Approximations with vorticity bounds for the Ginzburg-Landau functional

Approximations with vorticity bounds for the Ginzburg-Landau functional F. Bethuel, G. Orlandi and D. Smets Abstract We propose an approximation scheme for complex-valued functions defined on a smooth domain Ω : the approximating functions have a Ginzburg-Landau energy of the same magnitude as the initial function, but they possess moreover improved bounds on vorticity. As an application, we obtain a variant of a Jacobian estimate first established by Jerrard and Soner. This variant was conjectured by Bourgain, Brezis and Mironescu.

1

Introduction

Let Ω be a smooth bounded domain in RN , N ≥ 2. Consider complex-valued functions uε : Ω → C and the Ginzburg-Landau energy defined by Z Z |∇uε |2 (1 − |uε |2 )2 + Eε (uε ) = eε (uε ) = 2 4ε2 Ω Ω where 0 < ε < 1 is a small parameter. In order to analyze possible concentration phenomena in the asymptotic limit ε → 0, an important quantity is the Jacobian X Juε = ∂i uε × ∂j uε dxi ∧ dxj i 0 and 0 < α < 1 (depending only on N) such that Z Juε ∧ ϕ ≤ C Eε (uε ) kϕkL∞ + Cεα (1 + Eε (uε ))(1 + |Ω|2 )kϕkC 0,1 . (3) |log ε| Ω

For a given ϕ, the main contribution for small ε in the r.h.s. of (3) is the first term. Comparing with (1) one sees that a factor |log ε|−1 has been gained. Notice however that the integral estimate (3) involves also the derivative of ϕ whereas (1) is verified pointwise. Although the coefficient in front of kϕkC 0,1 is small, it cannot be removed, and an estimate of the form Z C eε (uε ) (4) kJuε kL1 (Ω) ≤ |log ε| Ω does not hold in general. The purpose of this paper is to show that even if (4) is not verified, such an estimate holds for an approximation (in a suitable norm) vε of uε , provided uε verifies an energy bound of the form Eε (uε ) ≤ ε−γ

(Hγ )

for some 0 < γ < 1. More precisely, we have Theorem 2. Let 0 < γ < 1. There exist constants C > 0 and α > 0, depending only on γ, N, and Ω, such that that if uε verifies (Hγ ) then there exists a smooth function vε : Ω → C such that |vε | ≤ 1 Eε (vε ) ≤ CEε (uε ) Eε (uε ) kJvε kL1 (Ω) ≤ C |log ε| 2 2 (1 − |vε | ) Eε (uε ) k kL1 (Ω) ≤ C 2 4ε |log ε| α kuε − vε kL2 (Ω) ≤ Cε Eε (uε )1/2 kJuε − Jvε kC 0,1 (Ω)∗ ≤ Cεα Eε (uε ) . 0

2

(5) (6) (7) (8) (9) (10)

Comments. 1) Observe that (4) is verified with uε replaced by vε . 2) Notice that Theorem 1 does not involve any bound on Eε (uε ), whereas in Theorem 2 our proof requires the energy bound (Hγ ). However, for large energies, the proof of (3) in Theorem 1 is fairly elementary. Indeed, assume for instance that Eε (uε ) ≥ ε−γ (0 < γ < 1), then Z Z Juε ∧ ϕ = (uε × duε) ∧ dϕ Ω ZΩ |uε | |duε| kϕkC 0,1 ≤ Ω Z Z ≤ (|uε | − 1) |duε| + |duε | kϕkC 0,1 Ω Ω p ≤ Eε (uε )|Ω| + εEε (uε ) kϕkC 0,1 p ≤ |Ω|εγ/2 Eε (uε ) + εEε (uε ) kϕkC 0,1 . The conclusion then follows with α = γ/2. 3) Combining the previous argument (in case Eε (uε ) ≥ ε−1/2 ) with Theorem 2 for γ = 1/2 (in case Eε (uε ) ≤ ε−1/2 ) one recovers estimate (3) of Theorem 1. 4) For extremely large energies, the conclusion of Theorem 2 holds similarly. More precisely, let δ > 0 and assume Eε (uε ) ≥ ε−2−δ , then vε ≡ 0 verifies all the conclusions in Theorem 2 with constants C and α depending only on δ, N, and Ω. However, we have not investigated the intermediate region + − ε−1 ≤ Eε (uε ) ≤ ε−2 . 5) In order to connect Theorem 2 with variational methods, it would be useful to improve the construction of vε so that the constant C in (6) can be replaced by 1, so that the Ginzburg-Landau energy of vε is smaller than the Ginzburg-Landau energy of uε . In dimension 2, a construction using parabolic regularization is given in [2], and yields this additional property (under a more restrictive energy bound). 6) Improved estimates for the potential term analogous to (8) have been established for critical points uε of Eε (see [4] for N = 2, and [6]). 3

7) Notice that if uε verifies the stronger bound Eε (uε ) ≤ |log ε|k , k ≥ 1, then interpolating (6) and (9) we see that kuε − vε kH s → 0

as ε → 0 ∀ 0 < s < 1 .

Although we expect the results in this paper will find some other applications, the main motivation for Theorem 2 came from a conjecture raised by Bourgain, Brezis and Mironescu in [8]. Our solution, given in [5], relies on Theorem 2 in an essential way, as well as on a new beautiful linear estimate of [8] (see Theorem 5.1 for a precise statement). In this paper, we present the following extension of the results of [5], which should be compared with Theorem 1. Theorem 3. Assume N ≥ 3, and let 0 < s ≤ 1 and p be such that sp = N. In the case N = 3, assume moreover that there exists q > 2∗ = 6 such that uε verifies kuε kLq (Ω) ≤ Cq . (Hq ) Then, for any ϕ ∈ C0∞ (Ω; ΛN −2 RN ), Z Juε ∧ ϕ ≤ C Eε (uε ) + εα0 Eε (uε )α1 kϕk ˙ s,p , W (Ω) |log ε|

(11)

Ω

for some constants C > 0 and 0 < α0 < 1, 0 < α1 < 1, depending only on Ω, s, N, q and Cq . Here kϕkW˙ s,p denotes the semi-norm defined, for s < 1, by Z Z |ϕ(x) − ϕ(y)|p p kϕkW˙ s,p ≡ (1 − s) dxdy , (12) N +sp Ω Ω |x − y| R while kϕkpW˙ 1,p ≡ Ω |∇ϕ|p . [See [9] for a motivation of the factor 1 − s in (12): with this choice of norms, kϕkW˙ s,p remains bounded as s → 1]. Remark 1. Observe that (11) holds true also for sp > N, since in this case kϕkW˙ s,p ≤ CkϕkW˙ s, Ns . However, if sp > N, then W s,p ,→ C 0,θ , so that in this case (11) is a direct consequence of (3). For the case sp = N, W s,p does not embed in L∞ : we just have the embedding W s,p ,→ BMO. Remark 2. Inequality (11) is not true for N = 2. As a counterexample, consider Ω = D 2 , the unit disk in R2 , and take (uε )ε>0 such that Eε (uε ) ≤ π|log ε| + C, and Juε * δ0 as ε → 0. Since δ0 ∈ / (W s,p)∗ for sp = 2, this contradicts (11). 4

