Vortex dynamics for the two dimensional non homogeneous Gross-Pitaevskii equation

0.33 MB·English

by  Robert L. Jerrard

Vortex dynamics for the two dimensional non homogeneous
Gross-Pitaevskii equation
Robert L. Jerrard & Didier Smets
3
1
0
2 Abstract

n
WederivetheasymptoticaldynamicallawforGinzburg-Landauvorticesinaninhomo-
a
J geneousbackgrounddensityundertheSchr¨odingerdynamics,whentheGinzburg-Landau
2 parameter goes to zero. New ingredients involve across the cores lower bound estimates
2 and approximations.

]
P 1 Introduction
A
.
h We are interested in the two dimensional Gross-Pitaevskii equation
t
a
1
m i∂ u ∆u+ V(x)+ u2 u = 0 (GP)
t − ε2 | |
[
1 for u :R2 R+ C, where 0 < ε 1 and V(cid:0) :R2 R+(cid:1)is a smooth potential such that
v × → ≪ →
3
V(x) + as x + .
1 → ∞ | | → ∞
2
5 The Gross-Pitaevskii equation is a widely used model to describe the dynamics of a Bose-
1. Einstein condensate in a trapping potential V. The equation on R2 arises via dimension
0 reduction from 3 dimensions; this has been justified for particular choices of V in [1] for
3
example.
1
: Equation (GP) is Hamiltonian, with Hamiltonian given by
v
i
X u2 1 u2 u4
(u) = |∇ | + V(x)| | + | | .
ar Eε,V ZR2 2 ε2 (cid:18) 2 4 (cid:19)
Another quantity which is preserved by the flow associated with (GP) is the total mass M,
given by
M(u) = u2.
R2| |
Z
For each m > 0, there exists1 at least one positive ground state η η : R2 R+ of total
ε,m
≡ →
mass equal to m. By definition, a ground state η realizes the infimum
(η) = inf (g), g H1(R2,C), M(g) = m ,
ε,V ε,V
E {E ∈ }
1We refer to Section 9 for the details on a number of statements regarding the ground states which we
state without justification in this introduction.
1
and satisfies the Euler-Lagrange equation
1 1
∆η+ (V +η2)η = λη,
− ε2 ε2
where we write the Lagrange multiplier as 1 λ for some λ λ .
ε2 ≡ ε,m
In the limit ε 0, we have
→
η2 ρ in L2(R2),
→ TF
where the function ρ , known as the Thomas-Fermi profile in the physics literature, is given
TF
byρ (x) := (λ V)+(x)wherethenumberλ isuniquelydeterminedbythemasscondition
TF 0− 0
(λ V(x))+dx = m.
0
R2 −
Z
We will study the behaviour of solutions of (GP) which correspond, in a sense to be made
precise later, to perturbations of the ground state η by a finite number of quantized vortices,
each carrying a single quantum of vorticity. Our goal is to prove that these vortices persist,
and to describe their evolution in time.
We will show that to leading order the vortices do not interact, and that each one evolves
(in a renormalized time scale) by the orthogonal gradient flow for the function logρ , with
TF
a sign depending on the winding number of the given vortex. More precisely, let
Ω := x : ρ (x) > 0
TF { TF }
be the interior of the limiting support2 of the ground state, let b0 l be distinct points in
{ i}i=1
Ω , and let d ,...,d 1,+1 . For each i 1, ,l , we denote by b (t) the solution of
TF 1 l ∈ {− } ∈ { ··· } i
the ordinary differential equation
⊥ρ
b˙ (t) = d ∇ TF(b (t)), (1)
i i i
ρ
TF
where ⊥ = ( ∂ ,∂ ), with initial datum b (0) = b0.
