λε}

|u|2|∇ρ|2

ρ2 ≤ k∇ρk2L

Z

{x/δ(x)λε}2 |u|2 δ(x)2

2k∇ρk2L

Z

|∇u|2<∞ where we have used the Hardy inequality.

Hence, we deduce that∇v∈L2ρ2 andv∈Hρ12 withkvkH1

ρ2 ≤C(ε)kukH01.

Step 2:We prove the splitting of the energy. This proof is very similar to that of [13] and [4], and was suggested by Shafrir. For anyt >0, we denote Ωt={x∈Ω/δ(x)> t}. For anyt >0 sufficiently small, we have

Z

t

1

2|∇(ρv)|2 + 1

2(1−ρ2|v|2)2+ω2r2

2 ρ2|v|2= Z

t

1

2|v|2|∇ρ|2+1

4∇ρ2.∇(|v|21)

+ 1

2ρ2|∇v|2+ Z

t

1

2(1−ρ2+ρ2(1− |v|2))2+ω2r2

2 ρ2+ω2r2

2 ρ2(|v|21)

= Z

t

1

2|∇ρ|2+ 1

2(1−ρ2)2+ω2r2

2 ρ2+ + 1

2ρ4(1− |v|2)2+1 2ρ2|∇v|2 +

Z

t

(|v|21)

1

4∆(ρ2) +ω2r2

2 ρ2 1

2ρ2(1−ρ2) +1 2|∇ρ|2

+ 1

2 Z

∂Ωt

ρ∂ρ

∂n(|v|21).

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Now, sinceρsatisfies (2.4), we have

∆ρ2= 2ρ(∆ρ)2|∇ρ|2=2r2ρ2+ 2

ε2(1−ρ222|∇ρ|2, so that

Z

t

1

2|∇u|2+ 1

2(1− |u|2)2+ω2r2

2 |u|2 = Z

t

1

2|∇ρ|2+ 1

2(1−ρ2)2 +ω2r2

2 ρ2+ Z

t

1

2ρ2|∇v|2+ ρ4

2(1− |v|2)2 +1

2 Z

∂Ωt

ρ∂ρ

∂n(|v|21). (2.15)

But, from the properties stated onρ, we have Z

∂Ωt

ρ∂ρ

∂n|v|2

Z

∂Ωt

∂ρ

∂n |u|2

ρ ≤Cεk∇ρkL Z

∂Ωt

|u|2 δ(x)· From the Hardy inequality,

Z

|u|2 δ(x)2 ≤C

Z

|∇u|2<∞, hence, we can find a sequencetn 0 such that

Z

∂Ωtn

|u|2

δ(x)2 C tn|log tn|, therefore,

Z

∂Ωtn

|u|2 δ(x) =

Z

∂Ωtn

|u|2

tn C

|logtn| −→0 asn→ ∞. On the other hand,

tlim0

Z

∂Ωt

ρ∂ρ

∂n= 0.

Applying (2.15) tot=tn, and passing to the limitn→ ∞, we obtain Z

1

2|∇u|2+ 1

2(1− |u|2)2+ω2r2

2 |u|2=J(ρ) + Z

1

2ρ2|∇v|2+ ρ4

2(1− |v|2)2.

There remains to deal with Z

(iu, x2ux1−x1ux2).

But replacingubyρv, we obtain that this term is equal to Z

ρ2(iv, x2vx1−x1vx2) + Z

ρ(iv, v)(x2ρx1−x1ρx2),

where the second term vanishes identically. Hence we have the desired result.

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3. Branches of vortex solutions in the case of the disc

In this section Ω =B(0, R) =BR. We consider rotation speeds ω≤Cεα

forα sufficiently small, to be specified in the proof, and obtain similar results as those of [16–18] concerning branches of solutions of (G.P.).