Remark 3. One may wonder if assumption (Hq ) for N = 3 is necessary in order to derive (11). Here we present an example which shows that (11) does not hold without any additional assumption on uε . Let 0 < s ≤ 1 and p such that sp = 3. Let B = B 3 be the unit ball in R3 , and ϕ ∈ C0∞ (B; Λ1 R3 ), u ∈ C0∞ (B; C) be such that Z Ju ∧ ϕ = 1 . (13) B

p

Define vε,R (x) := 1 + R|log ε|u(Rx), and ϕR (x) := ϕ(Rx), for R > 1. By scaling, Z Jvε,R ∧ ϕR = |log ε| , ||ϕR ||W˙ s,p (B) = ||ϕ||W˙ s,p(B) , B

and Z

2

B

|∇vε,R| = |log ε|

Z

2

B

|∇u| ,

1 ε2

Z

B

(1 − |vε,R |2 )2 ≤ C(u)

|log ε|2 . Rε2

Therefore, choosing R = R(ε) suitably large, we see that (11) does not hold for the sequence uε = vε,R . [Notice that kuε kLq → +∞ as ε → 0 for any q ≥ 2∗ = 6.] We would like to emphasize that the fact that an additional condition of type (Hq ) is needed only in dimension N = 3 is directly related to the growth at infinity of the potential term of Eε . In our case, the potential grows like |u|4 . This power has to be compared with the Sobolev exponent 2∗ . In particular, for N ≥ 4, we have 2∗ ≤ 4, and this fact, together with the bound of the potential given by the energy, is exploited in our estimates. Remark 4. In another direction, the next example shows that the term εα0 Eε (uε )α1 cannot be removed in formula (11). Let 0 < s ≤ 1 and p such that sp = N. Let B = B N be the unit ball in RN , and ϕ ∈ C0∞ (B; ΛN −2 RN ), u ∈ C0∞ (B; C) be such that (13) holds. Define vε,R (x) := 1 + u(Rx), and ϕR (x) = ϕ(Rx), for R > 1. By scaling, Z Jvε,R ∧ ϕR = 1 , ||ϕR ||W˙ s,p (B) = ||ϕ||W˙ s,p (B) RN −2 B

and

Z

2

B

|∇vε,R | =

1

RN −2

Z

B

1 ε2

2

|∇u| , 5

Z

B

(1 − |vε,R|2 )2 ≤ C(u)

1

R N ε2

.

Therefore, choosing for instance R = 1ε , we see that (11) with the term εα0 Eε (uε )α1 removed does not hold for the sequence uε = vε,R . Remark 5. In dimension N = 3, the fact that (11) does not hold for sp < 3 has been proved in [8]. We present an alternative simple proof here, which easily extends to any dimension. Let Γ be a unit circle centered in the origin of R3 and let ϕ be a smooth vector field with compact support such that Z σ := ϕ · ~t 6= 0. Γ

For τ > 0, we define the vector field ϕτ by ϕτ (x) := ϕ(τ x), and we denote by Γτ the homothetic of Γ having radius τ instead of one. By scaling, Z ϕτ · ~t = τ σ and kϕτ kW˙ s,p = τ sp−N kϕkW˙ s,p . Γτ

If N − sp < 0, we therefore obtain R ϕ · ~t Γτ τ → +∞ |Γτ | kϕτ kW˙ s,p

as

τ → 0.

(14)

On the other hand, for fixed τ > 0 it is proved in [1] that there exist, for ε small enough depending on τ , a function uε such that Z Z Eε (uε ) ∼ π|Γτ ||log ε| and Juε ∧ ϕτ ∼ π ϕτ · ~t. Γτ

The last construction combined with (14) clearly contradicts (11). For general dimension N, one replaces the circle Γ by an N − 2 dimensional unit sphere and the vector field ϕ by an N − 2 form; the analysis in [1] is used the same way. Remark 6. The conjecture in [8] and the results in [5] correspond to the case s = 1, p = N = 3 in Theorem 3. The key idea in the proof of Theorem 2 (which is rather close to the strategy of [1]) is a dimension reduction argument. Indeed, many of the essential aspects of Ginzburg-Landau theory appear already in dimension two and one (see however the discussion in Section 5). Therefore, the first part 6

of the proof (Section 2) will consist in a suitable choice of a grid with appropriate bounds on 2- and 1-dimensional faces. This part borrows ideas from [1, 3, 8, 16]. The second part (Section 3) deals with 2-dimensional boundary value problems. Here, we rely on constructions in [4, 18, 15] and lower bound estimates in [13, 17]. This part can be read independently of the rest of the paper, and we tried to keep it as expository as possible. The proof is completed in Section 4. First we construct vε on the 2-skeleton of the grid relying on the results of Section 3. Then vε is extended on the whole domain by filling the higher dimensional faces of the grid in a pyramidal way. In Section 5, we provide the proof of Theorem 3. In the next parts, 0 < γ < 1 will be a fixed parameter. We will also assume throughout Sections 2, 3 and 4 that uε verifies (Hγ ) and is smooth. This is not a restriction since we may always approximate uε by mollification by smooth maps. Finally, in some places we will suppose ε < ε0 , ε0 being some constant depending on Ω. When ε ≥ ε0 the statements in Theorem 2 are obvious. Acknowledgements. The authors wish to thank J. Bourgain, H. Brezis and P. Mironescu for providing them with an early version of the manuscript [8]. They also thank H. Brezis, A. Cohen and R. Danchin for enlightning discussions. The authors are partially supported by UE Grant RTN HPRNCT-2002-00274 “Fronts, Singularities”.

2

Restriction on a good grid

The main part in the construction of vε in Theorem 2 is to determine a onedimensional grid of appropriately small mesh size (depending on ε and γ) on which |uε | ≥ 21 . As we will see, this is only possible if an energy bound of the form (Hγ ) holds true. In the course of the proof we will also need energy bounds on this one-dimensional grid as well as on the corresponding two-dimensional skeleton. Throughout the construction, we will fix the value (x) of vε on the one-dimensional grid as being vε (x) = |uuεε (x)| .

