A NNALES DE L ’I. H. P., SECTION C
Y ISONG Y ANG
The existence of Ginzburg-Landau solutions on the plane by a direct variational method
Annales de l’I. H. P., section C, tome 11, n
o5 (1994), p. 517-536
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The existence of Ginzburg-Landau solutions
on
the plane by
adirect variational method
Yisong
YANG(*)
Department ot Mathematics and Center tor Nonlinear Analysis,
Carnegie Mellon University, Pittsburgh, PA 15213, USA.
Vol. 11, n° 5, 1994, p. 517-536 Analyse non linéaire
ABSTRACT. - This paper studies the existence and the minimization
problem
of the solutions of theGinzburg-Landau equations
inR2 coupled
with an external
magnetic
field or a source current. The lack of a suitable Sobolevinequality
makes it necessary to consider a variationalproblem
over a
special
admissible space so that the space norms of the gauge vector fields of a minimization sequence can be controlledby
thecorresponding
energy upper bound and a solution may be obtained as a minimizer of a
modified energy of the
problem. Asymptotic properties
and fluxquantization
are established for
finite-energy
solutions.Besides,
it is shown that the solutions obtained also minimize theoriginal Ginzburg-Landau
energy when the admissible space isproperly
chosen.Key words: Minimization, Sobolev inequalities.
RESUME. - Le but de ce travail est d’étudier 1’ existence et le
probleme
de minimisation des solutions des
equations
deGinzburg-Landau
dansR2, couplees
avec unchamp magnetique
externe ou une source de courant.L’absence d’une
inegalite
convenable de Sobolev amene a considerer unprobleme
variationnel sur un espace admissiblespecial.
Sur cet espace, les normes deschamps
de vecteurs de la gauge d’une suite minimisantepeuvent
etre controlees par la bornesuperieure
del’énergie correspondante.
A.M.S. Classification : 35 Q 20, 81 J 05.
(*) Present address: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA.
Annales de l ’lnstitut Henri Poincaré - Analyse non linéaire - 0294-1449
Aussi,
une solutionpeut
etre obtenue comme un minimiseur d’uneenergie
modifiee du
probleme.
Desproprietes asymptotiques
et unequantification
de flux sont etablies par des solutions
d’energie
finie. Entre autres, il estprouve
que les solutions obtenues minimisent aussiI’ énergie originale
deGinzburg-Landau lorsque 1’ espace
admissible est convenablement choisi.I. INTRODUCTION
Let ~ : complex
scalar field and A : Rn a vectorfield,
where n =2, 3,
4. TheGinzburg-Landau (GL)
energydensity
inthe presence of an external
magnetic
field =1, 2,
...,n)
takesthe form
where
8j Ak - lik Aj
is themagnetic
field induced from A =(Aj)-
called the gauge vector
field, DAj 03C6 = ~j03C6 -
z is thegauge-covariant
derivative of
~),
* denotes thecomplex conjugate,
and the summation convention is observed onrepeated
indices. There is a local gaugesymmetry
in(1):
When n =
2, 3, ( 1 )
defines the well-known GL model ofsuperconductivity, while,
when n =4,
it defines the Euclidean abelianHiggs
model in classical fieldtheory [DH]. Varying
the energy E(~, A) _
~
A)
dx leads to thefollowing
GLequations
~
B -GL
system (3) assuming
that the external field carries a finite energyr
ana, snow mat mere are solutions wnicn numnuze me ~i~ energy.
Annales de l ’lnstitut Henri Poincaré - Analyse non linéaire
In the earlier work
[YI, Y2],
existence and minimizationtheorems
have been established for the dimensions n =3,
4. Our solution of theproblem
there
depends
on a modification of the energydensity ( 1 )
and thefollowing
Poincare
type inequality
in Rn(n
>2) (see [B, GT]):
When n =
2, (5)
fails and thetechniques
of[Yl, Y2]
break down.However,
the case n = 2 isactually
a moreinteresting
situation of the model because itgives
rise to vortex-like or mixed-state solutions which isa fundamental
phenomenon
insuperconductivity physics.
