- Email: [email protected]
Asymptotic behavior of critical points for a Gross–Pitaevskii energy
Nonlinear Analysis 74 (2011) 4274–4291
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Asymptotic behavior of critical points for a Gross–Pitaevskii energy Ling Zhou a , Haifeng Xu a,∗ , Zuhan Liu b a
School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu, 225002, PR China
b
Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu, 221116, PR China
article
abstract
info
Article history: Received 6 August 2009 Accepted 2 April 2011 Communicated by Ravi Agarwal
In this paper, we study the asymptotic behavior of critical points of a Gross–Pitaevskii energy, which is proposed as a model for rotationally forced Bose–Einstein condensate. We prove that the limiting singularity set is one-dimensional rectifiable. We also establish the convergence result for critical points away from limiting singularities. © 2011 Elsevier Ltd. All rights reserved.
MSC: primary 35J55 35Q40 Keywords: Gross–Pitaevskii energy Bose–Einstein condensate Rectifiable
1. Introduction In this paper, we are interested in characterizing the vortices of solutions of the following equation:
−∆uε + 2iωε (x⊥ · ∇)uε =
uε = 0,
on ∂ Ω
1
ε2
(a(x) − |uε |2 )uε ,
in Ω
(1.1)
which is the necessary condition for the uε to be the critical points of the Gross–Pitaevskii energy
∫ 1 2 ⊥ 2 2 Gε (u) = |∇ u| − 2ωε x · ⟨iu, ∇ u⟩ + 2 a(x) − |u| dx. (1.2) 2 Ω 2ε √ Here i is the imaginary unit −1. ωε means the rotational velocity, ωε = ω| ln ε| for a small parameter ε > 0 and some constant ω > 0. The trapping potential a(x) is a real function given by a(x) = a0 − x21 − α 2 x22 − β 2 x23 for certain positive constants α and β . It describes the geometry of the trap, which is a smooth bounded three-dimensional simply connected 2 domain denoted by Ω := {x ∈ R3 : a(x) > 0}. For x = (x1 , x2 , x3 ) ∈ R3 , we let x⊥ := (−x2 , x1 , 0), x⊥ α := (−α x2 , x1 , 0). 1 ,2 1 u is a complex-valued function in the space W0 (Ω ; C) = H0 (Ω ; C). The operators ∇ and ∆ are the usual gradient and Laplace operators acting on real functions, that is ∇ u = ∇ℜu + i∇ℑu, ∆u = ∆ℜu + i∆ℑu. By regarding C as a two-real√ dimensional vector space,we define the real inner product on it, ⟨u, v⟩ := (uv¯ + u¯ v)/2. And we define |u| := ⟨u, u⟩. ℜu ℜv Note that ⟨iu, v⟩ = det ℑu ℑv . We simply write j(u) = ⟨iu, ∇ u⟩ := (⟨iu, ux1 ⟩, ⟨iu, ux2 ⟩, ⟨iu, ux3 ⟩) ∈ R3 , where 1
∗
Corresponding author. E-mail addresses: [email protected] (L. Zhou), [email protected] (H. Xu), [email protected] (Z. Liu).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.04.010
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4275
u u uxj = ∂∂xu = ∂ℜ + i ∂ℑ . And we define J (u) := 12 ∇ × j(u) (where J (u) is the Hodge dual of the Jacobian Ju defined ∂ xj ∂ xj j in [1, Section 3.2]). ‘‘·’’ is the usual inner product on R3 . We also use ⟨·, ·⟩ to denote it. There is a total analogy between this model and the well-known Ginzburg–Landau model of superconductivity. The Ginzburg–Landau functional for superconductors is
G(u, A) =
∫
1 2
Ω
1
1
2
4ε 2
|∇ u − iAu|2 + |curl A − hex |2 +
(1 − |u|2 )2 ,
where hex is the intensity of the applied magnetic field, A = (A1 , A2 , A3 ) ∈ R3 is the vector potential of the magnetic field, and h = curl A is the induced magnetic field in the material. As we shall see, the role of the external field hex is taken by the rotational velocity ωε . The main tools for studying vortices in ‘‘Ginzburg–Landau type’’ problems have been developed in [2–5] etc. The main difference between the two problems is the boundary condition. Here u = 0 on ∂ Ω , whereas, for the Ginzburg–Landau case, all functions in H 1 were admissible, so no boundary conditions were imposed. Since u has to be small on a layer of size of the order of ε near ∂ Ω , the condition u = 0 induces a cost of Cε at least in the energy. This cost is very large compared to the Ginzburg–Landau energy. Hence, if we make comparisons with test maps, all the fine information on the behavior of u in Ω will be hidden by the energetic cost of the boundary layer. For each ε > 0, uε is supposed to be a solution of (1.1). In this paper, we study the asymptotic behavior of uε as ε → 0, and try to capture the codimension 2 singularities in a rigorous mathematical way. Before stating the main theorem, we give some notation first. Inspired by Lassoued and Mironescu [6] and Aftalion and Rivière [7], we define uε = fε eiωε S vε ,
(1.3)
where S=
α2 − 1 x1 x2 , α2 + 1
(1.4)
1 ,2
and fε ∈ W0 (Ω , R) is a real-valued minimizer of the energy Gε . The existence of such minimizers is standard, and the solution is unique by the result of Brezis and Oswald [8]. Using the Euler–Lagrange equation
−∆fε =
fε = 0,
1
ε2
(a − fε2 )fε ,
in Ω
(1.5)
on ∂ Ω
and the strong maximum principle, one can deduce that fε is positive in Ω . In this paper, we assume that the energy of vortices blows up like | ln ε|: Hε (vε ) :=
1
∫
2
Ω
2
2
fε |∇vε | +
fε4
2ε 2
(1 − |vε | )
2 2
⩽ C | ln ε|,
(1.6)
where C is a positive constant. Define the measures µε :
µε :=
eε (vε )
| ln ε|
dx,
(1.7)
and suppose that
σ⃗ε =
1
| ln ε|
2∇ fε fε
· ∇ vε , ∇vε
∇ fε fε2 (bε − |vε |2 )2 + , fε 2ε 2
(1.8)
where eε (vε ) =
1 2
|∇vε |2 +
fε2 (bε − |vε |2 )2 4ε
2
gε := (2x⊥ · ∇ S − |∇ S |2 )ωε2 =
,
bε = 1 −
ε2 fε2
gε ,
α2 − 1 2 (α + 3)x21 − (3α 2 + 1)x22 ωε2 . (α 2 + 1)2
(1.9)
(1.10)
In view of Hε (vε ) ⩽ C | ln ε| and Lemma 3.1(c), µε and σ⃗ε are bounded in K (K b Ω ). Therefore, up to a subsequence, we may assume that
µε ⇀ µ∗ ,
σ⃗ε ⇀ σ⃗∗ as measures.
(1.11)
4276
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
Recall that, for a Radon measure ν ∈ M (K ) and m > 0, the m-dimensional density of ν at x ∈ K is defined by
Θm (ν, x) := lim inf
ν(Br (x))
.
rm Suppose that Θ∗ (x) = Θ1 (µ∗ , x), and that r →0
Σµ∗ = {x ∈ K : Θ∗ (x) > 0}.
(1.12)
Our main theorem is: Theorem 1.1. If K is a compact subset in Ω and uε is a solution to (1.1) with the assumption (1.6), then the following properties hold. (i) (Limiting singularities). The set Σµ∗ is closed in K and 1-rectifiable. The varifold V := V (Σµ∗ , Θ1 ) satisfies the equation
⃗ ( x) = ⋆ H
2ω
α +1 2
x⊥ α ∧⋆
dJ∗ dµ∗
+
dσ⃗∗ dµ∗
for µ∗ -a.e. x ∈ Σµ∗ ,
(1.13)
⃗ (x) denotes the generalized mean curvature of V at x and is defined by where H ∫ K
⃗=− divΣµ∗ X
∫
⃗ · X⃗ H
⃗ ∈ C0∞ (K , R3 ), for all X
K
dσ⃗∗ dµ∗
is the Radon–Nikodym derivative of σ⃗∗ with respect to µ∗ . J∗ is the limit of J (vε ) (see Theorem 2 in [1]). (ii) (Measure decomposition). The measure µ∗ can be decomposed as
µ∗ = |∇ h∗ (x)|2 · H 3 + Θ∗ (x) · H 1 ⌊Σµ∗ ,
(1.14)
where h∗ satisfies
∆ h∗ +
∇a
· ∇ h∗ = 0 a (iii) (Convergence). We have
in K .
0 |uε |2 → a in Cloc (K \ Σµ∗ ) as ε → 0.
(1.15)
(1.16)
It is proved that |uε |2 converges uniformly to a(x) away from limiting singularities Σµ∗ (limiting vortex lines), which is one-dimensional rectifiable. Our proofs borrow many ideas from [3–5,9]. Our main difficulty is as regards how to prove the η-compactness theorem. To overcome the difficulty, we modify the proof of Theorem 1 in [3]. First, the Hodge–de Rham decomposition (4.17) of vε × dvε is different from that in [3]. Second, by improving Lemma III.1 in [3], we obtain Lemma 4.2 (estimate (4.5) is new). Using this, we obtain the complex estimate of the energy in Theorem 4.3 which is important in the proof of the main theorem. The paper is organized as follows. Section 1 is a general introduction of what we do and what result we get in this paper. Section 2 provides some background on Bose–Einstein condensates. In Section 3, the monotonicity formula is derived. The η-compactness theorem is shown in the next section. Then we prove the main result, Theorem 1.1, in Section 4. 2. Background The physical phenomenon called the Bose–Einstein condensate (we called it the new state of matter) is a macroscopic quantum effect which was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–1925. Bose assumed certain rules for photons which are now called Bose–Einstein statistics. Einstein guessed that the rules might apply to atoms. But actually not all atoms follow the rules of Bose–Einstein statistics. According to the theory of wave–particle duality of quantum theory, like light waves or water waves, microparticles of matter also show communication, interference and diffraction fluctuations in behavior, and thus form matter waves (known as the de Broglie waves). However, the wavelength of the atom is too short. For the usual state, the atomic fluctuations are not obvious. They can be observed only in a very cold state. At very low temperatures, most of the atoms are in the same quantum level. All the them are absolutely identical. That is, they are in the same place. This is just the Bose–Einstein condensate. Generally we must take the temperature of atoms cooled enough (near µK) to achieve atomic Bose–Einstein condensation in the laboratory. And the atom system is hoped to be almost in a ground state with no interaction. With the inventions of atomic laser cooling techniques and atomic trap techniques, these requirements are realized. The inventors, Steven Chu, Phillips and Cohen-Tannoudji were awarded the Nobel Prize in Physics in 1997. The first gaseous condensate was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder NIST–JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) (1.7 × 107 K). For their achievements, Cornell, Wieman, and Wolfgang Ketterle at MIT received the 2001 Nobel Prize in Physics.1 1 This paragraph is quoted from http://en.wikipedia.org/wiki/Bose-Einstein~condensate.
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4277
From then on, many properties of BEC have been studied, both experimentally and theoretically, with the development of new techniques and methods. The existence and structure of vortices are of particular interest. The group of Dalibard at the ENS in Paris [10,11] and the group of Ketterle at MIT [12,13] have produced vortices by rotation of the trapping potential. The condensate is rotated along the Z -axis at velocity ωε . When the velocity is sufficiently high, vortices appear. For a prolate trap, it has been observed [10,11] that when a single vortex exists, the vortex line is bending. The ENS group [14] also studies configurations with a single vortex line. In the mathematical model, a condensate is described by a wavefunction uε . After nondimensionalization, the wavefunction is expected to be close to a critical point of the Gross–Pitaevskii energy (1.2) (see [15,16] and their references) 1 ,2 in the space W0 (Ω ; C). Above a critical velocity, uε exhibits vortices, zeros of uε around which there is a circulation of phase. For a(x) ≡ 1 and for a disc in R2 , Serfaty [17] proves the existence of local minimizers having vortices for different ranges of rotational velocity. For a(x) = a0 − (x21 + λ2 x22 ), Ω = {x ∈ R2 : a(x) > 0}, Aftalion and Du [18] investigate an asymptotic development of the energy, the critical rotational velocities of nucleation of vortices and the location of vortices. In [19], Ignat and Millot investigate a model corresponding to the experiments on a two-dimensional rotating BEC. They estimate the critical rotational velocity for vortex existence and give some fundamental energy estimates. Aftalion et al. [20] analyze the global minimizers of the energy for an annular region in R2 . For Ω a solid torus of revolution in R3 with star-shaped cross-section, for ωε = O(| ln ε|), Alama et al. [21] prove that there exist a family of local minimizers with a vortex of the energy for ε sufficiently small. For a(x) = a0 − α 2 x21 − x22 − β 2 x23 , Aftalion and Jerrard [22] analyze the properties of the vortex line using the line energy, which is the simplified energy for a vortex line derived by Aftalion and Rivière in [7] from the Gross–Pitaevskii energy. The stability and instability of the straight vortex were studied. And they proved that when the condensate has a cigar shape, the first vortex is bent, while when it is a pancake, the first vortex is straight and lies on the axis of rotation. Jerrard [1] gives a rigorous derivation of the line energy derived in [7] and proves the existence of local minimizers with vortex filaments for the Gross–Pitaevskii functional. 3. The monotonicity formula The following estimates will be essential at several steps of our analysis. Lemma 3.1 ([19–21]). For ε sufficiently small, we have (a) Gε (fε√ ) ⩽ C | ln ε|, √ √ (b) 0 ⩽ √ a(x) − fε (x) ⩽ C ε 1/3 a(x), for x ∈ Ω with |x| < a0 − ε 1/3 , (c) ‖fε − a‖C 1 (K ) ⩽ CK ε 2 , for any compact subset in Ω . Using the techniques introduced in [6,19], we have the following energy decomposition result. Lemma 3.2 ([19]). Suppose that uε (1.4) respectively. Then we have
∈ W01,2 (Ω , C). Define vε via uε = fε eiωε S vε , where fε and S satisfy (1.5) and
Gε (uε ) = Gε (fε eiωε S ) + Hε (vε ) −
+
1 2
∫ Ω
2ωε
∫
α2 + 1
Ω
fε2 x⊥ α · ⟨ivε , ∇vε ⟩
fε2 (|vε |2 − 1)(ωε2 |∇ S |2 − 2ωε2 x⊥ · ∇ S ),
(3.1)
where Hε (vε ) is defined in (1.6). Since uε = 0 on ∂ Ω , we restrict our analysis to the interior of Ω . Let K be a compact subset in Ω . Then vε satisfies the associated Euler–Lagrange equation
−∆ v ε +
2iωε
α2 + 1
(x⊥ α · ∇)vε +
iωε
vε
2
α 2 + 1 fε
fε
2 (x⊥ α · ∇)fε − 2
∇ fε · ∇vε =
fε2
ε2
(1 − |vε |2 )vε − gε vε in K ,
(3.2)
2 where gε isdefined in (1.10). Obviously, gε = O(| ln ε| ) since α is a constant and x1 , x2 are bounded. We also observe that
fε2 /ε 2 = O ε12
by Lemma 3.1(c). Then limε→0
gε fε2 /ε 2
= 0. So the term gε vε is infinitesimal compared with fε2 vε /ε2 as ε is
2 small enough. Writing bε := 1 − ε2 gε , (3.2) can be rewritten as
fε
−∆ v ε +
2iωε
(x⊥ α · ∇)vε +
iωε
vε
2 (x⊥ α · ∇)fε − 2
α2 + 1 α 2 + 1 fε For x0 ∈ K and Br (x0 ) = B(x0 , r ) ⊂ K , define ∫ 1 1 Eε (vε , x0 , r ) := Eε (vε , x0 , r ) ≡ eε (vε ), r
r
2 fε
∇ fε · ∇vε =
fε2
ε2
(bε − |vε |2 )vε in K .
Br (x0 )
where eε (vε ) is given in (1.9). Sometimes, we simply write Eε (x0 , r ) or Eε (r ) instead of Eε (vε , x0 , r ).
(3.3)
(3.4)
4278
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
We now begin to prove the monotonicity formula for the energy Eε in K . First, by the maximum principles, we have the following lemma. Lemma 3.3 ([5]). Let vε be a solution to (3.3). Then
‖vε ‖L∞ (K ) ⩽ 1 + C ε 2 | ln ε|2 + C
ε2 , dist(K , ∂ Ω )2
‖∇vε ‖L∞ (K ) ⩽
CK
ε
.