7

2.1

The canonical grid

For x = (x1 , ..., xN ) ∈ RN , we set kxkk = min max |xi |, S is a k-element subset of {1, 2, ..., N} , i∈S

and, for h > 0, m ∈ N∗ , consider the standard m-dimensional cube of size 2h Qm (h) ≡ [−h, h]m . If m = N we simply write Q(h) ≡ QN (h) . For k ∈ {0, ..., N − 1}, consider the k-skeleton [Q(h)]k of Q(h) defined by [Q(h)]k = {x ∈ Q, kxkk+1 = h} . In particular, for k = N −1, [Q(h)]N −1 = ∂Q(h), and it is therefore the union of the 2N faces of Q(h), which are isometric to QN −1 (h). More generally, [Q(h)]k is the union of 2(N!)/(k − 1)! cubes isometric to Qk (h), and which have at most one (k − 1)-face in common. By translation, we define a k-skeleton of RN by Sk = 2hZN + [Q(h)]k . Note that Sk is a countable union of k-dimensional affine subspaces, and that Sk−1 ⊂ Sk for 1 ≤ k ≤ N − 1. Notice also that the previous construction corresponds to a partition of RN into cubes (Qi (h))i∈J isometric to Q(h) and having at most one face in common, and that SN −1 = ∪i ∂Qi (h). Finally, for a ∈ Q(h), we consider the translated grid Sk,a = a + Sk . By Fubini theorem, we have Lemma 2.1. Let f ∈ L1 (RN ). Then, the following equality holds Z Z Z k k f dHk . f dH da = (2h) Q(h)

Sk,a

Sk,a

8

2.2

Choice of a grid

In order to handle the boundary ∂Ω, we extend first uε to a larger smooth domain. To that aim, consider for µ > 0 the set Ωµ = {x ∈ RN , dist(x, Ω) < µ}. If µ is sufficiently small, then Ωµ is smooth. Moreover, we may extend uε to Ωµ by a standard reflection, so that the new map u˜ε verifies u˜ε = uε on Ω and Z eε (˜ uε ) ≤ CEε (uε ) (15) Eε (˜ uε ) ≡ Ωµ

where the constant C depends only on Ω and µ, which is from now on considered fixed. Applying Lemma 2.1 with f = eε (˜ uε )χΩµ and an averaging argument, we derive

Lemma 2.2. There exists a ∈ Q(h) such that, for every k ∈ {1, ..., N} Z eε (˜ uε )dHk ≤ Chk−N Eε (˜ uε ), (16) Sk,a ∩Ωµ

where C is some universal constant. Throughout the paper, a will be fixed according to Lemma 2.2, for the choice of h given by (22) in the next section. Translating possibly the origin, we will assume without loss of generality that a = 0,

and in particular Sk,a = Sk .

(17)

In order to work with cubes only, we will restrict ourselves to an intermediate set Q, which is the union of those Qi , i ∈ J, which intersect Ω. More precisely, [ Q≡ (b + Q(h)) ≡ ∪ Qi (h), b∈2hZ, (b+Q(h))∩Ω6=∅

i∈I

where the set of indices I ⊂ J is finite and ]I ≤ ChN , C depending only on Ω. Clearly Ω ⊂ Q (by definition) and if h is sufficiently small, then Q ⊂ Ωµ . We will also consider the k-skeleton Qk of Q, defined by Qk = Sk ∩ Q = ∪i∈I [Qi (h)]k . Note in particular that Qk is a finite union of k-dimensional cubes isometric to Qk (h). 9

2.3

Determining the mesh size

Here we show that if h is sufficiently large (compared to ε) then the condition u˜ε ≥ 12 is verified on S1 ∩ Q (after translation of the origin by −a, for a given by Lemma 2.2). The starting point is the following classical lemma for line energies. Lemma 2.3. Let [a, b] be a line segment. Assume |b − a| ≥ ε. Then there exists a constant c0 > 0 such that for any wε : [a, b] → C verifying Eε (wε ) =

Z

b a

|wε0 |2 (1 − |wε |2 )2 c0 + ≤ 2 2 4ε ε

we have |wε | ≥

1 2

on [a, b].

Proof. From the inequality αβ ≤ 21 (α2 + β 2 ), we obtain Z

b

a

|wε0 | |1 − |wε |2 | ≤ 2εEε (wε ) ≤ 2c0 ,

so that osc |χ(wε )| ≤ [a,b]

Z

b a

|(χ ◦ wε )0 | ≤ 2c0 ,

(18)

where χ(t) = t − t3 /3. By a mean value argument, there exists x0 ∈ [a, b] such that Z b 4c0 ε 1 2 2 ≤ 4c0 . (19) (1 − |wε |2 )2 ≤ |1 − |wε (x0 )| | = b−a a b−a In particular, if c0 is sufficiently small, then 3/4 ≤ |wε (x0 )| ≤ 5/4. On the other hand, choosing c0 possibly even smaller, (in such a way that 2c0 ≤ |χ(wε (x0 ))| − χ(1/2)), from (18) and the definition of χ we conclude that χ(1/2) ≤ |χ(wε (x))| for any x ∈ [a, b], and therefore |wε | ≥ 1/2 on [a, b]. In particular, since S1 ∩ Q is a union of line segments of size h, if h ≥ ε and if Z c0 eε (˜ uε )dH1 ≤ , (20) ε S1 ∩Q then it follows from Lemma 2.3 that

10

1 on S1 ∩ Q. (21) 2 Comparing (20) with (16) for k = 1, we see that (21) is verified provided |˜ uε | ≥

Ch1−N Eε (uε ) ≤

c0 ε

and therefore in view of (Hγ ) we deduce Lemma 2.4. We have |˜ uε | ≥

1 2

on S1 ∩ Q provided 1−γ

h ≥ c1 ε N−1 , where c1 is some constant depending only on Ω. In view of Lemma 2.4, we choose throughout 1−γ

1−γ

h = ε N−1 |log ε| ≥ c1 ε N−1 ,

(22)

provided ε is sufficiently small. This choice of h determines the translation of the origin by −a (via Lemma 2.2).

3

Boundary value problems in dimension two

As already mentioned, in this section, we derive general results in dimension N = 2, some of which are already available in the literature in a close form. We restrict ourselves first to the unit cube Q ≡ Q(1).

3.1

Extension of S 1 -valued maps defined on ∂Q

Let gε : ∂Q → S 1 and set 1 Eε (gε , ∂Q) = 2

Z

∂Q

|∂τ g|2 dτ.

