Improved estimates for the Ginzburg-Landau equation : the elliptic case

Improved estimates for the Ginzburg-Landau equation:

the elliptic case

FABRICEBETHUEL, GIANDOMENICOORLANDI ANDDIDIERSMETS

Abstract. We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the G L-energyE_εand the parameterε. These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

Mathematics Subject Classification (2000):35J60 (primary); 35B40, 35Q40, 31B35, 46E35 (secondary).

1.

Introduction

Letbe a smooth bounded domain in

R

^N, N ≥2.For 0< ε <1 we consider the complex elliptic Ginzburg-Landau equation

−u_ε= 1

ε²u_ε(1− |u_ε|²) on (GL)_ε for maps u_ε : →

C

. This equation is the Euler-Lagrange equation for the Ginzburg-Landau (G L-) energy functional

E_ε(u_ε)=

e_ε(u_ε)≡

|∇u_ε|²

2 + (1− |u_ε|²)² 4ε² , and we will be concerned only with solutionsu_εwith finiteG L-energy.

The purpose of this paper is to derive estimates for various quantities which are of interest in the analysis of(GL)_ε, and which will be bounded in terms of theG L- energyE_εand the parameterε. These estimates are of local nature, and in particular independent of any boundary condition. Most of them extend and generalize earlier results on the subject.

The G L-energy has two components: the kinetic energy ^|∇₂^u|2 on one side, and the potential energy V_ε(u) ≡ ^(1−|_4ε^u2^|2)2 on the other. Our first result yields an

Pervenuto alla Redazione il 17 settembre 2004 e in forma definitiva l’8 aprile 2005.

“improved” local bound for the integral of the potential, and shows that, for suitable energy regimes, it is of lower order.

Theorem 1.1. Let= B₁and u_εbe a solution to(GL)_ε. We have

B_1/2

V_ε(u_ε)≤C₀E_ε(u_ε)

|logε| log

2+ E_ε(u_ε)

|logε|

, (1.1)

where C₀is some constant depending on N .

Estimate (1.1) provides some non trivial information only if the energy is not too large, more precisely only if

E_ε(u_ε)≤ε^−1/C0|logε|.

On the other hand, it is well-known⁽¹⁾that there exist real-valued solutions to(GL)_ε such that the “energy balance”

|∇u_ε|²

2 = (1− |u_ε|²)² 4ε²

holds pointwise, and such thatE_ε(u_ε)is of orderε⁻¹. In this case, log(2+^E_|log^ε(u_ε|^ε))∼

|logε|, and therefore estimate (1.1) is optimal in this respect⁽²⁾.

In another direction, the factor|logε|appearing on the right-hand side of in- equality (1.1) is the typical energy of a vortex solution⁽³⁾ in dimension 2. It is known⁽⁴⁾that for these solutions the integral of the potential remains bounded in- dependently of the parameter ε: Theorem 1.1 gives therefore a generalization of this fact for local integrals of the potential in arbitrary dimension, without impos- ing any boundary datum. More precisely, if for some constant M₀ >0, E_ε(u_ε) ≤ M₀|logε|, then (1.1) yields the bound

B_1/2

V_ε(u_ε)≤C₀M₀log(2+M₀),

which is uniform in ε. We would also like to emphasize that Theorem 1.1 is an improvement of earlier results given in [15, 36, 33, 29, 8]⁽⁵⁾.

(1)Take for instanceu_ε(x₁, ...,x_N)=tanh(√^x1 2ε)

(2)However, the optimal value of the constantC₀is not known.

(3)A typical example is provided by a solution of the formu_ε(x)= f(|x|)exp(iθ)onB₁⊂C, with f(0) = 0, f(1) = 1 and f verifying the ordinary differential equation−f + ^f_r =

ε1² f(1− f²)(see e.g. [25]).

(4)This is a consequence of Pohozaev identity (see e.g. [7]).

(5)In particular, in [8] it was proved

B_1/2∩{|u_ε|≤1/2}V_ε(u_ε)≤C

E_ε(u_ε)

|logε|

₂ .

Remark 1.2. Using the various tools presented in this paper, one may actually de- rive a better estimate for the potential when the energy is small. More precisely, there exist positive constantsη1, βandCsuch that, if

B₁

e_ε(u_ε)≤η1|logε|, (1.2) then

B_1/2

V_ε(u_ε)≤Cε^βE_ε(u_ε), (1.3) so that estimate (1.1) is far from being optimal for low energy regimes. The proof of (1.3) can be obtained combining the clearing-out property (see Theorem 2.7 in Section 2) with Proposition A.4 of the Appendix. We leave it to the reader.