(3.5)
A computation directly gives the following Pohozaev identity. Lemma 3.4 (Pohozaev Identity). Let vε be a solution to (3.3). Then, for Br (x0 ) ⊂ K , we have 1 2
∫
fε2
∫
3
(bε − |vε |2 )2 ε2 ∫ 2 ∫ |∇T vε |2 2ωε 1 ∂vε fε2 2 2 =r − J (vε ), x⊥ α × ( x − x0 ) + 4ε 2 (bε − |vε | ) + 2 +1 2 2 ∂ n α ∂ Br (x0 ) Br (x0 ) ∫ 2 ∫ vε fε ωε 1 2 i 2 x⊥ (bε − |v|2 )(x − x0 ) · ∇ bε + − α · ∇ fε , (x − x0 ) · ∇v 2 fε 2 Br ε 2 Br α + 1 ∫ ∫ 2 1 − ⟨∇ fε · ∇vε , (x − x0 ) · ∇vε ⟩ − 2 fε (bε − |vε |2 )2 (x − x0 ) · ∇ fε , f 2 ε Br (x0 ) ε Br (x0 ) Br (x0 )
where J (vε ) =
|∇vε |2 +
1 2
4
B(x0 ,r )
(3.6)
∇ × j(vε ) = 12 ∇ × ⟨ivε , ∇vε ⟩.
Lemma 3.5. Assume that vε satisfies (3.3) and BR (x0 ) is a subset of K ; then, for 0 < r < R, d dr
1 ( Eε (r )) =
∫
r
∂ Br (x0 )
∫ ∂vε 2 + 1 ∂n r2
fε2 (bε − |vε |2 )2
Br (x0 )
2ε 2
−
2
∫
r2
Br (x0 )
ωε J (vε ), x⊥ α × (x − x0 ) α2 + 1
∫ (bε − |vε |2 )2 2 1 + 2 fε ( x − x ) · ∇ f + ⟨∇ fε · ∇vε , (x − x0 ) · ∇vε ⟩ 0 ε 2 2 2r ε r Br (x0 ) Br (x0 ) fε ∫ ∫ 2 ωε vε ⊥ fε 1 1 2 i 2 xα · ∇ fε , (x − x0 ) · ∇v + 2 (bε − |v|2 )(x − x0 ) · ∇ bε . − 2 2 +1 2 r α f 2r ε Br Br ε ∫
1
(3.7)
Proof. Note that d dr
(Eε (r )) =
∫ ∂ Br (x0 )
fε2 2 2 + |∇T vε | + (bε − |vε | ) . 2 2 ∂n 4ε 2
1
2
2 1 ∂vε
Hence, d dr
( Eε (r )) = − +
1 r2 1 r
∫ Br (x0 )
∫ ∂ Br (x0 )
1 2
2
|∇vε | +
3fε2
4ε
(bε − |vε | ) 2
2 2
+
1 2ε 2 r 2
∫ Br (x0 )
fε2 (bε − |vε |2 )2
fε2 2 2 + |∇T vε | + (bε − |vε | ) . 2 2 ∂n 4ε 2 1
2
2 1 ∂vε
Combining this relation with Lemma 3.4, the conclusion follows.
Lemma 3.6. There exists C > 0, depending only on K , such that for BR (x0 ) ⊂ K ,
Λ = C (ωε + 1),
(3.8)
and for any vε satisfying (3.3), we have d dr
(eΛr Eε (r )) ⩾
∫ ∂vε 2 fε2 (bε − |vε |2 )2 + 1 ⩾ 0, 2 r ∂ Br (x0 ) ∂ n r 2ε 2 Br (x0 )
1
∫
for 0 < r < R. In particular, eΛr Eε (r ) is increasing in r.
(3.9)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4279
Proof. From Lemma 3.5, we need to estimate the last five terms in the r.h.s. of (3.7). Note that J (vε ) ⩽ C |∇vε |2 , |∇ bε | ⩽ C ε 2 | ln ε|2 ; hence
∫ ∫ 2 ωε 1 ⊥ J (v ), x · ( x − x ) |∇vε |2 ⩽ C ωε Eε (r ), ⩽ C ω ε 0 ε α r2 2 r Br (x0 ) Br (x0 ) α + 1 ∫ ∫ 1 C (bε − |vε |2 )2 fε2 (bε − |vε |2 )2 f ( x − x ) · ∇ f ⩽ C Eε (r ), ⩽ ε 0 ε r2 2 ε r Br (x0 ) ε2 Br (x0 ) ∫ ∫ 2 C 1 ⟨∇ f · ∇v , ( x − x ) · ∇v ⟩ ⩽ |∇vε |2 ⩽ C Eε (r ), ε ε 0 ε r2 f r ε Br (x0 ) Br (x0 ) ∫ 2 ∫ 1 C fε fε2 (bε − |vε |2 )2 2 ( b − |v| )( x − x ) · ∇ b ⩽ C Eε (r ). ⩽ ε 0 ε 2r 2 ε2 r ε2 Br (x0 )
Br
Since
x⊥ α
· ∇ a = 0, in view of Lemma 3.1(c) we have ∫ ∫ 1 ωε vε ⊥ 1 ωε vε ⊥ 2 2 i xα · ∇ fε , (x − x0 ) · ∇v = 2 i 2 xα · ∇(fε − a), (x − x0 ) · ∇v 2 r 2 Br α 2 + 1 fε2 r fε Br α + 1 ⩽ C Eε (r ).
Using Lemma 3.5, we have d dr
(eΛr Eε (r ))
=e
Λr
d Λ Eε (r ) + ( Eε (r ))
dr
∫ ∂vε 2 1 fε2 (bε − |vε |2 )2 + 1 ⩾ ⩾ 0. r ∂ Br (x0 ) ∂ n r 2 Br (x0 ) 2ε 2 ∫
This implies the conclusion. The proof of Lemma 3.6 is completed.
In order to establish the monotonicity formula, refined estimates of Jacobian integrals are needed. They were conjectured by Bourgain et al. [23], and were proved by Bethuel et al. [24]. 1 Lemma 3.7 ([23,24]). Suppose that w ∈ Hloc (Ω , C), ϕ ∈ C0∞ (Ω , ∧1 R3 ) and κ := supp ϕ . Moreover, assume that there exists q > 6 such that w satisfies
‖w‖Lq (Ω ) ⩽ Cq ; then,
∫ ∫ α1 ∫ 1 α0 ⟨J w, ϕ⟩ ⩽ C eε (w) + ε eε (w) ‖ϕ‖W 1,3 (Ω ) , | ln ε| κ
Ω
(3.10)
κ
for some constants 0 < α0 < 1, 0 < α1 < 1, and C > 0 which are dependent only on Ω , q and Cq . Now we prove the following monotonicity formula. Proposition 3.8 (Monotonicity Formula). There exist C > 0, β > 0, which are independent of ε , such that for BR (x0 ) ⊂ K b Ω and for vε satisfying (3.3), we have
Eε (vε , x0 , θ r ) ⩽ C Eε (vε , x0 , r ) + ε β , for 0 < θ <
1 2
(3.11)
and 0 < r < R.
Proof. We split the proof into three steps and drop the subscript ε for simplicity. Step 1. Define a cut-off function f :
1, a f ( r , a) = 2 − , r 0,
if a ⩽ r if r < a < 2r
(3.12)
if a ⩾ 2r
where r > 0. Set E ε (x0 , r ) :=
1 r
∫ B2r (x0 )
eε (v)f (r , |x − x0 |)dx.
(3.13)
4280
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
Then for 0 < r < d dr
R , 2
(E ε (x0 , r )) =
2 ∫ ∂v fε2 (bε − |v|2 )2 dt + 1 f (r , |x − x0 |) r 1 r 2 B2r (x0 ) 2ε 2 ∂ Btr (x0 ) ∂ n ∫ ∫ 2 ωε 1 (bε − |v|2 )2 ⊥ − 2 J (v), x × ( x − x ) f ( r , | x − x |) + fε 0 0 α 2 2 r 2r ε2 B2r (x0 ) α + 1 B2r (x0 ) ∫ 1 2 ⟨∇ fε · ∇v, (x − x0 ) · ∇v f (r , |x − x0 |)⟩ × (x − x0 )∇ fε f (r , |x − x0 |) + 2 2
∫
1
∫
t
r
− +
∫
1 r2
Br
1
∫
2r 2
B (x ) fε
2r 0 ωε vε ⊥ 2 x · ∇ f , ( x − x ) · ∇v f ( r , | x − x |) i 0 0 ε α 2 + 1 fε2 α
fε2
Br
ε2
(bε − |v|2 )(x − x0 ) · ∇ bε f (r , |x − x0 |).
Step 2. There exists a constant C > 0 such that for 0 < r <
E ε (x0 , θ r ) ⩽ C e
Cr
(3.14)
R , 2
εα0 | ln ε|α1 +1 E ε (x0 , r ) + , θr
(3.15)
for 0 < θ ⩽ 1. In fact, we need to estimate the last three terms in the r.h.s. of (3.14). The last four terms are treated as before:
∫ 1 (bε − |v|2 )2 fε (x − x0 ) · ∇ fε f (r , |x − x0 |) ⩽ C E ε (x0 , 2r ), 2r 2 2 ε B2r (x0 ) ∫ 1 2 ⩽ C E ε (x0 , 2r ), ⟨∇ f · ∇v, ( x − x ) · ∇v f ( r , | x − x |)⟩ ε 0 0 r2 B2r (x0 ) fε ∫ 1 ωε vε ⊥ 2 ⩽ C E ε (x0 , 2r ), i x · ∇ f , ( x − x ) · ∇v f ( r , | x − x |) 0 0 α ε r2 2 fε2 Br α + 1 ∫ 2 1 fε 2 ⩽ C E ε (x0 , 2r ). ( b − |v| )( x − x ) · ∇ b f ( r , | x − x |) ε 0 ε 0 2r 2 2 Br ε
(3.16)
(3.17)
(3.18)
(3.19)
For the first term, note that
‖ωε x⊥ α × (x − x0 )f (r , |x − x0 |)‖W 1,3 (B2r (x0 )) ⩽ Cr | ln ε|.
(3.20)
Hence, using Lemma 3.7, we have
∫ 2 r2
B2r (x0 )
∫ α 1 e (v) ωε C B2r (x0 ) ε ⊥ α0 J (v), xα × (x − x0 )f (r , |x − x0 |) ⩽ +ε eε (v) | ln ε| α2 + 1 r | ln ε| B2r (x0 ) ⩽ C E ε (x0 , 2r ) +
1 α0 ε | ln ε|α1 +1 . r
(3.21)
Combining (3.16)–(3.19), (3.21) with (3.14), we obtain d dr
(E ε (x0 , r )) ⩾ −C E ε (x0 , r ) −
C α0 ε | ln ε|α1 +1 . r
(3.22)
Inequality (3.15) then follows from a version of Gronwall’s lemma given in [5, Lemma A.7]. Step 3. First we consider the case
θ r < ρ :=
1
Λ
=
1 C (ωε + 1)
<
r 2
.
(3.23)
Using Lemma 3.6, we have
Eε (θ r ) ⩽ C Eε (ρ).
(3.24)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4281
Next, using (3.15) and the definition of ρ , we have
Eε (ρ) ⩽ E ε (ρ) ⩽ C E ε
ε α0 | ln ε|α1 +1 + 2 ρ
r
⩽C Eε (r ) + ε α0 | ln ε|α1 +2 .
(3.25)
α
Suppose that β = 40 ; combining (3.24) and (3.25), we get the desired formula (3.11). If θ r ⩾ ρ , (3.11) is inferred directly from (3.15). This completes the proof. 4. The η-compactness theorem In this section, we will show the η-compactness theorem by using the method of [3,4] with some modification. In this theorem, |vε | is bounded away from zero if the local energy is bounded by η| ln ε| with η small. Here, the Hodge–de Rham decomposition of vε × ∇vε is different from that in [3]. Theorem 4.1 (η-Compactness Theorem). Assume that vε satisfies (3.3) and σ > 0 is given. There exist η > 0 and ε0 > 0 depending only on σ such that if B2R (x0 ) ⊂ K , R ⩾ ε
α0 4
, ε ⩽ ε0 and
Eε (x0 , R) ⩽ η| ln ε|,
(4.1)
|vε (x0 )| ⩾ 1 − σ .
(4.2)
then
Proof. We follow closely the lines of [3]. Let 0 < δ < 1/400 be a constant to be determined later. In the sequel, we will use C to denote generic constants independent of the choice of δ . The proof of Theorem 4.1 is divided into the following three steps. Step 1. Choose ‘‘good’’ radii. The same argument as in [3] gives the following lemma. Lemma 4.2. Assume that 0 < ε
α0 4
< δ , and that there exist a constant C > 0 and radii r0 , rc ∈
δ r0 < rc < 2δ r0 such that ∫ 1 fε2 (bε − |vε |2 )2 ⩽ C (η| ln δ| + ε β ), r0 Br (x0 ) ε2 0
α0 ε 4 R, 41 R with
(4.3)
Eε (x0 , r0 ) − 2 Eε (x0 , δ r0 ) ⩽ C (η| ln δ| + ε β ), ∫ 2 ∫ ∂vε 2 ⩽ C (η| ln δ| + ε β ). t 1 ∂ Btrc (x0 ) ∂ n
(4.4) (4.5)
Step 2. δ -energy decay. Let 0 < γ < 1/8 be a constant to be determined later. Theorem 4.3. There exist εN > 0 and C > 0 such that for any 0 < ε < εN , we have
β
Eε (x0 , δ r0 ) ⩽ C δ r0 (η| ln δ| + ε ) + (γ + δ )Eε (x0 , r0 ) + (γ 2
3
Proof. By the mean-value inequality, there exists some r1 ∈
∫ ∂ Br1 (x0 )
∫ ∂ Br1 (x0 )
|∇vε |2 ⩽ C
∫ Br0 (x0 )
1 r 64 0
−2
+
γ −4 Eε (x0 , r0 ))
fε2 (bε − |vε |2 )2
∫ Br0 (x0 )
∫ Br0 (x0 )
.
(4.6)
1 , 32 r0 such that
|∇vε |2 ,
fε2 (bε − |vε |2 )2 ⩽ C
ε2
(4.7) fε2 (bε − |vε |2 )2 .
(4.8)
Let 1r1 denote the characteristic function of the set Br1 (x0 ). Consider a 2-form ψ in R3 defined by
[ ψ = −G ⋆ d 1r1 vε × dvε − 1r1
] ωε 2 3 ( 1 − |v | ) Σ c ( x ) dx , ε i i i=1 α2 + 1
in R3
(4.9)
4282
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
2 where c (x) = x⊥ α = (−α x2 , x1 , 0), G(x) = −C3 /|x| denotes the Green’s function of the Laplace operator in dimension 3. Note that
[ ] ωε −∆ψ = d 1 v × dv − 1 2 3 (1 − |vε | )Σi=1 ci (x)dxi , ε r1 2 r1 ε α +1 |ψ(x)| → 0 as |x| → ∞.
in R3
(4.10)
Since −∆ = dd∗ + d∗ d, it follows that
[
d vε × dvε −
] ωε 2 3 ∗ ( 1 − |v | ) Σ c ( x ) dx − d ψ = d∗ dψ , ζ in Br1 (x0 ). ε i i i =1 α2 + 1
(4.11)
We observe that
∆(dψ) = 0 in Br1 (x0 ).
(4.12)
Indeed, we have
[ ∆(dψ) = d(∆ψ) = −d2 vε × dvε −
] ωε 2 3 ( 1 − |v | ) Σ c ( x ) dx = 0 in Br1 (x0 ). ε i i i=1 α2 + 1
It follows that the 2-form ζ = d∗ dψ is closed, since dζ = d(d∗ dψ) = dd∗ (dψ) = −∆(dψ) − d∗ d(dψ) = 0.
(4.13)
By the Poincaré Lemma, there exists a 1-form ξ defined on B 1 r (x0 ), such that 2 1
dξ = ζ ,
in B 1 r (x0 )
2 1
(4.14)
in B 1 r (x0 )
d ξ = 0, ∗
2 1
and
‖ξ ‖L2 (B4δr
0
(x0 )) ⩽
C ‖ζ ‖L2 (B5δr (x0 )) . 0
(4.15)
Going back to (4.11), we may write
] ωε 2 3 ∗ d vε × dvε − 2 (1 − |vε | )Σi=1 ci (x)dxi − d ψ − ξ = 0 in B 1 r1 (x0 ). 2 α +1 [
(4.16)
Using again the Poincaré lemma, we deduce that there exists some function ϕ uniquely defined in B 1 r (x0 ) (up to a constant) 2 1
such that
vε × dvε =
ωε (1 − |vε |2 )Σi3=1 ci (x)dxi + d∗ ψ + dϕ + ξ in B 1 r1 (x0 ). 2 α +1 2
(4.17)
This is precisely the Hodge–de Rham decomposition of vε × dvε which best fits our needs. We are going to estimate the L2 -norm of each of the four terms on the r.h.s. of (4.17). The estimate for ξ . Since dψ is harmonic in Br1 (x0 ) by (4.12), we have for any k ∈ N,
‖dψ‖
C k B 11
δr 2 0
⩽ (x0 )
C ‖dψ‖L2 (B6δr (x0 )) ⩽ C ‖∇ψ‖L2 (B6δr (x0 )) . 0 0
(4.18)
On the other hand, since ζ = d∗ dψ , it follows that
‖ζ ‖L2 (B5δr
0
(x0 )) ⩽
C ‖∇ψ‖L2 (B6δr (x0 )) , 0
(4.19)
and going back to (4.15) we obtain the estimate
‖ξ ‖L2 (B4δr
0
(x0 )) ⩽
C ‖∇ψ‖L2 (B6δr (x0 )) . 0
(4.20)
The estimate for ϕ . Since d∗ ξ = 0, due to the Hodge–de Rham decomposition (4.17), we have d∗
vε × dvε −
ωε (1 − |vε |2 )Σ ci dxi 2 α +1
= −∆ϕ in B r1 (x0 ). 2
(4.21)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4283
On the other hand, vε satisfies (3.3); we have d
∗
ωε (1 − |vε |2 )Σ ci dxi vε × dvε − 2 α +1
2 ωε 2 = − ∇ fε · (vε × ∇vε ) + 2 |vε |2 x⊥ α · ∇ fε fε α +1 2 ωε = − ∇ fε · dϕ + d∗ ψ + ξ + 2 (1 − |vε |2 )Σ ci dxi fε α +1 ωε 2 + 2 |vε |2 x⊥ α · ∇ fε . α +1
(4.22)
Thus we obtain the equation for ϕ : 2
−∆ ϕ +
fε
2
∇ fε · ∇ϕ = − ∇ fε · (d∗ ψ + ξ + ωε (1 − |vε |2 )Σ ci dxi ) + fε
In view of −∆ϕ +
−div
2 fε
∇ fε · ∇ϕ = −fε2 div
1
=−
∇ϕ 2
fε
2 fε3
1 fε2
ωε 2 |vε |2 x⊥ α · ∇ fε . α2 + 1
(4.23)
∇ϕ , we rewrite (4.23) as
∇ fε · (d∗ ψ + ξ + ωε (1 − |vε |2 )Σ ci dxi ) +
ωε 2 |vε |2 x⊥ α · ∇ fε . α2 + 1
(4.24)
For simplicity, define L , −∆ +
2 fε
∇ fε · ∇.
