Concentration on curves for nonlinear Schrödinger
equations
Manuel del Pino
Universidad de Chile
Michal Kowalczyk
Kent State University and Universidad de Chile
AND
Jun-Cheng Wei
Chinese University of Hong Kong
Abstract
We consider the problem
!#"$&%'(
where ) #* , '+ is a small parameter and is a uniformly positive, smooth potential. Let , be a closed curve, non-degenerate geodesic relative to the weighted
" " . We prove the existence of a solution 87
arclength -. 0/ , where 1 3$24
56"
concentrating along the whole of , , exponentially small in at any positive distance from it, provided that is small and away from certain critical numbers.
In particular this establishes the validity of a conjecture raised in [3] in the two9
dimensional case. c 2000 Wiley Periodicals, Inc.
1 Introduction and statement of main result
We consider standing waves for a nonlinear Schrödinger equation in :<;
form
B6C
(1.1)
=?>A@ B6DFE
@(GIH
C
=FJLKM6N
C
of the
CPORQ CQ SUTWVXC
D_^
TWV D
NedfKM6Ng Assumwhere Y[Z]\ , namely solutions of the form K M`N E exp K>bac@
C
ing that the amplitude dKM`N is positive and vanishes at infinity, we see that this
satisfies (1.1) if and only if d solves the nonlinear elliptic problem
(1.2)
where iKM6N E
function with
O
S
^
^
V
^
Elk dmZ k
@(G$Hhd=jiKM`Ned
d
dPno
K: ; N
O
JLKM6N
a . In the rest of this paper we will assume that i
prq`s
tuv`w imKM6NxZ
k g
Communications
on Pure and Applied Mathematics, Vol. 000, 0001–0033 (2000)
y
c 2000 Wiley Periodicals, Inc.
is a smooth
2
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
Considerable attention has been paid in recent years to the problem of construction of standing waves in the so-called semi-classical limit of (1.1) @{z k . In the
pioneering work [14], Floer and Weinstein constructed positive solutions to this
problem when Y El| and } E \ with concentration taking place near a given point
M8~ with i€KM8~N E‚k , i# ƒKM8~NR
„E k , being exponentially small in @ outside any neighborhood of M8~ . More precisely, they established the existence of a solution dc… such
that
‹Œ
dc…(KM6N‡†ˆiPKM8~NŠ‰
where
(1.3)

TWV
KiŽKM8~N w ‰ @
‰I
KM=ŽM8~NN
is the unique solution of

 =
O

S

Elk
^
^
k
dPZ
^
 K k N Elk

K€‘ˆN E’k g

This result has been subsequently extended to higher dimensions to the construction of solutions exhibiting high concentration around one or more points of space
under various assumptions on the potential and the nonlinearity by many authors.
We refer the reader for instance to [2], [4], [7]-[13], [15, 16, 17, 18, 21, 31, 32,
34, 36, 37, 38]. An important question is whether solutions exhibiting concentration on higher dimensional sets Q exist.
In [3], Ambrosetti, Malchiodi and Ni have
Q
considered the
case
of
i E iPK M N , also treated in [5, 6], and constructed radial
Q Q
Q Q
solutions d … K M N exhibiting concentration on a sphere M ER“ ~ in the form
‹_Œ
d … K “ NW†ˆimK “ ~ N‚‰
KiPK “ ~ N w‰ @
‰ 
TWV
K“ =
“ ~ NXN
^
under the assumption that “ ~”Z k is a non-degenerate critical point of
TWV
•–K “ N ER“ ;
(1.4)
where
˜
(1.5)
E
O
Y
\
Y=™\
=
i#—'K “ N
š
\ ^
and is again the unique solution of (1.3). The conjecture is raised in [3] that this

type of phenomenon
takes place, at least along a sequence @ E @›#z k , whenever
Q Q
the sphere M ER“ ~ is replaced by a closed hypersurface œ , which is stationary and
non-degenerate for the weighted area functional - i — . In this paper we prove the
š
validity of this conjecture in dimension } E .
š
For } E , the functional above, defined on closed Jordan curves œ , has a simple geometrical meaning: itO corresponds to the arclength of œ measured with rež8M G N in : G . Thus we will establish the concentration
spect to the metric i — K$ž8M VG
G
phenomenon on œ , provided that this curve is a non-degenerate closed geodesic for
this metric in : G .
We do not prove the result for all small values @
Z k but only for those which
lie away from certain critical numbers. More precisely, there is an explicit number
a'ŸZ k such that given ¡Z k , if @ is sufficiently small and satisfies the gap condition
(1.6)
Q¢
GI@G=ja'Ÿ
Q¤£
$@
^
for all
¢
nL¥
^
CONCENTRATION ON CURVES
3
then a solution dc… with the required concentration property indeed exists. In other
words,
this will be the case whenever @ is small and away from the critical numbers
¦ §(¨
© , in the sense that for fixed and arbitrarily small «ª’¬ a'Ÿ ,
@[n®
„ ­
¬
¢
a Ÿ
¢
=
^ ¬
G
¢
a Ÿ
O
¢
¯ ^
¢
for all
G
n¥®g
To state our main result, we need to make precise the concept of a curve œ being
stationary and non-degenerate for the weighted length functional -  i — .
Q Q
Let œ be a closed smooth curve in : G and ° E œ its total length. We consider
natural parameterization ±Kƒ²³N of œ with positive orientation, where ² denotes arclength parameter measured from a fixed point of œ . Let ´‡Kƒ²³N denote outer unit
normal to œ . Points M which are µ$~ -close œ , for sufficiently small µ$~ can be represented in the form
M E
(1.7)
OFD
±Kƒ²³N
^
´Kƒ²³N
Q DIQ
^
ªRµ$~
^
²nF­ k
^
°N
D_^
where map M€¶z·K ²³N is a local diffeomorphism. Any curve sufficiently close to œ
can be parameterized as
±¹¸ºKƒ²³N E
Oˆ»
±¡Kƒ²³N
Kƒ²¹N¼´‡Kƒ²³N
»
where is a smooth, ½ -periodic function with small ¾«¿ -norm.
Call œ ¸ the curve
D^
defined this way. By slight abuse of notation we denote iPK ²³N to actually mean
iPKM6N for M in (1.7). Then weighted length of this curve is given by the functional
»
of
À
K
»
N‡ÁˆÂ
Q
Q
ÂŽÄ i — Kű¹¸³Kƒ²³NXN ± ¸  Kƒ²³N žº²
~
E
i —
UÃ
E
Q
Q
Â
~
¢
Ä i — K
»
^
Q Oj»
Oj»
Q
 ´ ž³²`g
Kƒ²³N ³
² N ± 
´ 
¢
Kƒ²³NA±' where Kƒ²³N denotes curvature of œ , we get that the
Since ±' E \ and ´` E
above quantity becomes
(1.8)
À
K
»
N E
»
ÂŽÄ i — K
~
^
O[¢³»
O
Kƒ²³N ³
² N$­XK¼\
N G
»
K
N G
¯
w ‰ ž³²`g
is said to be stationary
for the weighted length -  i — if the first variation of
»
the functional (1.8) at EÆk is equal to zero. That is, for any smooth, ° -periodic
function ÇKƒ²³N
œ
À
kPE
¯
 K k N$­ÈÇ
E
ÂmÄ ­Ki — NeÉAÇ
~
O
i —
¢
Ç
¯
^
žº²
which is equivalent to the relation
(1.9)
˜i6ÉK k
^
²³N E
=
¢
Kƒ²³NXiK k
^
²¹N
^
for all ²+nÊK k °N0g
4
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
We assume the validity of this relation at œ . Let us consider now the second variation quadratic form
À
^
  K k N$­ÈÇ
Ç
¯
E
\
š Â
E
\
š Â
~
~
Q Q O
Ä ­i — Ç  G
Ä ­i —
Q
O
­Ki — NeÉÉ
Q O
Ç  G
š
š
­Ki — NeÉÉË=
¢ ¯
Ki — NeÉ
¢
i —
¯
G
¯
Ç G
Ç G
¯
žº²
žº²
We say that œ is non-degenerate
if so is this quadratic form in the space of all
^
V
functions Çmnxo K k °N with ÇK k N E ÇKÅ°N . This is equivalent to the statement that
the differential equation
š
Ki — Ç  N  =Ì­Ki — N ÉÉ =
has only the ° -periodic solution ÇjÁ
problem
Ç 
(1.10)
O
˜i
TWV
icÎIÇ  Ï
= ­ƒ˜i
^
ÇKÅ°N Ç  K k N E
ÇK k N E
k
¢
i —
G
¯
Ç
EÍk
, or using (1.9), that the boundary value
TWV
i`Éɳ=ÏKƒ˜
^
Ç  KÅ°N
TWV O
\N
¢
G
¯
EÍk
Ç
^
has only the trivial solution. As an example, let us consider the radial case i E
iPK “ N . Then we see that œ EÑГÒEl“ ~¤Ó is stationary precisely if •ÔK “ ~N Eˆk where
•
 K “ ~INZ k , namely if “ ~ is a non-degenerate
is defined by (1.4). If in addition •]
O š
¢
local minimizer,À we have
that
Ki — NeÉÉ
Ki — NeÉ Z k . This makes automatically the
^ ¯
quadratic form  K k N$­ÈÇ Ç positive definite, hence non-singular. A non-degenerate
stationary curve close to œ will still be present if the radial potential is modified by
small non-radial perturbations. The geometric interpretation allows
the construcQ Q
V
tion of other examples. If i — KM6NՆ Ö Ve×Ø t Ø wXÙÅw up to a large value of M then the metric i — žºM G represents approximately that of aQ sphere
embedded in :<Ú . If eventually
Q
i increases so that iPKM`N4†®\ for very large M , the whole metric will resemble that
of a “globe attached to a plane”, the presence of at least two geodesics thus being clear: one on the globe, the other on the connecting neck. Non-degeneracy of
these geodesics is not true in general, but may be generically expected in a suitably
strong ÛÜ -topology.
We need a further element to describe the gap condition (1.6). Let denote the

unique positive solution of problem (1.3). We consider the associated linearized
eigenvalue problem,
(1.11)
Ç`Ý Ý=jÇ
O
Y
SUTWV

Ç E
a'Ç
^
^”^
in :
ÇK€‘ˆN Elk g
It is well known that this equation possesses a unique positive eigenvalue ac~ in
V
o
K:N , with associated eigenfunction even and positive Þ which we normalize so
that - v Þ G E \ (this follows for instance from the analysis in [30]). In fact, a simple
computation shows that
(1.12)
ac~ E
O
\
ßKàY=á\N$KàY
| N
^
Þ
E
‹Xã
\
â
- v