Using various elliptic estimates, we are able to relate, in the same spirit, local estimates for the gradient of the modulus to the potential as follows.

Proposition 1.3. Let u_εbe a solution of(GL)_εon B₁. We have

B_1/2|∇|u_ε||²≤C₁

B₁

V_ε(u_ε)+ε

B₁

V_ε(u_ε) _1/2

(1.4) where C₁is some constant depending only on N .

In contrast with Theorem 1.1, the result in Proposition 1.3 remains valid for vector-valued solutionsu_ε : B₁ →

R

^d, for arbitrary integerd ( the proof carries over word for word). In the course of the proof, we invoke a bound on theL⁴-norm of∇u_ε (stated in Lemma 3.1), which we hope is of independent interest.

Combining Theorem 1.1 and Proposition 1.3 with covering and scaling argu- ments, we deduce the global bound

Corollary 1.4. Letbe a smooth bounded domain in

R

^{N, N} ≥ 2, and u_ε be a solution of(GL)_ε. We have, for a constant C depending on,

| ∇|u_ε| |²+V_ε(u_ε)

|1+log(dist(x, ∂))|⁻¹d x ≤C

E_ε(u_ε)

|logε|

, (1.5) where, for t ≥0, we have set(t)=tlog(2+t).

As a consequence of the above analysis, we see that in some sense the main contribution to theG L-energy stems from the gradient of the phase, at least in the appropriate energy regime.

In a slightly different direction, a highly involved result of Bourgain, Brezis and Mironescu yields improved interior integral estimates for|∇u_ε|^p, withp<2 (i.e. p subcritical), in dimension three and for the energy regime E_ε(u_ε)≤M₀|logε|(see [18], Theorem 8). Our purpose is to obtain a similar result in arbitrary dimension

and for possibly larger energy regimes. Our approach is of somewhat different nature and relies heavily on results in harmonic analysis which state that measures with suitable growth over all balls are elements of the dual of W^1,p (see [38] and references therein).

In view of possible oscillations in the phase, which essentially propagate in the domain through an elliptic equation, it is not possible to expect a general improved estimate involving only the energy (see e.g. [20, 12]). However, these oscillations can be controlled by a weaker norm, say for instance the L¹-norm of the gradient.

More precisely, we will show, as a consequence of Theorem 1.1 and [1, 2, 38], that for every 1≤ p<2, the following estimate holds⁽⁶⁾

B_1/2

|∇u_ε|^p _1/p

≤C_p

E_ε(u_ε)

|logε|

+

B1

|∇u_ε|

. (1.6)

Estimate (1.6) can ever be slightly improved if instead of Theorem 1.1 we invoke Jacobian estimates introduced by Jerrard and Soner [28] and Alberti, Baldo and the second author [3]. Relying on our result in [13], and the aforementioned results in [38], we obtain

Theorem 1.5. Let1< p< 2. There exist C_p > 0such that if u_ε is a solution to (GL)_ε, then

B_1/2

|∇u_ε|^p 1/p

≤C_p

E_ε(u_ε)

|logε| +

B₁

|∇u_ε|

. (1.7)

Remark 1.6. For boundary value problems, the term||∇u_ε||_L¹ can be controlled by the trace on ∂. More precisely, it has been shown in [7, 29, 8, 18, 5, 9, 3]

that global bounds for||∇u_ε||L^p()withp< _N^N₋₁could be obtained under various restrictive assumptions ong_ε =u_ε|∂andu_ε. The caseg_ε belonging toH^1/2with the additional assumption⁽⁷⁾ |g_ε| = 1 was considered in [17] and then in [5, 9].

The case p = _N^N₋₁ was settled in [18] assuming a conjecture which we proved in [13]⁽⁸⁾.

More precisely, the results in [3, 18, 13] yield, for |g_ε| = 1 and smooth bounded and simply connected⁽⁹⁾,

|∇u_ε|^{NN−1 N−1}

N ≤C

E_ε(u_ε)

|logε| + ||g_ε||²_H1/2

. (1.8)

(6)Inequality (1.6) yields a nontrivial result only ifE_ε(u_ε) _|log^|log_|log^ε|2_ε||2.