The purpose of thepresent
paper is to solve the n = 2 case. Ourapproach
here is basedon an
inequality
which can control the localL~
normby
the Dirichlettype
norm on the
right-hand
side of(5),
of a functionvanishing
in a small ball.We are led to
introducing
aspecial
but natural function space for the vector gauge vector A so that theminimizing
sequence of the modified GL energy isweakly compact
in the space(f1)
for any bounded domain f1 inR2.
Thus themajor difficulty
in theearly attempts
insolving
theproblem
in
R2
is overcome. A solution over the fullplane
can then be obtainedby passing
to the limit from bounded domains. The solutions found areall in the
global
Coulomb gauge. The fluxquantization problem, typical
in two
dimensions,
will also bebriefly
studied. It is shownthat,
when theexternal field has
power-type decay
estimate atinfinity,
so will the excitedfields,
which leads to aquantized magnetic
flux as in[JT].
We will also show that the solutions obtained minimize theoriginal
GL energy among all fieldconfigurations
in the Coulomb gauge when the admissible space of the gauge vector fields isproperly
chosen to allow a convenientdisposal
of the
boundary
termsarising
from our energycomparison.
II. EXISTENCE BY THE CALCULUS OF VARIATIONS
We shall first find a solution of
(3)
inR2 by looking
for a criticalpoint
of an energy
stronger
than E(see
the discussion tofollow). Analysis
showsthat the failure of
(5)
in the case n = 2 makes it difficult to control the local norms of apair
of fieldconfigurations
in terms of its energy.However,
it turns out that we can use thefollowing inequality
to tackleour
problem
here.Vol.
LEMMA 1. - Let u
(x) (x
ER2 )
be adifferentiable function
with supportcontained in B~ _
~x
E Thenis assumed to be
compactly supported
in B~.However,
this result is also valid for functions in ourgenerality
which will be used in thestudy
of(3)
in two dimensions. We
proceed
as follows.First,
it iseasily
checked that there holds theidentity
inus follows trom taking R ~ oo m tne above inequality. LJ Let B be the set of vector fields on
R2
with differentiablecomponents
andfinite ~ ~H-norm. Here,
for A EB,
The completion or v unaer me norm ~ ~H is aenotea oy /-t. it is clear
that H is a Hilbert space.
LEMMA 2. - For any A =
(Aj )
EH,
there holds~ .t") I ~) i -
Proof. -
For A EH,
there is asequence {An}
c B so that- 0 as n - ~.
By
Lemma1,
we know that(8 )
is validfor any element of B. Thus
_ n .. ___ ..n « .
Letting
00, we obtain_ . -
As a consequence of the Minkowski
inequality
and(9),
we haveTaking ~ 2014~
oo, we arrive at(8).
We are now
ready
to solve(3).
Forsimplicity,
we assume that the external field is smooth. We rewrite the GL energydensity (1)
in the formThe main result of this section is
THEOREM 3. - In
R 2,
the GL system(3)
has afinite-energy
smoothsolution
A)
in the Coulomb gaugeo~~ Aj
= 0 andEo A) _
~o A)
dx 00.Proof -
Theoriginal
energydensity ( 10)
is notgood enough
to workon. As in
[Y 1 J,
we start from the modifieddensity
functions~ ~ ,
which are
obviously stronger
than E and~o .
Set I =
Idx
andIo = ) Io dx
overR2 .
Consider the variationalproblem
r . f r / i .. ’B 1/ I i B - l i ~ B
where J is the set ot
complex-valued
tunctions on it" with componentslying
in(R2 ) .
Since EL2 (R2 ) [see (4)],
it iseasily
checkedthat
Imin
in(12)
is finite.Next,
it is clear that the functions in( 11 )
are nolonger
invariant under thegeneral
gaugesymmetry (2). However, they
are invariant if úJ in(2)
is a linear function of x 1, ~2.