(4.25)
Now we consider the boundary value problem for some r > 0:
Lu = h, in Br (x0 ) ∂u = p, on ∂ Br (x0 ) ∂n ∫ u = 0.
(4.26)
∂ Br (x0 )
We have the following estimate. Proposition 4.4. Assume that 0 < r < 1 and u satisfies (4.26); then there exists some constant C such that
∫
∫
2
Br (x0 )
|∇ u| ⩽ C
∫
2
h +r
p
Br (x0 )
.
2
∂ Br (x0 )
(4.27)
Proof. Multiplying the first equation of (4.26) by (x − x0 ) · ∇v and integrating by parts, we easily get Pohozaev’s identity for the operator L: 1
∫
2
|∇ u|2 +
∫
Br (x0 )
Br (x0 )
(x − x0 ) · ∇ uh =
r
∫
2 ∂ Br (x0 )
∫ +
|∇ u|2 − r 2
Br (x0 ) fε
∫
2 ∂u ∂ Br (x0 ) ∂ n
⟨∇ fε · ∇ u, (x − x0 ) · ∇ u⟩.
(4.28)
Note that
∫ Br (x0 )
(x − x0 ) · ∇ uh −
∫ ⩽
|∇ u|2
r
2 Br (x0 ) fε
⟨∇ fε · ∇ u, (x − x0 ) · ∇ u⟩
12 ∫
21 h2
r
Br (x0 )
Br (x0 )
∫ ⩽ Cr
∫
2
|∇ u|2
Br (x0 )
∫
|∇ u| + Br (x0 )
∫ + Cr
h Br (x0 )
2
.
Combining this with (4.28) we obtain
∫
2 ∫ ∫ ∂u 1 2 |∇T u| ⩽ + + C |∇ u | + C h2 . r ∂ Br (x0 ) ∂ Br (x0 ) ∂ n Br (x0 ) Br (x0 ) 2
∫
(4.29)
4284
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
On the other hand, standard elliptic estimates [25] yield
∫
∫
2
u ⩽C r
Br (x0 )
4
2
h +r Br (x0 )
3
∫ p
.
2
∂ Br (x0 )
(4.30)
We rewrite (4.26) as
−div
1 fε2
∇u =
1 fε2
h.
(4.31)
Multiplying (4.31) by u and integrating by parts in Br (x0 ) we obtain
∫
∫
2
|∇ u| ⩽ C
Br (x0 )
h
2
12 ∫
12 u
Br (x0 )
2
∫ +
Br (x0 )
p
2
21 ∫
∂ Br (x0 )
21 u
2
∂ Br (x0 )
.
(4.32)
In view of having ∂ B (x ) u = 0, by the Poincaré–Wirtinger inequality we have r 0
∫
2
∂ Br (x0 )
u ⩽
r2 2
∫ Br (x0 )
|∇T u|2 .
Inserting this inequality in (4.29), we thus have
∫ ∂ Br (x0 )
u2 ⩽
r2 2
∫ ∂ Br (x0 )
p2 + C ( r + r 2 )
∫
|∇ u|2 + Cr 2 Br (x0 )
∫ Br (x0 )
h2 .
Inserting (4.30) and (4.33) in (4.32), we complete the proof of Proposition 4.4. Recall that for 0 < r <
(4.33)
, ϕ satisfies Lϕ = h, in Br (x0 ) ∂ϕ = p, on ∂ Br (x0 ) ∂n 1 r 2 1
(4.34)
where
ωε ωε 2 ∗ 2 2 (1 − |vε | )Σ ci dxi + 2 |vε |2 x⊥ h = − ∇ f ε · d ψ + ξ + 2 α · ∇ fε , fε α +1 α +1 ωε ∂v 2 ∗ p = vε × ε − ( 1 − |v | ) Σ c dx + d ψ + ξ · n. ε i i ∂n α2 + 1
(4.35)
Note that (4.34) involves only the gradient of ϕ ; we may assume that
∫ ∂ Br (x0 )
ϕ = 0.
(4.36)
In Lemma 4.2 we obtain rc ∈ ε 2
∫
∫ t
1
∂ Btrc (x0 )
α0 4
R, 41 R with δ r0 < rc < 2δ r0 and
∂vε 2 β ∂ n ⩽ C (η| ln δ| + ε ),
and thus there exists r2 ∈ (rc , 2rc ) ⊂ (δ r0 , 4δ r0 ) such that
∂vε 2 β ∂ n ⩽ C (η| ln δ| + ε ), ∂ Br2 (x0 ) 2 ∫ ∫ 2 2 ωε C ε 2 ωε2 2 2 (1 − |vε | ) ( 1 − |v | ) Σ c dx ⩽ f ε i i ε 2 rc ε2 ∂ Br2 (x0 ) α + 1 B2rc (x0 )\Brc (x0 ) β ⩽ C (η| ln δ| + ε ), ∫ ∫ ∫ C C |∇ψ|2 ⩽ |∇ψ|2 ⩽ |∇ψ|2 , r δ r c 0 ∂ Br2 (x0 ) B2rc (x0 )\Brc (x0 ) B4δ r (x0 ) ∫
0
(4.37)
(4.38) (4.39)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4285
and
∫
C
|ξ |2 ⩽
∂ Br2 (x0 )
∫
rc
B2rc (x0 )\Brc (x0 )
∫
C
|ξ |2 ⩽
δ r0
B4δ r (x0 )
|ξ |2 .
(4.40)
0
Choosing r = r2 in (4.34) and (4.36), we obtain
∫ r2
∂ Br2 (x0 )
p2 ⩽ C δ r0 (η| ln δ| + ε β ) + C
∫
|∇ψ|2 +
∫
B4δ r (x0 )
B4δ r (x0 )
0
|ξ |2 .
(4.41)
0
By the definition of h, we have
∫
∫
2
Br2 (x0 )
∫
2
|ξ | +
|∇ψ| +
h ⩽ C
B4δ r (x0 )
B4δ r (x0 )
B4δ r (x0 )
0
0
0
∫
∫
2
B4δ r (x0 )
|ξ | + B4δ r (x0 )
0
(1 − |vε |2 )2 + ε2
fε2 (bε − |vε |2 )2
∫
2
|∇ψ| +
⩽ C
∫
2
ε2
Br0 (x0 )
0
∫
ωε |∇(fε − a)| 2
B4δ r (x0 )
2
2
0
+ε
β
.
(4.42)
Using Proposition 4.4, and (4.41) and (4.42), we have
∫
β
|∇ϕ| ⩽ C δ r0 (η| ln δ| + ε ) + C 2
Br2 (x0 )
∫
∫
2
|∇ψ| +
fε2 (bε − |vε |2 )2
∫
2
|ξ | +
B4δ r (x0 )
B4δ r (x0 )
0
Br0 (x0 )
0
ε2
,
which yields
∫
β
|∇ϕ| ⩽ C δ r0 (η| ln δ| + ε ) + C 2
Bδ r (x0 ) 0
∫
2
∫
|∇ψ| + B4δ r (x0 )
|ξ | + B4δ r (x0 )
0
fε2 (bε − |vε |2 )2
∫
2
ε2
Br0 (x0 )
0
.
(4.43)
The estimate for ψ . As in [3], we easily get the L2 -norm estimate for ∇ψ :
∫
|∇ψ| ⩽ C (γ + δ ) 2
B6δ r (x0 )
2
3
0
∫
|∇vε | + C (γ 2
Br0 (x0 )
−4 β
ε +
γ −4 Eε (x0 , r0 ))
fε2 (1 − |vε |2 )2
∫ Br0 (x0 )
ε2
.
(4.44)
Estimates for ∇(|vε |2 ) and (1 − |vε |2 )|∇vε |2 . Note that
∆(|vε |2 ) = 2|∇vε |2 −
2
f 2 (bε − |vε |2 )|vε |2 − 2 ε
ε ωε ⊥ −4 2 x · j(vε ). α +1 α
2 fε
∇ fε · ∇(|vε |2 ) + 2
ωε |vε |2 (x⊥ · ∇)fε2 (α 2 + 1)fε2 α (4.45)
Multiplying (4.45) by (bε − |vε |2 ) and integrating in Br1 (x0 ), we get
∫
[ ] 2fε2 2 2 2 2 2 |∇|vε | | + 2 (bε − |vε | ) |vε | ε Br1 (x0 ) ∫ ∫ ∫ ∂|vε |2 ωε =2 (bε − |vε |2 )|∇vε |2 + (bε − |vε |2 ) −4 2 x⊥ · j(vε )(bε − |vε |2 ) ∂n α + 1 Br1 (x0 ) α Br1 (x0 ) ∂ Br1 (x0 ) ∫ ∫ 1 −2 ∇ fε · ∇(|vε |2 )(bε − |vε |2 ) + ∇|vε |2 · ∇ bε Br1 (x0 ) fε
+
2ωε
α +1 2
Br1 (x0 )
|vε |2
∫ Br1 (x0 )
fε2
2 (bε − |vε |2 )(x⊥ α · ∇)fε .
(4.46)
Note that
∫ 12 ∫ 21 ∂|vε |2 fε2 (bε − |vε |2 )2 2 (bε − |vε | ) |∇vε | , ⩽ Cε ∂n ε2 (x ) Br0 (x0 ) Br0 (x0 ) 1 0
∫ ∂ Br
2
(4.47)
4286
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
∫ Br
∫ ∫ fε2 (bε − |vε |2 )2 2 2 (bε − |vε | )|∇vε | ⩽ C γ 2 |∇vε |2 + C γ −2 ε2 (x ) Vγ Wγ 1 0 ∫ ∫ fε2 (bε − |vε |2 )2 ⩽ Cγ 2 |∇vε |2 + C γ −2 , ε2 Br0 (x0 ) Br0 (x0 )
12 ∫ 21 ∫ 2 2 2 f ( b − |v | ) ε ε ε (bε − |vε |2 )∇ fε · ∇(|vε |2 ) ⩽ C ε |∇|vε |2 |2 ε2 Br1 (x0 ) Br1 (x0 ) (x ) fε 1 0 ∫ ∫ 2 1 fε (bε − |vε |2 )2 ⩽ , |∇|vε |2 |2 + C 2 Br (x0 ) ε2 Br0 (x0 ) 1 ∫ 1∫ 2 |∇|vε |2 |2 + C ε β , ∇|vε | · ∇ bε ⩽ Br (x0 ) 4 Br (x0 ) 1 1 ∫ 2ω ∫ 2 fε2 (bε − |vε |2 )2 |vε | ε 2 ⊥ 2 )( x · ∇) f ( b − |v | ⩽C + C εβ , 2 ε ε α ε α + 1 Br (x0 ) fε2 ε2 Br0 (x0 ) 1 ∫ 2 Br
(4.48)
1
12 ∫ 21 ∫ 2 2 2 f ( b − |v | ) ε ε ε 2 2 x⊥ |∇vε |2 α · j(vε )(bε − |vε | ) ⩽ C ε ωε ε2 (x ) Br1 (x0 ) Br1 (x0 ) 1 0 ∫ ∫ 2 fε (bε − |vε |2 )2 . ⩽ Cε |∇vε |2 + C ε2 Br0 (x0 ) Br0 (x0 )
(4.49)
(4.50)
(4.51)
∫ 4ωε Br
(4.52)
Combining (4.46)–(4.52) we have
∫ Br1 (x0 )
|∇|vε |2 |2 ⩽ C (ε + γ 2 )
∫ Br0 (x0 )
|∇vε |2 + C γ −2
fε2 (bε − |vε |2 )2
∫ Br0 (x0 )
ε2
.
(4.53)
Recalling the identity 4|vε |2 |∇vε |2 = 4|vε × ∇vε |2 + |∇|vε |2 |2 , by (4.17), we thus have 4|∇vε |2 = 4|vε × ∇vε |2 + 4(1 − |vε |2 )|∇vε |2 + |∇|vε |2 |2 ⩽ C (|∇ψ|2 + |∇ϕ|2 + |ξ |2 + ωε2 (1 − |vε |2 )2 |Σ ci dxi |2 ) + 4(1 − |vε |2 )|∇vε |2 + |∇|vε |2 |2 . Note that
∫
ωε (1 − |vε | )|Σ ci dxi | ⩽ C ε 2
Bδ r (x0 ) 0
2
2
fε2 (1 − |vε |2 )2
∫ Br0 (x0 )
ε2
.
Combining (4.20), (4.43), (4.44), (4.48), (4.53) and (4.54), we have
∫ Bδ r (x0 ) 0
|∇vε |2 ⩽ C δ r0 (η| ln δ| + ε β ) + C (γ 2 + δ 3 ) fε2 (bε − |vε |2 )2
∫ × Br0 (x0 )
ε2
which completes the proof of Theorem 4.3.
∫ Br0 (x0 )
|∇vε |2 + C (γ −2 + γ −4 Eε (x0 , r0 ))
,
Step 3. Completing the proof of Theorem 4.1. Using (4.6), we have
γ2 2 + δ Eε (x0 , r0 ) δ ∫ 1 fε2 (bε − |vε |2 )2 + C (γ −2 + γ −4 Eε (x0 , r0 )) . δ r0 Br0 (x0 ) ε2
Eε (x0 , δ r0 ) ⩽ C (η| ln δ| + ε β ) + C
(4.54)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4287
By (4.3) and (4.4), we have
Eε (x0 , r0 ) ⩽ 2 Eε (x0 , δ r0 ) + C (η| ln δ| + ε β ) [ 2 ] γ −2 γ γ −4 ⩽ C (η| ln δ| + ε β ) + C (η| ln δ| + ε β ) + C + δ2 + (η| ln δ| + ε β ) Eε (x0 , r0 ) δ δ δ [ 2 ] γ −2 γ γ −4 ⩽ C 1+ (η| ln δ| + ε β ) + C + δ2 + (η| ln δ| + ε β ) Eε (x0 , r0 ). δ δ δ Now, choose δ such that C δ2 ⩽
1 4
,
and then choose γ such that C
γ2 1 ⩽ . δ 4
Hence, there exist ε0 and η0 such that if ε ⩽ ε0 and η ⩽ η0 , we have C
γ −4 1 (η| ln δ| + ε β ) ⩽ . δ 4
Thus
Eε (x0 , r0 ) ⩽ C (η| ln δ| + ε β ). By the monotonicity formula, we obtain 1
ε3
∫ Bε (x0 )
(1 − |vε |2 )2 ⩽ C Eε (x0 , ε) ⩽ C ( Eε (x0 , r0 ) + ε β ) ⩽ C (η| ln δ| + ε β ).
Using Lemma III.3 in [3], we have
1 − |vε (x0 )| ⩽ C
1
51
∫
ε3
1
⩽ C (η| ln δ| + ε β ) 5 ,
(1 − |vε | )
2 2
Bε (x0 )
which completes the proof of Theorem 4.1.
Corollary 4.5. Under the assumption of Theorem 4.1, suppose that 0 < σ < 1, η > 0 and ε0 > 0 are as given in Theorem 4.1. Supposing that B2r (x0 ) ⊂ K , r ⩾ ε
α0 4
, ε ⩽ ε0 and
1 Eε (x0 , r ) ⩽ η| ln ε|,
(4.55)
4
we have
|1 − |vε (x)|| ⩽ σ ∀x ∈ B 3r (x0 ).