∇ − x2 x1 i i
Theorem 1. Let (u0) be a family of initial data for (GP) such that
ε ε>0
M(u0) = m,
ε
l
(u0) (η)+π ρ (b0)logε +o(logε),
Eε,V ε ≤ Eε,V TF i | | | |
i=1
X
and
j(u0) l
curl ε 2π d δ in W−1,1(Ω ),
ρ −→ i b0i loc TF
TF i=1
(cid:0) (cid:1) X
as ε 0. Then, as long as the points b (t) l remain distinct,
→ { i }i=1
j(ut) l
curl ε 2π d δ in W−1,1(Ω ),
ρ −→ i bi(t) loc TF
TF i=1
(cid:0) (cid:1) X
as ε 0, where ut := u (,t logε).
→ ε ε · | |
2Note that we do not assume that ΩTF is simply connected or that its boundary is smooth.
2
Here, j(ut) := (iut, ut) where (z,w) := Im(zw¯). Therefore,
ε ε ∇ ε
1 j(ut) 1 ut ut
curl ε = curlj ε = J ε
2 ρ 2 √ρ √ρ
TF TF TF
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
is the Jacobian determinant of ut/√ρ . It is widely recognized, in the present regime for
ε TF
the Ginzburg-Landau energy, that the notion of a vortex of winding number d located at
i
the point b (t) is appropriately described by the presence of the term 2πd δ in the limit
i i bi(t)
of the vorticity field curlj ut/√ρ .
ε TF
Remark 1. Note that th(cid:0)e ordinar(cid:1)y differential equations (1) are decoupled. Also, since
ρ (b (t)) = ρ (b0) for any t R the points b (t) l remain distinct for all times unless
TF i TF i ∈ { i }i=1
two of them are located on the same level line of ρ and have opposite circulations.
TF
Results of this sort in the homogeneous case η 1 were first proved in the late 1990s,
≡
see [5, 6], and have subsequently been developed by a number of authors, see for example
[15, 2, 11]. The point of this paper is thus to understand the effect of the inhomogeneity on
the dynamical law for the vortices.
We remark that a number of authors have studied questions about vortex dynamics in in-
homogeneous backgrounds for parabolic equations [14, 12], or more recently [19] for a quite
general class of equations of mixed parabolic-Schro¨dinger type. The case of pure Schro¨dinger
dynamics presents distinct difficulties and as far as we know has not been treated until now.
The sequel of this introduction is devoted to the presentation of the strategy leading to
Theorem 1. We notice that will actually prove a result (Theorem 2 below) which is stronger
in two respects than Theorem 1: first it describes the dynamics of vortices at small but fixed
value of ε, rather than asymptotically as ε 0 in Theorem 1, and second it applies to a
→
broader class of inhomogeneous equations (see (NHG) below) where η need not necessarily
be the profile of a ground state.
1.1 Perturbation equation and Theorem 2
For the class of initial data which we consider in Theorem 1, it is convenient to rewrite the
corresponding solutions of (GP) in the form
u(x,t) = η(x)w(x,t) (2)
and to study the evolution equation for w. One easily checks that if u is a solution to (GP),
then w solves
1 λ
iη2∂ w div(η2 w)+ η4(w 2 1)w = η2w.
t − ∇ ε2 | | − −ε2
In particular, the change of phase and time scale
λ t t
v(x,t) = exp i w x,
ε2 logε logε
(cid:18) | |(cid:19) (cid:18) | |(cid:19)
leads to the equation
1
ilogεη2∂ v div(η2 v)+ η4(v 2 1)v = 0 (NHG)
| | t − ∇ ε2 | | −
3
for v. Note that the change of time scale is related to thefact that the phenomenon which we
wish to describe, namely vortex motion, arises in times of order one in that new time scale
(see the definition of ut in the statement of Theorem 1).
ε
Our analysis will henceforth focus on equation (NHG). Equation (NHG), like (GP), is
Hamiltonian, with Hamiltonian given by the weighted Ginzburg-Landau energy
v 2 (v 2 1)2
E (v) e (v) = η2|∇ | +η4 | | − . (3)
ε,η ≡ R2 ε,η R2 2 4ε2
Z Z
As a matter of fact, using the Euler-Lagrange equation for η, one realizes that
λ
(u) = (η)+E (v)+ M(u) M(η) . (4)
Eε,V Eε,V ε,η 2ε2 −
(cid:16) (cid:17)
In the sequel, we enlarge our framework and consider equation (NHG) where η : R2 R
→
is any smooth positive function such that the corresponding Cauchy problem is globally
well-posed for initial data in H1(R2,ηdx) and such that the corresponding solutions can be
approximated by smooth solutions3. In particular, under those assumptions the energy E
ε,η
is preserved along the flow of (NHG).