S×V 
w ‰ g
CONCENTRATION ON CURVES
5
We define the number a'Ÿ as
a'Ÿ E
(1.13)
\
ac~ ß8ä
Â
Gå
^
~
Ä iPK k
G
²³N w‰ žº²æ
g
Now we can state our main result.
Theorem 1.1. Let œ be a non-degenerate, stationary curve for the weighted length
functional -  i — , as described above. Then given ÒZ k there exists @(~{Z k such
that for all @{ªŠ@(~ satisfying the gap condition
Q
(1.14)
@ G
¢
Q'£
G =’a'Ÿ
^èç¢
Ë@
^
n¥
where acŸZ k is the number in (1.13), problem (1.2) has a positive solution d4é
which near œ , for M given by (1.7), takes the form
d é KM6N E
(1.15)
i Kk
P
^
‹_Œ
²³N
‰
^
i Kk
P
‰6ëê
D
² N w‰
³
For some number ~ Z k , d … satisfies globally, pró3ô
d … KM6NÒîÌï$ð¹ñfK¤={ ~ @
KU\
@'ì
TWVò
KM
^
O®í
K¼\NN0g
œfNºN¡g
To explain in few words the difficulties encountered in the construction of these
š ä
solutions, let us assume for the moment
that iÑÁõ
. Then in
\ on œ and that ° E
^÷
_
D
^
TWV
K ²³N the equation would look near
terms of the stretched coordinates Kö N E @
the curve approximately like
øù3ù O
š ä
øúbú O
ø S
÷
TWV
=
ø E’k
Kö
^÷
^
NfnL: G
where ø is @ -periodic in the variable. The effect of curvature and of variations of i are here neglected. The linearization of this problem around the profile
Kö¤N thus becomes û
û
û
û

ùù O
û
with û
V
š ä
E
TWV
@

ú
úeú O
÷
Y
STWV

=
ERk
Kö
û
-periodic
of the
form
ó3prq in . Functions
ó
Kö¤N$­ ü
¢
@
÷<O[ý¹þ$ÿ
¢
@
÷ ¯ ^
G E
Þ KöUN$­ ü
^÷
^
NnL:¡G
óprq
¢
@
÷0OÊý¹þ$ÿ
¢
ó
¢
@
÷ ¯ ^
¢
are eigenfunctions associated to eigenvalues respectively = G @ G and a6~0= G @ G .
Many of these numbers are small and “near non-invertibility” of the linear operator thus occurs. Therefore the use of a fixed point argument after inverting the
linear operator in the actual nonlinear problem is a very delicate matter. Worse
than this, these two effects combined, in principle orthogonal because of the ¾ G orthogonality of Þ and ú , are actually coupled through the smaller order terms

û
neglected. In [1, 19, 20, 33] related singular perturbation problems involving the
V
Allen-Cahn equation in phase transitions exhibiting only the translation effect
have been successfully treated through successive improvements of the approximation and fine spectral analysis of the actual linearized operator. The principle is
simple: the better the approximation, higher the chances of a correct inversion of
the linearized operator to obtain a contraction mapping formulation of the problem.
6
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
û
In [24, 25, 28, 35] resonance phenomena similar to the “ G -effect” has been faced
in related problems. In [24, 25] a Neumann problem involving whole boundary
concentration, widely treated for point concentration after the works [23, 29, 30].
Recently in [27, 26] this boundary concentration on a geodesic of the boundary in
the 3d-case has been treated via arbitrarily high order approximations. Our method,
closer in spirit to that of Floer and Weinstein, provides substantial simplification
and flexibility to deal with larger noise and coupling of the two effects inherent to
this problem. The solution to the full problem in the above idealized situation is
roughly decomposed in the form
û
ø
Kö
^÷
N E

š ä
Kö=
Kƒ@
÷
NXN
O
@ºKƒ@
÷
NXÞ Kö¡=
÷
Kƒ@
NXN
O
Kö û
^÷
N
^÷
Kö
N is ¾ G Kƒž³öUN where and are ÷ -periodic functions
left
as parameters, while
÷
÷
û
Kƒ@ NXN and to Þ Kö=
Kƒ@ NXN . Solving first in
orthogonal
for each both to ú Kö=

a natural projected problem where the linear operator is uniformly invertible, the
resolution of the full problem becomes reduced
to a nonlinear, nonlocal second
^
order system of differential equations in K (N which turns to be directly solvable
thanks
to the assumptions made. This approach is familiar when the parameters
^
K
¤N lie in a finite-dimensional space, corresponding this time to adjusting infinitely many parameters. To stress
out the difference with the radial case: the
parameter is not present, and is just a single number. The analysis we make
takes special advantage through Fourier analysis of the fact that the objects to be
adjusted are one-variable functions, while we still believe that the current approach
may be modified to the higher dimensional case. We also believe the gap condition
may be improved to size ö(@ , any ZÆ\ .
In the rest of the paper we carry out the program outlined above which leads to
the proof of Theorem 1.1.
2 The set up near the curve
Let œ be the curve in the statement of the theorem. We shall use the notation
introduced in the previous section.
Stretching variables, absorbing @ from Laplace’s operator replacing dfKM6N by
dKƒ@M6N , equation (1.2) becomes
(2.1)
Let Kö
@
TWV
œ
^÷
N E
O
S
^
^
V
Elk dZ k
H d=FiLKƒ@M6Ned
d
dnPo
K:<G³N<g
TWV D_^
@
K ²³N be natural stretched coordinates associated to
, now defined for
÷
(2.2)
n[­ k
^
@
TWV
°N
^
ö{n[Kb={@
TWV
µ$~
^
@
TWV
µ$~Ng
Equation (2.1) for d expressed in these coordinates becomes
(2.3)
d
ù3ù O
d
^ ÷
O
úeú O
V
KdWN`=FiPKƒ@(ö @ Ned
d
S
Elk
the curve œ … E
CONCENTRATION ON CURVES
7
in the region (2.2), where
V KdWN E
d
ùù
­ \=
O
K¼\
\
¢
@
¯ O
÷
Kƒ@
@
N¼öUN G
\
¢
Kƒ@
O
@
÷
Ned
¢
Kƒ@
ú
÷
O
N¼ö
K¼\
¢
@ G ö
O
¢
 Kƒ@
@
Kƒ@
÷
ù
Ned
÷
g
N¼öUN Ú
For further reference, it is convenient to expand this operator in the form
(2.4)
V KdÕN E
¢
Kƒ@
÷
Kƒ@
¢
N`=x@(Gö
÷
GKƒ@
ú O
NXNed
^
~8KdÕN
where
(2.5)
^ ÷
ù O
@ Ned
@(Göü V Kƒ@(ö
~8KdÕN E
@¤öü
^
D_^
Kƒ@(ö
G
^ ÷
ù3ù O
@ Ned
@ Ú öG$ü
^ š ^
Kƒ@(ö
Ú
^ ÷
ú
@ Ned
| . Observe that all terms in the operfor certain
smooth functions ü8K ²³N E \
ator V have @ as a common factor.
We consider now a further change of variables in equation (2.3) with the property that replaces at main order the potential i by \ . Let
«Kƒ²³N E
(2.6)
^
iK k
²³N
‹Œ
^
‰
Kƒ²³N E
‰
^
iPK k
²³N w‰
^ ÷
and fix a twice differentiable, ° -periodic function Kƒ²³N . We define ø K N by the
relation
dKö
(2.7)
^÷
÷
”Kƒ@
N E
ø
N
K
^U÷
^
N
E
÷
Kƒ@
N$Kö=
Kƒ@
÷
NXN0g
We want to express equation (2.3) in terms of these new coordinates. We compute:
ú
d
(2.8)
ù
d
(2.9)
d
ùù
E
(2.10)
ø ^
E
ø O
@ 
E
d
úbú
E
G
ø
ù O
ø K K ö=
NXN
øù
ù O ø ù ¯
ø O š
ø @ G 
@  ­
K K ö=
NXN
O
Q
Q
O
š ø
ù
ù K K ö=
ø K K ö¡=
XN N G
O ø ùù O øUù ù ¯
K Kö¡=
XN N
g
NXN
ù
We also have
K
Kö‡=
NXN
ù
E
@¹­
 Kö‡=
¯ ^

N³=
K
Kö‡=
NXN
ù3ù
E
@G8­
  Kö=
N³=
š 
 =

¯
g
In order to write down the equation it is convenient to also expand
(2.11)
iPKƒ@(ö
^ ÷
@ N E
iK k
for a smooth function üºK
by (2.7) solves:
(2.12)
K ø NÁ
^
÷
@
D^
O
N
i6ÉK k
^
÷
@
Nb@(ö
O
^ ÷
O
\
i ÉÉK k @ Nb@ G ö G
š `
ü8Kƒ@(ö
^ ÷
@ Nb@ Ú ö Ú
. It turns out that d solves (2.3) if and only if ø defined
²¹N
Ú
K
ø
N
O¡T
G
øù3ù O
ø O
ø S
=
ø E’k ^
8
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
where
K
Ú
Ú
K
ø
ø
N
is a linear differential operator defined by
¡TWV ¢
N E
@
¢
=x@G
G
ê
O" T

G$# @ G&% %
O
G
<T
#@
=
@ G
ê
š
%
Gi`É
ê
%

=
š 
@
G ø O
 %
%
O
 ø
@ G
%
ø ì!
=
%
O
O @
G
O @

ì

øù
ì
ø

ì
ø O

@ G <T
Gi`ÉÉ
š
O

=
ê
 =
=
ê
š
øù
ì
ê
O G' ø
O
G
ì
ø
K
N
and
(2.13)
ø
K
G
K(
N E
TWV G N
O
~KdÕN
K(
G N
TWV
ü8Kƒ@(ö
^ ÷
@ Nb@ Ú ö Ú
~KdÕN is the operator in (2.5) where derivatives are expressed in terms of formulas
TWV O .
(2.8)-(2.10), ü is given by (2.11) and ö is replaced by
Let KWN denote the unique positive solution of (1.3). Then, taking KÕN as


a first approximation, the error produced is @ -times a function with exponential
decay. Let us be more precise. We need to identify both the terms of order @ and
those of order @ G :
K

N E
K
Ú
TWV ¢
N E

@
O" T

G)# @ G % %
O
%
@ G
=
#
G
T

š
O
G @¤i6É
ê
ê
% 
G
O %
%
@
ê
O
ì
O ì! 
G
 %
@

G
=
%
¢
=x@ G
O

=
@ G
@ G T
G i6ÉÉ
š

ê


ê
ì
š =

 =

ì

O O
ì
G'