Thus,
for apair (~, A)
EJ
x 1-{ withIo ( ~, A)
oo, define(~, A’)
so thatThen the average of A’ on
{~
E3} vanishes,
~. e.,1 f
because A’ is a translation of A
by
a constant vector.be a
minimizing
sequence of theproblem ( 12).
TheSchwarz
inequality implies
that_ _ ....
Here, ana m tne sequei, o. > u is an irrelevant constant. From (13) ana tne definition of we see the finiteness of the
quantity
~ ~
n ’n ’B / r,... ’n. ’B , ~ .. ,By the above discussion, we may also assume that A’" == U, n ==
1, 2
...Thus the Poincaré
inequality applied
on(x
E3} implies
f ~
Kecan mat, in me definition or me norm II me truncatlon function
satisfies
p (x)
= 1for |x|
> 2. As a consequence, the combination of(14)
and(15) gives
us the boundedness of in H.Using
Lemma 2 andAnnales de’ l’lnstitut Henri Poincaré - Analyse non linéaire
the definition
of ~ 111i,
we conclude that thecomponents
are bounded sequences in(Q)
for any bounded domain [2 CR2 .
Thus{~1~}
is also bounded in([2)
for anyp >_
1. On the otherhand,
the definition of thegauge-covariant
derivativegives
.- 1 - _ _
is a bounded sequence in
(Q)
as well for any bounded domain H inR2 . Using
thecompact embedding W 1 ~ 2 ( SZ )
-(H) (p >_ 1)
andby passing
to asubsequence
if necessary(a diagonal subsequence argument),
we may assume the existence of apair (~, A)
E(R2 )
so thatfor every bounded domain 1 1 E R’ and any p O 1. Ut course A is also the weak limit of
(A")
in the space H. Thus(§, A)
EJ
x H. We are left toshow that
(§, A)
solves the variationalproblem (12).
Let
Io,
I be as defined in(1 1)
andBR
==(z
ER) (R
>0).
The functionals10 (§, A; R)
=Io (§, A)
dx and 1(§, A; R)
=I (§, A)
dz areintegrals
overB R.
The condition(4)
says that forgiven
c >
0 there isRo
> 0 to ensureThus for R >
where M
>
0 is asgiven
in( 14). Using ( 1 6),
the weak lowersemicontinuity
of the terms in I
(.,. ; R),
and IAn )
-+Imin (as
n ~(0),
we arrive at. _.
namely,
Vol. 5-1994.
Setting
R - oo in theabove,
weget I (~, A) Imin by
the arbitrariness of ~ > 0. HenceA)
=Imin
and(12)
is solved.Finally, varying
the modified energy I inJ
x Hby compactly supported
test functions around the obtained minimizer
(~, A)
of(12)
andusing
thestandard
elliptic regularity theory,
we see that(~, A)
is a smooth solution of thesystem
B
we can use me two equations in H /) to snow that
a~ Aj
is harmonic inHowever, since aj A j
EL2 (R2),
the Liouville theorem says that 0.As a consequence, it follows that
~2 Aj = ~~ (8k ~4~) ==
8k Fkj, j
=1 , 2.
In otherwords, ( 17)
recovers the GLequations (3)
inR2.
Thus
(~, A)
solves(3)
as well and the theorem isproved.
DIII. THE SOURCE CURRENT CASE
In our discussion of the GL
theory
in Section2,
the external field isa
magnetic field,
A moregeneral
situation is that the external field may becoupled
into the modelthrough
the form of a smooth source currentdensity,
Jex= ~ J~ x~ (say).
In this case the energydensity
takesthe
following
formand tne
corresponding
GLequations
areB
m uruer w ooserve consistency we must impose me current conservation law on
4, me existence or somnons as mimmizers or me GL
energy has been established in
[Yl , Y2]
under the condition(20)
andr 2 n l D n B : i ,_. - - ? A
ilVIW r a vrnrvm v a aaauaJam aavu saaavwaar
An obvious reason for
imposing
the above condition is due to the Poincare typeinequality (5)
so that the GL energy is ensured to be bounded from below. For the case n =2,
since(5)
is nolonger valid,
we need somenew conditions on Jex to solve
(19).