The topological degree of gε is given by 1 d = deg(gε , ∂Q) = 2π 11

Z

∂Q

(igε , ∂τ gε ).

Clearly we have 1 |d| ≤ 2π

1 |∂τ gε | ≤ ( π ∂Q

Z

Z

∂Q

|∂τ gε |2 )1/2 ,

so that d and Eε are related by Lemma 3.1. Let gε : ∂Q → S 1 . Then, for d ≡ deg(gε , ∂Q), we have d2 ≤

2 Eε (gε , ∂Q). π2

The aim of the next construction is to extend gε inside Ω, with a control on the energy as well as on the Jacobian. Recall that if vε : Q → C and vε = gε on ∂Q, then Z Jvε = π deg(gε , ∂Q) = πd . Q

In our construction, Jvε will be of constant sign and confined in small regions. The method is related to the construction of canonical harmonic maps with prescribed singularities as it can be found in [4]. The difference here is that we allow for a large number of singularities (diverging with ε). Similar constructions were carried out in [14, 18]. Lemma 3.2. There exists a positive constant σ > 0 such that if gε : ∂Q → S 1 verifies σ Eε (gε , ∂Q) ≤ , (23) ε then we may construct a function vε : Q → C with the follwing properties : i) vε = gε

on ∂Q, |vε | ≤ 1 on Q,

ii) Eε (vε , Q) ≤ C (|d log ε| + Eε (gε , ∂Q)), R R iii) Q |Jvε | = Q Jvε = π|d|,

iv)

R

Q

(1−|vε |2 )2 4ε2

≤ C|d|,

where d ≡ deg(gε , ∂Q) and C is a universal constant. 12

Proof. In order to present the different steps in the construction, we begin with the simplest case d = 0. If d = 0, then we may write uε = exp(iϕε ),

(24)

where ϕε is some continuous real valued function defined on ∂Q. Then, we consider the harmonic extension Φε of ϕε inside Q, and set vε = exp(iΦε ), so that i) is fulfilled. Clearly |∇vε | = |∇Φε |, and since Φε is harmonic, Z Z 1 |∇ϕε | 2 2 |∇Φε | ≤ C = CEε (gε , ∂Q) , 2 Q 2 ∂Q so that ii) is verified. For iii) and iv) we verify that Jvε ≡ 0 and (1 −|vε |2 )2 ≡ 0, since |vε | ≡ 1. If d 6= 0, (24) is no longer true, and every smooth function extending gε must have zeroes (vortices). The first step in the construction is to impose the location of the vortices. We may assume without loss of generality that d > 0 (otherwise we may pass to complex conjugates). Let n ∈ N be such that (n − 1)2 < d ≤ n2

1 (if n > 1). We consider the sublattice of Q( 21 ) of mesh δ and set δ = n−1 centered at 0. This yields n2 distinct points. We choose d arbitrary distinct points a1 , · · · , ad in this sublattice. Note in particular that

1 min dist(ai , aj ) ≥ δ ≥ √ i6=j d

(25)

and

1 min dist(aj , ∂Q) ≥ . j 2 Consider next the elliptic boundary value problem  d X    −∆Ψ = π δai in Q i=1

  

∂gε ∂Ψ = gε × ∂n ∂τ 13

on ∂Q,

(26)

(27)

which has a unique solution Ψ (up to an additive constant). There exists a multi-valued function φ such that ∂φ ∂Ψ = , ∂x1 ∂x2

∂Ψ ∂φ =− . ∂x2 ∂x1

on Q \ {a1 , · · · , ad }.

Moreover, locally, φ is defined up to an integer multiple of 2π. Therefore v = exp(iφ) is well-defined, smooth on Q \ {a1 , · · · , ad }, with values in S 1 . In view of the boundary condition in (27), we may assume (changing possibly φ by an additive constant) that v ≡ uε on ∂Q. (28) Let qε (t) = min{1, εt }, for t ∈ R+ , and set vε (x) = v(x)

d Y i=1

qε (|x − ai |),

so that vε is continuous on the whole of Q. We claim that vε satisfies the statements i) to iv). Statements i) and iv) are fairly elementary. Indeed, by (26), vε = v on ∂Q, and 1) follows by (28). For 4), notice that |vε | = 1 outside ∪di=1 B(ai , ε), |vε | ≤ 1 on Q, so that the conclusion follows. We turn next to the proof of ii) and iii). Proof of ii). We write Z Z 2 |∇vε | = Q

Qε

2

|∇vε | +

Z

∪di=1 B(ai ,ε)

|∇vε |2 ,

(29)

where Qε ≡ Q \ ∪di=1 B(ai , ε). Notice that, in view of assumption (23) and Lemma 3.1, 1 1 ≤ d ≤ Cε− 2 , (30) 2 δ so that for ε sufficiently small (say ε < ε0 , for a suitable constant ε0 ), the balls B(ai , ε) do not intersect and B(ai , ε) ⊂ B(ai , δ/2). For the first term in (29) we observe that vε = v on Qε , so that Z Z 2 |∇Ψ|2 , (31) |∇vε | = Qε

Qε

14

and we estimate the r.h.s. of (31) Pd using the elliptic equation (27). First we decompose Ψ as Ψ = Ψ0 + i=1 Ψi , where Ψ = − log |x − ai | (hence −∆Ψi = 2πδai on R2 ), and Ψ0 verifies the elliptic boundary value problem −∆Ψ0 = 0 in Q Pd ∂Ψi (32) ∂gε ∂Ψ0 = gε × ∂τ − i=1 ∂n on ∂Q. ∂n We write Z

Qε

|∇Ψ| ≤ 2

Z

=2

Z

2

|∇Ψ0 | + 2

Z

|∇Ψ0 |2 + 2

Z

2

Qε

Qε

(

d X

Qε i=1 d X

Qε i=1

∇Ψi )2

|∇Ψi |2 + 2

Z

d X

Qε i6=j

(33) ∇Ψi ∇Ψj .

For the first term, we remark that |∇Ψi | ≤ C on ∂Q, so that Z

∂Ψ0 2 2 ∂n ≤ C(Eε (gε , ∂Q) + d ) . ∂Q

Therefore, since Ψ0 is harmonic, Z |∇Ψ0 |2 ≤ C(Eε (gε , ∂Q) + d2 ) .

(34)

Q

An explicit computation shows that, for i = 1, ..., d, Z Z 2 1 ε 2 |∇Ψi | ≤ 2π rdr = 2π| log | 2 2 Qε ε r

(35)

≤ 2π|log ε| + C .