K N
G 
'
CONCENTRATION ON CURVES
G
K

N
?K
9
turns out to be of size @Ú . Gathering terms of order @ and @ G we get

N E
TWV ¢
@

=
¢
G ={@G # O
@G
G
O
¢
š

K N
O G 
@ G G
O
G
Q
 G
4G
G
=
š \
O

Ý Ý
O+ G Q  Q G

i6ÉÉ, G
G
O+
@ G .
Ú

š




 ^ ÷ @ N
G i É Kk
'
Q
O

š Ú
T
=R@

i`ÉÉ 
O-
N*
O

G
O
@
÷
@
O

G # =


@ V

G
¡T
O
E
 ^
i É Kk
O

š O
\
K

G
N³g


\
i ÉÉ G
š `
=
O


'

^
Let us observe that grouped this way, the quantities V are odd functions of
Ú
while , . are even. We want now to construct a further approximation to a
G
solution which eliminatesû the terms of order
û @ in theû error. We
û see that
O
K
û
where
O
N

û
¡T
N E
¾ ~ K
(2.14)
N E
} ~8K
(2.15)
û
We write
K
O
G
û

N
E
û
­ƒ@³K/ V
O
@ G N

N
OG
û
O
O
O
N
SUTWV
Y
} ~8K
û
=

Y
^
=
û
SUTWV
g

û
K N
G 
N
û

S
=
K
Ú
S TWV
O Y U
=

O
O T
+
@ G .
Ú
S
O
N
û
û
O
K
O
¾~K
3ù ù O
û
and
(2.16)
?K
N E

û
û
G
¯
3ù ù O+
û
K
Ú
N
O
û
} ~8K
Ng
V in order to eliminate the term between brackets in the above
We choose E
÷
expression. Namely
for û fixed , weû need a solution ofû
û
=
O
SUTWV
= Y
@³K/ V
E

O
G
N
^
K€‘ˆN ERk
As it is well known, this problem is solvable provided that
(2.17)
Â
¿
T
K/ V
O
¿
N
G 
ž0 Elk g
Furthermore, the solution is unique under
the constraint
û
(2.18)
Â
¿
T
¿

ž1 Elk g
g
10
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
We compute
Â
¿
T
O
K/ V
ž0 E
N
G 
¿
¿
Â
V
T
¿
TWV ¢
ž0 E

G =Fi TWV i6É(Â ¿
¿
Â
T
¿
T

¢
¿
Ò
TWV
g
i6É where ˜ is the
The assumption that œ is stationary (1.9) amounts to E ={˜i
constant that precisely makes the amount between brackets identically k . In fact,
š
G . The solution has the form
we have the validity of the identity
û - v û G ž1 E û ˜ - v
û
where
VXV E
(2.20)
Function
V

N
ü V
^
V
G
E
G
@
¡T
=

^
V
û
E
G
O
VXV
V KWN

<TWV3¢4^
ü VXV E
(2.21)
÷
@(ü VXV Kƒ@
with
÷
Kƒ@
^
Gi`É3K k
Nbü V
G
²³N E
÷
Kƒ@
^
N
TWV¢
 G
˜
g
the unique odd function satisfying
=
(2.22)
and

V E
(2.19)
O
V32 
SUTWV
V = Y


V E

O

\
˜
^
Â

v 
V
ž1 Elk ^

is the unique even solution satisfying
 G
=
(2.23)
O
2  G
=
 G
SUTWV
Y
E
 G

g

In fact, we can write
(2.24)
û
û
Substituting
K
(2.25)
O
E
 G
@ G V ûN E

\
š =
Y+=™\ 
g

in (2.16), we can compute the new
error K û
û
V
E û
\
=
O
O
@ G .
Ú
K N
G 
O T
O
V N ù3ù
G K
Ú

K
û
O
V N
O
V N
û
,
V Ng
} ~K
Observe that since V is of size 4LKƒ@UN all terms above carry @ G in front. We compute
û
û
û
for instanceû
(2.26)
Ú
K
V N E
@
¡TWV
¢
K
V N =
¡T
^ ÷
@ N*
V65 =l@
^÷
are expressed in K N
Gi6ÉK k
Observe that all functions involved
domain for those variables is the infinite strip
9
E
^
ª:PªR‘
Ð =
‘
k ª
÷
¡T
Gi6É3K k
^ ÷ @ N
V
O
@ Ú ü87
variables and the natural
ªá°;¤@Ó8g
9
We now want to measure the size of the error in ¾ G K N norm. A rather delicate
term in the cubic remainder is
the one carrying   since in reality we shall only
û
assume a uniform bound on <   <>= w Ö ~ 2 Ù . Observe that a similar one arises from the
computation of K û V N ù3ù . Both of thoseÄ terms have similar
form.
For instance
the
÷
÷
^
one arising from K V N ù3ù can be written as ? E @ Ú   Kƒ@ Nbü V Kƒ@ N 2 KÕN with ü V
G
G
 G
smooth. Observe that
Â@
Q
?
Q
G”îRÛ@
7
B Q
Â+A
~
  Kƒ@
÷
Q ÷
E
N G ž
@C0<
 < = w Ö~ 2 Ù g
Ä
CONCENTRATION ON CURVES
Hence
11
@
<>?D<>= w Ö Ù îlÛ@FwE <   <>= w Ö ~ 2 Ù g
Ä
While in total ¾ G -norm we have û
<û K
Ú
@
V N<>= w Ö û Ù îRÛ@FGw g
Q
Q
Let usQ consider
the term } ~K V N . Since V can be bounded by ۀ@ G KÕN for
Q

large we obtainû that
û
û
Q
}
Q
V N
~8K
Q
E
E
O
K
YfK

ûV N
OFD

hence
Q
S
STWV
û = Y
V
=


S T Q
Q
ORQ Q S
V N
V ”
G
G îRۀ@GK¼\
N
S
û

KWN
S
@
<} ~8K V N<>= w Ö Ù îlÛ@HGw g
In summary, we have
<K
(2.27)
û
O
@
V N<>= w Ö Ù îŠ@ Gw g

To improve the approximation for a solution still keeping terms of order @ G we need
to introduce a new parameter, additional to . We let Þ KÕN be the first eigenfunction of the problem
Þ 
O
SUTWV
Y

E
ÞÊ=FÞ
ac~IÞ
^
Þ
K€‘ˆN Elk g
Then, as it is well known, ac~+Z k , Þ KWN is one-signed and even in . We now
consider our basic approximation to a solution to the problem near the curve œ‡… to
be
û
O
I E
(2.28)
O
V

@ºKƒ@
÷
NXÞõg
In all what follows,
we will assume the validity of the following constraints on the
parameters and :
O
^
 < =8L Ö ~ 2 Ù
<   <>= w Ö ~ 2 Ù îÔ@ w‰
Ä
Ä
Ä
O
O
Ö
Ö
Ö
<NM<>OÁl@ G <N   <>= w ~ 2 Ù
@P<N  <>= w ~ 2 Ù
N< Q< =8L ~ 2 Ù î @ w‰ g
(2.30)
Ä
Ä
Ä
In reality, a posteriori, these parameters will turn û out to be smaller than stated here.
O
N E‚k , which can be expanded in
We set up the full problem in the form RK I
<
(2.29)
the following wayû
?KRI
O
<>J«ÁK<
O
< =ML Ö ~ 2 Ù
û
T
N E
O
ùù O
G
KRIcN
û
V E
KRI6N E
O
O
K
O
 O
@³­ YKXK û
O 
O
V
K

û
û
= û
O
Ú
The new error of approximationû is
S
<
K
û
SUTWV
û
û
YPI
O
O
S
S
SUTWV
N
RK I
N =DI = YHI
O
Elk
O
V N
@¾~UK,¤Þ+N
¯
O SUTWV
SUTWV
û N
V N
=
N$û K,(Þ+N
K,(Þ+
Ú