(4.56)
4
5. Proof of Theorem 1.1 We divide the proof of Theorem 1.1 into some lemmas. First using Corollary 4.5, as in [3,5,26], we have the following lemma. Lemma 5.1. There exists a constant η0 > 0 such that if x0 ∈ Σµ∗ , we have
Θ∗ (x0 ) ⩾ η0 . By the upper semicontinuity of Θ∗ , we have: Lemma 5.2. Σµ∗ is closed in K . From Lemma 5.1 and Corollary 4.5, we have the following uniform convergence result away from Σµ∗ . Lemma 5.3 ([4]). Let K1 ⊂ K \ Σµ∗ be any compact subset. For any σ > 0, there exists a constant ε2 > 0 depending on K1 and σ such that, for ε < ε2 ,
|1 − |vε || ⩽ σ on K1 .
(5.1)
4288
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
Lemma 5.4. We have
µ∗ = g (x)H 3 + h(x)H 1 ⌊Σµ∗ , where g , h ∈ L∞ loc (K ) and
η0 ⩽ Θ∗ (x) ⩽ h(x) ⩽ Θ ∗ (x) := lim sup
µ∗ (Br (x)) r
r →0
⩽ CM0 .
Proof. Since Σµ∗ is closed in K , Σµ∗ is hence measurable. We have
µ∗ = µ∗ ⌊Σµ∗ + µ∗ ⌊(K \ Σµ∗ ). By Corollary 4.5, the same argument as for Theorem VIII.1 in [3] yields
H 1 (Σµ∗ ) ⩽ CM0 . By the monotonicity formula of Proposition 3.8, we have for any x ∈ K ,
Θ ∗ (x) := lim sup
µ∗ (Br (x)) r
r →0
⩽ CM0 .
According to the Radon–Nikodym theorem, we obtain
µ∗ ⌊Σµ∗ = h(x)H 1 ⌊µ∗ ,
(5.2)
for Θ∗ ⩽ h(x) ⩽ Θ . Suppose that x0 ∈ K \ Σµ∗ and r > 0, such that B(x0 , 2r ) ⊂ K \ Σµ∗ . By Lemma 5.3, we obtain ∗
σ := ‖1 − |vε |‖L∞ (B(x0 ,2r )) = o(1) as ε → 0. On the other hand, we have the estimate
Eε
x0 ,
3 4
δr
⩽ C (δ 3 + σ + ε 2 | ln ε|2 + ε)Eε (x0 , r ) + C ε| ln ε|2 .
(5.3)
Note that σ = σ (ε) = o(1). Dividing both sides of (5.3) by | ln ε| and sending ε → 0 we obtain
3 µ∗ B x0 , δ r ⩽ C δ 3 µ∗ (B(x0 , r )), 4
which implies that µ∗ ⌊(K \ Σµ∗ ) is absolutely continuous with respect to the Lebesgue measure, and by the Radon–Nikodym theorem again we obtain
µ∗ = g (x)H 3 + h(x)H 1 ⌊Σµ∗ ,
(5.4)
for some locally bounded function g. The proof of Lemma 5.4 is completed.
The same proof as for Theorem A.(iv) in [4] gives: Lemma 5.5. We have g (x) = |∇ h∗ (x)|2
a.e. in K ,
where h∗ solves
∆ h∗ +
1 a
∇ a · ∇ h∗ = 0.
(5.5)
Moreover, we have
σ⃗ε =
1
2∇ fε
| ln ε|
fε
· ∇ vε , ∇vε +
fε2
2ε 2
0 in Cloc (K \ Σµ∗ ).
Lemma 5.6. Σµ∗ is rectifiable, and
¯ (x) = ⋆ H
2ω
α +1 2
for µ∗ -a.e. x in Σµ∗ .
x⊥ α ∧⋆
dJ∗ d µ∗
+
dσ⃗∗ d µ∗
,
(bε − |vε | )
2 2
∇ fε fε
−→
∇a a
· ∇ h∗ ∇ h∗
(5.6)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4289
⃗ ∈ D (K , R3 ) be a smooth vector field. Note that Proof. Let X ∫
⃗ =− eε (vε )divX
∫
1
K
2
K
+
∇(|∇vε |2 ) −
fε2 (bε − |vε |2 ) 2ε 2
fε2 (bε − |vε |2 )
ε2
⟨vε , ∇vε ⟩
(bε − |vε |2 )2 ⃗ ∇ bε + fε ∇ fε · X 2ε 2
(5.7)
and
∫ − ∫ ∫ ∂vε ∂vε ∂ X i 1 = − ∆vε ∇vε · X⃗ − ∇(|∇vε |2 ) · X⃗ . ∂ xi ∂ xj ∂ xj 2 K K K
(5.8)
Hence
∫
1
| ln ε|
eε (vε )δij −
K
∂vε ∂vε ∂ xi ∂ xj
∂Xi ∂ xj
∫
fε2 (bε − |vε |2 )vε fε2 (bε − |vε |2 ) (bε − |vε |2 )2 2 ⃗ ⃗ ⃗ = X · ∇vε , ∆vε + − ∇ bε · X − ∇ fε · X | ln ε| K ε2 2ε 2 4ε 2 ∫ 1 2iωε iωε vε ∇ fε2 ⊥ ⊥ 2 = ( x ( x · ∇v , ∇v · ∇)v + · ∇) f − ε ε ε ε | ln ε| K α 2 + 1 α (α 2 + 1)fε2 α fε2 f 2 (bε − |vε |2 ) (bε − |vε |2 )2 2 ⃗ − ε ∇ bε − ∇ fε · X 2 2ε 4ε 2 ∫ ∫ 2ω iωε vε 1 ⊥ ⊥ 2 ⃗ =− ⋆ x ∧ ⋆J (vε ) , X + (x · ∇)fε , ∇vε · X⃗ α2 + 1 α | ln ε| K (α 2 + 1)fε2 α K ∫ 2 ∫ fε (bε − |vε |2 ) 1 ( 2∇ fε · ∇)vε fε2 (bε − |vε |2 )2 ∇ fε 1 ∇ bε · X⃗ − , ∇vε + − · X⃗ , | ln ε| K 2ε 2 | ln ε| K fε 2ε 2 fε 1
where ⋆ refers to the Hodge duality. Set
αij,ε :=
1
| ln ε|
eε (vε )δij −
∂vε ∂vε ∂ xi ∂ xj
.
Write Aε = (αij,ε )n×n (where n = 3); then Aε is a symmetric matrix. By computation, we have: Proposition 5.7. Let Aε be the matrix given above; then: e (v )
ε (1) |aij,ε | ⩽ 3 |εln ε| ; n eε (vε ) (2) det(Aε ) = | ln ε| 1−
e (v )
n eε (vε ) 2 |∇v | ⩽ ; ε eε (vε ) | ln ε| 1
e (v )
ε ε (3) (n − 2) |εln ε| ⩽ tr(Aε ) ⩽ n |εln ε| ;
e (v )
ε (4) |εln ε|
1−
1 eε (vε )
|∇vε |2 is an eigenvalue of Aε , and the other n − 1 eigenvalues are all equal to
Proof. The first inequality is derived from the definition of aij,ε . Note that
|∂i vε ∂j vε | ⩽
|∂i vε |2 + |∂j vε |2 2
⩽ |∇vε |2 ⩽ 2eε (vε ).
Recall that eε (vε ) =
1 2
|∇vε |2 +
fε2 (bε − |vε |2 )2 4ε 2
.
The others are easily derived by using the following lemmas in linear algebra.
⃗ . Lemma 5.8. If A = In + α ⃗ β⃗ t , where we have α⃗ , β⃗ ∈ Fn and F is a field, then |A| = 1 + ⟨⃗ α , β⟩ Lemma 5.9. If α ⃗ = (a1 , . . . , an )t , then |⃗ α |2 is an eigenvalue of the matrix α⃗ α⃗ t .
eε (vε ) . | ln ε|
(5.9)
4290
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291 e (v )
ε If µε also denotes its coefficient |εln ε| , then we get
|αij,ε | ⩽ 3µε ,
|σ⃗ε | ⩽ C µε .
(5.10)
Passing to a subsequence, we may assume that
αij,ε ⇀ αij,∗ in the sense of measures. Then by (5.10), we obtain
|αij,∗ | ⩽ 3µ∗ ,
|σ⃗∗ | ⩽ C µ∗ .
Therefore we may write
αij,∗ (x) = Aij (x)µ∗ ,
⃗ µ∗ for µ∗ -a.e. x ∈ K , σ⃗ = m
where (Aij (x)) is a symmetric matrix, with trace less than or equal to 1 and eigenvalues less or equal to 1. From (5.10), we also have Aij ⩾ −3δij
for µ∗ -a.e. x ∈ K .
Now we deal with the second and third terms of the last expression in (5.9). For the second term, because (x⊥ α · ∇)a = 0 and in view of Lemma 3.1(c), we have
∫ 1/2 ∫ 1/2 ∫ 1 iωε vε ⊥ 2 2 ⊥ 2 2 ⃗ (xα · ∇)fε , ∇vε · X ⩽ C |∇vε | |(xα · ∇)(fε − a)| | ln ε| 2 2 K (α + 1)fε K K ⩽ C ε| ln ε|1/2 → 0
as ε → 0.
(5.11)
For the third term, recall that bε = 1 − ε 2 gε /fε2 and ‖∇ bε ‖L∞ ⩽ C ε 2 | ln ε|2 ; we get
1/2 ∫ ∫ 2 1 fε2 (bε − |vε |2 )2 fε (bε − |vε |2 ) ⃗ ∇ bε · X ⩽ C ε| ln ε| | ln ε| 2ε 2 2ε 2 K
K
⩽ C ε| ln ε|3/2 → 0
as ε → 0.
(5.12)
Therefore, passing to the limit in (5.9), we have
∫
∫ 2ω dJ∗ ⊥ ⋆ x ∧⋆ , X⃗ dµ∗ (x) α2 + 1 α d µ∗ K ∫ ∫ ∇a ⃗ , X⃗ ⟩dµ∗ (x)⌊Σµ∗ . − · ∇ h∗ ∇ h∗ , X⃗ dx − ⟨m
∂Xi dµ∗ (x) = − Aij (x) ∂ xj K
a
K
(5.13)
K
where J∗ is the limit of J (vε ); see Theorem 2 in [1]. We decompose the l.h.s. of (5.13) as
∫
∂Xi Aij (x) dµ∗ (x) = ∂ xj K
∫
∂Xi Aij (x) dµ∗ (x)⌊Σµ∗ + ∂ xj K
∫
|∇ h∗ |2
K
2
∂ h∗ ∂ h∗ ∂ X i δij − dx. ∂ xi ∂ xj ∂ xj
(5.14)
Since div(a∇ h∗ ) = 0, a direct computation yields
∫
|∇ h∗ |2
K
2
∫ ∂ h∗ ∂ h∗ ∂ X i ∇a ⃗ δij − dx = − · ∇ h∗ ∇ h∗ , X dx. ∂ xi ∂ xj ∂ xj a K
(5.15)
Combining (5.13)–(5.15), we obtain
∫
Aij (x) K
∂Xi dµ∗ (x)⌊Σµ∗ = − ∂ xj
∫ ∫ 2ω dJ∗ ⊥ ⃗ ⃗ , X⃗ ⟩dµ∗ (x)⌊Σµ∗ . ⋆ x ∧ ⋆ , X d µ ( x ) − ⟨m ∗ α2 + 1 α dµ∗ K K
Since the support of Tω is included in Σµ∗ , we have
∫
∂Xi Aij (x) j dµ∗ (x)⌊Σµ∗ = − ∂x K
∫ 2ω dJ∗ ⊥ ⃗ ⃗ , X dµ∗ (x)⌊Σµ∗ ⋆ x ∧⋆ +m α2 + 1 α d µ∗ K ∫ 2ω dJ∗ dσ⃗∗ ⊥ ⃗ =− x ∧ ⋆ + , X dµ∗ (x)⌊Σµ∗ . ⋆ α2 + 1 α d µ∗ d µ∗ K
(5.16)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4291
⃗ is arbitrary, (5.16) implies that the generalized 1-varifold (defined in [27]) Since X V := δAij (x) µ∗ ⌊Σµ∗ has a first variation. By (5.16), Lemma 5.1 and Theorem (3.8)(c) in [27], we obtain that V is a real rectifiable 1-varifold. In particular, Σµ∗ is rectifiable, which implies
Θ∗ (x) = Θ ∗ (x) for µ∗ -a.e. x ∈ Σµ∗ , and thus
µ∗ = |∇ h∗ |2 H 3 + Θ∗ (x)H 1 ⌊Σµ∗ and
V = V (Σµ∗ , Θ∗ ). By (5.16), we also have
⃗ =⋆ H
2ω
α2 + 1
x⊥ α ∧⋆
dJ∗ d µ∗
+
dσ⃗∗ d µ∗
for µ∗ -a.e. x ∈ Σµ∗ .
The proof of Theorem 1.1 is now completed.
Acknowledgements We wish to express our gratitude to the reviewers who carefully read and checked this paper. They gave us helpful suggestions and comments. References [1] R.L. Jerrard, Local minimizers with vortex filaments for a Gross–Pitaevsky functional, ESAIM Control Optim. Calc. Var. 13 (2007) 35–71. [2] F. Bethuel, H. Brezis, F. Hélein, Ginzburg–Landau Vortices, Birkhäuser, Boston, 1994. [3] F. Bethuel, H. Brezis, G. Orlandi, Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions, J. Funct. Anal. 186 (2001) 432–520; 188 (2002) 548–549 (erratum). [4] F. Bethuel, G. Orlandi, D. Smets, Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Ann. of Math. (2) 163 (2006) 37–163. [5] F. Bethuel, G. Orlandi, D. Smets, Vortex rings for the Gross–Pitaevskii equation, J. Eur. Math. Soc. 6 (2004) 17–94. [6] L. Lassoued, P. Mironescu, Ginzburg–Landau type energy with discontinuous constraint, J. Anal. Math. 77 (1999) 1–26. [7] A. Aftalion, T. Rivière, Vortex energy and vortex bending for a rotating Bose–Einstein condensate, Phys. Rev. A 64 (2001) 043611. [8] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986) 55–64. [9] F.H. Lin, T. Rivière, A quantization property for static Ginzburg–Landau vortices, Comm. Pure Appl. Math. 54 (2001) 206–228. [10] K. Madison, F. Chevy, J. Dalibard, W. Wohlleben, Vortex formation in a stirred Bose–Einstein condensate, Phys. Rev. Lett. 84 (2000) 806–809. [11] K. Madison, F. Chevy, J. Dalibard, W. Wohlleben, Vortices in a stirred Bose–Einstein condensate, J. Mod. Opt. 47 (2000) 2715. [12] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Observation of vortex lattices in Bose–Einstein condensate, Science 292 (2001) 476–479. [13] C. Raman, J.R. Abo-Shaeer, J.M. Vogels, K. Xu, W. Ketterle, Vortex nucleation in a stirred Bose–Einstein condensate, Phys. Rev. Lett. 87 (2001) 210402. [14] P. Rosenbusch, V. Bretin, J. Dalibard, Dynamics of a single vortex line in a Bose–Einstein condensate, Phys. Rev. Lett. 89 (2002) 200403. [15] E.H. Lieb, R. Seiringer, J. Yngvason, Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional, Phys. Rev. A 61 (2000) 043602. [16] E.H. Lieb, R. Seiringer, Derivation of the Gross–Pitaevskii equation for rotating Bose gases, Comm. Math. Phys. 264 (2006) 505–537. [17] S. Serfaty, On a model of rotating superfluids, ESAIM Control Optim. Calc. Var. 6 (2001) 201–238. [18] A. Aftalion, Q. Du, Vortices in a rotating Bose–Einstein condensate: critical angular velocities and energy diagrams in the Thomas–Fermi regime, Phys. Rev. A 64 (2001) 063603. [19] R. Ignat, V. Millot, The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate, J. Funct. Anal. 233 (2006) 260–306. [20] A. Aftalion, S. Alama, L. Bronsard, Giant vortex and the breakdown of strong pinning in a rotating Bose–Einstein condensate, Arch. Ration. Mech. Anal. 178 (2005) 247–286. [21] S. Alama, L. Bronsard, J.A. Montero, Vortices for a rotating toroidal Bose–Einstein condensate, Arch. Ration. Mech. Anal. 187 (3) (2008) 481–522. [22] A. Aftalion, R.L. Jerrard, On the shape of vortices for a rotating Bose–Einstein condensate, Phys. Rev. A 66 (2002) 023611. [23] J. Bourgain, H. Brezis, P. Mironescu, On the structure of the Sobolev space H 1/2 with values into the circle, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000) 119–124. [24] F. Bethuel, G. Orlandi, D. Smets, Approximations with vorticity bounds for the Ginzburg–Landau functional, Commun. Contemp. Math. 6 (2004) 803–832. [25] L.C. Evans, Partial Differential Equations, in: Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. [26] Z. Liu, Vortices set and the applied magnetic field for superconductivity in dimension three, J. Math. Phys. 46 (2005) 052111. [27] L. Ambrosetti, M. Soner, A measure theoretic approach to higher codimension mean curvature flow, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 25 (1997) 27–49.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Asymptotic behavior of critical points for a Gross–Pitaevskii energy Ling Zhou a , Haifeng Xu a,∗ , Zuhan Liu b a
School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu, 225002, PR China
b
Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu, 221116, PR China
article
abstract
info
Article history: Received 6 August 2009 Accepted 2 April 2011 Communicated by Ravi Agarwal
In this paper, we study the asymptotic behavior of critical points of a Gross–Pitaevskii energy, which is proposed as a model for rotationally forced Bose–Einstein condensate. We prove that the limiting singularity set is one-dimensional rectifiable. We also establish the convergence result for critical points away from limiting singularities. © 2011 Elsevier Ltd. All rights reserved.