Let ε> 0 and let Ω R2 be a boundedopen set. Let a0 l bedistinct points in Ω, and
⊂ { i}i=1
let d ,...,d 1,+1 . For each i 1, ,l , we denote by a (t) the solution, as long as
1 l i
∈ {− } ∈ { ··· }
it does not reach ∂Ω, of the ordinary differential equation
a˙ (t) = d ⊥logη2(a (t)), (5)
i i i
∇
where ⊥ = ( ∂ ,∂ ), with initial datum a (0) = a0.
∇ − x2 x1 i i
We assume that
η := inf η(x) > 0,
min
x∈Ω
and we fix a time T > 0 such that
col
1
ρ := min min d(a (t),a (t)) d(a (t),∂Ω) 1 > 0. (6)
min i j i6=j i i
t∈[0,Tcol] {2 } ∪{ } ∪{ }
(cid:8) (cid:9)
Finally, we consider a finite energy solution v of (NHG), we set vt := v(,t) and we define,
·
for t [0,T ],
col
∈
l
rt := Jvt π d δ , (7)
a k − i ai(t)kW−1,1(Ω)
i=1
X
and
E (vt) l
Σt := ε,η π η2(a (t)). (8)
i
logε −
| | i=1
X
We will deduce Theorem 1 from
3Thiscanbeverifiedforawidevarietyofweightfunctionsη,butwewishnotconsiderthatdiscussionhere
since we already know by means of the changeof unknown(2) that it satisfied when η is a ground state.
4
Theorem 2. There exist positive constants ε , γ and C , depending only on l, η , ρ ,
0 0 0 min min
and η2 , such that if 0< ε ε and if Σ0+r0 γ , then
k∇ kL∞(Ω) ≤ 0 a ≤ 0
log logε
rat ≤ ra0+ Σ0+ra0+ lo|gε | eC0t−1 +C0ε12, (9)
(cid:16) | | (cid:17)
(cid:0) (cid:1)
as long as t T and Σ0+rt(t) γ .
≤ col a ≤ 0
Remark 2. i) As we shall discuss in Section 1.2 below, the quantity rt, which is a sort
a
of discrepancy measure, can be thought of as measuring the distances between the “actual
vortex locations” and the desired vortex locations. The quantity Σt, multiplied by logε,
| |
corresponds to the excess of energy of the solution v with respect to an energy minimizing
field possessingthe vortices at the points a (t). Notice that since E is preserved by the flow
i ε,η
for v and η2 is preserved by the flow for the a′s, we have
i
Σt Σ0, t [0,T].
≡ ∀ ∈
ii) Theorem 2 is interesting for initial data such that Σ0 +r0 is small. The existence of
a
such data is standard. For example, if we fix f : [0, ) [0,1] such that f′ 0, f(0) = 0,
∞ → ≥
and f(s) 1 as s , then for the initial data
→ → ∞
l z a z a0 di
v0(z) := f(| − i|) − i , x := (x ,x )= z = x +ix ,
ε z a0 1 2 ∼ 1 2
i=1 (cid:18)| − i|(cid:19)
Y
one can check that Σ0 C logε−1,r0 Cε. In any case, (9) contains the error term in
≤ | | a ≤
log logε/logε which implies that (9) only yield the inequality Σ0 +rt γ for times at
| | | | a ≤ 0
most of order log logε.
| |
iii) One could supplement the claims of Theorem 2 with closeness estimate for j(v) to a
reference field j of very simple form. This would follow from an application of Corollary 1
∗
below;howeveratthelevelofapproximationwhichwehaveadoptedhereitisonlymeaningful
in a neighborhood of size o(1/logε) of the vortex core.