S
O
S
O
SUTWV
^
V N = f
V N
@(Þ+N =ÏK
Y K
@¤Þ


12
O
where K

M. DEL PINO, M. KOWLCZYK, J.-C. WEI
û
V N
is given at (2.25). We decompose
S
where
S
(2.31)
S
V E
T
VXV Ál@¾~8K,(Þ+N E
and
S
(2.32)
V
S
E
G
S
O
VXV
V
G
O
G @ Ú  Þ
S
V =
^
a6~$@'Þ
VXV
û
û
û
In summary the problem takes
the form
near the curve,
(2.33)
where
S
V
¾ V K
O
N
K
Ú
û
was just described,
and û
¾ V K
(2.34)
N E
û
V K
}
(2.35)
T
O
KRI
N
O
V
û
ùù O
û
G
N E
S
O
N
V K
}
N ERk
û
O YHI SUTWV
S
S
=DI
YPI
=
û
=
STWV
^
û
g
We recall that the description here made is only local. We will be able however
to reduce the problem to one qualitatively similar to that of the above form in the
infinite strip.
3 The gluing procedure
Let I‡KM6N denote the first approximation constructed near the curve in the coorªÆµ$~;¹\ kk , be a fixed number. We consider smooth cut-off
dinate M in : D G . Let µ
D
D
D
D
š
function T1UUK N where
n : such that T1U¤K N E \ if ª™µ and EÏk if Z
µ . Denote as
Q
Q
well T U Kö¤N E T1UUKƒ@ ö N , where ö is the normal coordinate to œ‡… . We define our first
global approximation to be simply
^
w KM6N E T U KöUNVI
Ú
û
extended globally as k beyond
the ûWµ;¤@ -neighborhood of œ‡… . Denote KdÕN û E h
H d{=
O
O+
OX
S
iPKƒ@M6Ned
d
for d E w
, now globally defined in : G . Then K w
N E’k if
and only if
û
û
where
S
û
û
}‚K
¾
K
N E
K
S O
N E
û
^
?K w N
E
and
We further separate
¾
K
(3.1)
w
û
N E
û
H
S
O
N
=
^
}RK
O
N
û
STWV
Y
w
S
û
=Fi
û
SUTWV
w = Y w
^
g
in the following
form:
û
û
where, in coordinates K
that we want
^÷
N
E
T U
Ú
OjC
û
, we assume that
¾
KT U
û
N
O
C
¾{K
N E
S O
is defined in the whole strip
}‚KT U
û
Ng
9
so
CONCENTRATION ON CURVES
13
û
C{^
N satisfies the following nonlinear coupled system:
We achieve this
û if the pairû K
OFC
O
SUTWV C{^
S O
¾ÒK N E T U }lK
N
T U
T UY w
(3.2)
C
CRO
SUTWV C
S O
û
š
û =YT U N Y w
E K¼û \=ZT U N
=’i
K¼\
@[\T U [
Ú
O
OˆC
š
(3.3)
@ G KH]T U N
K¼\=ZT U N }RKT U
N³g
Ú
Ú
9
Notice that the operator ¾ in the strip may be taken as any compatible extension
outside the Wµ ;¤@ -neighborhood of the curve.
û
What we want to do next
is to reduce the problem
to one in the strip. To do this,
C
H
O
we solve, given a small , Problem (3.3) for . This can be done in elementary
way: Let us
observe first that since i is uniformly positive and w is exponentially
Q Q
TWV
small for ö Z‚µ@ , where ö is the normal coordinate to œ‡… , then the problem
C
H
(3.4)
has a unique bounded solution
C
<
û
Assume now that
STWV C
N
=ÏKiR=ÏK¼\=ZT U N Y
C
whenever <Ç&<
<
îlÛD<Ç&<
¿
E
w
O
ª
¿
^
Ç
‘
. Moreover,
g
¿
satisfies
the following
decay condition
û
û
Q
(3.5)
[
KM6N
QeOlQ
Q
KM`N
T_^ B
îY
Q Q
for ö Z
µ ^
@
for certain constant ±mZ k . Since } has a power-like behavior with power greater
û
than one, a direct application of contraction
mapping principle yields that Problem
C
C
û (small) solution
û
E
K N with û
(3.3) has a unique
C
<
N<
K
¿
O
< =ML Ö Ø ú Ø ` U … Œ Ù
îlÛ@³­N<
¯ ^
< =ML Ö Ø ú Ø ` U … Œ Ù
Q Q ‰
<N[
‰
where with some abuse of notation by Ð ö Zlµ;¤@C Ó we denote the complement of
µ;¤@ -neighborhood of œ… . The nonlinear operator
satisfies a Lipschitz condition
û form û
û
û
û
û
of the
<
C
K
V N`=a<
C
K
G
N<
¿
îRÛ@¹­N<
V =
G
O
<>=ML Ö Ø ú Ø ` U … Œ Ù
<N[jK
‰
V =
G
N<>=ML Ö Ø ú Ø ` U … Œ
‰
¯
Ù g
The full
problem has been reduced to solving the (nonlocal) problem in the infinite
9
û
û
û
û
strip
(3.6) û
9
¾
G
K
T U R
} K
N E
OFC
K
NXN
O
T U
S O
SUTWV C
T UP
Y I
K
N
for a no G K N satisfying condition (3.5). Here ¾ denotes a linear operator that
Q Q
G
U
coincides with ¾ on the region ö ªÆ\ k … .
Q Q
U
û
¾ for ö ªÔ\ k … is given in
We shall define
this
operator
next.
The
operator
^÷
coordinates9 K N by formula
(2.34). We extend it for functions defined in the
^÷
û
û
entire strip , in terms of K û N , as follows:
(3.7)
b
¾
G
K
N E
¾ V K
N
Ob
Q Q Kƒ@ N
Ú
K
N
where K “ N is a smooth cut-off function which equals 1 for “ ªÑ\ k µ an vanishes
š
identically for for “ Z k µ and ¾ V is the operator defined in (2.34).
14
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
Rather than solving problem (3.1) directly,
we shall do it in steps. We consider
9
û
o G K N : given and satisfying bounds (2.29)the following projected
problem
in
^
^
9
û
(2.30), find functions
no G K û N , ž n¾ G K k °N such that
¾û
(3.8)
(3.9)
(3.10)
K
G
^
K
¿
Â
û T
N E
kû N E
K
¿
b S
û
V
K
^÷
O
^
}
Kû
°;¤@¤N
^G
O
ù
N
^
K
KÕN¤ž0 E
N
û 
OFC
û
Â
O
Kƒ@
÷
k N Eû
¿
T
¿
b
Nû
ù
O ž'Kƒ@ ÷ N b Þ

^÷
K
^
K
û
SUTWV C
°;¤@¤N
^
KÕN¤ž0
NÞ
in
9 ^
O
=
‘ͪ:Pª
^
Elk
÷
k ª
^
‘
°
ª
g
@
K NXN
T U YHI
K N .
Here } K N E T U }lK
G
We will prove that this problem has a unique solution whose norm is controlled
S
S
by the ¾ G norm, not of whole V but rather that of V .
G
After this has been done, our task is to adjust the parameters and in such
a way that and ž are identically zero. As we will see, this turns out to be equivalent to solving a nonlocal,
nonlinear coupled second order system of differential
^
equations for the pair K, ž¹N under periodic boundary conditions. As we will see
this system is solvable in a region where the bounds (2.29) and (2.30) hold.
We will carry out this program in the next sections. To solve (3.8)-(3.10) we
need to investigate invertibility of ¾ in ¾ G -o G setting under periodic boundary
G
conditions and orthogonality conditions.
4 Invertibility of ced
Let ¾ be the operator defined in o G K
G
linear problem û
(4.1)
(4.2)
(4.3)
¾û
K
N E
Çû
O
Kƒ@
÷
N
b
G ^
^
^
K k û N E
K °f;¤@UN
^ ÷
¿
KWN¤ž1
Â
K N
T

¿
b
Q
9
ÇnP¾ G K Ng Here Kƒ@ û

ù
E
Q
9
N
by (3.7). In this section we study the
O ž4Kƒ@ ÷ N û b Þ
^
K
Â
T
k N Eû
¿
ù
K
K
^÷
^
in
9 ^
^
°f;¤@UN = ‘·ªgª
NÞ KWN¤ž1
^
ERk
O
÷
k ª
^
‘
° ^
ª
@
¿
N is the cut-off introduced in the definition of ¾
for given
G
in (3.7). Our main result in this section is the following.
Proposition 4.1. If µ in the definition of ¾ is chosen sufficiently small then there
G
û
exists a constant ÛõZ k , independent
of @ such that for all small @ Problem (4.1)û h KÇ'N , which satisfies the estimate
(4.3) has a unique solution EY
<
@
@
<Ni w Ö Ù R
î ÛZ<Ç&<>= w Ö Ù g
For the proof of this result we need the validity of the corresponding assertion
û
for a simplerû operatorû which
does notû depend of µ . Let us consider the problem
(4.4)
(4.5)
(4.6)
jû
O
SUTWV
9 ^
E ûÇ
N E Ò
= Hû
= Y û
in
^
^
^  ù
^
^
^
ù
K k û N E
K ° ;¤@¤N
K k N E û
K °f;¤@UN
^ ÷
^÷
¿
K ÕN¤ž1 E  ¿
Â
K N
K
N Þ KÕN¤ž1
T
T

¿
¿
K
E’k
O
ª:ª
=+‘
^
k ª
÷
ª
^
‘
°
@
g
CONCENTRATION ON CURVES
15
Lemma 4.2. There exists a constant ÛÌZ k , independent of @ such that solutions
of (4.4)-(4.6) satisfy the a priori
û estimate
@
@
< i w Ö Ù îRÛZ<Ç&< = w Ö Ù g
<
û
Proof. Let us consider Fourier series decompositions for Ç and
û
K
ÇK
^÷
k¿
N E
^÷
û
k¿
N E
þ$ÿ
V © KWN
­
©l ~
ó
ê
÷
G
÷
@
°
(4.7)
j
© O
G @ G
ì
m
© N E
~UK
Ç
with orthogonality conditions
Â
(4.8)
ûFm
¿
T
©
ž1 E

¿
û.m
We have denoted here
j
ûFm
© N E
~K
=
V
K
CÒ^C
N E
Â
¿
T
­
© ^
Û­R<
°
¯
÷
ì
g
Elk g
SUTWV
û.m
© g

j
associated to the operator ~ , namely
SUTWV
K¼\= Y
© N < G w Ö v6Ù
=
^A¢
<(K
ê
@
ì
n:
© = Y
Since (4.8) holds we conclude
that û.m
ûFm
(4.9)
š ä ¢
¯ ^
÷
@
°
¿

¿
© < G w Ö v6Ù O
=
G
© ÞPž1
K:N
QC Q O
G
ê
© KÕN
û.m
© 2 O
Let us consider the bilinear form in o
T
© KWN
û.m
¿
Â
Ç
of the form
š ä ¢
ó3prq
O
Then we have the validity of û.the
m equations
û.m
¢
ó3prq
ì
š ä ¢
ê
û
O
@
°
ó
þ$ÿ
­Ç V © KÕN
©l ~
š ä ¢
¯
î
Q C+Q ¯
G ž0jg
N
ûFm
K
© ^
û.m
© N
for a constant Û Z k independent of ½ . Using this fact and equation (4.7) we
conclude the estimate
ûFm
û m
.
m
K¼\
O[¢ @
© < G w Ö v6Ù O <
=
ûFm
N<
© 2 < G w Ö v6Ù î
=
ÛD<Ç
© < G w Ö v6Ù g
=
©
In particular, we see from (4.7)
an equation of the form
û.m that û.m satisfies
m
© 2 O
© E
^
Ç ©
nL:”g
©
v Ù îˆÛD<Ç © >
v Ù . Hence it follows that additionally we have the
where < Ç >< = w Ö `
< = w Ö6
estimate
û.m
m
(4.10)
<(K © 2 < =G w Ö v`Ù î Z
Û <Ç © < G= w Ö v`Ù g
m=
m
Adding up estimates
û (4.9), (4.10)û in
<>n G
@
< G= w Ö Ù
which ends the proof.
O
<>n
¢
and ½ we
û conclude that
@ O
< G= w Ö Ù
<
@
< G= w Ö Ù î
@
ÛZ<Ç&< G= w Ö Ù
^
o
16
o
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
We
consider
now^ the following problem: given ÇPnP¾ G K
^
9
û ¾ G K k °N such that
N , ž n
G K
û
9
N
, find functions
n
÷
O ž4Kƒ@ ÷ XN Þ û in 9 ^
Kƒ@ N û

^
^
^
^
^
O
^
ù
Ö ^ k N E
K °;¤@¤N
K k N E û ù K °f;¤@UN =
‘ ª:ª
‘
û
^÷
^÷
^
÷
°
¿
¿
K WN¤1
Â
K
N
ž E Â
K
NÞ KWN¤ž1 ERk k ª
ª
g
(4.13)
T
T

@
¿
¿
û
Lemma 4.3. Problem (4.11)-(4.13)
possesses a unique solution. Moreover,
jû
(4.11)
(4.12)
K
Ç û
E
N
O
@
@
<Ni w Ö Ù îRÛZ<Ç&<>= w Ö Ù g
<
Proof. To establish existence, we assume that
ÇK
^÷
k¿
N E
ó
þ$ÿ
­Ç V © KWN
©l ~
š ä ¢
© O
G @ G
and
©
T
ž1 E

¿
Ç
ì
G
ê
@
°
¯ ^
÷
m ì m

û.m
¿
Â
š ä ¢
© KÕN
V
m K:N , andm constants © , ž © such that
O ž © Þ
©
Pnh:g
© m nLo
© O
Ç
ó3p q
O
ûFm
¿
Â
© N E
¾0~K
@
ê û.m °
and consider the problem
of finding
û.m
û.m
¢
÷
© ÞPž1
T
Elk g
¿
m this problem is solvablem with the choices
Fredholm’s alternative yields that
m
© E
- T¿
=
¿
- T¿
k¿
(4.14)
Q
©l ~
Finally define
û
K
^÷
û
k¿
N E
­
©l ~