For the third term in (33) (the interaction term), we write, for i 6= j, and denoting Bk = B(ak , ε), k = 1, ..., d, Z Z X Z ∇Ψ ∇Ψ ∇Ψ ∇Ψ ≤ ∇Ψ ∇Ψ + i j i j i j Qε \(Bi ∪Bj ) Bk Qε k=i,j Z (36) 2 ∂Ψ ε i ≤ Ψj + C 2 , ∂Bi ∪∂Bj ∪∂Q ∂n δ 15

where we integrated by parts in the last inequality. We have Z ∂Ψi Ψj ≤ C , ∂Q ∂n Z ∂Ψi Ψj ≤ 2π| log(|ai − aj | − ε)| ≤ C| log |ai − aj | | ∂Bi ∂n and

Z ε ∂Ψi Ψj ≤ C|log ε| ≤ C , ∂Bj ∂n δ

so that combining the previous inequalities with (36), we deduce Z ∇Ψi ∇Ψj ≤ C (| log |ai − aj | | + 1) . Qε

Summing for i 6= j, we are led to Z X ∇Ψi ∇Ψj ≤ C 2 Q i6=j

X i6=j

| log |ai − aj | | + d2

!

.

(37)

Finally, we claim that

X i6=j

| log |ai − aj | | ≤ Cd2 .

(38)

Indeed, for fixed j, we have Z X 2 C | log |ai − aj | | ≤ 2 log |x − aj | ≤ 2 ≤ Cd , δ B(aj ,2) δ i6=j and the conclusion (38) follows. Going back to (31), we obtain, combining (34), (35), (37) and (38), Z |∇vε |2 ≤ C d|log ε| + d2 + Eε (gε , ∂Q) . (39) Qε

Finally, to conclude, it remains to estimate the second term in the r.h.s. of (29). We have ! Z Z Z 2 |x − a | 1 i |∇Ψ|2 |∇vε |2 ≤ C dx + dx . (40) 2 ε2 B(ai ,ε) B(ai ,ε) B(ai ,ε) ε 16

We claim that |∇Ψ(x)| ≤

C |x − ai |

on B(ai , ε) .

(41)

Indeed, first we have in view of equation (27), the definition of Ψi and standard elliptic estimates, 1

||∇(Ψ − Ψi )||L1 (Q) ≤ Cd + Eε (gε , ∂Q) 2 . Similarly, since ∇(Ψ − Ψi ) is harmonic on B(ai , δ), it follows ||∇(Ψ − Ψi )||L∞ (B(ai ,δ/2)) ≤ Cδ −2 ||∇(Ψ − Ψi )||L1(Q) 1

≤ Cδ −2 Eε (gε , ∂Q) 2 ≤ CEε (gε , ∂Q) ≤ Cσε−1 ,

(42)

where we have used assumption (23), (30) and Lemma 3.1. In view of the definition of Ψ, the claim (41) follows. Combining (40) and (41) we obtain Z |∇vε |2 ≤ C . B(ai ,ε)

Going back to (29), we are led to Z |∇vε |2 ≤ C d|log ε| + d2 + Eε (gε , ∂Q) . Q

Invoking Lemma 3.1 once more and iv) (which has already been proved), statement ii) follows. Proof of iii). Since |vε | = 1, Jvε = 0 on Qε . We are going to show that Jvε ≥ 0 in the ball B(ai , ε). We have, using polar coordinates centered at ai , ∂vε ∂vε × ∂r ∂θ ∂v ∂v ∂q ∂v = rq 2 × + rq v × ∂r ∂θ ∂r ∂θ ∂q ∂v = rq v × , ∂r ∂θ

Jvε = r

17

(43)

since the first term in the second equality vanishes, because |v| = 1. We claim that ∂v ≥0 on B(ai , ε). v× ∂θ Indeed, on B(ai , ε), v×

∂v ∂Ψi ∂(Ψ − Ψi ) =− − ∂θ ∂r ∂r 1 Cσ ≥ − r ε ≥ (1 − Cσ)ε−1 ,

(44)

and the conclusion follows if σ is choosen sufficiently small. Hence Z Z |Jvε | = Jvε = πd Q

Q

and the proof of iii) is complete in the case ε < ε0 . Otherwise, for ε ≥ ε0 , we may lower the value of σ so that (23) implies, by Lemma 3.1, that d = 0, case for which iii) is already proved for any ε > 0. Working on cubes of size h, we deduce similarly, by scaling, Corollary 3.1. Assume that gε : ∂Q(h) → S 1 verifies σ Eε (gε , ∂Q(h)) ≤ . ε Then there is a function vε : Q → C such that i) vε = gε

(45)

on ∂Q(h), |vε | ≤ 1 on Q(h),

ii) Eε (vε , Q(h)) ≤ C |d log( hε )| + hEε (gε , ∂Q(h)) , R R iii) Q(h) |Jvε | = Q(h) Jvε = π|d|,

iv)

R

Q(h)

(1−|vε |2 )2 4ε2

≤ C|d|.

Proof. Consider := gε (hx), so that

ε h

and the function g : ∂Q(h) → S 1 defined by g (x) :=

E (g , ∂Q) = hEε (gε , ∂Q(h)) ≤ Then apply Lemma 3.2 to g to conclude. 18

hσ σ ≤ . ε

3.2

Bounds for the degree

The construction in Lemma 3.2, in particular ii), shows that the infimum of the Ginzburg-Landau energy, among maps with boundary values gε is at most C(Eε (gε , ∂Q) + |d| |log ε|). The next results provide a lower bound on the Ginzburg-Landau energy for any map vε with boundary gε , which is of similar nature. Such results have already played a central role in the theory (see [4, 12]). The best results in that direction are due independently to [13] and [17]. The following statement in [17] is particularly well adapted to our needs. Theorem 3.1. ([17], Proposition 1) Let Ω be a simply connected bounded domain in R2 , and ω ⊂ Ω a smooth subdomain. For every v : Ω → S 1 the following inequality holds Z dist(ω, ∂Ω) 2 , |∇v| ≥ π|d| log (46) |∂ω| Ω\ω

where d = deg(v, ∂Ω).