MSC: primary 35J55 35Q40 Keywords: Gross–Pitaevskii energy Bose–Einstein condensate Rectifiable
1. Introduction In this paper, we are interested in characterizing the vortices of solutions of the following equation:
−∆uε + 2iωε (x⊥ · ∇)uε =
uε = 0,
on ∂ Ω
1
ε2
(a(x) − |uε |2 )uε ,
in Ω
(1.1)
which is the necessary condition for the uε to be the critical points of the Gross–Pitaevskii energy
∫ 1 2 ⊥ 2 2 Gε (u) = |∇ u| − 2ωε x · ⟨iu, ∇ u⟩ + 2 a(x) − |u| dx. (1.2) 2 Ω 2ε √ Here i is the imaginary unit −1. ωε means the rotational velocity, ωε = ω| ln ε| for a small parameter ε > 0 and some constant ω > 0. The trapping potential a(x) is a real function given by a(x) = a0 − x21 − α 2 x22 − β 2 x23 for certain positive constants α and β . It describes the geometry of the trap, which is a smooth bounded three-dimensional simply connected 2 domain denoted by Ω := {x ∈ R3 : a(x) > 0}. For x = (x1 , x2 , x3 ) ∈ R3 , we let x⊥ := (−x2 , x1 , 0), x⊥ α := (−α x2 , x1 , 0). 1 ,2 1 u is a complex-valued function in the space W0 (Ω ; C) = H0 (Ω ; C). The operators ∇ and ∆ are the usual gradient and Laplace operators acting on real functions, that is ∇ u = ∇ℜu + i∇ℑu, ∆u = ∆ℜu + i∆ℑu. By regarding C as a two-real√ dimensional vector space,we define the real inner product on it, ⟨u, v⟩ := (uv¯ + u¯ v)/2. And we define |u| := ⟨u, u⟩. ℜu ℜv Note that ⟨iu, v⟩ = det ℑu ℑv . We simply write j(u) = ⟨iu, ∇ u⟩ := (⟨iu, ux1 ⟩, ⟨iu, ux2 ⟩, ⟨iu, ux3 ⟩) ∈ R3 , where 1
∗
Corresponding author. E-mail addresses: [email protected] (L. Zhou), [email protected] (H. Xu), [email protected] (Z. Liu).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.04.010
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4275
u u uxj = ∂∂xu = ∂ℜ + i ∂ℑ . And we define J (u) := 12 ∇ × j(u) (where J (u) is the Hodge dual of the Jacobian Ju defined ∂ xj ∂ xj j in [1, Section 3.2]). ‘‘·’’ is the usual inner product on R3 . We also use ⟨·, ·⟩ to denote it. There is a total analogy between this model and the well-known Ginzburg–Landau model of superconductivity. The Ginzburg–Landau functional for superconductors is
G(u, A) =
∫
1 2
Ω
1
1
2
4ε 2
|∇ u − iAu|2 + |curl A − hex |2 +
(1 − |u|2 )2 ,
where hex is the intensity of the applied magnetic field, A = (A1 , A2 , A3 ) ∈ R3 is the vector potential of the magnetic field, and h = curl A is the induced magnetic field in the material. As we shall see, the role of the external field hex is taken by the rotational velocity ωε . The main tools for studying vortices in ‘‘Ginzburg–Landau type’’ problems have been developed in [2–5] etc. The main difference between the two problems is the boundary condition. Here u = 0 on ∂ Ω , whereas, for the Ginzburg–Landau case, all functions in H 1 were admissible, so no boundary conditions were imposed. Since u has to be small on a layer of size of the order of ε near ∂ Ω , the condition u = 0 induces a cost of Cε at least in the energy. This cost is very large compared to the Ginzburg–Landau energy. Hence, if we make comparisons with test maps, all the fine information on the behavior of u in Ω will be hidden by the energetic cost of the boundary layer. For each ε > 0, uε is supposed to be a solution of (1.1). In this paper, we study the asymptotic behavior of uε as ε → 0, and try to capture the codimension 2 singularities in a rigorous mathematical way. Before stating the main theorem, we give some notation first. Inspired by Lassoued and Mironescu [6] and Aftalion and Rivière [7], we define uε = fε eiωε S vε ,
(1.3)
where S=
α2 − 1 x1 x2 , α2 + 1
(1.4)
1 ,2
and fε ∈ W0 (Ω , R) is a real-valued minimizer of the energy Gε . The existence of such minimizers is standard, and the solution is unique by the result of Brezis and Oswald [8]. Using the Euler–Lagrange equation
−∆fε =
fε = 0,
1
ε2
(a − fε2 )fε ,
in Ω
(1.5)
on ∂ Ω
and the strong maximum principle, one can deduce that fε is positive in Ω . In this paper, we assume that the energy of vortices blows up like | ln ε|: Hε (vε ) :=
1
∫
2
Ω
2
2
fε |∇vε | +
fε4
2ε 2
(1 − |vε | )
2 2
⩽ C | ln ε|,
(1.6)
where C is a positive constant. Define the measures µε :
µε :=
eε (vε )
| ln ε|
dx,
(1.7)
and suppose that
σ⃗ε =
1
| ln ε|
2∇ fε fε
· ∇ vε , ∇vε
∇ fε fε2 (bε − |vε |2 )2 + , fε 2ε 2
(1.8)
where eε (vε ) =
1 2
|∇vε |2 +
fε2 (bε − |vε |2 )2 4ε
2
gε := (2x⊥ · ∇ S − |∇ S |2 )ωε2 =
,
bε = 1 −
ε2 fε2
gε ,
α2 − 1 2 (α + 3)x21 − (3α 2 + 1)x22 ωε2 . (α 2 + 1)2
(1.9)
(1.10)
In view of Hε (vε ) ⩽ C | ln ε| and Lemma 3.1(c), µε and σ⃗ε are bounded in K (K b Ω ). Therefore, up to a subsequence, we may assume that
µε ⇀ µ∗ ,
σ⃗ε ⇀ σ⃗∗ as measures.
(1.11)
4276
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
Recall that, for a Radon measure ν ∈ M (K ) and m > 0, the m-dimensional density of ν at x ∈ K is defined by
Θm (ν, x) := lim inf
ν(Br (x))
.
rm Suppose that Θ∗ (x) = Θ1 (µ∗ , x), and that r →0
Σµ∗ = {x ∈ K : Θ∗ (x) > 0}.
(1.12)
Our main theorem is: Theorem 1.1. If K is a compact subset in Ω and uε is a solution to (1.1) with the assumption (1.6), then the following properties hold. (i) (Limiting singularities). The set Σµ∗ is closed in K and 1-rectifiable. The varifold V := V (Σµ∗ , Θ1 ) satisfies the equation
⃗ ( x) = ⋆ H
2ω
α +1 2
x⊥ α ∧⋆
dJ∗ dµ∗
+
dσ⃗∗ dµ∗
for µ∗ -a.e. x ∈ Σµ∗ ,
(1.13)
⃗ (x) denotes the generalized mean curvature of V at x and is defined by where H ∫ K
⃗=− divΣµ∗ X
∫
⃗ · X⃗ H
⃗ ∈ C0∞ (K , R3 ), for all X
K
dσ⃗∗ dµ∗
is the Radon–Nikodym derivative of σ⃗∗ with respect to µ∗ . J∗ is the limit of J (vε ) (see Theorem 2 in [1]). (ii) (Measure decomposition). The measure µ∗ can be decomposed as
µ∗ = |∇ h∗ (x)|2 · H 3 + Θ∗ (x) · H 1 ⌊Σµ∗ ,
(1.14)
where h∗ satisfies
∆ h∗ +
∇a
· ∇ h∗ = 0 a (iii) (Convergence). We have
in K .
0 |uε |2 → a in Cloc (K \ Σµ∗ ) as ε → 0.
(1.15)
(1.16)
It is proved that |uε |2 converges uniformly to a(x) away from limiting singularities Σµ∗ (limiting vortex lines), which is one-dimensional rectifiable. Our proofs borrow many ideas from [3–5,9]. Our main difficulty is as regards how to prove the η-compactness theorem. To overcome the difficulty, we modify the proof of Theorem 1 in [3]. First, the Hodge–de Rham decomposition (4.17) of vε × dvε is different from that in [3]. Second, by improving Lemma III.1 in [3], we obtain Lemma 4.2 (estimate (4.5) is new). Using this, we obtain the complex estimate of the energy in Theorem 4.3 which is important in the proof of the main theorem. The paper is organized as follows. Section 1 is a general introduction of what we do and what result we get in this paper. Section 2 provides some background on Bose–Einstein condensates. In Section 3, the monotonicity formula is derived. The η-compactness theorem is shown in the next section. Then we prove the main result, Theorem 1.1, in Section 4. 2. Background The physical phenomenon called the Bose–Einstein condensate (we called it the new state of matter) is a macroscopic quantum effect which was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–1925. Bose assumed certain rules for photons which are now called Bose–Einstein statistics. Einstein guessed that the rules might apply to atoms. But actually not all atoms follow the rules of Bose–Einstein statistics. According to the theory of wave–particle duality of quantum theory, like light waves or water waves, microparticles of matter also show communication, interference and diffraction fluctuations in behavior, and thus form matter waves (known as the de Broglie waves). However, the wavelength of the atom is too short. For the usual state, the atomic fluctuations are not obvious. They can be observed only in a very cold state. At very low temperatures, most of the atoms are in the same quantum level. All the them are absolutely identical. That is, they are in the same place. This is just the Bose–Einstein condensate. Generally we must take the temperature of atoms cooled enough (near µK) to achieve atomic Bose–Einstein condensation in the laboratory. And the atom system is hoped to be almost in a ground state with no interaction. With the inventions of atomic laser cooling techniques and atomic trap techniques, these requirements are realized. The inventors, Steven Chu, Phillips and Cohen-Tannoudji were awarded the Nobel Prize in Physics in 1997. The first gaseous condensate was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder NIST–JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) (1.7 × 107 K). For their achievements, Cornell, Wieman, and Wolfgang Ketterle at MIT received the 2001 Nobel Prize in Physics.1 1 This paragraph is quoted from http://en.wikipedia.org/wiki/Bose-Einstein~condensate.
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4277
From then on, many properties of BEC have been studied, both experimentally and theoretically, with the development of new techniques and methods. The existence and structure of vortices are of particular interest. The group of Dalibard at the ENS in Paris [10,11] and the group of Ketterle at MIT [12,13] have produced vortices by rotation of the trapping potential. The condensate is rotated along the Z -axis at velocity ωε . When the velocity is sufficiently high, vortices appear. For a prolate trap, it has been observed [10,11] that when a single vortex exists, the vortex line is bending. The ENS group [14] also studies configurations with a single vortex line. In the mathematical model, a condensate is described by a wavefunction uε . After nondimensionalization, the wavefunction is expected to be close to a critical point of the Gross–Pitaevskii energy (1.2) (see [15,16] and their references) 1 ,2 in the space W0 (Ω ; C). Above a critical velocity, uε exhibits vortices, zeros of uε around which there is a circulation of phase. For a(x) ≡ 1 and for a disc in R2 , Serfaty [17] proves the existence of local minimizers having vortices for different ranges of rotational velocity. For a(x) = a0 − (x21 + λ2 x22 ), Ω = {x ∈ R2 : a(x) > 0}, Aftalion and Du [18] investigate an asymptotic development of the energy, the critical rotational velocities of nucleation of vortices and the location of vortices. In [19], Ignat and Millot investigate a model corresponding to the experiments on a two-dimensional rotating BEC. They estimate the critical rotational velocity for vortex existence and give some fundamental energy estimates. Aftalion et al. [20] analyze the global minimizers of the energy for an annular region in R2 . For Ω a solid torus of revolution in R3 with star-shaped cross-section, for ωε = O(| ln ε|), Alama et al. [21] prove that there exist a family of local minimizers with a vortex of the energy for ε sufficiently small. For a(x) = a0 − α 2 x21 − x22 − β 2 x23 , Aftalion and Jerrard [22] analyze the properties of the vortex line using the line energy, which is the simplified energy for a vortex line derived by Aftalion and Rivière in [7] from the Gross–Pitaevskii energy. The stability and instability of the straight vortex were studied. And they proved that when the condensate has a cigar shape, the first vortex is bent, while when it is a pancake, the first vortex is straight and lies on the axis of rotation. Jerrard [1] gives a rigorous derivation of the line energy derived in [7] and proves the existence of local minimizers with vortex filaments for the Gross–Pitaevskii functional. 3. The monotonicity formula The following estimates will be essential at several steps of our analysis. Lemma 3.1 ([19–21]). For ε sufficiently small, we have (a) Gε (fε√ ) ⩽ C | ln ε|, √ √ (b) 0 ⩽ √ a(x) − fε (x) ⩽ C ε 1/3 a(x), for x ∈ Ω with |x| < a0 − ε 1/3 , (c) ‖fε − a‖C 1 (K ) ⩽ CK ε 2 , for any compact subset in Ω . Using the techniques introduced in [6,19], we have the following energy decomposition result. Lemma 3.2 ([19]). Suppose that uε (1.4) respectively. Then we have
∈ W01,2 (Ω , C). Define vε via uε = fε eiωε S vε , where fε and S satisfy (1.5) and
Gε (uε ) = Gε (fε eiωε S ) + Hε (vε ) −
+
1 2
∫ Ω
2ωε
∫
α2 + 1
Ω
fε2 x⊥ α · ⟨ivε , ∇vε ⟩
fε2 (|vε |2 − 1)(ωε2 |∇ S |2 − 2ωε2 x⊥ · ∇ S ),
(3.1)
where Hε (vε ) is defined in (1.6). Since uε = 0 on ∂ Ω , we restrict our analysis to the interior of Ω . Let K be a compact subset in Ω . Then vε satisfies the associated Euler–Lagrange equation
−∆ v ε +
2iωε
α2 + 1
(x⊥ α · ∇)vε +
iωε
vε
2
α 2 + 1 fε
fε
2 (x⊥ α · ∇)fε − 2
∇ fε · ∇vε =
fε2
ε2
(1 − |vε |2 )vε − gε vε in K ,
(3.2)
2 where gε isdefined in (1.10). Obviously, gε = O(| ln ε| ) since α is a constant and x1 , x2 are bounded. We also observe that
fε2 /ε 2 = O ε12
by Lemma 3.1(c). Then limε→0
gε fε2 /ε 2
= 0. So the term gε vε is infinitesimal compared with fε2 vε /ε2 as ε is
2 small enough. Writing bε := 1 − ε2 gε , (3.2) can be rewritten as
fε
−∆ v ε +
2iωε
(x⊥ α · ∇)vε +
iωε
vε
2 (x⊥ α · ∇)fε − 2
α2 + 1 α 2 + 1 fε For x0 ∈ K and Br (x0 ) = B(x0 , r ) ⊂ K , define ∫ 1 1 Eε (vε , x0 , r ) := Eε (vε , x0 , r ) ≡ eε (vε ), r
r
2 fε
∇ fε · ∇vε =
fε2
ε2
(bε − |vε |2 )vε in K .
Br (x0 )
where eε (vε ) is given in (1.9). Sometimes, we simply write Eε (x0 , r ) or Eε (r ) instead of Eε (vε , x0 , r ).
(3.3)
(3.4)
4278
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
We now begin to prove the monotonicity formula for the energy Eε in K . First, by the maximum principles, we have the following lemma. Lemma 3.3 ([5]). Let vε be a solution to (3.3). Then
‖vε ‖L∞ (K ) ⩽ 1 + C ε 2 | ln ε|2 + C
ε2 , dist(K , ∂ Ω )2
‖∇vε ‖L∞ (K ) ⩽
CK
ε
.
(3.5)
A computation directly gives the following Pohozaev identity. Lemma 3.4 (Pohozaev Identity). Let vε be a solution to (3.3). Then, for Br (x0 ) ⊂ K , we have 1 2
∫
fε2
∫
3
(bε − |vε |2 )2 ε2 ∫ 2 ∫ |∇T vε |2 2ωε 1 ∂vε fε2 2 2 =r − J (vε ), x⊥ α × ( x − x0 ) + 4ε 2 (bε − |vε | ) + 2 +1 2 2 ∂ n α ∂ Br (x0 ) Br (x0 ) ∫ 2 ∫ vε fε ωε 1 2 i 2 x⊥ (bε − |v|2 )(x − x0 ) · ∇ bε + − α · ∇ fε , (x − x0 ) · ∇v 2 fε 2 Br ε 2 Br α + 1 ∫ ∫ 2 1 − ⟨∇ fε · ∇vε , (x − x0 ) · ∇vε ⟩ − 2 fε (bε − |vε |2 )2 (x − x0 ) · ∇ fε , f 2 ε Br (x0 ) ε Br (x0 ) Br (x0 )
where J (vε ) =
|∇vε |2 +
1 2
4
B(x0 ,r )
(3.6)
∇ × j(vε ) = 12 ∇ × ⟨ivε , ∇vε ⟩.