¿
Observe in particular that
m
÷
ž'K
÷
N E
N E
k¿
©l ~
k¿
©l ~
ž © E
- T¿
=
- T¿
© ÞҞ0
Ç
¿
g
Þ G ž0
¿
m
Q l
O Q © Q
@
© G
ž
G îRÛ@P<Ç&< G= w Ö Ù
ó
þ$ÿ
V © KÕN
and correspondingly
K
m
ž0
^

G ž1
Ç ©
þ$ÿ
­ƒ V ©
þ$ÿ
­ƒž V ©
÷
š ä ¢
ê
ó
ê
°
š ä ¢
ê
@
°
š ä ¢
ó
÷
°
û
O
ó3prq
G
ì
÷
ó3p q
O
ì
÷
O
ì
÷
©
G
ê
G
©
ê
š ä ¢
°
š ä ¢
óprq
ž
š ä ¢
© KÕN
ê
°
@
°
¯ ^
÷
ì
¯ ^
÷
ì
¯
÷
ì
9
g
Estimate (4.14) gives that Kƒ@ N and ž'Kƒ@ NXÞ have their ¾ G K N norms controlled

û
by
that of Ç . The a priori
estimates of the previous lemma tell us that the series for
9
is convergent in o G K N and defines a unique solution for the problem with the
desired bounds.
o
CONCENTRATION ON CURVES
17
Proof of Proposition 4.1 . We will reduce Problem (4.1)-(4.3) to a small perturbation of a problem of the form (4.4)-(4.6) in which Lemma 4.2 is applicable. We
will
achieve thisû by introducing a change of variables that eliminates the weight
T
ù3ù
G in front of û . We let
K
^÷
^
The map ürq­ k … N‡z
Ä û
then
^
E m p K
N ^1s
N
­k
ù E
üÕK
÷
NXN
^
÷
üÕK
N E
@
ù
TWV
Â
K “ Nbž “ g
~
is a diffeomorphism, where t½ E - ~ Ä
û
^
p ù
ùù E
Ý
G Kƒ@
÷
ù ù O
N p
Ý Ý
@
 Kƒ@
÷
ù
N p
K “ Nbž “
. We denote
Ý
while differentiation in does not change. The equation in terms of p now reads
O
S TWV
U
SUTWV
O
ù E
K p N
YfKRI
=
N p
@ p
Çt
Ú

Ý
O
÷
O
÷
^
9
in t
ž4
t Kƒ@  NXÞ
t Kƒ@  N 
^
^
^
^
^
^
O
^
ù
ù
p K k N EXp K t½V;¤@N p K k N X
E p
K t½V;¤@¤N =
‘·ª:ª
‘
Ý
Ý
^÷
^÷
^
÷
t½
¿
¿
N
 NÞ KÕN¤ž0 Elk k ª
 ª
p K
p K Â
KÕN¤ž0 E Â
g
T
T

@
¿
¿
^÷
^ TWV ÷
rN E ÇK ü
K  NXN and the operator t
Here Ç
is defined by using
the above
t K
÷
÷
÷
Ú
TWV ÷
K N
formulas to replace the derivatives by  derivatives and the variable by ü
H
STWV
p = Y
in the operator

Ú
O
p
Ob p
t
. The key point is the following: the operator
b ºK p N E
is small in the sense that
<
t
Ú
O
K p N
@
O
p ù
Ý
SUTWV
YfKRI
=
SUTWV

N p
ºK p N < = w Ö @ s Ù îlۀµP< p < i w Ö @ s Ù g
Q
Q
This last estimate is a rather straightforward consequence of the fact that @(ö ª
š
TWV
k µ(@
wherever the operator t is supported, and the other terms are even smaller
Ú
when @ is small. Thus by reducing µ if necessary, we apply the invertibility result
û
of Lemma 4.2. The result thus follows by transforming the estimate for p into
similar one for via change of variables. This concludes the proof.
o
5 Solving the nonlinear intermediate problem
In this section
û we willû solve problem (3.8)-(3.10)
û
¾
û
û
G
K
N
O
Ú
K
N E
b S
V
O
}
G
K
O
N
Kƒ@
÷
N
b
O ž'Kƒ@ ÷ N b Þ ^

9 û
û
under
periodic
boundary conditions, and orthogonality conditions in . Here } K N E
b
OáC
G
} V K
K NXN whenever this operator is well defined, namely for
satisfying
(3.5). A first elementary, but crucial observation is that the term
S
VXV E
­@ Ú
¡T
G  
O
¯
@Ua6~ Þ
18
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
S
÷
in the decomposition of V , (2.31)-(2.32), has precisely the form ž4Kƒ@ NXÞ and can
therefore be absorbed for now in that term. Thus, the equivalent problem we will
look at is
û
û
û
¾
K
G
O
N
Ú
b S
N E
K
O
V
}
G
The big difference between the terms
S
<
V
S G
O
N
VXV
and
÷
S
UKƒ@
V
G
^
@
Ö
w
<>= w Ù R
î ۀ@ G
G
S
K
N
b
O ž'Kƒ@ ÷ N b Þõg

is their sizes. Notice that
b S
S
V .
while VXV is a priori only of size 4LKƒ@ w‰ N . We call
Á
G
G
For furtherS reference, it is useful to point out the Lipschitz dependence of the
on the parameters and for the norms defined in (2.29)-(2.30).
term of error
G
We have the validity of the estimate
S
<
(5.1)
V
K
G
^
V V N6=
S
V
K
G
@
N<>= w Ö Ù R
î ۀ@ Gw R­ < V =
^
G
G
O
< J
G
<N V =u
G
¯
<>O
Let h be the operator defined by Proposition 4.1. Then the equation is equivalent to the fixed point problem
û
û
û
EYh
(5.2)
K
S
O
}
G
û
NXN‡ÁavxK
K
G
Ng
Q
The operator h has a useful property: Assume Ç has support contained in ~ U
G … . Then EYh KÇ'N satisfies the estimate
Q
(5.3)
û
^÷
K
N
QbORQ
û
[
^÷
K
Q
N
Q
û
îw<
Q
<
~ U
T w x
B
¿
Q
for Q
ß
k µ
Z
@
Q
î
g
û since
In fact,
is supported on ª G … and so do the terms involving and ž ,
Ú
Q Q¤£
~ U
then satisfies for of the formû
G … û an equation
û
¡T
with
í
form p
=ÏK¼\ OFí K¼\NXN Elk
Q QU£
~ U
û
k uniformly
as @?z k . For K¼\Nfz
G … we can then use a barrier of the
Ö
(
w
(
y
x
^÷
Q Q
T
T B Ù
3 ~ U
K N E < <
w‰
to conclude
that for Z … we have
û
û
¿
^÷
T y(x
Kû Nfîw< <
‰B g
û
G
ù3ù O
¿
The remaining inequalities for are found in the same way. The bound forC [ û
follows simply by local elliptic estimates. Now we recall that the operator K N
satisfies, as seen directly
from its definition,
û
û
û
<
(5.4)
C
N< = L
K
î
Q
Û­N< [
and also the Lipschitz
û
û condition
(5.5)
<
C
K
V N6=
C
K
G
N< =8L
QeOlQ
Q
ÛL­N< [jK
î
Q
û
< =8L Ö Ø Ø ` (w y(B x Ù
û
V =
G
N
Q¤O]Q
û
O
û
V =
T xB ¯ ^
Q
< =ML Ö Ø Ø ` (w B y Ù
G
¯
g
These facts will allow us to construct a region where contraction mapping principle
applies. As we have said,
<
S
G
@
<>= w Ö Ù îR۔Ÿ_@FGw g
CONCENTRATION ON CURVES
19
for certain
constant ۔Ÿ”Z k . We consider the following closed, bounded subset of
9
o G K N : z
û
^
@
w Ö û Ù îgnm@ Gw
û
% < û <Ni
no G K N % Q QeOlQ
Q
@ T x
%
[
< =ML Ö Ø Ø `€ (yB x هîw< < i w Ö Ù B‚„ ƒ
%
%
%
we claim that if the constant
n is fixed sufficiently large then the map v
z
}~
E|{
û
9
defined
in (5) is a contraction from into itself.
Let us analyze the Lipschitz character of the nonlinear operator involved in v
û
û
û
for functions in :
}
û
where
D
V K
}
^
b
N E
K
G
9
9†
so thatû denoting U E
<}
G
@
K N<>= w Ö Ù î
û
Q
Ð N
î
û
Q
ÛÏ­
û
Using Sobolev’sû embedding we
get
S
<
< = w‹ Ö@ Ù
O
@
< G=  Ö Ù î
<
ۊ
û ­N<
<
S
O
@
K N < = w‹ Ö x Ù
<
C
@
K N <=G  Ö x Ù î
û
It thus follows that
<}
(5.6)
G
K
NXN
û
Q
û
STWV ¯
^
G
z
Q ¯ ^
G
û
û for
have that
­N<
û
C
n
S
@
N< = w ‹ Ö x Ù
K
S
û
O
< i w Ö@ Ù
<
,
O
û
C
<
K
@ ¯ ^
< Gi w Ö Ù
@
N< G=  Ö x Ù
¯
g
9
, (5.3), that the area of U is of order
while using estimate (5.4), the facts that n
4Kµ;¤@UN and
Sobolev’s embedding,
we get
û
û
C
z
K
=DI
QS O
ªÆ\ k µ û ;¤@Ó , we
O
@ O
< = w‹ Ö@ Ù
< < G=  Ö Ù
S
ÛÏ­N<
}
Q
V K
Q
SUTWV
N
û
for n[K k \N . From here it follows
that
Q
OjC
V K
û
OFD
Y
­XKRI
N E
}
T
Ûu
@
N< = w Ö Ù î
xB
 ­¼\
ÛÏK$@ G w
û
O
O
< i w Ö@ Ù
<
‹ O
û
S
@ ¯
<Gi w Ö Ù g
<
@ Ú Ng
Besides, asû for Lipschitz
condition, we
estimate,
û
û find after a direct
û
<} V K
V N6=x}
V K
G
@
N<>= w Ö Ù
îÍ­N<
O
SUTWV @ O
ûV < = w‹ Ö Ù
@
< û V <>=  Ö Ù
SUTWV
¯
@ O
<
>< û =  Ö Ù û
<
< = w ‹ Ö@ Ùû
G
G
@ O
‡
V <>= w ‹ Ö Ù
­N<
=
<
G
G
û
=
with constants Û independent of the bound a priori assumed on <
û
û
û
û
û
then conclude
for } û ,
<}
G
K
V N6=Ž}
G
G
K
G
@
N<>= w Ö Ù
îˆ<}
O
Vû
OFC
V K
<} û V K
G
û
K
OFC
V û NXN`=x}
V NXN`=x}
K
O
@
V = û
>< =  Ö x û Ù
G
C
C
ʭN<
O
‰ ­N<
á
O
C
<
K
Kû
V N6=
V N`=
C
K
Kû
G
G
<
N<>=
N<>=
<Ni w Ö ~ 2 Ù . We
Ä
@
Ö
K û
K V û NXN<>= w x Ù
G
OFC
@
ûV K
û
K
NXN< = w Ö x Ù
G
G @
¯
V =
>< = w ‹ Ö x Ù
@Ö Ù G
 x
@ ¯
w‹ Ö Ù
OFC
V
@ ¯ ^
V <>=  Ö Ù
x
20
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
where
‰
withm
‰
û.m
SUTWV
O
< = w‹ Ö@ x Ù
<
E
C
û.m
<
‰
ûFm
SUTWV
O
N < = w ‹ Ö @ x Ù
K
O
‰ V
E
G
<
SUTWV
O
< = w‹ Ö@ x Ù
C
<
Arguing as before,
the conclusion
of these estimates is
û
û
û <}
(5.7)
Now, let
n
G
V N`=x}
K
then pjE
G
K
G
v K
x
@
Nû < = w Ö Ù î
N
Ö SUTWV Ù O
ÛLKƒ@HG w
ûFm
K
û
@HGw N<
@ ^
N<>=  Ö x Ù ½ E
û
V =
G
^ š
\
g
@
< i w Ö Ùg
satisfies thanks to (5.6),
O
S
S
@
< p <Ni w Ö Ù î‹< h < Р۔ŸI@ Gw
یn
@ Gw 8
Ó g
ZRÛ Ÿ < h
Choosing any number n
On the other hand we have
< we get that for small @
@
< p <Ni w Ö Ù îgP
n @ Gw g
@
@
< p < = L Ö Ù îRÛZ< p <Ni w Ö Ù g
z
z
But p satisfies an equation of the form ¾ K p N E Ç with Ç compactly supported.
G
Hence p belongs to thanks to the discussion above. v is clearly a contraction
mapping thanks to (5.7). We conclude that (5) has a unique fixed point in .
S
and the operator h itself carry the functions and
We recall that the error
G
as parameters. A tedious but straightforward analysis of all terms involved in
the differential operator and in the error yield that this dependence is indeed Lipschitz with respect to the o G -norm (for each fixed @ ). In the operator, consider for
: û
instance the following only term involving
û
Œ
N E
K
@(G
÷
  Kƒ@
Then we have û
<
Œ
Let p K N E - T ¿
¿
û
Q K
ó>Ž
^÷
N G ž1
ù ñ p K
÷
ù ñ Â T
. Then
û
Q ¿
K
¿
û
û
û
Q Q O š
Q Q Q ù Q
N+îÔ@Ր
 