As an immediate corollary we obtain Corollary 3.2. Let ω be a smooth subdomain of Q(h) and vε : Q(h)\ω → S 1 . Then, we have Z |∇vε |2 ≥ π|d| |log |∂ω| | − ChEε (vε , ∂Q(h)) , (47) Q(h)\ω

where C > 0 is some universal constant. Proof. Construct an extension of vε to Q(2h) with values in S 1 such that Eε (vε , Q(2h) \ Q(h)) ≤ ChEε (vε , ∂Q(h)) and apply Theorem 3.1 to that extension. In the context of the Ginzburg-Landau energy Eε , this yields Lemma 3.3. Assume h ≤ 1. Let uε : Q(h) → C be such that |uε | ≥ 1/2 on ∂Q(h) and assume that, for some constant C0 > 0, Eε (uε , ∂Q(h)) ≤ 19

C0 . ε

(48)

Then, we have |d| ≤

C (Eε (uε , Q(h)) + hEε (uε , ∂Q(h))) , |log ε|

(49)

where d = deg(uε , ∂Q(h)), and C is some constant depending possibly on C0 . Proof. We may assume without loss of generality that uε is smooth. For t ∈ R+ , consider the set ωt = {x ∈ Q(h), |uε (x)| ≤ t}, so that ∂ωt = {x ∈ Q(h), |uε (x)| = t} for a.e. t and it is a smooth curve by Sard’s Theorem. By the coarea formula, we have Z

1/2

1/4

|∂ωt |dt =

Z

1

ω1/2 \ω1/4

1

|∇|uε| | ≤ Eε (uε ) 2 |ω1/2 | 2 1 2

≤ CEε (uε , Q(h)) ε ≤ CεEε (uε , Q(h)) .

Z

Q(h)

(1 − |uε |2 )2 ε2

12

(50)

Therefore, there exists t0 ∈ (1/4, 1/2) such that |∂ωt0 | ≤ CεEε (uε , Q(h)) ,

(51)

and ∂ωt0 is smooth and does not intersect ∂Q(h) (since |uε | ≥ 1/2 on ∂Q(h)), and therefore it is a closed curve. On Q(h) \ ωt0 , |uε | ≥ 41 , so that we may consider the function wε = |uuεε | , which is S 1 -valued, and verifies wε = uε on ∂Q(h). By Corollary 3.2 we have 1 |∇uε | ≥ 16 Q(h)

Z

2

Z

Q(h)\ωt0

|∇wε |2 ≥ C|d| | log |∂ωt0 | | − ChEε (uε , ∂Q(h))

≥ C|d| | log(CεEε (uε , Q(h)))| − ChEε (uε , ∂Q(h)) .

(52)

Hence |d| ≤

C (Eε (uε , Q(h)) + hEε (uε , ∂Q(h))) . | log(CεEε (uε , Q(h)))|

(53)

Next we distinguish two cases: 3

Case 1. CEε (uε , Q(h)) ≤ ε− 4 . Then | log(CεEε (uε , Q(h)))| ≥ 14 |log ε|, and the conclusion follows. 20

3

Case 2. CEε (uε , Q(h)) ≥ ε− 4 . Then Lemma 3.1 implies p 1 |d| ≤ C hEε (uε , ∂Q(h)) ≤ Cε− 2 3

C Cε− 4 ≤ Eε (uε , Q(h)) , ≤ |log ε| |log ε|

(54)

provided ε is sufficiently small. This establishes (49).

3.3

Comparing uε and vε

Here we combine the results of the two previous sections, and apply them to the following situation. For 0 < ε < h < 1, we consider a map uε : Q(h) → C and assume σ0 Eε (uε , ∂Q(h)) ≤ , (55) ε where σ0 ≡ min{c0 , σ4 }, and c0 , σ are the constants appearing respectively in Lemma 2.3 and Lemma 3.2. By Lemma 2.3, we have |uε | ≥ 1/2 on ∂Q(h), and the map gε : ∂Q(h) → 1 S given by uε gε = |uε | is therefore well-defined. By (55), assumption (23) is verified and we may therefore consider the map vε given by Corollary 3.1, verifying i), ii), iii), iv). Estimates ii), iii) and iv) depend on d = deg(gε , ∂Q(h)), which can be bounded thanks to Lemma 3.3. In the special case where h is of order εα , in particular if h as in (22), this yields

Proposition 3.1. Assume h verifies (22) and uε verifies (55). Then the map vε constructed in Corollary 3.1 for gε = |uuεε | verifies i) vε =

uε |uε |

on ∂Q(h), |vε | ≤ 1 on Q(h),

ii) Eε (vε , Q(h)) ≤ C (Eε (uε , Q(h)) + hEε (uε , ∂Q(h))), R C (Eε (uε , Q(h)) + hEε (uε , ∂Q(h))), iii) Q(h) |Jvε | ≤ |log ε|

iv)

R

Q(h)

(1−|vε |2 )2 4ε2

≤

C |log ε|

(Eε (uε , Q(h)) + hEε (uε , ∂Q(h))),

where C is some constant depending only on c1 and γ. 21

4

Proof of Theorem 2

In the last sections we have introduced the main tools in the proof of Theorem 2. Here we complete the proof: first, we explicitely construct vε . Requirements (5), (6), (7), (8) are a direct consequence of the construction. Then we give the proof of (9) and (10), which requires a separate treatment.

4.1

Construction of vε: filling the grid

As already mentioned, the starting point of the construction is to set vε =

u˜ε |˜ uε |

on Q1 = S1 ∩ Q .

(56)

[Recall that u˜ε is an extension of uε on Ωµ , so that u˜ε = uε on Ω]. In view of Lemma 2.4 and the choice (22) of h, |˜ uε | ≥

1 2

on Q1 ,

(57)

so that vε is well-defined on Q1 and |vε | = 1 there. Recall that Q1 is a finite union of 1-dimensional cubes (segments) Q1i , i ∈ I1 , (]I1 ≤ ChN ), which are isometric to Q1 (h). The main principle of our construction is then to determine the value of vε on Qk , inductively on k, for k = 1, . . . , N. The extension of vε from Q1 to Q2 relies on the analysis of Section 3, whereas the extension from Qk to Qk+1 , for k = 2, . . . , N − 1 follows a more standard pyramidal procedure. Step 1: defining vε on Q2 . As above Q2 is a finite union of 2-dimensional cubes Q2i , i ∈ I2 , (]I2 ≤ ChN ), which are isometric to Q2 (h). Clearly, for i ∈ I2 , ∂Q2i ⊂ Q1 and on the other hand, an elementary argument shows that, for j ∈ I1 , ]{i ∈ I2 , Q1j ⊂ ∂Q2i } ≤ 2(N − 1) , (58)

that is, every segment Q1j of Q1 belongs to at most 2(N − 1) 2-dimensional cubes Q2i composing Q2 . In particular, we deduce X Eε (˜ uε , ∂Q2i ) ≤ 2(N − 1)Eε (˜ uε , Q1 ) . (59) i∈I2