Lemma 3.5. Assume that vε satisfies (3.3) and BR (x0 ) is a subset of K ; then, for 0 < r < R, d dr
1 ( Eε (r )) =
∫
r
∂ Br (x0 )
∫ ∂vε 2 + 1 ∂n r2
fε2 (bε − |vε |2 )2
Br (x0 )
2ε 2
−
2
∫
r2
Br (x0 )
ωε J (vε ), x⊥ α × (x − x0 ) α2 + 1
∫ (bε − |vε |2 )2 2 1 + 2 fε ( x − x ) · ∇ f + ⟨∇ fε · ∇vε , (x − x0 ) · ∇vε ⟩ 0 ε 2 2 2r ε r Br (x0 ) Br (x0 ) fε ∫ ∫ 2 ωε vε ⊥ fε 1 1 2 i 2 xα · ∇ fε , (x − x0 ) · ∇v + 2 (bε − |v|2 )(x − x0 ) · ∇ bε . − 2 2 +1 2 r α f 2r ε Br Br ε ∫
1
(3.7)
Proof. Note that d dr
(Eε (r )) =
∫ ∂ Br (x0 )
fε2 2 2 + |∇T vε | + (bε − |vε | ) . 2 2 ∂n 4ε 2
1
2
2 1 ∂vε
Hence, d dr
( Eε (r )) = − +
1 r2 1 r
∫ Br (x0 )
∫ ∂ Br (x0 )
1 2
2
|∇vε | +
3fε2
4ε
(bε − |vε | ) 2
2 2
+
1 2ε 2 r 2
∫ Br (x0 )
fε2 (bε − |vε |2 )2
fε2 2 2 + |∇T vε | + (bε − |vε | ) . 2 2 ∂n 4ε 2 1
2
2 1 ∂vε
Combining this relation with Lemma 3.4, the conclusion follows.
Lemma 3.6. There exists C > 0, depending only on K , such that for BR (x0 ) ⊂ K ,
Λ = C (ωε + 1),
(3.8)
and for any vε satisfying (3.3), we have d dr
(eΛr Eε (r )) ⩾
∫ ∂vε 2 fε2 (bε − |vε |2 )2 + 1 ⩾ 0, 2 r ∂ Br (x0 ) ∂ n r 2ε 2 Br (x0 )
1
∫
for 0 < r < R. In particular, eΛr Eε (r ) is increasing in r.
(3.9)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4279
Proof. From Lemma 3.5, we need to estimate the last five terms in the r.h.s. of (3.7). Note that J (vε ) ⩽ C |∇vε |2 , |∇ bε | ⩽ C ε 2 | ln ε|2 ; hence
∫ ∫ 2 ωε 1 ⊥ J (v ), x · ( x − x ) |∇vε |2 ⩽ C ωε Eε (r ), ⩽ C ω ε 0 ε α r2 2 r Br (x0 ) Br (x0 ) α + 1 ∫ ∫ 1 C (bε − |vε |2 )2 fε2 (bε − |vε |2 )2 f ( x − x ) · ∇ f ⩽ C Eε (r ), ⩽ ε 0 ε r2 2 ε r Br (x0 ) ε2 Br (x0 ) ∫ ∫ 2 C 1 ⟨∇ f · ∇v , ( x − x ) · ∇v ⟩ ⩽ |∇vε |2 ⩽ C Eε (r ), ε ε 0 ε r2 f r ε Br (x0 ) Br (x0 ) ∫ 2 ∫ 1 C fε fε2 (bε − |vε |2 )2 2 ( b − |v| )( x − x ) · ∇ b ⩽ C Eε (r ). ⩽ ε 0 ε 2r 2 ε2 r ε2 Br (x0 )
Br
Since
x⊥ α
· ∇ a = 0, in view of Lemma 3.1(c) we have ∫ ∫ 1 ωε vε ⊥ 1 ωε vε ⊥ 2 2 i xα · ∇ fε , (x − x0 ) · ∇v = 2 i 2 xα · ∇(fε − a), (x − x0 ) · ∇v 2 r 2 Br α 2 + 1 fε2 r fε Br α + 1 ⩽ C Eε (r ).
Using Lemma 3.5, we have d dr
(eΛr Eε (r ))
=e
Λr
d Λ Eε (r ) + ( Eε (r ))
dr
∫ ∂vε 2 1 fε2 (bε − |vε |2 )2 + 1 ⩾ ⩾ 0. r ∂ Br (x0 ) ∂ n r 2 Br (x0 ) 2ε 2 ∫
This implies the conclusion. The proof of Lemma 3.6 is completed.
In order to establish the monotonicity formula, refined estimates of Jacobian integrals are needed. They were conjectured by Bourgain et al. [23], and were proved by Bethuel et al. [24]. 1 Lemma 3.7 ([23,24]). Suppose that w ∈ Hloc (Ω , C), ϕ ∈ C0∞ (Ω , ∧1 R3 ) and κ := supp ϕ . Moreover, assume that there exists q > 6 such that w satisfies
‖w‖Lq (Ω ) ⩽ Cq ; then,
∫ ∫ α1 ∫ 1 α0 ⟨J w, ϕ⟩ ⩽ C eε (w) + ε eε (w) ‖ϕ‖W 1,3 (Ω ) , | ln ε| κ
Ω
(3.10)
κ
for some constants 0 < α0 < 1, 0 < α1 < 1, and C > 0 which are dependent only on Ω , q and Cq . Now we prove the following monotonicity formula. Proposition 3.8 (Monotonicity Formula). There exist C > 0, β > 0, which are independent of ε , such that for BR (x0 ) ⊂ K b Ω and for vε satisfying (3.3), we have
Eε (vε , x0 , θ r ) ⩽ C Eε (vε , x0 , r ) + ε β , for 0 < θ <
1 2
(3.11)
and 0 < r < R.
Proof. We split the proof into three steps and drop the subscript ε for simplicity. Step 1. Define a cut-off function f :
1, a f ( r , a) = 2 − , r 0,
if a ⩽ r if r < a < 2r
(3.12)
if a ⩾ 2r
where r > 0. Set E ε (x0 , r ) :=
1 r
∫ B2r (x0 )
eε (v)f (r , |x − x0 |)dx.
(3.13)
4280
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
Then for 0 < r < d dr
R , 2
(E ε (x0 , r )) =
2 ∫ ∂v fε2 (bε − |v|2 )2 dt + 1 f (r , |x − x0 |) r 1 r 2 B2r (x0 ) 2ε 2 ∂ Btr (x0 ) ∂ n ∫ ∫ 2 ωε 1 (bε − |v|2 )2 ⊥ − 2 J (v), x × ( x − x ) f ( r , | x − x |) + fε 0 0 α 2 2 r 2r ε2 B2r (x0 ) α + 1 B2r (x0 ) ∫ 1 2 ⟨∇ fε · ∇v, (x − x0 ) · ∇v f (r , |x − x0 |)⟩ × (x − x0 )∇ fε f (r , |x − x0 |) + 2 2
∫
1
∫
t
r
− +
∫
1 r2
Br
1
∫
2r 2
B (x ) fε
2r 0 ωε vε ⊥ 2 x · ∇ f , ( x − x ) · ∇v f ( r , | x − x |) i 0 0 ε α 2 + 1 fε2 α
fε2
Br
ε2
(bε − |v|2 )(x − x0 ) · ∇ bε f (r , |x − x0 |).
Step 2. There exists a constant C > 0 such that for 0 < r <
E ε (x0 , θ r ) ⩽ C e
Cr
(3.14)
R , 2
εα0 | ln ε|α1 +1 E ε (x0 , r ) + , θr
(3.15)
for 0 < θ ⩽ 1. In fact, we need to estimate the last three terms in the r.h.s. of (3.14). The last four terms are treated as before:
∫ 1 (bε − |v|2 )2 fε (x − x0 ) · ∇ fε f (r , |x − x0 |) ⩽ C E ε (x0 , 2r ), 2r 2 2 ε B2r (x0 ) ∫ 1 2 ⩽ C E ε (x0 , 2r ), ⟨∇ f · ∇v, ( x − x ) · ∇v f ( r , | x − x |)⟩ ε 0 0 r2 B2r (x0 ) fε ∫ 1 ωε vε ⊥ 2 ⩽ C E ε (x0 , 2r ), i x · ∇ f , ( x − x ) · ∇v f ( r , | x − x |) 0 0 α ε r2 2 fε2 Br α + 1 ∫ 2 1 fε 2 ⩽ C E ε (x0 , 2r ). ( b − |v| )( x − x ) · ∇ b f ( r , | x − x |) ε 0 ε 0 2r 2 2 Br ε
(3.16)
(3.17)
(3.18)
(3.19)
For the first term, note that
‖ωε x⊥ α × (x − x0 )f (r , |x − x0 |)‖W 1,3 (B2r (x0 )) ⩽ Cr | ln ε|.
(3.20)
Hence, using Lemma 3.7, we have
∫ 2 r2
B2r (x0 )
∫ α 1 e (v) ωε C B2r (x0 ) ε ⊥ α0 J (v), xα × (x − x0 )f (r , |x − x0 |) ⩽ +ε eε (v) | ln ε| α2 + 1 r | ln ε| B2r (x0 ) ⩽ C E ε (x0 , 2r ) +
1 α0 ε | ln ε|α1 +1 . r
(3.21)
Combining (3.16)–(3.19), (3.21) with (3.14), we obtain d dr
(E ε (x0 , r )) ⩾ −C E ε (x0 , r ) −
C α0 ε | ln ε|α1 +1 . r
(3.22)
Inequality (3.15) then follows from a version of Gronwall’s lemma given in [5, Lemma A.7]. Step 3. First we consider the case
θ r < ρ :=
1
Λ
=
1 C (ωε + 1)
<
r 2
.
(3.23)
Using Lemma 3.6, we have
Eε (θ r ) ⩽ C Eε (ρ).
(3.24)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4281
Next, using (3.15) and the definition of ρ , we have
Eε (ρ) ⩽ E ε (ρ) ⩽ C E ε
ε α0 | ln ε|α1 +1 + 2 ρ
r
⩽C Eε (r ) + ε α0 | ln ε|α1 +2 .
(3.25)
α
Suppose that β = 40 ; combining (3.24) and (3.25), we get the desired formula (3.11). If θ r ⩾ ρ , (3.11) is inferred directly from (3.15). This completes the proof. 4. The η-compactness theorem In this section, we will show the η-compactness theorem by using the method of [3,4] with some modification. In this theorem, |vε | is bounded away from zero if the local energy is bounded by η| ln ε| with η small. Here, the Hodge–de Rham decomposition of vε × ∇vε is different from that in [3]. Theorem 4.1 (η-Compactness Theorem). Assume that vε satisfies (3.3) and σ > 0 is given. There exist η > 0 and ε0 > 0 depending only on σ such that if B2R (x0 ) ⊂ K , R ⩾ ε
α0 4
, ε ⩽ ε0 and
Eε (x0 , R) ⩽ η| ln ε|,
(4.1)
|vε (x0 )| ⩾ 1 − σ .
(4.2)
then
Proof. We follow closely the lines of [3]. Let 0 < δ < 1/400 be a constant to be determined later. In the sequel, we will use C to denote generic constants independent of the choice of δ . The proof of Theorem 4.1 is divided into the following three steps. Step 1. Choose ‘‘good’’ radii. The same argument as in [3] gives the following lemma. Lemma 4.2. Assume that 0 < ε
α0 4
< δ , and that there exist a constant C > 0 and radii r0 , rc ∈
δ r0 < rc < 2δ r0 such that ∫ 1 fε2 (bε − |vε |2 )2 ⩽ C (η| ln δ| + ε β ), r0 Br (x0 ) ε2 0
α0 ε 4 R, 41 R with
(4.3)
Eε (x0 , r0 ) − 2 Eε (x0 , δ r0 ) ⩽ C (η| ln δ| + ε β ), ∫ 2 ∫ ∂vε 2 ⩽ C (η| ln δ| + ε β ). t 1 ∂ Btrc (x0 ) ∂ n
(4.4) (4.5)
Step 2. δ -energy decay. Let 0 < γ < 1/8 be a constant to be determined later. Theorem 4.3. There exist εN > 0 and C > 0 such that for any 0 < ε < εN , we have
β
Eε (x0 , δ r0 ) ⩽ C δ r0 (η| ln δ| + ε ) + (γ + δ )Eε (x0 , r0 ) + (γ 2
3
Proof. By the mean-value inequality, there exists some r1 ∈
∫ ∂ Br1 (x0 )
∫ ∂ Br1 (x0 )
|∇vε |2 ⩽ C
∫ Br0 (x0 )
1 r 64 0
−2
+
γ −4 Eε (x0 , r0 ))
fε2 (bε − |vε |2 )2
∫ Br0 (x0 )
∫ Br0 (x0 )
.
(4.6)
1 , 32 r0 such that
|∇vε |2 ,
fε2 (bε − |vε |2 )2 ⩽ C
ε2
(4.7) fε2 (bε − |vε |2 )2 .
(4.8)
Let 1r1 denote the characteristic function of the set Br1 (x0 ). Consider a 2-form ψ in R3 defined by
[ ψ = −G ⋆ d 1r1 vε × dvε − 1r1
] ωε 2 3 ( 1 − |v | ) Σ c ( x ) dx , ε i i i=1 α2 + 1
in R3
(4.9)
4282
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
2 where c (x) = x⊥ α = (−α x2 , x1 , 0), G(x) = −C3 /|x| denotes the Green’s function of the Laplace operator in dimension 3. Note that
[ ] ωε −∆ψ = d 1 v × dv − 1 2 3 (1 − |vε | )Σi=1 ci (x)dxi , ε r1 2 r1 ε α +1 |ψ(x)| → 0 as |x| → ∞.
in R3
(4.10)
Since −∆ = dd∗ + d∗ d, it follows that
[
d vε × dvε −
] ωε 2 3 ∗ ( 1 − |v | ) Σ c ( x ) dx − d ψ = d∗ dψ , ζ in Br1 (x0 ). ε i i i =1 α2 + 1
(4.11)
We observe that
∆(dψ) = 0 in Br1 (x0 ).
(4.12)
Indeed, we have
[ ∆(dψ) = d(∆ψ) = −d2 vε × dvε −
] ωε 2 3 ( 1 − |v | ) Σ c ( x ) dx = 0 in Br1 (x0 ). ε i i i=1 α2 + 1
It follows that the 2-form ζ = d∗ dψ is closed, since dζ = d(d∗ dψ) = dd∗ (dψ) = −∆(dψ) − d∗ d(dψ) = 0.
(4.13)
By the Poincaré Lemma, there exists a 1-form ξ defined on B 1 r (x0 ), such that 2 1
dξ = ζ ,
in B 1 r (x0 )
2 1
(4.14)
in B 1 r (x0 )
d ξ = 0, ∗
2 1
and
‖ξ ‖L2 (B4δr
0
(x0 )) ⩽
C ‖ζ ‖L2 (B5δr (x0 )) . 0
(4.15)
Going back to (4.11), we may write
] ωε 2 3 ∗ d vε × dvε − 2 (1 − |vε | )Σi=1 ci (x)dxi − d ψ − ξ = 0 in B 1 r1 (x0 ). 2 α +1 [
(4.16)
Using again the Poincaré lemma, we deduce that there exists some function ϕ uniquely defined in B 1 r (x0 ) (up to a constant) 2 1
such that
vε × dvε =
ωε (1 − |vε |2 )Σi3=1 ci (x)dxi + d∗ ψ + dϕ + ξ in B 1 r1 (x0 ). 2 α +1 2
(4.17)
This is precisely the Hodge–de Rham decomposition of vε × dvε which best fits our needs. We are going to estimate the L2 -norm of each of the four terms on the r.h.s. of (4.17). The estimate for ξ . Since dψ is harmonic in Br1 (x0 ) by (4.12), we have for any k ∈ N,
‖dψ‖
C k B 11
δr 2 0
⩽ (x0 )
C ‖dψ‖L2 (B6δr (x0 )) ⩽ C ‖∇ψ‖L2 (B6δr (x0 )) . 0 0
(4.18)
On the other hand, since ζ = d∗ dψ , it follows that
‖ζ ‖L2 (B5δr
0
(x0 )) ⩽
C ‖∇ψ‖L2 (B6δr (x0 )) , 0
(4.19)
and going back to (4.15) we obtain the estimate
‖ξ ‖L2 (B4δr
0
(x0 )) ⩽
C ‖∇ψ‖L2 (B6δr (x0 )) . 0
(4.20)
The estimate for ϕ . Since d∗ ξ = 0, due to the Hodge–de Rham decomposition (4.17), we have d∗
vε × dvε −
ωε (1 − |vε |2 )Σ ci dxi 2 α +1
= −∆ϕ in B r1 (x0 ). 2
(4.21)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4283
On the other hand, vε satisfies (3.3); we have d
∗
ωε (1 − |vε |2 )Σ ci dxi vε × dvε − 2 α +1
2 ωε 2 = − ∇ fε · (vε × ∇vε ) + 2 |vε |2 x⊥ α · ∇ fε fε α +1 2 ωε = − ∇ fε · dϕ + d∗ ψ + ξ + 2 (1 − |vε |2 )Σ ci dxi fε α +1 ωε 2 + 2 |vε |2 x⊥ α · ∇ fε . α +1
(4.22)
Thus we obtain the equation for ϕ : 2
−∆ ϕ +
fε
2
∇ fε · ∇ϕ = − ∇ fε · (d∗ ψ + ξ + ωε (1 − |vε |2 )Σ ci dxi ) + fε
In view of −∆ϕ +
−div
2 fε
∇ fε · ∇ϕ = −fε2 div
1
=−
∇ϕ 2
fε
2 fε3
1 fε2
ωε 2 |vε |2 x⊥ α · ∇ fε . α2 + 1
(4.23)
∇ϕ , we rewrite (4.23) as
∇ fε · (d∗ ψ + ξ + ωε (1 − |vε |2 )Σ ci dxi ) +
ωε 2 |vε |2 x⊥ α · ∇ fε . α2 + 1
(4.24)
For simplicity, define L , −∆ +
2 fε
∇ fε · ∇.