G
û
>ó Ž
÷ O ß TWV
Q \
ù QG g
@
 
î š
ù ñ p K N
Hence
(5.8)
ê
~
Q
g
ó6Ž
Q
Q
@
  Kƒ²³N G ž³²
K N < G= w Ö Ù Š
î @ Ú ÂmÄ
÷
N
÷
p K NfîlÛ@
so that finally
<
Œ
K
û
TWV
<
û
@ ^
< Gi w Ö Ù
@
N<>= w Ö Ù îŠ@Q< < J g
^÷
Q
N G ž0
ì
CONCENTRATION ON CURVES
21
For the other
terms the analysis follows in simpler way. Emphasizing the depen
dence on what we find for the Linear operator h is the Lipschitz dependence

< h

h
=
‰
w <0îlÛ@Q<
V =
< J g
G
z
û
We recall that we have the Lipschitz dependence
(5.1). Moreover, the operator }
^
also has Lipschitz dependence on K (N . It is easily checked that for n
we
have, with obviousû notation, û
O


@
<} Ö 2 ‘ Ù K N6=x} Ö w 2 ‘ w Ù K N< = w Ö Ù îRۀ@FwE ­R< V =
< J
G
‰
‰
Hence from û the fixed point
characterization we then see that
û
Ö  2 ‘ Ù <Ni w Ö @ Ù îRÛ@FGw ­R< V =
w w
Ö  2‘ Ù =
‰ ‰
<
(5.9)
O
< J
G
¯
<N V =’
<N V =’
G
<>O
¯
G
<>O g
g
We summarize the result we have obtained in the following:
Proposition
5.1. There is a number n Z k such that for all sufficiently small @
^
û
û
and all K ^ (N satisfying (2.29)-(2.30), problem (3.8)-(3.9) has a unique solution
E
K
¤N which satisfies
û
Q
û
<
û
Besides
QbORQ
^
@
< i w Ö Ù Š
î nmû @HGw
<
û
Q
[
@ T x
<Ni w Ö Ù B
î <
< =ML Ö Ø Ø `€ (yB x Ù ˆ
and in the sense of estimate (5.9).
depends Lipschitz-continuously on
Next
we carry out the second part of the program
which is to set up equations
^
for and ž which are equivalent to making ž identically zero. These equations
are obtained by simply integrating the equation (only in ) against respectively 
andS Þ . It is therefore
of crucial importance to carry out computations of the terms
S
V ž1 and - v
V ÞLž0 . We do that in the next section
- v

6 Estimates for projections of the error
S
S
In this section we carry out some estimates for the terms - v V ž0 and - v V ÞPž1 ,
S

where V , we recall, was defined in (2.31)-(2.32), and is an odd function. InteS

gration against all even terms in V therefore just vanish. We have
Â
S
v
V

O
Â
v 
O
We recall
E
Â
Â
S
V
v
# @3Yf­K
v 
#K
G 
Â
û
O
V N

O

E
û
V
O
v
STWV
?K
O
V N

SUTWV ¯
=
@(Þ+N
û
S

=ÏK
K,¤Þ
N
O


O
û
V N
S
@
=
Ú
YfK
K,(Þ+N

O
'
û
V N
STWV
@(Þ
'
g
22
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
K
û
O
@ G V N E

O
O Ú
O
@ G .û O
­K
S
V N

K
G
=
N
S
= Y

û
O“
û
K ûV N
¯ O’<T
S Ú TWV
ù3ù
V
GUK V N

where is an odd function, ” is an even function and
G
we see that
Â
?K
v
û
O
V N

E

TWV>
={@ GM•
O š O š
 Â
¡T

G
TWV6¡TWV
TWV ¢
O ê
O
O
Â
(6.1)
ý
v 
m  v 
Here and below we denote by … , ½ E
functions of the form m
m
ý
ý
ý
E
÷`^ … K
Kƒ@
\
÷


V N
S
O
V N
K

N

G OX T Ú i6Éɤ v
Ò 엖
v 
û
û
Ú
^ š
G O Ý Ý Â v G

O
­K
K
v
G Â
is of order @ Ú . Thus
N

G
v 
 Â
K
G
S
=

@ Ú
ý
V
= Y
SUTWV
V

O

û
@ Ú
ý
¯
cg
G
, generic, uniformly bounded continuous
^
÷ ^ ÷ ^
÷
N º
Kƒ@ N  Kƒ@ N @  Kƒ@ NXN
where additionally V … is uniformly Lipschitz in its four last arguments.
÷
The coefficient in front of Kƒ@ N in (6.1) can be computed as
<TWV¢
T
¡
E ¡TWV¢ 6
G O+¡T Ú i ÉÉ Â G x
G
Ú i ÉÉ Â
G Â
= ˜
v
v 
v  ì
{ ì
ê
E = V - vû
G .
û - v where we have used the fact that
= ˜ - v
G E {
G`Â
ê
v 
O
{ S
S
 ¯
SUG TWV

û
û
û
V N =
V is to main
= Y
that
the term ­K
order
ofû the form



O
û
VXV
V where
G G V and it is therefore of the order 4Kƒ@ G N . We have V E
G

G
û
we recall VXV is odd, V is even, and with their sizes respectively
proportional to @
Ö
Ù
S
U
S
W
T
V
G
SUT
û
and to @ . Thus in the expansionû of the square
term
G G V asymptotically

G
X
V
V
V
only the mixed product between
and
gives rise to a nonzero term after the
G
integration against û . We have
û
We see
S Ö SUTWV Ù U
S Tû
Â
v 
­K
O

V N

E
YKàY+=™\N
Â
š
E
@(GIü VXV Kƒ@
(6.2)
÷
S
S
=
v 
Nbü V
=

G
Kƒ@
SUT

÷
N
Y
G V ž1
G
Kƒ@
¯
V 0
ž SUTWV
û
÷
O
@ Ú
ý
N YfKàY=™\NÂ
G
v 
SUT

G

V
 G
ž1
O
@ Ú
ý
G
g
CONCENTRATION ON CURVES
Now, let us consider
÷
p Kƒ@
N E
û
Â
23
v
V N
K
Ú
g

÷
All terms in this expression carry in ¾ G -norm as functions
of ² E @ powers three
or higher with the exception of the terms of size @ in . Thus we find
÷
p Kƒ@
N E
@
¢
û
Since
V E
with
Ú
û
^
÷
V 5
K V N =
Â
Gi6ÉK k @ N*
v

û
T
¡
^ ÷ O
V ={@
Gi É K k @ N Â
4LKƒ@ Ú Ng
v

V KWN

p Kƒ@
÷
odd and
N E
@ G
÷
Kƒ@
KÕN
Nbü V
=Ò@G
(6.3)
÷
Kƒ@
G
÷
Kƒ@
N
¡T
O
V KWN

@
÷
Kƒ@
Nbü V
÷
Kƒ@
G
N
KWN
 G
even we obtain
TWV
N˜
<T
N
÷
@ G ü VXV Kƒ@
E
÷
@(ü VXV Kƒ@
 G
û
<TWV
÷
Kƒ@
G
K
v
÷
GIü VXV Kƒ@
Nbü V
¢
Â
^ ÷
@ NÂ
NXi`É_K k
N
÷
Kƒ@
= T G i6ÉXK k ^ @ ÷ N*
N
 G
NÂ
#
v
V
v 
 G
2
š™

O 4Kƒ@ Ú N

O

5
 G
\
˜
O

 G
V

' g

Note that by differentiating the equation (2.23) and using equation (2.22), we obtain
Â
(6.4)
YfKàY=™\N
v
SUT
G