Using Lemma 2.2 and (15) we have Eε (˜ uε , Q1 ) ≤ Ch1−N Eε (˜ uε ) ≤ Ch1−N Eε (uε ) . 22

(60)

Combining (59), (60) with assumption (Hγ ) and the definition (22) of h, we also derive X σ0 C ≤ , (61) Eε (˜ uε , ∂Q2i ) ≤ N −1 ε|log ε| ε i∈I 2

if ε is sufficiently small. Next, consider an arbitrary cube Q2i ∈ Q2 . In view of (61) we may apply Proposition 3.1 to u˜ε on this cube. This determines the construction of vε on Q2 . It agrees with the previously defined value on Q1 , it is continuous on Q2 and verifies |vε | ≤ 1. Moreover, summing relations ii), iii) and iv) in Proposition 3.1 over all cubes Q2i , we deduce

Z

Q2

Eε (vε , Q2 ) ≤ C (Eε (˜ uε , Q2 ) + hEε (˜ uε , Q1 )) , Z C (Eε (˜ uε , Q2 ) + hEε (˜ uε , Q1 )) , |Jvε | ≤ |log ε| Q2 C (1 − |vε |2 )2 ≤ (Eε (˜ uε , Q2 ) + hEε (˜ uε , Q1 )) . 2 4ε |log ε|

(62)

Invoking Lemma 2.2 and (15) we are led to

Z

Q2

Eε (vε , Q2 ) ≤ Ch2−N Eε (uε ) Z C h2−N Eε (uε ), |Jvε | ≤ |log ε| Q2 2 2 C (1 − |vε | ) h2−N Eε (uε ). ≤ 2 4ε |log ε|

(63)

Step 2: pyramidal extension. Assume that vε has been constructed on Qk (for 2 ≤ k ≤ N − 1), and verifies condition (Pk ) below  |vε | ≤ 1 on Qk       Eε (vε , Qk ) ≤ Ck hk−N Eε (uε )   Z  Ck k−N (Pk ) (64) |Jvε | ≤ h Eε (uε ),  |log ε|  Qk  Z    (1 − |vε |2 )2 Ck k−N   ≤ h Eε (uε ),  4ε2 |log ε| Qk

where Ck is a constant depending on k, γ and Ω. We then construct vε on , for , and for that purpose it suffices to define vε on Qk+1 Qk+1 = ∪i∈Ik+1 Qk+1 i i 23

every i ∈ Ik+1 . As above, we notice that ∂Qk+1 ⊂ Qk , and that, for j ∈ Ik , i ]{i ∈ Ik+1 , Qkj ⊂ ∂Qk+1 } ≤ 2(N − k) . i

Identifying Qk+1 with the standard cube Qk+1 (h), we set i x vε (x) = vε h , for x ∈ Qk+1 ' Qk+1 (h) , i kxk ∞

(65)

(66)

where, for x = (x1 , ..., xk+1 ) ∈ Qk+1 (h), kxk∞ = max{|xj |, j = 1, ..., k + 1}. We clearly have |vε | ≤ 1, and elementary computations show that vε ∈ H 1 (Qk+1 ) (the fact that k + 1 > 2 is crucial!) and i Eε (vε , Qk+1 ) ≤ ChEε (vε , ∂Qk+1 ) i i Z Z |Jvε | |Jvε | ≤ Ch Qk+1 i

(67)

∂Qk+1 i

Qk+1 i

Z

(1 − |vε |2 )2 ≤ Ch 4ε2

Z

∂Qk+1 i

(1 − |vε |2 )2 . 4ε2

Summing inequalities (67) over i ∈ Ik+1 , and using (65) we derive Eε (vε , Qk+1 ) ≤ Ck hEε (vε , Qk ) Z Z |Jvε |, |Jvε | ≤ Ck h

Qk+1

(68)

Qk

Qk+1

Z

2 2

(1 − |vε | ) ≤ Ck h 4ε2

Z

Qk

2 2

(1 − |vε | ) . 4ε2

Assuming (Pk ), we see that vε constructed as above on Qk+1 verifies (Pk+1 ) for some constant Ck+1 depending only on k, γ and Ω. Since P2 is already established, iterating the procedure for k = 2, ..., N − 1 we ultimately obtain a map vε defined on the whole of Q. Moreover, since PN holds, properties (5), (6), (7) and (8) are automatically verified. We finally turn to estimates (9) and (10). Proof of (9). In order to establish (9), we invoke the following linear estimate. Lemma 4.1. There exists a constant C > 0 depending only on N such that Z Z Z 2 2 2 2 w ≤C h w +h |∇w| , (69) Qm (h)

∂Qm (h)

for any v ∈ H 1 (Qm (h)), 1 ≤ m ≤ N. 24

Qm (h)

We apply Lemma 4.1 to wε = uε − vε on the cubes Qm i , i ∈ Im . Summing over i ∈ Im , we are led to Z Z Z 2 2 2 2 (70) |∇wε | . wε + h wε ≤ C h Qm

Qm−1

Qm

For the gradient term, we simply write |∇wε | ≤ |∇uε | + |∇vε |. Invoking Lemma 2.2 and property (Pm ) we deduce Z Z 2 2+m−N 2 w +h Eε (uε ) , for m = 2, ..., N , (71) wε ≤ C h Qm

Qm−1

while for m = 1 we have by definition wε = 0 on Q1 , so that follows therefore that Z wε2 ≤ Ch2+m−N Eε (uε ) , for m = 2, ..., N .

R

Q1

wε2 = 0. It (72)

Qm

In particular, for m = N, we obtain Z |uε − vε |2 ≤ Ch2 Eε (uε ) .

(73)

Q

This establishes estimate (9) in view of the definition (22) of h. Proof of (10). Since Ju = 21 d(u × du), we have the identity 1 Juε − Jvε = d(uε × duε − vε × dvε ) 2 1 = d ((uε − vε ) × (duε + dvε )) . 2 Let ϕ ∈ C0∞ (Ω; ΛN −2 RN ). We have Z Z 1 (Juε − Jvε ) ∧ ϕ = (u − v ) × (du + dv ) ∧ dϕ ε ε ε ε 2 N B Ω 1 ≤ ||uε − vε ||L2 (Ω) ||duε + dvε ||L2 (Ω) ||dϕ||L∞(Ω) 2 ≤ ChEε (uε )||dϕ||L∞(Ω) ,

(74)

(75)

where we have used (6) and (73) for the last inequality. This yields (10), in view of (22). The proof of Theorem 2 is therefore complete. 25