(4.25)
Now we consider the boundary value problem for some r > 0:
Lu = h, in Br (x0 ) ∂u = p, on ∂ Br (x0 ) ∂n ∫ u = 0.
(4.26)
∂ Br (x0 )
We have the following estimate. Proposition 4.4. Assume that 0 < r < 1 and u satisfies (4.26); then there exists some constant C such that
∫
∫
2
Br (x0 )
|∇ u| ⩽ C
∫
2
h +r
p
Br (x0 )
.
2
∂ Br (x0 )
(4.27)
Proof. Multiplying the first equation of (4.26) by (x − x0 ) · ∇v and integrating by parts, we easily get Pohozaev’s identity for the operator L: 1
∫
2
|∇ u|2 +
∫
Br (x0 )
Br (x0 )
(x − x0 ) · ∇ uh =
r
∫
2 ∂ Br (x0 )
∫ +
|∇ u|2 − r 2
Br (x0 ) fε
∫
2 ∂u ∂ Br (x0 ) ∂ n
⟨∇ fε · ∇ u, (x − x0 ) · ∇ u⟩.
(4.28)
Note that
∫ Br (x0 )
(x − x0 ) · ∇ uh −
∫ ⩽
|∇ u|2
r
2 Br (x0 ) fε
⟨∇ fε · ∇ u, (x − x0 ) · ∇ u⟩
12 ∫
21 h2
r
Br (x0 )
Br (x0 )
∫ ⩽ Cr
∫
2
|∇ u|2
Br (x0 )
∫
|∇ u| + Br (x0 )
∫ + Cr
h Br (x0 )
2
.
Combining this with (4.28) we obtain
∫
2 ∫ ∫ ∂u 1 2 |∇T u| ⩽ + + C |∇ u | + C h2 . r ∂ Br (x0 ) ∂ Br (x0 ) ∂ n Br (x0 ) Br (x0 ) 2
∫
(4.29)
4284
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
On the other hand, standard elliptic estimates [25] yield
∫
∫
2
u ⩽C r
Br (x0 )
4
2
h +r Br (x0 )
3
∫ p
.
2
∂ Br (x0 )
(4.30)
We rewrite (4.26) as
−div
1 fε2
∇u =
1 fε2
h.
(4.31)
Multiplying (4.31) by u and integrating by parts in Br (x0 ) we obtain
∫
∫
2
|∇ u| ⩽ C
Br (x0 )
h
2
12 ∫
12 u
Br (x0 )
2
∫ +
Br (x0 )
p
2
21 ∫
∂ Br (x0 )
21 u
2
∂ Br (x0 )
.
(4.32)
In view of having ∂ B (x ) u = 0, by the Poincaré–Wirtinger inequality we have r 0
∫
2
∂ Br (x0 )
u ⩽
r2 2
∫ Br (x0 )
|∇T u|2 .
Inserting this inequality in (4.29), we thus have
∫ ∂ Br (x0 )
u2 ⩽
r2 2
∫ ∂ Br (x0 )
p2 + C ( r + r 2 )
∫
|∇ u|2 + Cr 2 Br (x0 )
∫ Br (x0 )
h2 .
Inserting (4.30) and (4.33) in (4.32), we complete the proof of Proposition 4.4. Recall that for 0 < r <
(4.33)
, ϕ satisfies Lϕ = h, in Br (x0 ) ∂ϕ = p, on ∂ Br (x0 ) ∂n 1 r 2 1
(4.34)
where
ωε ωε 2 ∗ 2 2 (1 − |vε | )Σ ci dxi + 2 |vε |2 x⊥ h = − ∇ f ε · d ψ + ξ + 2 α · ∇ fε , fε α +1 α +1 ωε ∂v 2 ∗ p = vε × ε − ( 1 − |v | ) Σ c dx + d ψ + ξ · n. ε i i ∂n α2 + 1
(4.35)
Note that (4.34) involves only the gradient of ϕ ; we may assume that
∫ ∂ Br (x0 )
ϕ = 0.
(4.36)
In Lemma 4.2 we obtain rc ∈ ε 2
∫
∫ t
1
∂ Btrc (x0 )
α0 4
R, 41 R with δ r0 < rc < 2δ r0 and
∂vε 2 β ∂ n ⩽ C (η| ln δ| + ε ),
and thus there exists r2 ∈ (rc , 2rc ) ⊂ (δ r0 , 4δ r0 ) such that
∂vε 2 β ∂ n ⩽ C (η| ln δ| + ε ), ∂ Br2 (x0 ) 2 ∫ ∫ 2 2 ωε C ε 2 ωε2 2 2 (1 − |vε | ) ( 1 − |v | ) Σ c dx ⩽ f ε i i ε 2 rc ε2 ∂ Br2 (x0 ) α + 1 B2rc (x0 )\Brc (x0 ) β ⩽ C (η| ln δ| + ε ), ∫ ∫ ∫ C C |∇ψ|2 ⩽ |∇ψ|2 ⩽ |∇ψ|2 , r δ r c 0 ∂ Br2 (x0 ) B2rc (x0 )\Brc (x0 ) B4δ r (x0 ) ∫
0
(4.37)
(4.38) (4.39)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4285
and
∫
C
|ξ |2 ⩽
∂ Br2 (x0 )
∫
rc
B2rc (x0 )\Brc (x0 )
∫
C
|ξ |2 ⩽
δ r0
B4δ r (x0 )
|ξ |2 .
(4.40)
0
Choosing r = r2 in (4.34) and (4.36), we obtain
∫ r2
∂ Br2 (x0 )
p2 ⩽ C δ r0 (η| ln δ| + ε β ) + C
∫
|∇ψ|2 +
∫
B4δ r (x0 )
B4δ r (x0 )
0
|ξ |2 .
(4.41)
0
By the definition of h, we have
∫
∫
2
Br2 (x0 )
∫
2
|ξ | +
|∇ψ| +
h ⩽ C
B4δ r (x0 )
B4δ r (x0 )
B4δ r (x0 )
0
0
0
∫
∫
2
B4δ r (x0 )
|ξ | + B4δ r (x0 )
0
(1 − |vε |2 )2 + ε2
fε2 (bε − |vε |2 )2
∫
2
|∇ψ| +
⩽ C
∫
2
ε2
Br0 (x0 )
0
∫
ωε |∇(fε − a)| 2
B4δ r (x0 )
2
2
0
+ε
β
.
(4.42)
Using Proposition 4.4, and (4.41) and (4.42), we have
∫
β
|∇ϕ| ⩽ C δ r0 (η| ln δ| + ε ) + C 2
Br2 (x0 )
∫
∫
2
|∇ψ| +
fε2 (bε − |vε |2 )2
∫
2
|ξ | +
B4δ r (x0 )
B4δ r (x0 )
0
Br0 (x0 )
0
ε2
,
which yields
∫
β
|∇ϕ| ⩽ C δ r0 (η| ln δ| + ε ) + C 2
Bδ r (x0 ) 0
∫
2
∫
|∇ψ| + B4δ r (x0 )
|ξ | + B4δ r (x0 )
0
fε2 (bε − |vε |2 )2
∫
2
ε2
Br0 (x0 )
0
.
(4.43)
The estimate for ψ . As in [3], we easily get the L2 -norm estimate for ∇ψ :
∫
|∇ψ| ⩽ C (γ + δ ) 2
B6δ r (x0 )
2
3
0
∫
|∇vε | + C (γ 2
Br0 (x0 )
−4 β
ε +
γ −4 Eε (x0 , r0 ))
fε2 (1 − |vε |2 )2
∫ Br0 (x0 )
ε2
.
(4.44)
Estimates for ∇(|vε |2 ) and (1 − |vε |2 )|∇vε |2 . Note that
∆(|vε |2 ) = 2|∇vε |2 −
2
f 2 (bε − |vε |2 )|vε |2 − 2 ε
ε ωε ⊥ −4 2 x · j(vε ). α +1 α
2 fε
∇ fε · ∇(|vε |2 ) + 2
ωε |vε |2 (x⊥ · ∇)fε2 (α 2 + 1)fε2 α (4.45)
Multiplying (4.45) by (bε − |vε |2 ) and integrating in Br1 (x0 ), we get
∫
[ ] 2fε2 2 2 2 2 2 |∇|vε | | + 2 (bε − |vε | ) |vε | ε Br1 (x0 ) ∫ ∫ ∫ ∂|vε |2 ωε =2 (bε − |vε |2 )|∇vε |2 + (bε − |vε |2 ) −4 2 x⊥ · j(vε )(bε − |vε |2 ) ∂n α + 1 Br1 (x0 ) α Br1 (x0 ) ∂ Br1 (x0 ) ∫ ∫ 1 −2 ∇ fε · ∇(|vε |2 )(bε − |vε |2 ) + ∇|vε |2 · ∇ bε Br1 (x0 ) fε
+
2ωε
α +1 2
Br1 (x0 )
|vε |2
∫ Br1 (x0 )
fε2
2 (bε − |vε |2 )(x⊥ α · ∇)fε .
(4.46)
Note that
∫ 12 ∫ 21 ∂|vε |2 fε2 (bε − |vε |2 )2 2 (bε − |vε | ) |∇vε | , ⩽ Cε ∂n ε2 (x ) Br0 (x0 ) Br0 (x0 ) 1 0
∫ ∂ Br
2
(4.47)
4286
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
∫ Br
∫ ∫ fε2 (bε − |vε |2 )2 2 2 (bε − |vε | )|∇vε | ⩽ C γ 2 |∇vε |2 + C γ −2 ε2 (x ) Vγ Wγ 1 0 ∫ ∫ fε2 (bε − |vε |2 )2 ⩽ Cγ 2 |∇vε |2 + C γ −2 , ε2 Br0 (x0 ) Br0 (x0 )
12 ∫ 21 ∫ 2 2 2 f ( b − |v | ) ε ε ε (bε − |vε |2 )∇ fε · ∇(|vε |2 ) ⩽ C ε |∇|vε |2 |2 ε2 Br1 (x0 ) Br1 (x0 ) (x ) fε 1 0 ∫ ∫ 2 1 fε (bε − |vε |2 )2 ⩽ , |∇|vε |2 |2 + C 2 Br (x0 ) ε2 Br0 (x0 ) 1 ∫ 1∫ 2 |∇|vε |2 |2 + C ε β , ∇|vε | · ∇ bε ⩽ Br (x0 ) 4 Br (x0 ) 1 1 ∫ 2ω ∫ 2 fε2 (bε − |vε |2 )2 |vε | ε 2 ⊥ 2 )( x · ∇) f ( b − |v | ⩽C + C εβ , 2 ε ε α ε α + 1 Br (x0 ) fε2 ε2 Br0 (x0 ) 1 ∫ 2 Br
(4.48)
1
12 ∫ 21 ∫ 2 2 2 f ( b − |v | ) ε ε ε 2 2 x⊥ |∇vε |2 α · j(vε )(bε − |vε | ) ⩽ C ε ωε ε2 (x ) Br1 (x0 ) Br1 (x0 ) 1 0 ∫ ∫ 2 fε (bε − |vε |2 )2 . ⩽ Cε |∇vε |2 + C ε2 Br0 (x0 ) Br0 (x0 )
(4.49)
(4.50)
(4.51)
∫ 4ωε Br
(4.52)
Combining (4.46)–(4.52) we have
∫ Br1 (x0 )
|∇|vε |2 |2 ⩽ C (ε + γ 2 )
∫ Br0 (x0 )
|∇vε |2 + C γ −2
fε2 (bε − |vε |2 )2
∫ Br0 (x0 )
ε2
.
(4.53)
Recalling the identity 4|vε |2 |∇vε |2 = 4|vε × ∇vε |2 + |∇|vε |2 |2 , by (4.17), we thus have 4|∇vε |2 = 4|vε × ∇vε |2 + 4(1 − |vε |2 )|∇vε |2 + |∇|vε |2 |2 ⩽ C (|∇ψ|2 + |∇ϕ|2 + |ξ |2 + ωε2 (1 − |vε |2 )2 |Σ ci dxi |2 ) + 4(1 − |vε |2 )|∇vε |2 + |∇|vε |2 |2 . Note that
∫
ωε (1 − |vε | )|Σ ci dxi | ⩽ C ε 2
Bδ r (x0 ) 0
2
2
fε2 (1 − |vε |2 )2
∫ Br0 (x0 )
ε2
.
Combining (4.20), (4.43), (4.44), (4.48), (4.53) and (4.54), we have
∫ Bδ r (x0 ) 0
|∇vε |2 ⩽ C δ r0 (η| ln δ| + ε β ) + C (γ 2 + δ 3 ) fε2 (bε − |vε |2 )2
∫ × Br0 (x0 )
ε2
which completes the proof of Theorem 4.3.
∫ Br0 (x0 )
|∇vε |2 + C (γ −2 + γ −4 Eε (x0 , r0 ))
,
Step 3. Completing the proof of Theorem 4.1. Using (4.6), we have
γ2 2 + δ Eε (x0 , r0 ) δ ∫ 1 fε2 (bε − |vε |2 )2 + C (γ −2 + γ −4 Eε (x0 , r0 )) . δ r0 Br0 (x0 ) ε2
Eε (x0 , δ r0 ) ⩽ C (η| ln δ| + ε β ) + C
(4.54)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4287
By (4.3) and (4.4), we have
Eε (x0 , r0 ) ⩽ 2 Eε (x0 , δ r0 ) + C (η| ln δ| + ε β ) [ 2 ] γ −2 γ γ −4 ⩽ C (η| ln δ| + ε β ) + C (η| ln δ| + ε β ) + C + δ2 + (η| ln δ| + ε β ) Eε (x0 , r0 ) δ δ δ [ 2 ] γ −2 γ γ −4 ⩽ C 1+ (η| ln δ| + ε β ) + C + δ2 + (η| ln δ| + ε β ) Eε (x0 , r0 ). δ δ δ Now, choose δ such that C δ2 ⩽
1 4
,
and then choose γ such that C
γ2 1 ⩽ . δ 4
Hence, there exist ε0 and η0 such that if ε ⩽ ε0 and η ⩽ η0 , we have C
γ −4 1 (η| ln δ| + ε β ) ⩽ . δ 4
Thus
Eε (x0 , r0 ) ⩽ C (η| ln δ| + ε β ). By the monotonicity formula, we obtain 1
ε3
∫ Bε (x0 )
(1 − |vε |2 )2 ⩽ C Eε (x0 , ε) ⩽ C ( Eε (x0 , r0 ) + ε β ) ⩽ C (η| ln δ| + ε β ).
Using Lemma III.3 in [3], we have
1 − |vε (x0 )| ⩽ C
1
51
∫
ε3
1
⩽ C (η| ln δ| + ε β ) 5 ,
(1 − |vε | )
2 2
Bε (x0 )
which completes the proof of Theorem 4.1.
Corollary 4.5. Under the assumption of Theorem 4.1, suppose that 0 < σ < 1, η > 0 and ε0 > 0 are as given in Theorem 4.1. Supposing that B2r (x0 ) ⊂ K , r ⩾ ε
α0 4
, ε ⩽ ε0 and
1 Eε (x0 , r ) ⩽ η| ln ε|,
(4.55)
4
we have
|1 − |vε (x)|| ⩽ σ ∀x ∈ B 3r (x0 ).