V

E
 G
=mÂ
v 
O

V
Â
K
v
O

Â
v 
÷
Nbü V
O
 G
E
E
(6.5)
¡TWV
@(G
=Ò@ G
˜
TWV
2
TWV¢
˜
S
V N

@ G ü VXV Kƒ@
E
û
O
G
=
÷
Kƒ@
S
N
O

\
TWV ¢
Â
G
­
v
Â
÷
Kƒ@
˜
G
=


š
2
 G
v 
Y
NÂ
# YfKàY=á\N
v

O

¯ O
V

 G
G O
û
SUTWV

V

Â
SUT
@ Ú ­ V
V

Â
v  G

E
š
= K
Y=á\
O
\
š NÂ
v 
2 g
 G
G
 G 
OÊý ¯

G
N
¯ O @ Ú ­ ý V …   OÊý … ¯
G
where we have used (1.9) and the following integral identities
2
 G
V N
K
Ú

N
' O @ Ú ­ ý V …   OÊý … ¯
O ˜ TWV fK
ý
G

û
v 
˜
Adding (6.2) and (6.3) and using (6.4), we have
­K
\
G ^ ˜ TWV Â
v  G 
E
\
K š =
š
Y=™\
NÂ
v 
G g
24
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
In summary, we have established that
û
O
(6.6) Â v K

V N
ž0 E

@ G ­
÷
  Kƒ@
O
N
÷
± V Kƒ@
N
O

±
÷
Kƒ@
G
¯ O
N
ý
OÊý ¯
@ Ú ­ V … 
G
where ± V is given by
and ±
± V Kƒ²³N E
(6.7)
G
T
K
G
O
Ý
š
TWV TWV
݃N E
TWV
˜i
icÎ
^
is given by
±
(6.8)
Kƒ²³N E
G
TWV
={˜i
O
i`ÉÉ
TWV`O
Kƒ˜
¢
\N
GUg
Let us estimate now the term
Â
v 
Â
#K
v 
(6.9) Â v

O
E
Â
v 
û
O
û
V N

Ð K
@XYKàY+=™\NÂ
O
V

O
SUTWV
V N

We find now that
Ð @XY­K
û
O
# @3Yf­K
O

SUT
@(Þ+N
û
S
V N

Ú
'
û
O
= YfK
û
O
=ÏK
S
K,(Þ+N
V N

û
O
= YfK
SUTWV
STWV
V N

@(Þ
@(Þ
'
g
¯
Ó
ST
O @ G YfKàY=á\NÂ
O @ Ú ý … g
G G Þ G
v
G



V (Þ
G
v 

@
V N
Ó³K,(Þ+N
S
O
û
O
=ÏK

O
V
S
SUTWV ¯
=
K,(Þ+N

@¤Þ+N
SUTWV
û
SUTWV ¯
=
Theû second integral vanishes, while in the first only the term carrying the odd part
of V is non-zero. Thus we find that this terms equals
÷
@ G ü VXV Kƒ@
N›4Â
Þ
v
V

ž0 O @ Ú ý … g
G

Let us compute now @ - v
K,¤Þ
N¤ž0jg In this term, we have to consider compo
Ú

nents of order 4LKƒ@UN in the coefficients of
which are odd functions. We obtain
@4Â
v 
Ú
Ú
K,¤Þ
N¤ž0
¢
@ G E
÷
Kƒ@
O š
NÂ
@

­ Þ O V ÕÞ ¯
v 
¢
÷
Kƒ@
NÂ
v
Þ

ý V
O[ý
OÊý ¯
@ Ú ­ V … 
g
GV …  
G
O
Summarizing, we have proven that
Â
(6.10)
v
S
V

ž0 E
@(G­  
O
O
± V
ýV
@ Ú ­ V … 

O
±
OÊý
G
GV
Kƒ@
÷
N

¯ O
@(GU­ ±
¯
OÊý
G
g
Ú
Kƒ@
÷
N›
O
@(G$±f  
¯
CONCENTRATION ON CURVES
25
The next computations, rather analogous,
correspond to the projection onto Þ of
S
the error. We compute now - v V Þ . The main component in this expression is
given by
T
@¹­ @ G
We also have a term like û
p Kƒ@
÷
N E
¡TWV
@
E
¢8¡TWV
@(G
O
E
K
v
@G
¢8 TWV
@ G
÷
ü VXV Kƒ@
NÂ
O
V N ­K
v
÷
ü VXV Kƒ@
NÂ
v
­K
Kƒ@
G
O
V N 
÷

NÂ
V ¯
Â
Þ G
v
û
O
V>5 Þ
<T
@
^ ÷ Gi`É3K k @ N Â
û
V Þ
v
¯
V Þ
V 
^ ÷ Gi É K k @ N G ü V
¢8¡T
a6~
<T
^ ÷
V N =
GIi6ÉK k @ N*
¢
Â
O
G 

v  G
¯
O
V Þ
Þ
ý
@ Ú
G
because of the assumption on .
There are also terms of order @ G coming from the second order expansion of
?K N . We recall, from the decomposition (2.31)-(2.32) that the error carries either

terms accompanied
by
@ G as a factor or by @ Ú . The terms with @ Ú produce functions
÷
^
E
k
with ¾ G K °N -norms of order @ Ú . The terms of order @ G in the decompoof ²
@
S
sition of V are either even or odd in the variable . Those which are odd do not
contribute to the integral since the function Þ is even. Taking also into account that
and  are uniformly controlled by @ w‰ we just
need to consider
û
÷
ž Kƒ@
ž
÷
C
Kƒ@
÷
ž78Kƒ@
<T
N E
@
Gci É K k
N E
@ G ü G VXV Kƒ@
N E
@ G Â
÷
N YKàY+=™\NÂ
T
=
v
OD TWV =
^ ÷ ÷
@ N Kƒ@ NÂ
š
¢
G
 \ T G O

SUT
­
v
V ޞ1
v

G

GV
¯
ÞPž1
O T Q  Q G G

TWV T
i`ÉÉ, G
G 
O

O
š

TWV T
Ú 

ÞPž1xg

It is easy to see that also ž0 E 4LKƒ@ÚN , The common pattern of ž and ž07 is that
÷
C
even though they have size @ G in ¾ G norm, they define smooth functions of ² E @ ,
which is a very important fact to obtain the desired result.
For the term parallel
to (6.9) we get
û
SUT
@3YfKàY=™\NÂ
v 
G
V (ގÞPž1
O
@ G YfKàY=™\NÂ
SUT
v 
G G Þ Ú ž1
O
@ Ú
ý
V
E
We consider now another component:
@4Â
v
Ú
K,(Þ+NXÞ
E
={@GP³Kƒ@
÷
N
Kƒ@
÷
N
¡T
GUi`ÉÂ
v
ÞÒG
O
@ Ú E
4Kƒ@ Ú N<g
@ Ú
ý
V …”g
26
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
Additionally, we also need to consider some higher order terms in . The ones
involving first derivative are
š
@ Ú 

Â
G
O
ÞÒG
v
š
Ý
@ Ú Ý Â
Ú
ÕÞ Þg
v
Only one term (in
K,(Þ+N ) involving   carries also @ Ú . But this term is also
G
N G 'ž0 Elk . There also is a term of the form
accompanied by - v Þ KÕ
T
@ Ú ­@
G ±
÷
Kƒ@
C
¯
O
N
÷
4Kƒ@ G N   Kƒ@
N
with 4Kƒ@ G N uniform in @ .
÷
In summary, we have established that, as a function of ² E @ ,
S
Â
(6.11)
^
^
v
V ÞPž1
@ Ú ­\
E
O
O
@
@ Ú ±M7 
^*Ÿ
O
±
O
¯ T
4LKƒ@ G N
C
G 
÷
@Ua6~
=x@ G ±MœKƒ@
O
N
4Kƒ@ Ú N³g
where ± > E+ž gXgXg are smooth functions of their argument. Explicit expression
for the coefficient ±87 , which we will need later, is
š
±M7 E
(6.12)

G
=
Ý
g
Ú
7 Projections of terms û involving û
in (3.8)-(3.10)
integrated against
We will estimate next the terms that involve
and Þ . Concerning , we call the sum of them p K N , which can be decom
û
posed as pjEX¡ Ú l V p  below.
÷
û
. We make the following observation: all terms ûin
p V Kƒ@ N E - v
K N
Let
^
Ú

K N carry @ and involve powers of times derivatives of k \ or two orders of .
The conclusion is that since has exponential decay then

Q
û
Q

< Gi w Ö Ù g
ÂŽÄ p Kƒ²³N G ž³²îRÛ@ Ú <
~
Hence
< p V <>= w Ö ~ 2 Ù îRÛ@ Ú g
û
Ä
In K N we single out two less regular terms.
The one whose coefficient depends
Ú
on   explicitly has the form
û
p VŸ E
E
@ G
û
={@G
 Â
Þ K¼\ OÊ¢ @ K= NXN T G
v
 Â
v
û
Ð Þ
K¼\
O[¢
@
K=
NXN
T
G¤Ó
g
^
Since has Lipschitz dependence on K ¤N in the form (5.9), we see that this is
transmitted from
Sobolev’s
embedding into
û
û
<
Ö  2‘ Ù =
‰ ‰
Ö  2 ‘ Ù < =ML Ö @ Ù î‚Û€@HGw ­R< V =
w w
G
< J
O
<N V =’
G
<>O
¯ ^
CONCENTRATION ON CURVES
27
from where it follows
^
< p V Ÿ8K V V N6= p V ŸK
^
G
×
N<>= w Ö ~ 2 Ù îRۀ@ Ú w‰ ­R< V =
û
G
Ä
÷
The one arising from second derivative in for
û
p V ŸXŸ E
Â
ù3ù
v
Þ
O[¢
­ \=ÏK¼\
@
G
O
< J
¯
<N V =u
G
<>O g
is
K=
¯
ž g
G 0
T
NXN
We readily see that
^
< p V ŸXŸ8K V V N6= p V ŸXŸK
N<>= w Ö ~ 2 Ù î‚Û@ Ú ­R< V =
^
G
G
G
Ä
< J
O
<N V =’
G
<>O
¯
g
The remainder p V = p V Ÿ‡= p V ŸXŸ actually defines for fixed @ a compact operator of
^
^
the pair K (N into ¾ G K k °N . This is a consequence of the fact that weak convergence
9
V 9
o
K N , and the same is the case for
in o G ^ K N implies local
strong
convergence
in
¯
^
^
û
V
o G K k °N and Û
­ k ° . If and > are weakly convergent sequences in o G K k °N
Ö ¢ ¢
V 9
2 ‘ Ù constitute a bounded sequence in o K N . In
then clearly the functions
the above remainder one can integrate by parts if necessary once in . Averaging
against which decays exponentially localizes the situation and the desired fact

û
follows.
÷
We observe alsoû that p Kƒ@ N E - v }RK N can be estimated similarly. Using
G

the definition of }lK N and the exponential decay of we obtain

û
< p
@
< Gi w Ö Ù î
<>= w Ö ~ 2 Ù îRÛ@ w‰ <
G
Û@ Ú g
Ä
Let us consider now
Since I E