Remark 4.1. In the statement of Theorem 2 the map vε is required to be smooth, whereas the map vε constructed here above is not. However, one may recover the smoothness of vε by a standard mollification argument. Remark 4.2. Let q > N and σ > 1 be such that 1q + σ1 + 12 = 1. In particular, 2q we have σ = q−2 < 2∗ = N2N , i.e. the Sobolev exponent in dimension N. We −2 have the following variant of (10): kJuε − Jvε kW 1,q (Ω)∗ ≤ Chβ (Eε (uε ) + 1) ,

(76)

where β = σ2 ( 22∗−σ ) = 1 + Nσ − N2 . Indeed, going back to (75), we have, by −2 Young’s inequality, Z (Juε − Jvε ) ∧ ϕ ≤ ||uε − vε ||Lσ (Ω) ||duε + dvε ||L2 (Ω) ||dϕ||Lq (Ω) . (77) ∗

Ω

By Sobolev embedding and (6), we have

kuε − vε kL2∗ (Ω) ≤ C(Eε (uε )1/2 + 1),

(78)

so that by interpolation, we derive 1+ N − N 2

σ kuε − vε kLσ (Ω) ≤ kuε − vε kL2 (Ω)

N

−N

σ kuε − vε kL22∗ (Ω) ,

(79)

so that (76) follows from (77) and (73).

5

Application to Jacobian estimates

The purpose of this Section is to provide variants of the Jerrard and Soner estimates stated in Theorem 1 : the main point is to replace the norms for the test function ϕ there by Sobolev norms. As mentioned, the first result in that direction was given in [5], answering a question of [8]. The analysis in this Section follows closely the arguments of [5] (where we worked only in dimension 3). The starting point is to decompose Juε as Juε = Jvε + κε ,

(80)

so that κε = Juε − Jvε . As already noticed (see (9) and Remark 4.2), κε is small for suitable (weak) norms. We will further discuss this property in Section 5.2. First, we analyse the contribution of Jvε . 26

5.1

Estimates for Jvε

We clearly have Z Jvε ∧ ϕ ≤ kJvε kL1 (Ω) kϕkL∞ (Ω) , Ω

∀ϕ ∈ C0∞ (Ω; ΛN −2 RN ),

so that, in view of (7), we obtain Z Jvε ∧ ϕ ≤ C Eε (uε ) kϕkL∞ (Ω) , |log ε| Ω

(81)

(82)

which should be compared with the first term in the r.h.s. of inequality (3). In dimension N ≥ 3, we have the following variants of (81) and (82). Proposition 5.1. Assume N ≥ 3, and and v ∈ H 1 (Ω; C) be given. Let 0 < s ≤ 1 and p ≥ 1 be such that sp = N. There exists a constant C depending only on s, N and Ω, such that Z Jv ∧ ϕ ≤ CkJvkL1 (Ω) kϕk ˙ s,p , ∀ ϕ ∈ C0∞ (Ω; ΛN −2 RN ) . (83) W Ω

In particular, if uε and vε are as in Theorem 2, then Z Jvε ∧ ϕ ≤ C Eε (uε ) kϕk ˙ s,p , ∀ ϕ ∈ C0∞ (Ω; ΛN −2 RN ) . W |log ε| Ω

(84)

Remark 5.1. In the case Ω = BR , a ball of radius R, the constant C in (83) does not depend on R. This follows by scaling arguments, in particular the fact that the semi-norm k · kW˙ s,p is scale invariant if sp = N, i.e. kϕR kW˙ s,p := kϕkW˙ s,p ,

for ϕR (x) := ϕ(Rx) .

Remark 5.2. Inequality (83) is not true for N = 2. As a counterexample, consider Ω = D 2 , the unit disk in R2 , and take {vn }n∈N such that Jvn * δ0 as n → +∞. Since δ0 ∈ / (W s,p)∗ for sp = 2, this contradicts (83). Remark 5.3. One may wonder if one may replace kϕkW˙ s,p with kϕkBM O in (83). Recall that Z Z 1 1 |ϕ − ϕ¯x,r |, ϕ¯x,r = ϕ. kϕkBM O = sup |Bx,r | Bx,r x,r |Bx,r | Bx,r 27

However, this is not possible. Take for instance Ω = B 3 , the unit ball in R3 , and {vn }n∈N such that Jvn * µ = H1 [P− , P+ ] as n → +∞, where P± = (0, 0, ±1). Clearly, µ ∈ / (BMO)∗ . Concerning (84), see however Lemma 5.2. Remark 5.4. We have not investigated the extension of (83) to larger classes of maps v, for which Jv can only be defined in the distributional sense Jv = 1 d(v × dv) (e.g. v ∈ W 1,p (Ω), for p ≥ N2N ). 2 +1 As mentioned, inequality (83) was motivated by a new inequality derived in [8, 7]. More precisely, we have

Theorem 5.1. Let 0 < s ≤ 1 and p ≥ 1 be such that sp = N. There exists a constant C0 depending only on s and N such that for any closed, oriented, rectifiable curve Γ in RN , we have Z φ ≤ C0 |Γ| ||ϕ|| ˙ s,p N , ∀ ϕ ∈ C0∞ (RN ; Λ1 RN ) . (85) W (R ) Γ

Our proof of Proposition 5.1 relies on estimate (85) in an essential way. The case s = 1, p = N = 3 was already treated in [5], where the coarea formula entered also as an important ingredient: here we follow essentially the same strategy, combined with a slicing argument which allows to use (85). Proof of Proposition 5.1. First, we claim that it suffices to consider the case Ω =B, (86)

where B = B N is the unit ball in RN . Indeed, for an arbitrary domain Ω we may extend v to Ωµ (defined in Section 2.2) by a standard reflection, provided µ is sufficiently small. We still denote v this extension. A simple computation shows that kJvkL1 (Ωµ ) ≤ CkJvkL1 (Ω) , for some constant C depending only on Ω. Next we consider a smooth partition P of unity (ξi )1≤i≤n such that ξi has compact support in a ball Bi ⊂ Ωµ , and ni=1 ξi = 1 on Ω. We have Z Z n X Jv ∧ ϕ = Jv ∧ ϕ = Jv ∧ (ξiϕ) . Ω

Ωµ

i=1

If the result (83) is established for B, and hence for Bi (see Remark 5.1), then Z ≤ CkJvkL1 (Ω) kξiϕk ˙ s,p ≤ CkJvkL1 (Ω) kϕk ˙ s,p , Jv ∧ (ξ ϕ) i W W Bi

28

and the conclusion (83) follows for Ω summing over i = 1, ..., n. Assume therefore (86). It is convenient to write the integral on the l.h.s. of (83) using coordinates. Set X ϕ= ϕij ? dxi ∧ dxj . 1≤i