(7)The proof in [5] actually carries over to the case|g_ε|is bounded away from zero, i.e.β >

|g_ε|> α >0.

(8)Using a new linear estimate given in [18], Proposition 4, see also [37, 16].

(9)In [9], the assumption|g_ε| = 1 is replaced by an assumption on theG L-energy on the boundary.

Notice however that in the global estimate (1.8) and the case g_ε ∈ H^1/2 the ex- ponent _N^N₋₁ is the best possible (see [18]), in contrast with the result of Theorem 1.5.

Combining (1.8) and Theorem 1.5 one obtains the announced generalization of [18], Theorem 8.

Going back to Theorem 1.5, notice that estimate (1.7) is immediate ifE_ε(u_ε)≥

|logε|², since by H¨older’s inequality,

|∇u_ε|^p _1/p

≤C

|∇u_ε|² _1/2

≤C E_ε(u_ε)^1/2.

In view of Theorem 1.5 as well as the mentioned Jacobian estimates, it is tempting to believe that estimate (1.1) can also be improved replacing possibly (^E_|log^ε(u_ε|^ε)) by ^E_|log^ε(u_ε|^ε) for suitable energy ranges⁽¹⁰⁾. In another direction, it would be interest- ing to extend our estimates to boundary value problems (see e.g. [22]) and to the corresponding parabolic equation.

ACKNOWLEDGEMENTS. The third author wishes to thank warmly the Centro di Ricerca Matematica Ennio De Giorgi of the Scuola Normale Superiore di Pisa for its kind hospitality during the preparation of this work.

2.

Estimates for the potential

V_ε(u_ε)

The purpose of this section is to provide the proof of Theorem 1.1. The vorticity set

V

_ε = {x ∈, |u_ε| ≤1−σ0},

enters in an essential way in our discussion. The constant 0< σ0≤1/2 appearing in the definition of

V

ε depends only on the dimension N, and will be defined at a later stage of our analysis (see Appendix).

The main point in the proof is to provide integral estimates of the potential V_ε(u_ε)on small ballsB(x,r), distinguishing carefully the case where the balls are included in\

V

ε from the case where they intersect the vorticity set

V

ε.

2.1. Potential estimates off the vorticity set We first have:

Theorem 2.1. Let x ∈and r > 0such that B(x,r)⊂ \

V

_ε.There are some constants0< α0<1and0≤σ0≤1/2such that if r ≥εand

E˜_ε(u_ε,x,r)≡ 1 r^N−2

B(x,r)e_ε(u_ε)≤ε r

_−α0

, (2.1)

(10)For instance, forE_ε(u_ε)≤ε^−γ, for a given 0< γ <1.

then

|u_ε| ≥1−Cε²

r² (1+ ˜E_ε(u_ε,x,r)) on B(x,r/2). (2.2) The proof of Theorem 2.1 follows some arguments in [11, 14]. The necessary adaptation will be exposed in a separate Appendix. Theorem 2.1 provides a lower bound for|u_ε|off the vorticity set. On the other hand, we recall some important well-known global upper bounds.

Proposition 2.2. Letbe a smooth domain in

R

^{Nand u}εbe a solution of(GL)_εon . There exists a constant C >0depending only on N such that, ifdist(x, ∂) >

√ε,then

|u_ε(x)| ≤1+ Cε²

dist(x, ∂), (2.3)

|∇u_ε(x)| ≤ C

ε. (2.4)

Combining Proposition 2.2 and Theorem 2.1, we deduce immediately the following uniform estimate for the potential on balls included in\

V

_ε.

Proposition 2.3. Let x,r and u_ε be as in Theorem 2.1 and assume that r ≥εand dist(x, ∂) >√

ε.Then sup

B(x,r/2)V_ε(u_ε)≤Cε²

r⁴(1+ ˜E_ε(u_ε,x,r))². (2.5) In the proof of Theorem 1.1 we will invoke Proposition 2.3 for balls of fixed radius r =r₀, specifying throughout the choice ofr₀in a somewhat arbitrary way as

r₀= ε^1/4 2 . For this choice ofr₀, the factor ^ε2

r₀⁴ is equal to 16ε, and hence small.