As mentioned in the introduction, we cannot study zeros ofuclose to∂Ω because|u|vanishes at the boundary and is smaller than 12 on a layer of size λεnear the boundary. On the opposite, we can study vortices of v, which does not have to vanish on∂Ω, and vortices of v are vortices ofu. But it is difficult to get information onv and its vortices near the boundary, and anyway, it is not very relevant, sinceuhas something like a layer of vortices of sizeεnear the boundary. This is why we restrict to studying von the domain{x∈Ω, δ(x)≥εβ} whereβ is some constant <1 and close to 1. Furthermore, there are no boundary conditions on v, hence we can adjust the techniques of [16–18] tov. The rest of this section is just these adjustments.

3.1. Defining the domains of minimization

We perform, as in [16–18], local minimizations ofJ over well-chosen domains.

First, Lemma 2.2 has allowed us to separate the very strongly-divergent part Jε) (in Cε) from the rest which is very similar to the Ginzburg-Landau energy functionalJ of [16–18], and diverges at most in2. We use here the notation of [16]:

F(v) = 1 2 Z

|∇v|2+ 1

2(1− |v|2)2 (3.1)

for the energy functional studied in [5]. We shall also writeFV for the functionalFrestricted to any subdomain V of Ω.

Then, we denote

G(v) =1 2 Z

ρ2|∇v|2+ ρ4

2(1− |v|2)2, (3.2)

for the “weighted” BBH-functional that appears naturally in the splitting ofJ. We then define the following mappings:

H1−→Hρ12

u7−→v= u ρ

(3.3) is a continuous mapping, as proved in Section 2. So is

Hρ12−→Hρ12

v7−→T(v) =

v

ρ|v| if |v| ≥ 1 ρ votherwise.

(3.4)

In addition, we have the following lemma, whose proof is postponed to the end of this section:

Lemma 3.1. 1) For anyu∈H01(Ω,C),

|T(v)| ≤ 1

ρ· (3.5)

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2) There existsα >0such that, for any usatisfying J(u)≤J(ρ), ifω≤Cεα,

J(ρT(v))≤J(ρv) +o(1) asε→0. (3.6)

This lemma means that, if we replaceuby ρT(v), we get a function with values in B(0,1) and with a lower energy thanu, up to a small error term. Hence, we can make this replacement to find local minimizers.

We now define the domains of minimization. A real (large) positive constantM∈/πN (M> π) being set, we define (as in [16]),

DM=

u∈H01(BR,C)/G◦T u

ρ

<M|logε|

· (3.7)

This is going to be our largest domain of minimization. We shall also use smaller domains of minimization, of the form

Da,b =



u∈H01(BR,C)/a <

G◦T u

ρ

|log ε| < b



,

where a, b < M but may depend on ε. DM and Da,b are open domains in H01, from the continuity of the mappings (3.3) and (3.4).

3.2. Definition of the regularized map and its vortices

We wish to study minimizers ofJ in DM or Da,b. Considering any u in one of these domains, we write v=T(uρ), so that|v| ≤ 1ρ and, thanks to Lemma 3.1, we can studyv instead of uρ.

As in [16–18], we need to define vortices of v for any u DM. But, exactly as in these papers, this is impossible to do directly becauseuis nota priorisolution of (G.P.) hence there is no upper bound of the type

|∇u| ≤ Cε on its gradient. As in [16–18], in order to define vortices of v, we replace it, following the method of [1], by a regularized mapvγ which has well-defined vortices. First, we remove the boundary of the domain, by setting

B0=BR\{x/δ(x)≤εβ}, (3.8)

whereβ is some constant]0,1[. From (2.5),

01−ρ≤Cε1β+O(ε2ω2) =o(1) inB0, (3.9) hence we can considerρas being equal to 1 in B0. Then, our regularized mapvγ is defined from v to be the solution of the following problem:

min

H1(B0,C)

Z

B0

1

2|∇w|2+(1− |w|2)2

2 +|v−w|2

, (3.10)

where γis some constant in ]0,1[.

Exactly as in [16–18] and [1], thisvγ has the same behaviour as v at scales larger thanεγ (it is a parabolic regularization ofv), hence its vortex-structure and behaviour with respect toJ are going to be almost the same as those ofv, as we shall prove.

Lemma 3.2. Ifu∈DM andv=T(uρ), thenvγ satisfies

|∇vγ| ≤C

ε |vγ| ≤ 1 ρ,

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