| |
1.2 Elements in the proofs
Under the conditions that will prevail throughout most of this paper, we will be able to
identify points ξt,...ξt and a number rt such that
1 l ξ
l
Jvt π d δ rt ε1/2 rt. (10)
k − i ξitkW−1,1(Ω) ≤ ξ ≤ ≪ a
i=1
X
This is expressed in Proposition 2 below, and entitles us to think of ξt,i = 1,...,l as being
i
the “actual locations of the vortices” in vt, up to precision of order rt. Admitting this
≤ ξ
interpretation, then basic facts about the W−1,1 norm, recalled in Section 2, imply that
l
1
rt = (1+o(1)) ξt at (11)
a π | i − i|
i=1
X
5
is essentially the aggregate distance between the actual vortex locations and the desired
vortex locations, as remarked above.
Heuristic considerations also suggest that if vt is a function with vortices at points
ξt,...,ξt (or more precisely, if (10) holds), then
1 l
l
E (vt)' π logε(1 o(1)) η2(ξt). (12)
ε,η | | − i
i=1
X
Hence E (vt) π logε l η2(ξt) corresponds to energy that is not committed to the
ε,η − | | i=1 i
vortices, and this energy in principle can cause difficulties for our analysis. From (10), (11),
P
we have
l
1
E (vt) π logε η2(ξt) logε Σt+ (1+o(1)) η2 rt
ε,η − | | i ≤ | | π k∇ k∞ a (13)
i=1 (cid:18) (cid:19)
X
logε Σ0+Crt .
≤ | | a
(cid:0) (cid:1)
For our analysis, it suffices to use estimates in the spirit of (13) that are a little weaker than
those suggested in (13), these are established in Proposition 1. We expect from (12) and
(13) that control of rt should yield a good deal of information about vt. This is expressed in
a
Proposition 3, where we compare j(vt) to a reference field jt. An important feature of that
∗
approximation is that it holds across the vortex core.
Inordertocontroltheevolutionintimeofrt,werelyonsomeevolutionequationssatisfied
a
by smooth solutions of (NHG). Conservation of energy is a consequence of the identity
∂ e (v) = div(η2( v,v )), (14)
t ε,η t
∇
and the canonical equation for conservation of mass can be written
logε
| |∂ η2(v 2 1) = (η2j(v)). (15)
t
2 | | − ∇·
(cid:0) (cid:1)
Thevorticity Jv satisfies an evolution equation that it is convenient to write in integral form:
d 1 η2 η2
ϕJv = ǫ ϕ xk v v +δ (v 2 1)2 +ǫ ϕ v v (16)
dt logε lj xl η2 xj · xk jkε2 | | − lj xkxl xj · xk
ZΩ | | ZΩ (cid:20) (cid:21) !
where ϕ is any smooth, compactly supportedtest function and ε is the usual antisymmetric
lj
tensor. This follows from the fact that Jv = 1curlj(v) together with the equation for the
2
evolution of j(v), which is obtained from (NHG) after multiplying by v and rewriting the
∇
result.
Identity (16) is central to our analysis of vortex dynamics, as in previous works [5, 6, 15,
2, 11] on the homogeneous case (for which of course (16) still holds, with η 1). Under the
conditions that Jv is approximately a measure of the form π l d δ , wh≡ere ξ (t) are the
i=1 i ξi(t) i
vortex locations and d 1 their signs, one expects the left-hand side of (16) to satisfy
i
∈ {± } P
d d d
ϕJv ϕ(π d δ ) (π d ϕ(ξ (t))) = π d ϕ(ξ (t)) ξ˙(t).
dt ≈ dt i ξi(t) ≈ dt i i i∇ i · i
ZΩ ZΩ
X X X
6
Assuming that this holds, then to understand the vortex velocities ξ˙, it only suffices to
i
understand the right-hand side of (16). It turns out that it also suffices to consider test
functions ϕ that are linear near each vortex. For such test functions, in the homogeneous
case η2 0, the integrand on the right-hand side of (16) is supportedaway from the vortex
∇ ≡
locations, and one is thus able to control vortex dynamics by controlling terms of the form
v v away from the vortex cores. This argument is a key feature of all existing work on
xi · xj
vortex dynamics in the homogeneous case.