O
p
û
V
O
@¤Þ
Q
@ (Þ
Ú
÷
Kƒ@
N E
û
Â
v
SUTWV
Yf­ I
=
û
STWV ¯


and V can be estimated as
KÕN
QbOlQ
û
V K
^÷
Q
N
Q Q O
îlÛ@³K G
\N
TØ Ø
we easily see that for some ˜xZ k we have the uniform bound
Q SUTWV
I
=

SUTWV(QXQ

From here we readily find that
< p < = w Ö ~ 2 Ù îRÛ@FGw <
Ú
Ä
Q î
û
Û@
T
@
< i w Ö Ù î
—
Ø Ø
g
Û@ Ú g
This terms define compact operators similarly as before. We observe that exactly
the same estimates can be carried out in the terms obtained from integration against
Þ .
28
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
8 The system for £¥¤§¦¥¨1© : proof of the theorem
û
In this section we set up equations relating end such that for the solution
÷
of (3.8)-(3.9) predicted by Proposition 5.1 one has that the coefficient Kƒ@ N is
identically zero. To achieve this we multiply first the equation against and

Elk is then equivalent to the relation
integrate only in , The equation
û
û
û
S
Â
V
v
ž1 O Â

Kƒ}
v
O
K
G
N
Similarly ž Elk if and only if û
Â
S
V ÞҞ1
O
v
Â
Kƒ}
v
K
G
K
Ú
N
O
û
O
Ú
K
O
N
STWV ¯
SUTWV
Yf­ I
N
=
Yf­ I
SUTWV
N

=
STWV ¯
ž0 Elk g

û
NXÞҞ1

Elk g
Using the estimates in the previous sections we then find that these relations are
equivalent ^ to the following nonlinear, nonlocal system of differential equations for
the pair K (Nǻ
ª
V K
(8.1)
(8.2)
G
NWÁ
O

± V
E
m

O
±
E
G
a ~¬
c
±
Ú
O
O
^
@(G±  
@ • V
O
O
± 
@ ±Qœ
I
@ G •
g
C
G
T
O
O
E @ Ú
± 7¬  N
m G  M
^
• … K m ¤N can be decomposed
m
m in the following form:
^
^ O
­
^
• … K
N E ‰ … K (N
(
(N
K
^ m
^
k
K,(N‡Ál
m @ G K
The operators •
­
where … is uniformly bounded in ¾ G K °N for K @UN satisfying constraints (2.29)(2.30) and is
m also compact.
m The operator ‰ … is Lipschitz in this region,
^
^
V V N6=Z‰ … K
N< = w Ö ~ 2 Ù î‚Û€@³­R< V =
G
G
^
^
^*Ÿ
Ä
ª
The functions ±8 > E \ ggg are smooth.
<>‰
K
G
< J
O
<N V =u
G
<>O
¯
g
We will solve now system
(8.1)-(8.2). The first observation is that the operaª
tor V is invertible under ° -periodic boundary» conditions.
This follows from the
^
assumed non-degeneracy
condition
(1.10):
if
nF¾ G K k °N then there is a unique
^
»
solution nPo G K k °N of V K N E
which is ° -periodic andª satisfies
<
O
 < = w Ö~ 2 Ù
Ä
<
O
 < =8L Ö ~ 2 Ù
Ä
<
»
< =ML Ö ~ 2 Ù îRÛZ< < = w Ö ~ 2 Ù g
Ä
ª deal with invertibility of
We use now assumption (1.14) to
Ä
G
. We have:
^
Lemma 8.1. Assume that^ condition (1.14) holds. If žLnj¾ G K k °N then there is a
unique solution no G K k °N of K,¤N E ž which is ° -periodic and satisfies
G
O
O
TWV
Ö
w
Ù
2



= L Ö ~ 2 Ù îRۀ@
@ G <N >
< = ~
@P<N <>= w Ö ~ 2 Ù
N< M< M
Ä
Ä ^
Ä
Moreover, if ž is in o G K k ° N , then
O
O
@(G8<N   <>= w Ö ~ 2 Ù
<N  <>= w Ö ~ 2 Ù
<NM< = L Ö ~ 2 Ù î ÛLR­ <ž   <>= w Ö ~
O
Ä
Ä
Ä
Û <ž®< =
Z
<ž®<>= w Ö ~ 2 Ù g
Ä
2 Ù
O
wÄÖ~ 2 Ù g
Ä
<ž  <>= w Ö ~ 2 Ù
Ä
¯
CONCENTRATION ON CURVES
29
ª
Let us accept for the moment the validity of this result and let us conclude the
proof of the theorem.
O
We solve first K,~N E @±MœKƒ²³N and replace E ~ . Observe that by Lemma
G
8.1 we have
O
@ G <N ~  <>= w Ö ~ 2 Ù
<N ~ < = L Ö ~ 2 Ù îRۀ@8g
^
Ä
Ä
ª the same form as (8.1)-(8.2)
ª
The system resultingª on K U N has
except that now the
term @M
± œ disappears. Let us observe now that the linear operator
ª
^
^
^
K
(N E K V K N6=P± =R@ G ±  
,K ¤NXN
Ú
G
»c^
^
E
is invertible with bounds for K (N
K žËN given by
<
< J
ª
O
<NQ<>OîRÛD<
»
<
O
@
G
TWV
<ž¯< g
G
It then follows from contraction mapping principle that the problem
O
­
Kƒ@‰ V
^
@ G ‰
G
¯ ^
$N K
¤N E
^
K
»c^
žËN
×.°
»
is uniquely solvable for K (N satisfying (2.29)-(2.30) if < < ªõ@ w‰ , <ž®< ª
G
G
×.° ^
@ Gw
for some ±mZ k . The desired result for the full problem (8.1)-(8.2) then
follows directly from Schauder’s fixedO point
theorem. In fact, refining the fixed
point region, we can actually get <NM<>O
< < J E 4Kƒ@¤N for the solution.
o
ª
Proof of Lemma 8.1. We consider then the boundary value problem
(8.3)
K,(N E
G
^
ž
ºK k N E
ºKÅ°N
^
Kk N E
^
 KÅ°N
We make the following Liouville transformation c.f. [22]:
°I~ E
³
Â
~
Kƒ²³N E
D
MWK N E
Ä
Kƒ²¹Nbžº²
^LD
E
- ~
Î Kƒ²³Nbž³² ä ^²
ac~ E
°I~
Î
¡T w
^
\
exp
b
K
=
Â
G±878Kƒ²³Nbž³²³N
š
‰
~
³
^
D
³ TWV
° G~
ÝÝ
Kƒ²³N›º
Kƒ²¹N `
K N E ä [
³ g
G
³
ä
° G~
G
a6~
G
Observe
ª that is ° -periodic thanks to the explicit formula (6.12) for the coefficient
±M7 . Then (8.3) gets transformed into
(8.4)
G
@ G KM  
KM`N E
O
D
`K NeM`N
O-
a6~M E
ž
^
MÕK k N E
MÕK
ä
N
^
M  Kk N E
M  K
ä
N
^
and it then suffices to establish the estimates in Lemma 8.1 for the solution of this
problem in terms of corresponding norms of ž . It is standard that the eigenvalue
problem
(8.5)
M 
O
D
ËK NeM
O
a`M Elk
^
MWK k N E
MWK
ä
N
^
M Kk N E
M  K
ä
N
^
30
M. DEL PINO, M. KOWLCZYK, J.-C. WEI
¢x£
k , with associated orthonormal
has an infinite
sequence of eigenvalues a © ,
^ ä
by
eigenfunctions.
A result in [22] provides asbasis in ¾ G K k N , Ð M © Ó , constituted
¢
O
ymptotic expressions as z
‘ for these eigenvalues and eigenfunctions, which
turn out to correspond to those for ?Á k . We have:
š ¢”O
´ a © E
(8.6)
4µ ¢
\
ÚQ¶
g
¢
Problem (8.4) is then solvable if and only if a © @ G „E c
a ~ for all . In such a case, the
solution to (8.3) then can be written as
D
MWK N E
ž ©
k¿
D
M © K N
©l ~ a6~=ja © @ G
with this series convergent in ¾ G . Hence
Q © Q
ž
G
g
©l ~ K a6~=Fa © @ G N G
k¿
<3M·< =G w Ö ~ 2 ¸ Ù'E
We then choose @ such that
Qß ¢
(8.7)
G @ G =
QÕ£
ac~
Ë@
¢
for all , where is small. This corresponds precisely to the condition (1.14) in the
Q
Q£K¹
statement of the theorem. From (8.6) we then find that a6~‡=Ža © @ G
@ if @ is also
TWV G
Ö
Ö
sufficiently small. It follows that <3M·< = w ~ 2 ¸ Ù îRÛ@
< ž§< = w ~ 2 ¸ Ù . Observe also that
<3M  < G= w Ö ~ 2 ¸ Ù îRÛ
Hence
k¿
©l ~
Q © Q
ž
G
OlQ © Q
a
K ac~=Fa © @ G N G
\
î‚Û
k¿
©l ~
K¼\
OÊ¢ Q Q
N ž © G g
O
T V W
= L Ö ~ 2¸ Ù R
@P<3M  < = w Ö ~ 2 ¸ Ù
3< M·< 8
î Û@
< ž¯< = w Ö ~ 2 ¸ Ù g
^ ä
ä
ä
o G Kk
N with ž'K k N E ž4K N , ž  K k N E ž  K N , then the sum ¡
Besides, if ž is in
is finite and bounded by the o G -norm of ž . This automatically implies
O
@G<3M   <>= w Ö ~ 2 ¸ Ù
and the proof is complete.
O
<3M  <>= w Ö ~ 2 ¸ Ù
©
¢ ž ©G
^
<3M·< = L Ö ~ 2 ¸ Ù îRÛD< ž¯N< i w Ö ~ 2 ¸ Ù
o
Acknowledgment. The first author has been partly supported by grants Fondecyt 1030840 and FONDAP, Chile. He is grateful to the Department of Mathematics
of The Chinese University of Hong Kong for its hospitality. The second author has
been supported by grant NSF OISE-0405626, and Fondecyt grants 1050311 and
7040142. The research of the third author is partially supported by an Earmarked
Grant (CUHK402503) from RGC of Hong Kong.
CONCENTRATION ON CURVES
31
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MANUEl DEL PINO
Departamento de Ingenierı́a Matemática and CMM,
Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile.
Email: delpino@dim.uchile.cl
MICHAL KOWALCZYK
Department of Mathematical Sciences,
Kent State University, Kent, OH 44242 and
Departamento de Ingenierı́a Matemática,
Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile.
Email:kowalcyk@math.kent.edu
JUN-CHENG WEI
Department of Mathematics
The Chinese University of Hong Kong,
CONCENTRATION ON CURVES
Shatin, N.T., Hong Kong.
Email: wei@math.cuhk.edu.hk
Received Month 200X.
33