In thissection,
we assume in addition to(20)
thatTHEOREM 4. - Assume that
(20)-(22 )
hold. The GLequations (19)
overR2 subject
to thegeneral
source current Jex ==(JJx)
have afinite-energy
smooth solution
(~, A)
in the Coulomb gauge,~~
O.Proof. - Again
the energydensity ( 18)
is notgood enough
to work onand we have to use the modified energy
Io
=I0
dx whereIo
is asdefined in
( 11 ).
Set I =Io
+Aj Jexj dx.
Consider theproblem
Since for A E
Pi!,
theretore the Schwarz
inequality
and(21) imply
thatAj
EWe first show that I is invariant under the gauge transformation
(2)
when 03C9
(x)
= cxl x1 + a2 X2 is a linear function. Infact,
we havealready
observed the invariance of
Io.
On the otherhand,
from(20)
and thedivergence theorem,
we have for A’ = A +r r
Thus, using (22)
andletting
00, we see thatp r
Namely,
the invariance of I holds as well.We next show that
Imin
in(23)
is finite. Forgiven (03C6, A)
in theadmissible set of
(23),
we obtain as in Section II a modifiedpair (§’, A’)
so that A’ =
A +
is a linearfunction,
and the averageot A’ on B3
= {x ~ R2~x f
0} is zero. using me Poincare inequalityon
B3
and Lemma2,
we haveOn the other
hand,
there holds for any E >0,
inserting ana imo yieias
particular,
the finiteness ofImin
follows.be a
minimizing
sequence of(23).
Our earlier discussionon the invariance of the terms in the energy functional allows us to assume, after a suitable gauge
transformation,
that An has zero average onB3.
Hence An verifies
(25). Consequently (27) implies
that is a boundedsequence in H. So it follows
by passing
to asubsequence
if necessary thatwe may assume
(16)
Given E >
0,
letRo
> 2 be such that/*
Then the
integral
Jsatisfies
Annales de l’Institut Henri Poincaré - Analyse non linéaire
where
which is finite
according
to(25)
and(27).
We can thenduplicate
thesteps
inproving
Theorem 3 to show that(~, A)
is a minimizer of(23)
whichalso solves the GL
equations (19)
because of(20).
DIV. ASYMPTOTIC BEHAVIOR AND FLUX
QUANTIZATION
In this
section,
we assume that(~, A)
is a solution of(3)
or(19)
withfinite modified energy
7o (~, A)
oo and in the Coulomb gauge.Using
themethods of
[Yl, Y2] ,
it is not hard to establish thefollowing L2-estimates.
THEOREM 5. -
Suppose
in(3) (( 19))
that theexternal field Fexik
lies in the space
(R2) (L2 (R2)) (j, ~
=l, 2~.
ThenD3
E£2 (R2), e~~(R~) (~ ~-1, 2)
and 1 -14>12
E(R2) for
any p > 1.
Besides, ( ~ ( 2
~ 1 ~ 00 1 inR2
or otherwise4~I
= 1.If furthermore, F~~
EW2.’ 2 (R2)
E(R2)) (~, ~ _
1, 2),
thenDt ~, 8j Ak
EW2’ 2 (R2) (j, k
=1, 2).
We
skip
theproof
here.From the well-known Sobolev
embedding (R2) ~
LP(R2 ) (‘d
p >1)
and the fact that functions in(R2) (p
>2)
vanish atinfinity,
we obtain
COROLLARY 6. -
W2’ 2 (R2)
G(R2)) (j~ ~ - l~ 21~
there hold
l Jd
We nave shown m section 11 (111) that A ) can be obtained as a
minimizer of the
problem (12) ((23))
which is afinite-energy
smoothsolution of the GL
equations (3) (( 19)).