When η2 = 0, it becomes necessary to control terms like v v across the vortex cores.
∇ 6 xi· xj
Carrying this out, in particular relying on the approximation given by Proposition 3, is the
main new point in our analysis. Once this is established, the whole argument is completed
by using a Gronwall type argument on a quantity related to rt, namely Σ0 +g(rt), where
a a
the function gb is defined in (19). This demonstrates in particular that the new information
found in Proposition 3 is strong enough to close the estimates and conclude the proof.
Acknowledgments. This research was partially supported by the National Science and
Engineering Research Council of Canada under operating Grant 261955, as well as by the
projects “Aroundthedynamics of the Gross-Pitaevskii equation” (JC09-437086) and “Schro¨-
dinger equations and applications” (ANR-12-JS01-0005-01) of the Agence Nationale de la
Recherche.
2 A useful lemma
We frequently use the W−1,1 norm. The specific convention we use is in our definition is
1,∞
µ := sup µ,ϕ : ϕ W (Ω),max ϕ , ϕ 1 .
k kW−1,1(Ω) {h i ∈ 0 {k k∞ k∇ k∞} ≤ }
In this paper, we will only use this norm on measures or more regular objects, although of
course it is well-defined for a somewhat larger class of distributions.
Thefollowinglemma, whichwewillusenumeroustimes, isaneasyspecialcaseofclassical
results (see [3] for example).
Lemma 1. Suppose that Ω is an open subset of Rn, and that a l are distinct points in
{ i}i=1
Ω. Define ρ := min 1 a a d(a ,∂Ω) 1 . Given any points ξ l in Ω
a {{2| i − j|}i6=j ∪{ i }i ∪{ }} { i}i=1
and d l 1 l, if
{ i}i=1 ∈ {± }
l
1
d δ(a ξ ) ρ , (17)
k i i − i kW−1,1(Ω) ≤ 4 a
i=1
X
then (after possibly relabelling the points ξ l ),
{ i}i=1
l l
d δ(a ξ ) = a ξ . (18)
i i i W−1,1(Ω) i i
k − k | − |
i=1 i=1
X X
In the remainder of this paper, we will always tacitly assume that under the conditions
of the lemma, the points ξ are labelled so that the conclusion holds.
i
We give the short proof for the reader’s convenience.
7
Proof. For i = 1,...,l, define ϕ (x) := d (1ρ x a )+. Then max( ϕ , ϕ ) =
i i 2 a −| − i| k ik∞ k∇ ik∞
max(1ρ ,1) = 1, for every i, so
2 a
l l
ρ ρ
d δ(a ξ ) d (δ δ ),ϕ = a d d ( a ξ a )+.
k i i − i kW−1,1(Ω) ≥ h j aj − ξj ii 2 − i j 2 −| j − i|
i=1 j=1 j
X X X
Then (17) implies that ξ l B(a ,ρ /2) is nonempty for every i. Since B(a ,ρ /2) l
{ j}j=1∩ i a { i a }i=1
arepairwisedisjoint,itfollows(afterpossiblyreindexing)that ξ l B(a ,ρ /2) = ξ for
{ j}j=1∩ i a { i}
alli. Now letϕ = ϕ . Thefunctions ϕ havedisjointsupport,somax( ϕ , ϕ ) =
i i { i} k k∞ k∇ k∞
1, and thus l d δ(a ξ ) l d (δ δ ),ϕ = l a ξ . On the
k i=P1 i i − i kW−1,1(Ω) ≥ h i=1 i ai − ξi i i=1| i − i|
other hand, if ψ is any compactly supported function such that max( ψ , ψ ) 1,
∞ ∞
P P Pk k k∇ k ≤
then
l l l
d (δ δ ),ψ ψ(a ) ψ(ξ ) a ξ .
h i ai − ξi i ≤ | i − i |≤ | i − i|
i=1 i=1 i=1
X X X
Hence l d δ(a ξ ) l a ξ .
k i=1 i i − i kW−1,1(Ω) ≤ i=1| i − i|
P P
3 Relating weighted and unweighted energy
In this section, we relate the weighted and unweighted energy under some localization as-
sumptions on the Jacobian. For a measurable subset A R2 and v H˙1(A,C) we set
⊂ ∈
E (v;A) := e (v) and E (v;A) := E (v,A).
ε,η ε,η ε ε,1
ZA
Define the function g on R+ by
x+ |logx| if x > 1
g(x) = |logε| |logε| (19)
1+log|logε|
( otherwise.
|logε|
We have
Proposition 1. Let Ω R2 an open set, a l distinct points in Ω, d l 1 ,
⊂ { i}i=1 { i}i=1 ∈ {± }
and η : Ω R a positive Lipschitz function such that inf η =: η > 0. Set ρ :=
Ω min a
→
min 1d(a ,a ) d(a ,∂Ω) 1 , and let ε exp( 8 ) and v H˙1(Ω,C) be such
{{2 i j } i6=j ∪{ i }i ∪{ }} ≤ −ρa ∈
that
+
l
E (v)
Σ := ε,η π η2(a ) < + . (20)
a i
logε − ∞
!
| | i=1
X
Assume also that
l
ρ
a
r := Jv π d δ . (21)
a k − i aikW−1,1(Ω) ≤ 8
i=1
X
Then there exists a constant C, depending only on l, η2 and η , such that
∞ min
k∇ k
E (v;B(a ,R))
ε˜ i
π+C(Σ +g(r )) for i = 1,...,l
a a
logε˜ ≤
| | (22)
E (v;Ω l B(a ,R))
ε˜ \∪i=1 i C(Σ +g(r ))
a a
logε˜ ≤
| |
8
where R = 4max(r , logε−1) ρa and ε˜:= ε , and the function g is defined in (19).
a | | ≤ 4 ηmin
Proof. Let r [r , ρa] be a number that will be fixed later. Then the balls B(a ,4r) l
∈ a 8 { i }i=1
are disjoint and contained in Ω. Let i 1, ,l , by monotonicity of the W−1,1 norms with
∈{ ··· }
respect to the domain, we deduce from (21) that
Jv πd δ r r.
k − i aikW−1,1(B(ai,4r) ≤ a ≤
It follows from the lower bounds estimates of Jerrard [8] or Sandier [18] that
4r
E (v,B(a ,4r)) πlog K , (23)
δ i 1
≥ δ −
for every δ > 0, where K is a universal constant. We next write
1
v 2 (v 2 1)2
E (v,B(a ,4r)) = η2(x)|∇ | +η4(x) | | −
ε,η i 2 4ε2
ZB(ai,4r)
v 2 (v 2 1)2
η2(x) |∇ | + | | −
≥ 2 2 (24)
ZB(ai,4r) 4 ε
(cid:2) ηmin (cid:3)
(cid:16) (cid:17)
inf η2(x) E (v,B(a ,4r)).
ε˜ i
≥ (cid:18)x∈B(ai,4r) (cid:19)
Therefore, from (23) with the choice δ = ε˜, and noting that logr log ρa log8 1, we
| |≥ | 8 | ≥ ≥
obtain
E (v,B(a ,4r)) η2(a )π logε K (r logε + logr ), (25)
ε,η i i 2
≥ | |− | | | |
where K depends only on η2 and η .
2 ∞ min
k∇ k
On the other hand, we deduce from (20) and (25) that
E (v,B(a ,4r)) E (v,Ω) E (v,B(a ,4r))
ε,η i ε,η ε,η j
≤ −
j6=i (26)
X
πη2(a )logε +Σ logε +(l 1)K (r logε + logr ).
i a 2
≤ | | | | − | | | |
Hence, going back to (24) we obtain
1
E (v,B(a ,4r)) E (v,B(a ,4r))
ε˜ i ε,η i
≤ infx∈B(ai,4r)η2(x) (27)
π logε˜ +K (Σ logε +r logε + logr ),
(cid:0) 3 a (cid:1)
≤ | | | | | | | |
where K depends only on l, η2 and η .
3 ∞ min
k∇ k
Concerning the energy outside the balls B(a ,4r), we have from (20) and (25)
i
E (v,Ω B(a ,4r)) = E (v,Ω) E (v,B(a ,4r))
ε,η i i ε,η ε,η j
\∪ −
i (28)
X
Σ logε +lK (r logε + logr ).
a 2
≤ | | | | | |
Hence,
1
E (v,Ω B(a ,4r)) E (v,Ω B(a ,4r))
ε˜ \∪i i ≤ infη2 ε,η \∪i i (29)
K (Σ logε +r logε + logr ),
4 a
≤ | | | | | |
9
where K depends only on l, η2 and η .
4 ∞ min
k∇ k
The function r r + logr /logε is minimized taking r := max(r , 1 ), in which
7→ | | | | a |logε|
cas r ρa by assumption on r and ε. The conclusions (22) follow with the choice C :=
≤ 8 a
max(K ,K ).
3 4
Remark 3. If we define Σ˜ := Eε,η(v) π l η2(a ) , then (25) implies that
a |logε| − i=1 i
P
E (v,B(a ,4r)) logr
Σ˜ ε,η j πη2(a ) lK (r+ | |)
a i 2
≥ logε − ≥ − logε
i (cid:18) | | (cid:19) | |
X
for every r [r , ρa]. Choosing r = max(r , 1 ) as above, we find that Σ˜ lK g(r ).
∈ a 8 a |logε| a ≥ − 2 a
In particular, Σ = (Σ˜ )+ Σ˜ +2lK g(r ). So all our estimates remain true if we replace
a a a 2 a
≤
C(Σ +g(r )) by C(Σ˜ +(2lK +1)g(r )).
a a a 2 a
4 Improved localization for Jacobians
In this section, we prove that if the Jacobian of a function v is known to be sufficiently
localized, then, provided the excess energy of v with respect to the points of localization is
not to big, the localisation is actually potentially much stronger. A result in the same spirit
was obtained by Jerrard and Spirn in [10] for the Ginzburg-Landau functional without a
weight. Our proof here below makes a direct use of Theorem 1.1 and Theorem 1.2′ in [10] by
relating the weighted and unweighted Ginzburg-Landau energies according to Section 3.
Proposition 2. Let Ω R2 be a bounded, open set, a l distinct points in Ω, d l
⊂ { i}i=1 { i}i=1 ∈
1 , and η : Ω R a positive Lipschitz function such that inf η =: η > 0. Set
Ω min
{± } →
ρ = min 1d(a ,a ) d(a ,∂Ω) 1 . Let ε exp( 8 ) and let v H˙1(Ω,C) be
a {2 i j } i6=j ∪{ i }i ∪{ } ≤ −ρa ∈
such that
(cid:8) (cid:9) +
l
E (v)
Σ := ε,η π η2(a ) < + . (30)
a i
logε − ∞
!
| | i=1
X
Also, assume that
l
ρ
a
r = Jv π d δ . (31)
a k − i aikW−1,1(Ω) ≤ 16
i=1
X
Then there exists C 1, depending only on a lower bound for ρ and η and on an
1 a min
≥
upper bound for l and η2 , and for each i 1, ,l there exists a point ξ B(a ,2r ),
∞ i i a
k∇ k ∈ { ··· } ∈
such that
l
Jv π d δ r r (Σ ,r ) εexp(C (Σ +g(r ))logε). (32)
k − i ξikW−1,1(Ω) ≤ ξ ≡ ξ a a ≡ 1 a a | |
i=1
X
where g is defined in Proposition 1.
Remark 4. Note that Lemma 1 and (31), (32) imply that
l
1
a ξ (r +r ). (33)
i i a ξ
| − |≤ π
i=1
X
10