(4.56)
4
5. Proof of Theorem 1.1 We divide the proof of Theorem 1.1 into some lemmas. First using Corollary 4.5, as in [3,5,26], we have the following lemma. Lemma 5.1. There exists a constant η0 > 0 such that if x0 ∈ Σµ∗ , we have
Θ∗ (x0 ) ⩾ η0 . By the upper semicontinuity of Θ∗ , we have: Lemma 5.2. Σµ∗ is closed in K . From Lemma 5.1 and Corollary 4.5, we have the following uniform convergence result away from Σµ∗ . Lemma 5.3 ([4]). Let K1 ⊂ K \ Σµ∗ be any compact subset. For any σ > 0, there exists a constant ε2 > 0 depending on K1 and σ such that, for ε < ε2 ,
|1 − |vε || ⩽ σ on K1 .
(5.1)
4288
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
Lemma 5.4. We have
µ∗ = g (x)H 3 + h(x)H 1 ⌊Σµ∗ , where g , h ∈ L∞ loc (K ) and
η0 ⩽ Θ∗ (x) ⩽ h(x) ⩽ Θ ∗ (x) := lim sup
µ∗ (Br (x)) r
r →0
⩽ CM0 .
Proof. Since Σµ∗ is closed in K , Σµ∗ is hence measurable. We have
µ∗ = µ∗ ⌊Σµ∗ + µ∗ ⌊(K \ Σµ∗ ). By Corollary 4.5, the same argument as for Theorem VIII.1 in [3] yields
H 1 (Σµ∗ ) ⩽ CM0 . By the monotonicity formula of Proposition 3.8, we have for any x ∈ K ,
Θ ∗ (x) := lim sup
µ∗ (Br (x)) r
r →0
⩽ CM0 .
According to the Radon–Nikodym theorem, we obtain
µ∗ ⌊Σµ∗ = h(x)H 1 ⌊µ∗ ,
(5.2)
for Θ∗ ⩽ h(x) ⩽ Θ . Suppose that x0 ∈ K \ Σµ∗ and r > 0, such that B(x0 , 2r ) ⊂ K \ Σµ∗ . By Lemma 5.3, we obtain ∗
σ := ‖1 − |vε |‖L∞ (B(x0 ,2r )) = o(1) as ε → 0. On the other hand, we have the estimate
Eε
x0 ,
3 4
δr
⩽ C (δ 3 + σ + ε 2 | ln ε|2 + ε)Eε (x0 , r ) + C ε| ln ε|2 .
(5.3)
Note that σ = σ (ε) = o(1). Dividing both sides of (5.3) by | ln ε| and sending ε → 0 we obtain
3 µ∗ B x0 , δ r ⩽ C δ 3 µ∗ (B(x0 , r )), 4
which implies that µ∗ ⌊(K \ Σµ∗ ) is absolutely continuous with respect to the Lebesgue measure, and by the Radon–Nikodym theorem again we obtain
µ∗ = g (x)H 3 + h(x)H 1 ⌊Σµ∗ ,
(5.4)
for some locally bounded function g. The proof of Lemma 5.4 is completed.
The same proof as for Theorem A.(iv) in [4] gives: Lemma 5.5. We have g (x) = |∇ h∗ (x)|2
a.e. in K ,
where h∗ solves
∆ h∗ +
1 a
∇ a · ∇ h∗ = 0.
(5.5)
Moreover, we have
σ⃗ε =
1
2∇ fε
| ln ε|
fε
· ∇ vε , ∇vε +
fε2
2ε 2
0 in Cloc (K \ Σµ∗ ).
Lemma 5.6. Σµ∗ is rectifiable, and
¯ (x) = ⋆ H
2ω
α +1 2
for µ∗ -a.e. x in Σµ∗ .
x⊥ α ∧⋆
dJ∗ d µ∗
+
dσ⃗∗ d µ∗
,
(bε − |vε | )
2 2
∇ fε fε
−→
∇a a
· ∇ h∗ ∇ h∗
(5.6)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4289
⃗ ∈ D (K , R3 ) be a smooth vector field. Note that Proof. Let X ∫
⃗ =− eε (vε )divX
∫
1
K
2
K
+
∇(|∇vε |2 ) −
fε2 (bε − |vε |2 ) 2ε 2
fε2 (bε − |vε |2 )
ε2
⟨vε , ∇vε ⟩
(bε − |vε |2 )2 ⃗ ∇ bε + fε ∇ fε · X 2ε 2
(5.7)
and
∫ − ∫ ∫ ∂vε ∂vε ∂ X i 1 = − ∆vε ∇vε · X⃗ − ∇(|∇vε |2 ) · X⃗ . ∂ xi ∂ xj ∂ xj 2 K K K
(5.8)
Hence
∫
1
| ln ε|
eε (vε )δij −
K
∂vε ∂vε ∂ xi ∂ xj
∂Xi ∂ xj
∫
fε2 (bε − |vε |2 )vε fε2 (bε − |vε |2 ) (bε − |vε |2 )2 2 ⃗ ⃗ ⃗ = X · ∇vε , ∆vε + − ∇ bε · X − ∇ fε · X | ln ε| K ε2 2ε 2 4ε 2 ∫ 1 2iωε iωε vε ∇ fε2 ⊥ ⊥ 2 = ( x ( x · ∇v , ∇v · ∇)v + · ∇) f − ε ε ε ε | ln ε| K α 2 + 1 α (α 2 + 1)fε2 α fε2 f 2 (bε − |vε |2 ) (bε − |vε |2 )2 2 ⃗ − ε ∇ bε − ∇ fε · X 2 2ε 4ε 2 ∫ ∫ 2ω iωε vε 1 ⊥ ⊥ 2 ⃗ =− ⋆ x ∧ ⋆J (vε ) , X + (x · ∇)fε , ∇vε · X⃗ α2 + 1 α | ln ε| K (α 2 + 1)fε2 α K ∫ 2 ∫ fε (bε − |vε |2 ) 1 ( 2∇ fε · ∇)vε fε2 (bε − |vε |2 )2 ∇ fε 1 ∇ bε · X⃗ − , ∇vε + − · X⃗ , | ln ε| K 2ε 2 | ln ε| K fε 2ε 2 fε 1
where ⋆ refers to the Hodge duality. Set
αij,ε :=
1
| ln ε|
eε (vε )δij −
∂vε ∂vε ∂ xi ∂ xj
.
Write Aε = (αij,ε )n×n (where n = 3); then Aε is a symmetric matrix. By computation, we have: Proposition 5.7. Let Aε be the matrix given above; then: e (v )
ε (1) |aij,ε | ⩽ 3 |εln ε| ; n eε (vε ) (2) det(Aε ) = | ln ε| 1−
e (v )
n eε (vε ) 2 |∇v | ⩽ ; ε eε (vε ) | ln ε| 1
e (v )
ε ε (3) (n − 2) |εln ε| ⩽ tr(Aε ) ⩽ n |εln ε| ;
e (v )
ε (4) |εln ε|
1−
1 eε (vε )
|∇vε |2 is an eigenvalue of Aε , and the other n − 1 eigenvalues are all equal to
Proof. The first inequality is derived from the definition of aij,ε . Note that
|∂i vε ∂j vε | ⩽
|∂i vε |2 + |∂j vε |2 2
⩽ |∇vε |2 ⩽ 2eε (vε ).
Recall that eε (vε ) =
1 2
|∇vε |2 +
fε2 (bε − |vε |2 )2 4ε 2
.
The others are easily derived by using the following lemmas in linear algebra.
⃗ . Lemma 5.8. If A = In + α ⃗ β⃗ t , where we have α⃗ , β⃗ ∈ Fn and F is a field, then |A| = 1 + ⟨⃗ α , β⟩ Lemma 5.9. If α ⃗ = (a1 , . . . , an )t , then |⃗ α |2 is an eigenvalue of the matrix α⃗ α⃗ t .
eε (vε ) . | ln ε|
(5.9)
4290
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291 e (v )
ε If µε also denotes its coefficient |εln ε| , then we get
|αij,ε | ⩽ 3µε ,
|σ⃗ε | ⩽ C µε .
(5.10)
Passing to a subsequence, we may assume that
αij,ε ⇀ αij,∗ in the sense of measures. Then by (5.10), we obtain
|αij,∗ | ⩽ 3µ∗ ,
|σ⃗∗ | ⩽ C µ∗ .
Therefore we may write
αij,∗ (x) = Aij (x)µ∗ ,
⃗ µ∗ for µ∗ -a.e. x ∈ K , σ⃗ = m
where (Aij (x)) is a symmetric matrix, with trace less than or equal to 1 and eigenvalues less or equal to 1. From (5.10), we also have Aij ⩾ −3δij
for µ∗ -a.e. x ∈ K .
Now we deal with the second and third terms of the last expression in (5.9). For the second term, because (x⊥ α · ∇)a = 0 and in view of Lemma 3.1(c), we have
∫ 1/2 ∫ 1/2 ∫ 1 iωε vε ⊥ 2 2 ⊥ 2 2 ⃗ (xα · ∇)fε , ∇vε · X ⩽ C |∇vε | |(xα · ∇)(fε − a)| | ln ε| 2 2 K (α + 1)fε K K ⩽ C ε| ln ε|1/2 → 0
as ε → 0.
(5.11)
For the third term, recall that bε = 1 − ε 2 gε /fε2 and ‖∇ bε ‖L∞ ⩽ C ε 2 | ln ε|2 ; we get
1/2 ∫ ∫ 2 1 fε2 (bε − |vε |2 )2 fε (bε − |vε |2 ) ⃗ ∇ bε · X ⩽ C ε| ln ε| | ln ε| 2ε 2 2ε 2 K
K
⩽ C ε| ln ε|3/2 → 0
as ε → 0.
(5.12)
Therefore, passing to the limit in (5.9), we have
∫
∫ 2ω dJ∗ ⊥ ⋆ x ∧⋆ , X⃗ dµ∗ (x) α2 + 1 α d µ∗ K ∫ ∫ ∇a ⃗ , X⃗ ⟩dµ∗ (x)⌊Σµ∗ . − · ∇ h∗ ∇ h∗ , X⃗ dx − ⟨m
∂Xi dµ∗ (x) = − Aij (x) ∂ xj K
a
K
(5.13)
K
where J∗ is the limit of J (vε ); see Theorem 2 in [1]. We decompose the l.h.s. of (5.13) as
∫
∂Xi Aij (x) dµ∗ (x) = ∂ xj K
∫
∂Xi Aij (x) dµ∗ (x)⌊Σµ∗ + ∂ xj K
∫
|∇ h∗ |2
K
2
∂ h∗ ∂ h∗ ∂ X i δij − dx. ∂ xi ∂ xj ∂ xj
(5.14)
Since div(a∇ h∗ ) = 0, a direct computation yields
∫
|∇ h∗ |2
K
2
∫ ∂ h∗ ∂ h∗ ∂ X i ∇a ⃗ δij − dx = − · ∇ h∗ ∇ h∗ , X dx. ∂ xi ∂ xj ∂ xj a K
(5.15)
Combining (5.13)–(5.15), we obtain
∫
Aij (x) K
∂Xi dµ∗ (x)⌊Σµ∗ = − ∂ xj
∫ ∫ 2ω dJ∗ ⊥ ⃗ ⃗ , X⃗ ⟩dµ∗ (x)⌊Σµ∗ . ⋆ x ∧ ⋆ , X d µ ( x ) − ⟨m ∗ α2 + 1 α dµ∗ K K
Since the support of Tω is included in Σµ∗ , we have
∫
∂Xi Aij (x) j dµ∗ (x)⌊Σµ∗ = − ∂x K
∫ 2ω dJ∗ ⊥ ⃗ ⃗ , X dµ∗ (x)⌊Σµ∗ ⋆ x ∧⋆ +m α2 + 1 α d µ∗ K ∫ 2ω dJ∗ dσ⃗∗ ⊥ ⃗ =− x ∧ ⋆ + , X dµ∗ (x)⌊Σµ∗ . ⋆ α2 + 1 α d µ∗ d µ∗ K
(5.16)
L. Zhou et al. / Nonlinear Analysis 74 (2011) 4274–4291
4291
⃗ is arbitrary, (5.16) implies that the generalized 1-varifold (defined in [27]) Since X V := δAij (x) µ∗ ⌊Σµ∗ has a first variation. By (5.16), Lemma 5.1 and Theorem (3.8)(c) in [27], we obtain that V is a real rectifiable 1-varifold. In particular, Σµ∗ is rectifiable, which implies
Θ∗ (x) = Θ ∗ (x) for µ∗ -a.e. x ∈ Σµ∗ , and thus
µ∗ = |∇ h∗ |2 H 3 + Θ∗ (x)H 1 ⌊Σµ∗ and
V = V (Σµ∗ , Θ∗ ). By (5.16), we also have
⃗ =⋆ H
2ω
α2 + 1
x⊥ α ∧⋆
dJ∗ d µ∗
+
dσ⃗∗ d µ∗
for µ∗ -a.e. x ∈ Σµ∗ .
The proof of Theorem 1.1 is now completed.
Acknowledgements We wish to express our gratitude to the reviewers who carefully read and checked this paper. They gave us helpful suggestions and comments. References [1] R.L. Jerrard, Local minimizers with vortex filaments for a Gross–Pitaevsky functional, ESAIM Control Optim. Calc. Var. 13 (2007) 35–71. [2] F. Bethuel, H. Brezis, F. Hélein, Ginzburg–Landau Vortices, Birkhäuser, Boston, 1994. [3] F. Bethuel, H. Brezis, G. Orlandi, Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions, J. Funct. Anal. 186 (2001) 432–520; 188 (2002) 548–549 (erratum). [4] F. Bethuel, G. Orlandi, D. Smets, Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Ann. of Math. (2) 163 (2006) 37–163. [5] F. Bethuel, G. Orlandi, D. Smets, Vortex rings for the Gross–Pitaevskii equation, J. Eur. Math. Soc. 6 (2004) 17–94. [6] L. Lassoued, P. Mironescu, Ginzburg–Landau type energy with discontinuous constraint, J. Anal. Math. 77 (1999) 1–26. [7] A. Aftalion, T. Rivière, Vortex energy and vortex bending for a rotating Bose–Einstein condensate, Phys. Rev. A 64 (2001) 043611. [8] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986) 55–64. [9] F.H. Lin, T. Rivière, A quantization property for static Ginzburg–Landau vortices, Comm. Pure Appl. Math. 54 (2001) 206–228. [10] K. Madison, F. Chevy, J. Dalibard, W. Wohlleben, Vortex formation in a stirred Bose–Einstein condensate, Phys. Rev. Lett. 84 (2000) 806–809. [11] K. Madison, F. Chevy, J. Dalibard, W. Wohlleben, Vortices in a stirred Bose–Einstein condensate, J. Mod. Opt. 47 (2000) 2715. [12] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Observation of vortex lattices in Bose–Einstein condensate, Science 292 (2001) 476–479. [13] C. Raman, J.R. Abo-Shaeer, J.M. Vogels, K. Xu, W. Ketterle, Vortex nucleation in a stirred Bose–Einstein condensate, Phys. Rev. Lett. 87 (2001) 210402. [14] P. Rosenbusch, V. Bretin, J. Dalibard, Dynamics of a single vortex line in a Bose–Einstein condensate, Phys. Rev. Lett. 89 (2002) 200403. [15] E.H. Lieb, R. Seiringer, J. Yngvason, Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional, Phys. Rev. A 61 (2000) 043602. [16] E.H. Lieb, R. Seiringer, Derivation of the Gross–Pitaevskii equation for rotating Bose gases, Comm. Math. Phys. 264 (2006) 505–537. [17] S. Serfaty, On a model of rotating superfluids, ESAIM Control Optim. Calc. Var. 6 (2001) 201–238. [18] A. Aftalion, Q. Du, Vortices in a rotating Bose–Einstein condensate: critical angular velocities and energy diagrams in the Thomas–Fermi regime, Phys. Rev. A 64 (2001) 063603. [19] R. Ignat, V. Millot, The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate, J. Funct. Anal. 233 (2006) 260–306. [20] A. Aftalion, S. Alama, L. Bronsard, Giant vortex and the breakdown of strong pinning in a rotating Bose–Einstein condensate, Arch. Ration. Mech. Anal. 178 (2005) 247–286. [21] S. Alama, L. Bronsard, J.A. Montero, Vortices for a rotating toroidal Bose–Einstein condensate, Arch. Ration. Mech. Anal. 187 (3) (2008) 481–522. [22] A. Aftalion, R.L. Jerrard, On the shape of vortices for a rotating Bose–Einstein condensate, Phys. Rev. A 66 (2002) 023611. [23] J. Bourgain, H. Brezis, P. Mironescu, On the structure of the Sobolev space H 1/2 with values into the circle, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000) 119–124. [24] F. Bethuel, G. Orlandi, D. Smets, Approximations with vorticity bounds for the Ginzburg–Landau functional, Commun. Contemp. Math. 6 (2004) 803–832. [25] L.C. Evans, Partial Differential Equations, in: Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. [26] Z. Liu, Vortices set and the applied magnetic field for superconductivity in dimension three, J. Math. Phys. 46 (2005) 052111. [27] L. Ambrosetti, M. Soner, A measure theoretic approach to higher codimension mean curvature flow, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 25 (1997) 27–49.