In order to calculate themagnetic
flux carried
by
thesolution,
some minimaldecay assumption
for the external field has to be made. The main reason fordoing
this is thatdecay
estimatesof the
quantities 1 2014 ~p, Dt ~,
andFj k
can thus be established so that undesiredboundary
terms will be eliminated in theintegral
of our fluxevaluation.
For this purpose, we
impose
thefollowing
powertype decay properties
for the external field:
/ I B -- ~ I ! 20142014 ~/ /""’’’’,
lLn°
where a,
p
>U, j, k
=1,
2,and x ~
islarge.
THEOREM 7. -
Suppose Fexjk
EW2, 2 ( R2 )
EW1, 2(R2)) (j
=1, 2) If Fexjk (Jexj) decays according
to(28),
thenasymptotically
If
inaddition, satisfies (29),
thenthe source current, Jex .
Set gj
=and
= gjg;.
It is crucial to derive thedecay
estimatefor Igl
first.Let 03C8
be anarbitrary complex
scalar function.By
the definition of thegauge-covariant derivatives,
weeasily verify
the useful curvatureidentity
_ ~ _ ~ ~
From this and the
equations (19),
we have after alengthy
calculationB B I 1
Applying (30)
inSince |03C6|
- 1, F12 ~ Uas Ixl
- oo(see Corollary 6)
and 03BB >0,
theabove
inequality
has the reductionwhere
Cl. C2
> 0 are constants and R > 0 issufficiently large.
Annales de l’Institut Henri Poincaré - Analyse non linéaire
Introduce the
comparison
functionThen ~203C3 = >
R). Inserting
thisequation
and(28) (for
Jex)
into(31) yields
where we have assumed that Rand
C3
are such that 4a2 R-2
+Ci. Obviously,
for fixedR,
we may chooseC3
> 0 in(32)
solarge
that
(6 -
> 0. From thisboundary condition,
theproperty
a - - 0 as -7 00,
(33),
and the maximumprinciple,
we see thatcr for
Ix1
> R.Namely Igl
= OSimilarly,
theasymptotic
estimate of 1 -1/J12
follows fromusing
theabove
argument
in theequation
while
treating
as adecaying
source term.Now suppose
(29)
holds in addition.Differentiating (19) gives
Thus, using (34)
and the aforementioned curvatureidentity,
it is seen thatF12
=81 A2 - 82 Al
satisfies theequation
Thus
where
C1, C2
> 0 are constants, R is alarge number, and 03B3
=min {2
a,03B2}
with
/3 being given
in(29).
Thus we can prove as before thatThe
proof
of the theorem iscomplete.
DConcerning
the totalmagnetic
flux of the solutionA ) ,
we have asin
[JT]
thefollowing
statement.THEOREM 8. - Assume the condition
(28)
where a > 1 and let H =F12
denote the induced
magnetic field.
Then the totalflux ~
isquantized according
to theexpression
p
order
parameter ~
on the circle atinfinity of
theplane.
Proof. -
LetRo
> 0 be such that|03C6| > - for |x (
>(Theorem
5 or7).
Then thewinding
number Nof ~
atinfinity obeys
On the other
hand,
thedivergence
theorem and Theorem 7imply
thatInserting (36)
into(37)
andletting
R ~ oo, we find 03A6 = 203C0 N asexpected.
DThus, although
the flux of the external field may take anyprescribed value,
the excited flux canonly
assumequantized
values which is atypical phenomenon
insuperconductivity theory.
V. MINIMIZATION OF ENERGY
In Sects. II and
III,
weproved
the existence offinite-energy
smoothsolutions of the GL
system (3)
or(19)
inR2 by obtaining
minimizers of theproblem ( 12)
or(23)
where the energy functional I takes a modified form and is notphysical.