2.2. Potential estimates on the vorticity set

In this section, we consider the case where B(x,r₀)intersects the vorticity set

V

ε, that is

dist(x,

V

_ε)≤ ε^1/4

2 . (2.6)

In this situation, we show

Proposition 2.4. Assume thatdist(x, ∂)≥ε^1/8and that(2.6)holds. Then there exists some radiusε^1/4≤r(x)≤ε^1/8depending only on u_εand x such that

B(x,r(x))V_ε(u_ε)≤ C

|logε|log

2+ E˜_ε(x,u_ε, ε^1/8)

|logε| B(x,r(x))e_ε(u_ε), (2.7) where C >0is some constant depending only on N .

Comment. 1) We would like to draw the attention of the reader to the fact that the l.h.s. of (2.7) is the potential integrated on thesamedomain B(x,r(x))as the energye_ε(u_ε)on the right-hand side of the inequality. However the scaled energy E˜_ε(x,u_ε, ε^1/8)is considered on the larger domain B(x, ε^1/8).

2) In contrast with Proposition 2.3, the radiusr(x)for which this inequality holds isnotfixed a priori, but depends onx.

The main tools in the proof are the monotonicity formula and the clearing-out theorem which we recall first.

Proposition 2.5 (Monotonicity formula⁽¹¹⁾). Assume u_ε is a solution of(GL)ε in B(x,R). Then we have

d

dr(E˜_ε(u_ε,x,r))= 1 r^N−2

∂B(x,r)

∂u_ε

∂n

²+ 1 r^N−1

B(x,r)

(1− |u_ε|²)² 2ε² , for0<r < R.

Remark 2.6. Note in particular that d

dr(E˜_ε(u_ε,x,r))≥ 1 r^N−1

B(x,r)

(1− |u_ε|²)²

2ε² ≥0, (2.8)

so that for fixedx, the functionF(r)≡ ˜E_ε(u_ε,x,r)is a nondecreasing function of the radiusr. Moreover, setting

G(r)=r^2−N

B(x,r)

(1− |u_ε|²)² 2ε² , (2.8) yields the differential inequality

r F ≥G, (2.9)

which relates in a simple way the energy integral and the potential integral on B(x,r). This relation will be exploited in the proof of Proposition 2.4.

(11)Here we consider the version given in [8]. Earlier versions were given in [36, 33, 29].

Theorem 2.7 (Clearing-out⁽¹²⁾for vorticity). Let u_εbe a solution to(GL)_ε, x ∈, r ≥εsuch that B(x,r)⊂. If

V

ε∩B(x,^r₂)= ∅, then E˜_ε(u_ε,x,r)≥η0|log(^ε_r)|, whereη0>0is some constant depending only on N .

Proof of Proposition2.4. Fors ∈ I_ε ≡ [¹₄logε,¹₈logε], set f(s) = F(exps), g(s)=G(exps), so that relation (2.9) writes, for f andg,

f (s)≥g(s) ∀s ∈I_ε. (2.10) For this ODE we invoke the next elementary lemma.

Lemma 2.8. Let I =[a,b]be a bounded interval of

R

, f and g two nonnegative absolutely continuous functions such that

f ≥g on I. Then there exists s₀∈I such that

g(s₀)≤ 1 b−alog

f(b) f(a)

f(s₀) . (2.11)

Applying Lemma 2.8 to (2.10) we deduce that there exists somer(x)∈[ε^1/4, ε^1/8] such that

G(r(x))≤ 1

|logε|log

F(ε^1/8) F(ε^1/4)

F(r(x)). (2.12)

On the other hand, by the clearing-out theorem, since B(x,^ε1/4₂ )∩

V

_ε = ∅, we have F(ε^1/4)= ˜E_ε(x,u_ε, ε^1/4)≥η0|log( ε

ε^1/4)| = 3η0

4 |logε|. (2.13) Combining (2.12) and (2.13), we deduce the conclusion (2.7).

Proof of Lemma 2.8 The argument is by contradiction. If (2.11) were false, we would have

f (s) > λf(s) ∀s ∈[a,b],

(12)This kind of result was proved under various assumptions in [15, 36, 33, 29, 30, 8, 10] and termedη-compactness in [33, 29, 30],η-ellipticity in [8, 10]. Here we follow the terminology in [27, 11], which was introduced by Brakke [19].

where we have set

λ= 1

b−alog f(b)

f(a)

, so that

d

ds(exp(−λs)f(s)) >0 on [a,b]. Hence, by integration

f(b) >exp(−λ(a−b))f(a)= f(b), a contradiction.

2.3. Proof of Theorem 1.1

Recall that in this section ≡ B₁. First, notice that we may assume throughout thatu_ε satisfies the energy bound

E_ε(u_ε)≤ε^−β0, (2.14)

whereβ0= ^α₂⁰,α0being the constant appearing in Theorem A.2. Indeed, if other- wiseE_ε(u_ε)≥ε^−β0, then, as already mentioned in the introduction, the conclusion is straightforward, choosingC₀= _β²₀.

We next consider the sets

_ε = x∈ B_1/2, dist(x,

V

ε)≤r₀= ε^1/4 2

and _ε = B_1/2\_ε, so that_ε∪_ε = B_1/2.

Step 1: bounds for the potential on_ε. If x ∈ _ε, then B(x,r₀) ⊂ B_3/4\

V

_ε provided ε is sufficiently small, and we may therefore invoke Proposition 2.3 to assert that

V_ε(u_ε(x))≤Cε(1+ ˜E_ε(u_ε,x,r₀))². (2.15) By monotonicity we have

E˜_ε(u_ε,x,r₀)≤ ˜E_ε(u_ε,x,r₀+ 1

4)≤C E_ε(u_ε). (2.16) Combining (2.15) with (2.16), together with the fact that E_ε(u_ε) ≤ε^−β ≤ε^−1/2, we deduce

sup

ε

V_ε(u_ε)≤Cε^1/2(1+E_ε(u_ε)). (2.17)

Step 2: integrating the potential on_ε.Ifx ∈_ε, then dist(x,

V

_ε)≤r₀, so that we may apply Proposition 2.4. Combining (2.7) with the monotonicity formula as above, we deduce

B(x,r(x))V_ε(u_ε)≤ C

|logε|

B(x,r(x))e_ε(u_ε)log

2+ E_ε(u_ε)

|logε|

. (2.18)

We next consider the covering∪x∈εB(x,r(x))of_εand apply Besicovitch cov- ering Theorem. This yields a finite set A= {x1, ...,xm} ⊂ε such that

ε ⊂ ∪x∈AB(x,r(x))

and the balls B(x_i,r(x_i)), i = 1, ...,m can be distributed in families

B

k, k = 1, ..., of disjoint closed balls, where is a constant depending only on N. We write

ε

V_ε(u_ε)≤ m i=1

B(x_i,r(x_i))V_ε(u_ε)

≤ C

|logε|log

2+ E_ε(u_ε)

|logε|

m i=1

B(x_i,r(x_i))e_ε(u_ε).

(2.19)

On the other hand, m i=1

B(x_i,r(x_i))e_ε(u_ε)≤

k=1

B(x_i,r(x_i))∈Bk

B(x_i,r(x_i))e_ε(u_ε)

≤E_ε(u_ε).

(2.20)

Combining (2.18), (2.19) and (2.20), we obtain

ε

V_ε(u_ε)≤ C

|logε|E_ε(u_ε)log

2+ E_ε(u_ε)

|logε|

. (2.21)

Step 3: Proof of Theorem 1.1 completed.We distinguish two cases.

Case A:E_ε(u_ε)≥ε^1/2|logε|. It follows from (2.17) that sup

ε

V_ε(u_ε)≤CE_ε(u_ε)

|logε|, (2.22)

so that the conclusion (1.1) follows from (2.22) and (2.21).

Case B:E_ε(u_ε)≤ε^1/2|logε|. In this case it follows from the clearing-out theorem that ifεis sufficiently small,

|u_ε| ≥1−σ0 on B_9/10,

so that_ε = ∅, and_ε= B_1/2. The conclusion (1.1) then follows from Proposition A.4 of the Appendix.

Remark 2.9. Letε < r < 1. The analogue of (1.1) for the ball B_r writes, by scaling,

B_r/2

V_ε(u_ε)≤ E_ε(u_ε,r)

|log^ε_r| log

2+ E_ε(u_ε,r) r^N−2|log^ε_r|

, (2.23)

whereE_ε(u_ε,r)≡

Bre_ε(u_ε)=r^N−2E˜_ε(u_ε,0,r).Indeed, settingu˜_ε(x˜)=u_ε(rx˜) forx˜ ∈ B₁, we verify thatu˜_εis a solution of(GL)_ε˜ onB₁, forε˜ =ε/r. Moreover, we have

E_ε(u˜_ε)= 1

r^N−2E_ε(u_ε,r),

B₁

V_˜ε(u˜_˜ε)= 1 r^N−2

B_r

V_ε(u_ε).

3.

Gradient estimates for the modulus

_|u_ε|

The aim of this section is to prove Proposition 1.3. The starting point is the elliptic equation for the functionζ ≡1−ρ_ε²≡1− |u_ε|²,

−ζ +2ρ_ε²

ε²ζ =2|∇u_ε|², (3.1)

which is a straightforward consequence of(GL)_ε. We introduce next a test function χ ∈

C

_c^∞(B₁)such thatχ ≡1 on B_1/2andχ =0 on B₁\B_3/4. Multiplying (3.1) byχ²ζ and integrating by parts, we obtain, after a few standard computations,

1 2

B_1/2

|∇ζ|²≤2

B_3/4

ζ|∇u_ε|²+2

B_3/4

ζ²|∇χ|². (3.2) On the other hand, we have the pointwise equality|∇ζ|²=|∇|u_ε|²|²=4|u_ε|²|∇|u_ε| |², so that

|∇|u_ε| |²= 1

4|∇ζ|²+(1− |u_ε|²)|∇|u_ε| |² ≤ 1

4|∇ζ|²+ |ζ||∇|u_ε| |², and hence (3.2) yields

B_1/2

|∇|u_ε| |²≤23 2

B_3/4

|ζ||∇u_ε|²+

B_3/4

ζ²|∇χ|². (3.3) To complete the proof of Proposition 1.3, we apply Cauchy-Schwarz inequality to the first integral on the right-hand side of (3.3). This involves the term||∇u_ε||_L⁴_(B3/4), which is handled by the following

Lemma 3.1. We have

B_3/4

|∇u_ε|⁴≤C

1+ 1 ε²

B_4/5

V_ε(u_ε)

, (3.4)

where C >0is a constant depending only possibly on N .

Comment. The factorε⁻² in (3.4) is somewhat optimal. This can be checked both for vortices and kinks. For vortices, i.e. planar complex solutions of the form u_ε = f_ε(r)exp(iθ), we have

B_3/4

|∇u_ε|⁴C _3/4

ε

1

r⁴r dr C ε² C

ε²

B₁

V_ε(u_ε), whereas for the standard kink tanh(^√x_2ε)we have

B_3/4

|∇u_ε|⁴C _ε

−ε

1

ε⁴d x C ε³ C

ε²

B₁

V_ε(u_ε).

Proof of Lemma3.1 On the ballB_4/5we decompose the functionu_εasu_ε≡v_ε+w_ε, wherew_ε is a harmonic function and verifiesw_ε = u_ε on the boundary∂B_4/5.In view of Proposition 2.2,u_ε, and hencew_ε are uniformly bounded on∂B_4/5, so that, sincew_εis harmonic,

|∇w_ε| ≤C on B_3/4, (3.5)

whereCdepends only on the dimension N.

Turning tovε, we notice that by constructionvε =0 on∂B_4/5, and that

|v_ε| ≤ C

εV_ε(u_ε)^1/2. Hence by standard elliptic theory

||v_ε||²_H2(B_4/5) ≤ C ε²

B_4/5

V_ε(u_ε). (3.6)

Moreover, sinceu_ε andw_ε are uniformly bounded onB_4/5, the same holds forv_ε, i.e.

||v_ε||L^∞(B_4/5)≤C, (3.7) where C > 0 is depending only on N. We next invoke a classical Gagliardo- Nirenberg type inequality which asserts that

||∇v_ε||²_L4(B_4/5) ≤C||v_ε||_H²||v_ε||L^∞, (3.8) whereC depends only onN. An elementary proof of (3.8) may actually be derived as follows. Since v ≡ v_ε is compactly supported in B_4/5, we may write, fori = 1, ...,N, integrating by parts,

||vx_i||⁴_L4(B_4/5) =

v_x³_ivx_i = −

(v³_xi)x_iv= −3

vx_ix_i(vx_i)²v

≤3||vx_ix_i||_L²||vx_i||²_L4||v||L^∞,

(3.9)

which yields (3.8). Combining (3.6), (3.7) and (3.8), we obtain

||∇v_ε||⁴_L4(B_4/5) ≤ C ε²

B_4/5

V_ε(u_ε). (3.10)

Invoking finally (3.5) together with the decompositionu_ε =v_ε+w_εwe complete the proof of Lemma 3.1

Remark 3.2. In view of the inequality |∇u_ε| ≤ C/ε, we deduce by (3.5) that

|∇v_ε| ≤C/εonB_3/4. It follows that for any p≥4 we have

||∇v_ε||_L^pp(B_3/4)≤ C_p ε^p−2

B_4/5

V_ε(u_ε), and hence, for every p≥4,

B_3/4

|∇u_ε|^p ≤C_p

1 ε^p−2

B_4/5

V_ε(u_ε)+1

. (3.11)

As in the case p = 4, the exponentε^−(p−2) is somewhat optimal. We conjecture actually that inequality (3.11) is true for every p>2.

Proof of Proposition 1.3completed. We go back to inequality (3.3). In view of Lemma 3.1, we have, by Cauchy-Schwarz inequality

B_3/4|ζ||∇u_ε|²≤ ||ζ||L²(B_3/4)||∇u_ε||²_L4(B_3/4)

≤C

B_3/4ε²V_ε(u_ε) 1/2

1+

B_4/5ε⁻²V_ε(u_ε) 1/2

≤C





B_4/5

V_ε(u_ε)+ε

B_4/5

V_ε(u_ε) 1/2

.

(3.12)

The conclusion follows.

Remark 3.3. Combining Theorem 1.1 and Proposition 1.3 we are immediately led to the inequality

B_1/2|∇|u_ε| |²≤C₂

E_ε(u_ε)

|logε|

. (3.13)

Indeed, as in the proof of Theorem 1.1 one distinguishes two cases:

Case 1:

B_4/5V_ε(u_ε)≥ε². In this caseε(

B_4/5V_ε(u_ε))^1/2 ≤

B_4/5V_ε(u_ε)and the conclusion follows directly from (1.4).

Case 2:

B_4/5V_ε(u_ε)≤ε². One concludes in this case using Proposition A.4.

Remark 3.4. As in Remark 2.9 we may argue by scaling to assert that, forε <r<1, we have

B_r/2

|∇|u_ε| |²+V_ε(u_ε)≤ E_ε(u_ε,r)

|log^ε_r| log

2+ E_ε(u_ε,r) r^N−2|log^ε_r|

. (3.14)

Proof of Corollary1.4. Letbe an arbitrary smooth domain in

R

^{N. Let}ε^1/2 ≤ r ≤1 and consider a ball B(x,r)⊂. As a direct consequence of (3.14) and the elementary inequality log(2+_rN−2^E )≤log(2+E)(1+ |logr|), it follows that

B(x,r/2)|∇|u_ε| |²+V_ε(u_ε)≤ C

|logε|log

2+ E_ε(u_ε)

|logε|

(1+ |logr|)

B(x,r)e_ε(u_ε).

(3.15) Using a covering argument, we deduce

{dist(x,∂)≥ε^1/4}

| ∇|u_ε| |²+V_ε(u_ε)

|1+log(dist(x, ∂))|⁻¹d x≤C

E_ε(u_ε)

|logε|

. On the other hand,

{dist(x,∂)≤ε^1/4}

| ∇|u_ε| |²+V_ε(u_ε)

|1+log(dist(x, ∂))|⁻¹d x

≤ E_ε(u_ε)

1+ ¹₄|logε| ≤CE_ε(u_ε)

|logε|.

Combining the last two inequalities, the conclusion (1.5) follows.

4. L^p

estimates for

∇u_ε

,

p<2

The purpose of this section is to prove (1.6) and its improvement (1.7) stated in Theorem 1.5. Throughout this section, our arguments follow closely methods in- troduced in the afore quoted literature. We first decompose the gradient into contri- butions of phase and modulus, writing (here and in the sequelu≡u_ε)

4|u|²|∇u|²=4|u× ∇u|²+ |∇|u|²|², (4.1) so that

4|∇u|²=4|u× ∇u|²+ |∇|u|²|²+4(1− |u|²)|∇u|². (4.2) For the two last terms on the right-hand side of (4.2) we invoke our previous esti- mate (3.13) and (3.12), so that

B_1/2

|∇|u|²|²+4|1− |u|²| |∇u|²≤C

E_ε(u_ε)

|logε|

,