Forexample,
the invariance under thegeneral
gaugesymmetry (2)
is nolonger
valid. Since the GL energy E isactually given
Annales de l ’lnstitut Henri Poincaré - Analyse non linéaire
in
(10)
or(18),
it will beimportant
to find solutions ot (3) or so tnatthey
also minimize the GL energy. Notethat,
ingeneral,
the GLequations
may have
multiple
solutions and thephysical
states arerepresented by
energy minimizers. The purpose of this section is to
get
solutions of the GLequations
that minimize the energy in a suitable admissible class.Our
starting point
now is to find acomparison
of the GL energy E and the modified energy I defined in(12)
or(23).
It appears that the space H used in Sects. II and III is not proper for such acomparison
and it isnecessary to consider a
subspace
ofH,
which is restrictiveenough
so thatwe can compare E and I
and,
at the sametime, large enough
so that theGL
equations
can be fulfilledby
energy minimizers of E. Thefollowing study
will follow this line.Let C c B
(see
Sect.II)
be the set of vector fields inR2 satisfying
thecondition
that,
for each A EC,
there is a constant vectorA°
ER2
such that A -A°
is ofcompact support.
The closure of C in H is denotedby
A useful
property
ofHi
is that there holds theidentity
In
fact,
it isstraightforward
toverify (38)
in C.However,
since bothsides of
(38)
are continuous withrespect
to the norm of7~,
we see that(38)
is true in ingeneral.
We are now
ready
to obtain a solutionpair
of theequations (3)
or(19)
as a GL energy minimizer in the
following
sense.THEOREM 9. -
suppose
that(4)
or(20)-(22)
hold. Then the GLequations (3)
or(19)
overR2
have a smooth solution(~, ~4)
which minimizes the GLenergy E = ~
dx among allfield configurations in
the admissibleset
y
x and in theglobal
Coulomb gauge. is asdefined in (10)
or(18).
Proof. -
We canproceed
to show as in Sect. II or III that theoptimization problem (12)
or(23),
with Hreplaced by
has a solution(~,
thatthis solution satisfies the GL
equations (3)
or( 19),
and that 0 inR~.
We claim that ,E(~ ~)
=min{E(~ A)~(~ A) e V
x8~ A~
=0~. (39)
Infact,
theidentity (38)
says that I(~ A)
= E(~, A)
for(~ A)
eJ
x~l
with9, A~
= 0. Thus E(~, A)
= I(~, A) _
I(~, A)
= E(~, A)
for
(03C6, A) ~ V
xH1
and8j Aj
= 0 asexpected.
DVol. 11, n° 5-1994.
Note. Another technical reason for the choice of
~C1 (or C)
is that the space is invariant under translationsby
constant vectors. Thisproperty
is crucial because it allows us to make gauge transformations(2)
in themodified energy I in which 03C9 is a linear function of the variables. The discussion in Sects. II and III showed the
importance
of such a feature inextracting
alocally weakly convergent minimizing
sequence.VI. THE GENERAL SCALAR POTENTIAL CASE
We have assumed in our
Lagrangian density ( 1 )
that thepotential
functionof the order
parameter 8 ~
takes the formV(!~)
== -( ~ "
factour method in the existence
proofs applies
to the moregeneral
situationthat the
non-negative
functionY (s~ (s > 0)
satisfiesextension is to accommodate in the
family
offinite-energy
solutionsthe normal
state ~
=0, Fjk
= Forexample,
we may chooseV
(|03C6|) = 03BB 8 2 (14)12
-1 ) 2
as in the self-dualChern-Simons-Hi gg s theory [HKP, JW, SY].
In this case thegeneralized
GLequations
inR2
are~ .x __ ’B
nere we consider only the source current case. yvc Cdll prove as 111 v,
for
example,
the existence of a smooth solution of theequations
which isalso a GL energy minimizer among all field
configurations
in the Coulomb gauge. In thissection,
we restrict our attention on thestudy
of theasymptotic
behavior of a
finite-energy
solution. In view of(40),
alengthy
calculation shows that gj = satisfiesFrom
(41)
we can prove thatD~ D: cjJ
EL2 (R~),
EW 1 ~ 2 (R2)
if
(03C6, A)
is of finite energy andJJx
EL2 (R2). Moreover,
since~k Aj
Annales de l’Institut Henri Poincaré - Analyse non linéaire
satisfies
(34),
we obtain8k Aj, DAj03C6
EW2, 2(R2)
if EW1, 2(R2).
Therefore the
simple inequality
ana tne 1 imply mai -1; (=
vt~ i ~ y ~ t~,~ ~
tor any p > i.As a consequence
(see
Sect.IV),
we can state thefollowing asymptotic
behavior of
4>,
THEOREM 10. -
Suppose Wl,2 (R2) for j
=1, 2
and(4), A) is a finite-energy
solutionof (40)
in the Coulomb gauge. ThenFurthermore, if fulfills (28)-(29),
then the solutionssatisfying 11>1
= 1at
infinity
have thedecay properties
described in Theorem 7. On the otherhand,
the solutionssatisfying 11>1
= 0 atinfinity obey
theexponential decay property
-
In this
situation I decays
as inI
---+ 1 case.Proof -
It suffices to elaborate on the secondpart
of the statement. As in Sect.IV, (41 ) gives
us theinequality
Since either
1p1
- 0 or1p1
- 1 asIxl
- 00, theinequality (43) always
leads to
(31).
ThusIgl
= 0Assume first that
~~~ I
- 1I
- 00. ThenBecause the coefficient of
1 -14J/2
on theright-hand
side of aboveequation
becomes
positive when |x|
islarge,
we see that1 - 14J12 decays like |g|2
as in Sect. IV.
Assume next that
14>1
- 0as x ~
- oJ. ThenÇ"Ir.... t"’} _ .. ~
As - oo, the coefficient ot on the
right-hand
side above goes to~ .
Hence it is standard that14> x 2
= 0 forlarge ~ .
DNote. The accurate statement of
(42)
is that forany 0
~l,
thereis
C (e)
>0,
so thatlt is
interesting
to see that thedecay
rateof |03C6 I ¿.
in this case isindependent
of theproperty
of the source term atinfinity.
This reveals a difference of solutions which areasymptotically
thesymmetric
vacuum, characterizedby 14>1 i
=0,
from the solutions which areasymptotically asymmetric
vacua,characterized
==1.
The former are called in the Chern-Simons modelcase
non-topological
solutions[JLW]
in contrast to thelatter, topological solutions,
for which theinteger
Ngiven
in the flux formula(35)
in Sect. IV is atopological
invariant.VII. A CONSTRAINED MINIMIZATION PROBLEM We now go back to the classical case that the bare energy
density So
is asdefined in
( 10)
so that afinite-energy
solution goes to theasymmetric
vacuaat
infinity.
From the discussion in Sect.IV,
we see that such a solution carries aquantized magnetic
fluxgiven
in(35).
Theinteger
N is unknownto us. In
particular,
it is not clear whether everyinteger
N can be realizedby
the flux of afinite-energy
solutionaccording
to theexpression (35).
Thus we are led to the
following question.
Given an
integer N,
can we find a solution of theproblem Minimize E
subject
toAnnales ae l’Institut Henri Poincare - Analyse non linéaire
There are
only
available results when the external field orJJx
is absent. The
problem
is best understood in the self-dual case[Bo]
whenA = 1 due to the work in
[Tl, T2, JT] :
For anyN, (44)
has a 2N-parameter family
of solutions with theparameters characterizing
the locations of thezeros of the scalar field
~.
These solutionsrepresent
Nnon-interacting
vortices. When
A # 1,
it is shown in[P, BC] that,
for eachN,
the energyE has a
radially symmetric
criticalpoint satisfying
the constraint in(44).
However,
it is not clear as to whether these radial solutions are absolute energy minimizers in the constraint class.ACKNOWLEDGMENT
The author would like to thank John Albert for many
stimulating
conversations.
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(Manuscript received June 23, 1993;
revised January 3, 1994. )
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire