on a singular perturbation problem involving the distance to a curve
TRANSCRIPT
ON A SINGULAR PERTURBATION PROBLEM INVOLVING THE
DISTANCE TO A CURVE
N. ANDR
�
E AND I. SHAFRIR
D�epartement de Math�ematiques
Universit�e de Tours
37200 Tours, Fran e
and
Department of Mathemati s
Te hnion, Israel Institute of Te hnology
32000 Haifa, Israel
June 17, 2002
1. Introdu tion
Let G and be two simply onne ted, bounded subsets of R
2
with boundaries of lass C
2
.
For a point z 2 R
2
we denote by d(z) the signed distan e of z to � with the onvention that
d(z) > 0 for z 2 and d(z) < 0 for z 62 . We shall assume, without loss of generality, that
0 2 . Let F : (�1;1)! [0;1) be a C
2
fun tion satisfying:
F (0) = 0;(1.1)
F
00
(0) > 0;(1.2)
F
0
< 0 on (�1; 0) and F
0
> 0 on (0;1).(1.3)
For a given smooth boundary ondition g : �G! R
2
and any " > 0 we onsider the energy
(1.4) E
"
(u) =
Z
G
jruj
2
+
F (d(u(x)))
"
2
;
for u 2 H
1
g
(G;R
2
) := fu 2 H
1
(G;R
2
); u = g on �Gg. Denoting for ea h " > 0 by u
"
a
minimizer for E
"
over H
1
g
(G;R
2
), we are interested in the asymptoti behavior of the family
fu
"
g
">0
as "! 0.
A parti ular well studied ase of the above is the Ginzburg-Landau energy,
(1.5)
Z
G
jruj
2
+
(1� juj
2
)
2
"
2
:
Indeed, it orresponds to the hoi e = B(0; 1) (i.e. the unit disk) and F (d) = d
2
(2 � d)
2
.
The spe ial ase of the Ginzburg-Landau energy (1.5) for g taking its values in S
1
= � was
1
studied in the fundamental work of Brezis, Bethuel and H�elein [7℄, see also Struwe [16℄. It was
generalized in our previous work [2℄ to the ase G : �G ! R
2
n f0g and in our re ent arti le
[3℄ we treated ases where g is allowed to have zeros.
A basi tool in the analysis of the minimizers for (1.5) in [2℄ was the following de omposition
formula for the energy, based on an argument of Lassoued and Mirones u (see [13℄),
(1.6)
Z
G
jruj
2
+
(1� juj
2
)
2
"
2
=
Z
G
jr�
"
j
2
+
(1� j�
"
j
2
)
2
"
2
+
Z
G
�
2
"
jrvj
2
+ �
4
"
(1� jvj
2
)
2
"
2
;
when we write ea h admissible u as u = �
"
v where �
"
is the unique minimizer for the s alar
problem of minimizing the energy
R
G
jr�j
2
+
(1�j�j
2
)
2
"
2
for s alar fun tions � satisfying � = jgj on
�G. It then follows that the energy of a minimizer u
"
for (1.5) splits into two parts. The �rst
term on the RHS in (1.6) ontributes a boundary layer energy of the order
0
(g)
"
+O(1) (with
0
(g) =
R
�G
(
4
3
� 2jgj+2
jgj
3
3
), see [2℄ for details) while the ontribution of the se ond one is of
the order 2�Dj log "j+O(1) where D is the winding number of the boundary ondition g with
respe t to 0, that we assume nonnegative, without loss of generality. Using this de omposition
we proved in [2℄ that for a subsequen e "
n
! 0 we have, u
"
n
! u
�
in C
k
lo
(Gnfa
1
; : : : ; a
D
g), 8k,
where u
�
is a singular S
1
-valued harmoni map with exa tly D singularities a
1
; : : : ; a
D
2 G.
When dealing with the more general energy (1.4) we also need to impose some restri tions
on the boundary ondition g. In the ase where the target domain is the unit dis , as for the
Ginzburg-Landau energy, we required in [2℄ that g has no zeros. The reason is that the origin
is the unique point whi h does not have a unique nearest point proje tion on the unit ir le.
An analog requirement for a general domain would be that ea h point in image(g) has a
unique nearest point proje tion on �. We shall make a slightly stronger assumption whi h
involves the notion of the skeleton (or medial axis, see [11℄) of a set that we shall now re all.
The skeleton S() of is de�ned as the set of enters of the maximal dis s in (i.e. x 2
S() if and only if there exists r > 0 su h that B(x; r) � but there are no y 2 and � > 0
su h that B(x; r) ( B(y; �) � ). The stru ture of the skeleton may be very ompli ated
even if � is of lass C
1
(see [11℄). However for � whi h is real analyti , it was shown in [11℄
that S() onsists of a onne ted geometri graph with a �nite number of real analyti edges.
Under our C
2
hypothesis it is enough to noti e that S() is a losed subset of , every point
x 2 n S() has a unique nearest point proje tion on � and the proje tion map s(x) is a
C
2
map on n S() (note however that some points in S() may have a unique proje tion
as well). Indeed, for x 2 n S() it follows from the in lusion B(x; d(x)) � B(y; �); for some
y 2 and � > d(x), that 1 � �(s(x))d(x) > 0 with �(s(x)) denoting the urvature of � at
s(x). The inequality 1� �(s(~x))d(~x) > 0 ontinues to hold for ~x in a neighborhood of x, and
the argument of [12, Se . 14.6℄ shows that s(x) is a C
2
-map on n S(). Sin e we allow g to
take values outside , the skeleton of the omplement
is relevant as well. Therefore, we
shall always make the following assumption in the sequel,
(1.7) image(g) � R
2
n S; where S := S() [ S(
):
2
For g satisfying (1.7) we de�ne the degree deg(g) as the winding number D around 0 of the
map g (re all that we assumed that 0 2 ). Equivalently, D is the topologi al degree of s Æ g
as a map from �G to �. We shall assume in the sequel, without loss of generality, that
D � 0.
Next we are going to state the main results of this paper. The �rst deals with the asymptoti
behavior of the energy E
"
(u
"
) as "! 0. Roughly speaking, there are two main ontributions,
as in the ase of the Ginzburg-Landau energy. Ea h is asso iated with a di�erent feature
of the boundary ondition g. The �rst, of the order O(
1
"
), is determined by d(g) only. The
se ond one, of the order O(j log "j), depends on the winding number of g. We denote by H(t)
the unique solution of the equation
(1.8)
(
H
0
(t) = sgn(t)
p
F (t); t 2 R;
H(0) = 0:
We then have,
Theorem 1. Let g : �G! � be a smooth map satisfying (1.7) of degree D � 0 . Then,
(1.9) E
"
(u
"
) =
2
"
Z
�G
H(d(g(�))) d� +
l
2
2�
Dj log "j+O(1); for all 0 < " � 1;
where l is the length of �.
Next we turn to a pre ise des ription of the minimizers fu
"
g. The next theorem shows
that they onverge lo ally (up to a subsequen e), as " ! 0, to a limit whi h is a singular
\�-valued harmoni map". By a (regular) \�-valued map" on a domain E we mean a map
v : E ! � whi h is a riti al point for the Diri hlet energy. Equivalently, if : [0; l℄ ! �
is an ar length parameterization of �, extended to a l-periodi map on R, and we represent
v = Æ �, then the fun tion � : E ! R is required to be harmoni . Using we de�ne a
C
2
-map � : S
1
! � by,
(1.10) �(e
i�
) = (
l�
2�
); � 2 (0; 2�℄;
that we shall use in the sequel. We shall also need the notion of the anoni al �-valued
harmoni map asso iated with a smooth map h : �G ! � of degree D > 0, and a on�g-
uration of D distin t points a = (a
1
; : : : ; a
D
) in G, whi h requires an obvious generalization
of the orresponding notion for S
1
-valued harmoni maps from [7℄. There it was shown
that for ea h
~
h : �G ! S
1
of degree D and a as above there exists a unique harmoni
map ~u 2 C(
�
G n fa
1
; : : : ; a
D
g; S
1
) \ C
1
(G n fa
1
; : : : ; a
D
g; S
1
), alled the anoni al S
1
-valued
harmoni map asso iated with a and
~
h, su h that ~u =
~
h on �G, deg(~u; a
j
) = 1; 8j and
lim
z!a
j
jz�a
j
j
z�a
j
~u(z) exists for all j. A tually, in [7℄ anoni al harmoni maps asso iated with
arbitrary on�gurations of degrees (d
1
; : : : ; d
k
), with
P
k
j=1
d
j
= D were de�ned, but here we
shall restri t ourselves to the ase were all the degrees equal to 1.
3
Given a on�guration a as above and a smooth h : �G ! � of degree D > 0 we shall
say that u 2 C(
�
G n fa
1
; : : : ; a
D
g; �) \ C
1
(G n fa
1
; : : : ; a
D
g; �) is the anoni al �-valued
harmoni map asso iated with a and h if ~u = �
�1
Æ u is the anoni al S
1
-valued harmoni
map asso iated with a and
~
h = �
�1
Æ h. Note that in parti ular we have u = h on �G and
deg(u; a
j
) = 1; 8j. Our main onvergen e result is the following.
Theorem 2. Assume the same hypotheses as in Theorem 1. Then, for a subsequen e "
n
! 0
we have u
"
n
! u
�
strongly in H
1
lo
(Gnfa
1
; : : : ; a
D
g) and in C
�
lo
(Gnfa
1
; : : : ; a
D
g), 8� 2 (0; 1),
for some D points a
1
; : : : ; a
D
2 G (assuming that D > 0), where u
�
is the anoni al singular
�-valued harmoni map asso iated with a = (a
1
; : : : ; a
D
) and the map s Æ g. Moreover, the
on�guration a = (a
1
; : : : ; a
D
) minimizes the renormalized energy W (�
�1
Æ s Æ g;b) over all
on�gurations of D distin t points b 2 G
D
.
The renormalized energy appearing in the statement of Theorem 2 was introdu ed and
studied in [7℄, see Se tion 6 for more details.
The proofs of Theorem 1 and Theorem 2 are more involved than those of the analogous
results in [2℄ due to the la k of an analogue to the de omposition formula (1.6). Nevertheless,
the solution of a ertain s alar problem (generalizing the �
"
appearing in (1.6)) plays an
important role here too. Some results on that s alar problem, presented in Se tions 2 and 3,
may be of independent interest. We mention in parti ular Proposition 2.1 and Proposition 3.3
whi h generalize results of Berger and Fraenkel ([5℄).
A knowledgment. We thank Laurent Veron for raising a question that motivated our study.
We are indebted to Haim Brezis for his onstant en ouragement and for interesting dis ussions
on the subje t. The resear h of I.S. was supported by the fund for the promotion of resear h
at the Te hnion.
2. Preliminary results
We begin with some notations. It will be onvenient to introdu e two new oordinate
systems asso iated with and G respe tively. We �rst asso iate with ea h x 2 R
2
n S the
oordinates (s; d) where s = s(x) is, as above, the unique nearest point proje tion of x on �
and d = d(x) is the signed distan e to �. As explained in the introdu tion it follows that
d 2 C
2
(R
2
n S) and that the map x 7! (s(x); d(x)) is a C
1
-di�eomorphism of R
2
n S on its
image. We shall also need a similar system in G. For x 2 G we denote by Æ(x) the distan e
from x to �G and for � > 0 we set
(2.1) G
�
= fx 2 G; Æ(x) < �g and �
�
= G nG
�
:
Sin e �G is of lass C
2
there exists b
0
> 0 su h that every x 2 G
b
0
has a unique nearest
point proje tion �(x) 2 �G, Æ 2 C
2
(G
b
0
) and that the map � : x 7! (�(x); Æ(x)) is a C
1
-
di�eomorphism of G
b
0
on its image, see [12, Se . 14.6℄ (this is true even on R
2
n S(G) but
we shall need only a lo al oordinate system near �G). For ea h 0 < t < b
0
, the mapping
4
P
t
:= �
�1
(t; �) from �G to fx 2 G; Æ(x) = tg is also a C
1
-di�eomorphism and its Ja obian
satis�es, for some > 0 :
(2.2) jJ(P
t
)(�) � 1j � t; 8� 2 �G;8t 2 (0; b
0
):
For every fun tion v 2 L
1
(G
�
), � 2 (0; b
0
), there holds (identifying a point x 2 G
�
with the
pair (�(x); Æ(x))),
(2.3)
Z
G
�
v dx =
Z
�
0
dt
Z
�G
v(P
t
(�); t)J(P
t
) d� :
We shall denote throughout this paper by C di�erent positive onstants (their value may vary
on ea h appearan e).
Next we prove two basi and simple properties of the minimizers fu
"
g.
Lemma 2.1. There exists a onstant R > 0 su h that
(2.4) ku
"
k
L
1
(G)
� R; 8":
Proof. Set diam() = sup
x;y2
jx � yj and re all that 0 2 by assumption. Let R > 0 be
any number satisfying [ image(g) � B(0; R). We shall show that (2.4) holds for this R. We
denote by P
R
the nearest point proje tion on the dis B(0; R), i.e.,
P
R
(z) =
8
<
:
z if jzj � R;
R
z
jzj
if jzj > R:
Given a minimizer u
"
we de�ne a new map v
"
by v
"
(x) = P
R
(u
"
(x)). Clearly v
"
2 H
1
g
(G;R
2
).
For ea h x 2 G su h that ju
"
(x)j � R we denote by �(x) a point on � su h that,
ju
"
(x)� �(x)j = jd(u
"
(x))j = �d(u
"
(x))
(we annot assert in general that su h a point is unique). Sin e P
R
is nonexpansive we have,
(2.5) �d(v
"
(x)) � jv
"
(x)� �(x)j = jP
R
(u
"
(x))� P
R
(�(x))j � ju
"
(x)� �(x)j = �d(u
"
(x)):
We learly have also jrv
"
(x)j � jru
"
(x)j; 8x 2 G and d(u
"
(x)) = d(v
"
(x)) if ju
"
(x)j � R. It
follows from the above and (1.3) that E
"
(v
"
) � E
"
(u
"
). Sin e P
R
is a stri t ontra tion on
fjzj > Rg we have a stri t inequality in (2.5) if ju
"
(x)j > R. Thus we would get a ontradi tion
to the minimizing property of u
"
unless image(u
"
) � B(0; R), as laimed. �
Remark 2.1. In view of Lemma 2.1, it is enough to require that (1.3) holds on (�R;R) only.
Lemma 2.2. There exists a onstant C
0
> 0 su h that for every " > 0, we have
(2.6) kru
"
k
L
1
(G)
�
C
0
"
:
5
Proof. The minimizer u
"
satis�es,
(2.7)
8
<
:
�u
"
=
1
2"
2
F
0
(d(u
"
)) � (rd)(u
"
) in G;
u
"
= g on �G:
The result now follows from standard ellipti estimates by a simple res aling argument as in
[6, 16℄ (using (2.4)). �
The oeÆ ient multiplying
1
"
in (1.9) depends on d(g) only. It measures how far the
boundary ondition is from being �-valued. In order to ompute it expli itly we need to
study the solution of a ertain s alar problem. This solution will play an essential role in our
de omposition of the energy later on. For ea h " > 0 we denote thus by d
0;"
a minimizer for
the problem,
(2.8) min
�
E
"
(h) :=
Z
G
jrhj
2
+
F (h)
"
2
; h 2 H
1
(G); h = d(g) on �G
:
The fun tion d
0;"
satis�es,
(2.9)
8
<
:
�d
0;"
=
1
2"
2
F
0
(d
0;"
) in G;
d
0;"
= d(g) on �G:
By a result of Angenent [1℄, see also [4, Proposition 7.1℄, it follows that there exists an "
0
> 0
su h that for every " � "
0
the minimizer to (2.8) is unique.
Our next obje tive is to get pointwise estimates for the fun tion d
0;"
. For that matter we
shall onstru t an expli it approximation for d
0;"
using the solution U
to the problem:
(2.10)
8
>
>
<
>
>
:
U
00
=
1
2
F
0
(U
) on [0;1);
U
(0) = 2 [�R;R℄;
U
(1) = 0;
where R is given by Lemma 2.1. We start with a simple lemma.
Lemma 2.3. For any 2 [�R;R℄ there exists a unique solution U
to (2.10). There exist two
onstants
1
;
2
> 0 su h that: if 2 [�R; 0) then �
1
e
�
2
t
� U
(t) < 0; 8t 2 [0;1), while if
2 (0; R℄ then 0 < U
(t) �
1
e
�
2
t
; 8t 2 [0;1) ( learly U
� 0 for = 0). Furthermore, we
have U
0
(t); U
00
(t) = O(e
�
2
t
).
Proof. Multiplying the equation in (2.10) by 2U
0
yields 2U
00
U
0
= F
0
(U
)U
0
. Hen e for some
onstant dC we have (U
0
(t))
2
= F (U
(t)) + C; 8t 2 [0;1): The assumption U
(+1) = 0
for es C = 0 and we obtain,
(2.11)
8
<
:
U
0
=
p
F (U
) if < 0;
U
0
= �
p
F (U
) if � 0:
6
Next we onsider, without loss of generality, the ase < 0. Denoting by G
the fun tion
G
(s) =
R
s
dt
p
F (t)
, we have learly
d
dt
G
(U
(t)) = 1. Therefore, G
(U
(t)) = t, i.e.,
(2.12) U
(t) = G
�1
(t):
Note that,
(2.13) U
(t) =
8
<
:
U
R
(t+ U
�1
R
( )) for 2 (0; R℄;
U
�R
(t+ U
�1
�R
( )) for 2 [�R; 0):
We have thus essentially two possible pro�les for ea h U
(besides the trivial solution U
0
� 0).
The exponential de ay of U
and its derivatives, uniformly in 2 [�R;R℄, follows from (2.12)
and (2.13), using (1.2). �
The next lemma provides some a priori estimates for d
0;"
.
Lemma 2.4. For some positive onstants a
1
; a
2
we have,
(2.14) jd
0;"
(x)j � a
1
e
�a
2
Æ(x)="
; 8x 2 G:
Proof. Sin e repla ing d
0;"
(x) by the fun tion min(R;max(d
0;"
(x);�R)) an only de rease the
energy E
"
(d
0;"
), we get that
(2.15) kd
0;"
k
L
1
(G)
� R; 8" > 0:
The proof of (2.14) relies on a onstru tion of families of sub and super solutions. We �rst
note that from our assumptions (1.1){(1.3) it follows that for some � > 0 we have,
(2.16)
p
F (t) � ��F
0
(t) on [�R; 0℄:
For a �xed x 2 G we denote for short Æ = Æ(x). Then, for any y 2 B(x; Æ) we denote
r = r(y) = jy � xj. For every � 2 (0; 1=4℄ we set,
w
�
(y) = U
�R
�
�
Æ
2
� r
2
Æ"
�
on B(x; Æ):
Next we ompute for y 2 B(x; Æ), using (2.10),(2.11) and (2.16),
��w
�
= �(w
�
)
rr
�
(w
�
)
r
r
= �
4�
2
r
2
Æ
2
"
2
U
00
�R
�
�
Æ
2
� r
2
Æ"
�
+
4�
Æ"
U
0
�R
�
�
Æ
2
� r
2
Æ"
�
= �
2�
2
r
2
Æ
2
"
2
F
0
(w
�
) +
4�
Æ"
p
F (w
�
) � �
2�
2
"
2
F
0
(w
�
) +
4�
Æ"
p
F (w
�
)
� �
�
2�
2
"
2
+
4��
Æ"
�
F
0
(w
�
) � �
1
2"
2
F
0
(w
�
);
provided that x satis�es Æ(x) � 4�" (as we may always assume). It follows that the family
fw
�
; � 2 (0; 1=4℄g onsists of subsolutions for the problem,
(2.17)
8
<
:
�w =
1
2"
2
F
0
(w) in B(x; Æ);
w = d
0;"
on �B(x; Æ):
7
By Serrin's sweeping prin iple ([15℄) we get that d
0;"
� w
1=4
in B(x; Æ). In parti ular for some
a
1
; a
2
> 0 we obtain, using Lemma 2.3, that
(2.18) d
0;"
(x) � w
1=4
(x) = U
�R
�
Æ
4"
�
� �a
1
e
�a
2
Æ(x)="
:
By an analogous argument to the above, involving a family of super solutions of the form
v
�
(y) = U
R
�
�
Æ
2
� r
2
Æ"
�
on B(x; Æ);
we obtain the omplementary inequality to (2.18), i.e.
d
0;"
(x) � a
1
e
�a
2
Æ(x)="
:
�
We are now ready to de�ne an approximating family for d
0;"
. We �x a fun tion � 2
C
1
(G) \C
2
(G) satisfying:
(i) 0 � �(x) � 1; 8x 2 G;
(ii) �(x) � 1; 8x 2 G
b
0
=2
;
(iii) supp �(x) � G
b
0
:
We de�ne the approximation
~
d
"
by,
(2.19)
~
d
"
(x) = �(x) � U
d(g(�(x)))
�
Æ(x)
"
�
; 8x 2 G:
The next proposition shows that
~
d
"
is indeed a good approximation for d
0;"
.
Proposition 2.1. There exists a onstant C
1
> 0 su h that,
(2.20) k
~
d
"
� d
0;"
k
L
1
(G)
� C
1
"; 8" > 0:
Proof. We apply an argument similar to the one used in [3℄. We begin with some preliminary
omputations. Using (1.2) we �x an � > 0 su h that
(2.21) F
00
(t) � � > 0 for t 2 [��; �℄:
By Lemma 2.3 and Lemma 2.4 it is lear that there exists K > 0 su h that
(2.22) j
~
d
"
(x)j; jd
0;"
(x)j � �; on �
K"
:
In G
b
0
we use the oordinates Æ and � and we de�ne the ve tor �elds: � = r� and n = rÆ.
Using these oordinates we an write,
(2.23) �
~
d
"
= (
~
d
"
)
ÆÆ
+ (
~
d
"
)
Æ
� divn+ (
~
d
"
)
��
+ (
~
d
"
)
�
� div� on G
b
0
;
( learly �
~
d
"
� 0 on G nG
b
0
). Using Lemma 2.3 we infer that,
I
"
(x) := �
~
d
"
�
1
2"
2
F
0
(
~
d
"
) =
8
>
>
<
>
>
:
(
~
d
"
)
Æ
� divn+ (
~
d
"
)
��
+ (
~
d
"
)
�
� div� on G
b
0
=2
;
O
�
1
"
2
e
�
2
b
0
2"
�
on G
b
0
nG
b
0
=2
;
0 on �
b
0
:
8
By Lemma 2.3 it follows that the fun tion I
"
(x) satis�es on G:
(2.24) jI
"
(x)j � C
�
1
"
e
�
2
Æ(x)
"
+
1
"
2
e
�
2
b
0
2"
�
:
After the above preparation we an now turn to the proof itself. Arguing by ontradi tion,
we assume that for some sequen e "
n
! 0 we have,
(2.25) lim
n!1
1
"
n
� k
~
d
"
n
� d
0;"
n
k
L
1
(G)
=1 :
Let x
n
2 G denote a point where the maximum of j
~
d
"
n
�d
0;"
n
j over G is a hieved. Assume �rst
that for an in�nite number of indi es n we have x
n
2 �
K"
n
, so by passing to a subsequen e we
an assume this is the ase for all n. Clearly we may assume that the sign of
~
d
"
n
(x
n
)�d
0;"
n
(x
n
)
is independent of n. We assume then that for all n we have, for example,
(2.26)
~
d
"
n
(x
n
)� d
0;"
n
(x
n
) > 0:
Then for some r
n
2 (d
0;"
n
(x
n
);
~
d
"
n
(x
n
)) we have, using (2.21) and (2.24),
0 � �(
~
d
"
n
� d
0;"
n
)(x
n
) = I
"
n
(x
n
) +
�
1
2"
2
n
�
F
00
(r
n
) (
~
d
"
n
� d
0;"
n
)(x
n
)
� (
�
2"
2
n
)(
~
d
"
n
� d
0;"
n
)(x
n
)� C
�
1
"
n
e
�
2
Æ(x
n
)
"
n
+
1
"
2
n
e
�
2
b
0
2"
n
�
:
Thus,
k
~
d
"
n
� d
0;"
n
k
L
1
(G)
�
�
2C
�
� �
"
n
e
�
2
K
+ e
�
2
b
0
2"
n
�
whi h learly ontradi ts (2.25) for n large enough. Obviously an analogue argument works
in the ase of a reversed inequality in (2.26).
We are left then with the remaining ase when x
n
2 G
K"
n
for n � n
0
. In the sequel we
shall write for short Æ
n
= Æ(x
n
) and �
n
= �(x
n
). Passing to a subsequen e we may assume
that x
n
! �� 2 �G and that the limit
(2.27)
�
t = lim
n!1
Æ
n
"
n
exists:
Identifying any point x 2 G
K"
n
with the pair (�(x); Æ(x)) we next de�ne two sequen es of
res aled fun tions on the domain (in the ~s
~
t-plane) D
"
n
= f(~s;
~
t); (�
n
+ "
n
~s; "
n
~
t) 2 Gg by
~w
"
n
(~s;
~
t) =
~
d
"
n
(�
n
+ "
n
~s; "
n
~
t) and w
"
n
(~s;
~
t) = d
0;"
n
(�
n
+ "
n
~s; "
n
~
t):
From standard ellipti estimates it follows that ~w
"
n
! ~w and w
"
n
! w in C
1
lo
(R
2
+
) (with
R
2
+
= f(~s;
~
t);
~
t > 0g) where ~w(~s;
~
t) and w(~s;
~
t) are both solutions of
(2.28)
8
<
:
�v =
1
2
F
0
(v) in R
2
+
;
v = d(g(��)) on �R
2
+
:
It follows from a result of Angenent [1℄ that the solution of (2.28) is unique. In fa t, it is a
fun tion of the variable
~
t only and it is given by the solution U
0
of (2.10) for
0
= d(g(��)).
9
Next we set,
V
"
n
(x) =
~
d
"
n
(x)� d
0;"
n
(x)
~
d
"
n
(x
n
)� d
0;"
n
(x
n
)
:
By assumption, jV
"
n
(x)j � 1 for all x and V
"
n
(x
n
) = 1. The equation satis�ed by V
"
n
is,
(2.29) �V
"
n
=
I
"
n
(x)
~
d
"
n
(x
n
)� d
0;"
n
(x
n
)
+
�
1
2"
2
n
�
F
00
(R
"
n
(x))V
"
n
;
where R
"
n
(x) is a point lying between
~
d
"
n
(x) and d
0;"
n
(x). De�ning a res aled sequen e by
~
V
"
n
(~s;
~
t) = V
"
n
(�
n
+ "
n
~s; "
n
~
t) as above, we may pass to the limit in (2.29), using (2.24), (2.25)
and the fa t that ~w = w = U
0
(see above), to infer that
~
V
"
n
!
~
V in C
1
lo
(R
2
+
), where
~
V
satis�es,
(2.30)
8
>
>
>
<
>
>
>
:
�
~
V =
1
2
F
00
(U
0
)
~
V in R
2
+
;
~
V = 0 in R
2
+
;
~
V (0;
�
t) = 1 (see (2.27)):
But by [1℄ there is no solution to (2.30). This ontradi tion ompletes the proof of the
proposition. �
We next prove an estimate for the energy of the fun tion d
0;"
.
Corollary 2.1. For ea h � 2 ("
9=10
; b
0
) we have,
(2.31) E
"
(d
0;"
) = E
"
(d
0;"
; G
�
) + o(1) =
2
"
Z
�G
H(d(g(�))) d� +O(1);
where o(1) and O(1) denote respe tively a quantity whi h goes to 0 and a quantity whi h stays
bounded, as "! 0, both uniformly in � 2 ("
9=10
; b
0
).
Proof. Using (2.9) and (2.14) and a simple res aling argument we dedu e, as in [2, Prop. 2.1℄,
that,
(2.32) jrd
0;"
(x)j �
C
Æ
�
(Æ=")
2
+ 1
�
e
�a
2
Æ=(2")
; 8x 2 G:
By (2.14) and (2.32) we obtain that E
"
(d
0;"
;�
�
) = o(1), and the �rst equality in (2.31) follows.
For the proof of the se ond equality in (2.31) we start with the lower bound,
(2.33) E
"
(d
0;"
) �
2
"
Z
�G
H(d(g(�))) d� +O(1):
The proof, analogous to the one of [2, Prop. 2.2℄, is based on an idea of Modi a [14℄. Fixing a
ve tor �eld V 2 C
1
(
�
G;R
2
) whi h satis�es jV (x)j � 1; 8x 2 G; and V (x) = �n(x) = �rÆ(x)
on �G, we obtain, using the Cau hy-S hwarz inequality and Green's formula,
(2.34)
E
"
(d
0;"
) �
2
"
Z
G
q
F (d
0;"
) jrd
0;"
j �
2
"
Z
G
rH(d
0;"
) � V
=
2
"
Z
�G
H(d(g(�))) d� �
2
"
Z
G
H(d
0;"
) div V;
10
where H is de�ned in (1.8). It follows from (1.1){(1.3) and (1.8) that
(2.35) H(t) � F (t); 8t 2 [�R;R℄ for some > 0:
Therefore, we infer from (2.34) that,
(1 + C")E
"
(d
0;"
) �
2
"
Z
�G
H(d(g(�))) d�;
whi h learly implies (2.33).
To prove the upper bound for E
"
(d
0;"
) we �rst note that by the minimizing property of d
0;"
we have E
"
(d
0;"
) � E
"
(
~
d
"
), where
~
d
"
is de�ned in (2.19). Next, (2.19),(2.11) and (2.13) yield,
(2.36)
E
"
(
~
d
"
) = E
"
(
~
d
"
; G
b
0
=2
) + o(1) =
Z
G
b
0
=2
�
�
�
�U
d(g(�))
(Æ=")
�Æ
�
�
�
2
+
1
"
2
F (U
d(g(�))
(Æ=")) +O(1)
= �
2
"
Z
G
b
0
=2
rH(U
d(g(�))
(Æ=")) � n+O(1)
=
2
"
Z
�G
H(d(g(�))) d� �
2
"
Z
fÆ=b
0
=2g
H(U
d(g(�))
(b
0
=(2"))) d�
+
2
"
Z
G
b
0
=2
H(U
d(g(�))
(Æ=")) divn+O(1) := I
1
+ I
2
+ I
3
+O(1):
By Lemma 2.3 it is lear that I
2
= o(1). From (2.35) we get as above that jI
3
j � C"E
"
(
~
d
"
).
Therefore, (2.36) implies that,
(1� C")E
"
(
~
d
"
) �
2
"
Z
�G
H(d(g(�))) d� +O(1);
and the proof is omplete. �
We lose this se tion with an upper bound for E
"
(u
"
), whi h is the \easy part" of Theorem 1.
Proposition 2.2.
(2.37) E
"
(u
"
) �
2
"
Z
�G
H(d(g(�))) d� +
l
2
2�
Dj log "j+O(1); 8" > 0;
where l = j�j and H is de�ned in (1.8).
Proof. It will be suÆ ient to onstru t a family fw
"
g
"<"
0
� H
1
g
(G;R
2
) su h that,
(2.38) E
"
(w
"
) =
2
"
Z
�G
H(d(g(�))) d� +
l
2
2�
Dj log "j+O(1); 8" 2 (0; "
0
℄:
We hoose D distin t points a
1
; : : : ; a
D
in G with Æ(a
j
) > b
0
; 8j; and then we �x r
0
su h that,
2r
0
< min
�
min
i 6=j
ja
i
� a
j
j;min
j
Æ(a
j
)� b
0
�
:
11
For ea h " < r
0
we de�ne a map w
"
as follows. On G
b
0
we set,
(2.39) w
"
(x) =
8
<
:
s(g(�(x))) + d
0;"
(x)�(g(�(x))) for Æ(x) 2 [0; b
0
=2);
s(g(�(x))) +
2
b
0
(b
0
� Æ(x))d
0;"
(x)�(g(�(x))) for Æ(x) 2 [b
0
=2; b
0
℄;
where s(g(�(x))) is the nearest point proje tion of g(�(x)) on � and �(g(�(x))) is the inward
unit normal to � at the point s(g(�(x))). On A
0
:= �
b
0
n
D
S
i=1
B(a
i
; r
0
), let w
"
= �w, where
�w 2 C
2
(A
0
; �) satis�es,
�w(x) =
8
<
:
s(g(�(x))) on ��
b
0
;
�
�
x�a
i
jx�a
i
j
�
on �B(a
i
; r
0
); i = 1; : : : ;D:
Finally, on ea h B(a
i
; r
0
), i = 1; : : : ;D, we set,
w
"
(x) =
8
<
:
�
�
x�a
i
jx�a
i
j
�
on B(a
i
; r
0
) nB(a
i
; ");
W
�
x�a
i
"
�
on B(a
i
; ");
where W : B(0; 1) :! is a C
2
map su h that W = � on S
1
(� is de�ned in (1.10)). Next we
estimate the energy of w
"
. From (2.2),(2.3) we dedu e as in the proof of Corollary 2.1 that,
(2.40) E
"
(w
"
; G
b
0
) = E
"
(d
0;"
) +O(1) =
2
"
Z
�G
H(d(�)) d� +O(1):
Clearly,
(2.41) E
"
(w
"
; A
0
) =
Z
A
0
jr �wj
2
= C; for some C independent of ":
Finally, an easy omputation yields,
(2.42) E
"
(w
"
;
[
i
B(a
i
; r
0
)) = D
l
2
2�
log
r
0
"
+O(1):
Combining (2.40){(2.42) we are led to (2.38). �
3. A stability result for d
0;"
This se tion is devoted to the proof of the following proposition whi h establishes a strong
minimizing property of d
0;"
in a neighborhood of the boundary.
Proposition 3.1. Let the fun tion F satisfy onditions (1.1){(1.3) and in addition,
(3.1) F
00
(t) � a
0
for jtj � L; for some a
0
; L > 0:
Then, there exist onstants "
0
;K;
0
> 0 su h that for every " � "
0
; T � K" and v 2 H
1
0
(G)
we have,
Z
G
T
jrvj
2
+
1
"
2
(F (d
0;"
+ v)� F (d
0;"
)� F
0
(d
0;"
)v) �
0
Z
G
T
jrvj
2
:
12
In the next se tion we will apply Proposition 3.1 to fun tions v satisfying kvk
1
� 2R. In
this setting the additional ondition (3.1) auses no restri tion sin e we may modify the values
of the original F near �1 in order to satisfy (3.1).
The main ingredient of the proof of Proposition 3.1 is the next result whi h an be viewed
as a one dimensional version of the proposition.
Proposition 3.2. There exist T
0
> 0 and �
0
> 0 su h that for all 2 [�R;R℄ and T � T
0
,
there holds,
Z
T
0
�
(v
0
)
2
+ F (U
+ v)� F (U
)� F
0
(U
)v
�
� �
0
Z
T
0
(v
0
)
2
; 8v 2 C
1
[0; T ℄ with v(0) = 0;
where U
is de�ned in (2.10).
The proof of Proposition 3.2 requires several lemmas. The �rst one is a simpli�ed version of
Proposition 3.2 when we use the approximation F (U
+ v)� F (U
)� F
0
(U
)v
�
=
1
2
F
00
(U
)v
2
.
Lemma 3.1. There exist onstants T
1
> 0 and �
1
> 0 su h that for all 2 [�R;R℄ and
T � T
1
there holds,
(3.2)
Z
T
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
� �
1
Z
T
0
(v
0
)
2
; 8v 2 C
1
[0; T ℄ with v(0) = 0:
Proof. We shall onsider only the ase 2 [�R; 0), the ase 2 (0; R℄ being similar. By (1.2)
there exist �
1
> 0 and
0
2 (0; R) su h that
(3.3) F
00
(s) � �
1
> 0 for s 2 [�
0
;
0
℄:
For 2 (�
0
; 0) we have U
(t) 2 (�
0
; 0); 8t 2 [0;1); and it is lear that (3.2) holds
with �
1
= 1; 8T: Therefore, it remains to �nd T
1
and �
1
su h that (3.2) is satis�ed for all
2 [�R;�
0
℄.
Let v
0
= U
0
, so that v
0
> 0 on [0; T ℄ and v
00
0
=
1
2
F
00
(U
)v
0
. Next we ompute,
Z
T
0
(v
0
�
v
0
0
v
0
v)
2
=
Z
T
0
(v
0
)
2
�
�
v
2
v
0
�
0
v
0
0
= �
v
2
(T )
v
0
(T )
v
0
0
(T ) +
Z
T
0
(v
0
)
2
+
v
2
v
0
v
00
0
= �
v
2
(T )
v
0
(T )
v
0
0
(T ) +
Z
T
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
:
Hen e,
(3.4)
v
2
(T )
v
0
(T )
v
0
0
(T ) +
Z
T
0
v
2
0
�
�
(v=v
0
)
0
�
�
2
=
Z
T
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
:
By Cau hy-S hwarz inequality we get,
v
2
(T )
v
2
0
(T )
=
�
Z
T
0
(v=v
0
)
0
�
2
�
Z
T
0
v
2
0
�
�
(v=v
0
)
0
�
�
2
�
Z
T
0
1
v
2
0
;
whi h yields,
(3.5)
v
2
(T )
v
0
(T )
jv
0
0
(T )j � v
0
(T )jv
0
0
(T )j
Z
T
0
v
2
0
�
�
(v=v
0
)
0
�
�
2
�
Z
T
0
1
v
2
0
:
13
From (3.4) and (3.5) we infer that,
(3.6)
Z
T
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
�
�
1� v
0
(T )jv
0
0
(T )j
Z
T
0
1
v
2
0
�
�
Z
T
0
v
2
0
�
�
(v=v
0
)
0
�
�
2
:
Next we ompute,
lim
T!1
v
0
(T )jv
0
0
(T )j
Z
T
0
dt
v
2
0
= lim
T!1
1
2
U
0
(T )jF
0
(U
(T ))j
Z
T
0
dt
(U
0
)
2
= lim
r!0
1
2
p
F (r)jF
0
(r)j
Z
r
ds
F
3
2
(s)
;
(3.7)
where in the last equality we substituted s = U
(t) and r = U
(T ). Denoting A =
p
F
00
(0)=2
we have,
p
F (r) ' Ajrj;
1
2
jF
0
(r)j ' A
2
jrj and
Z
r
ds
F
3
2
(s)
'
Z
r
ds
A
3
jsj
3
'
1
2A
3
r
2
as r! 0 :
Plugging these estimates in (3.7) we �nally obtain that,
(3.8) lim
T!1
v
0
(T )jv
0
0
(T )j
Z
T
0
dt
v
2
0
=
1
2
:
Sin e lim
T!1
U
(T ) = 0 uniformly on 2 [�R; 0) (see (2.13)), we get further that the limit
in (3.8) is uniform in 2 [�R;�
0
℄ and there exists thus T
1
> 0 su h that,
(3.9) v
0
(T
1
)jv
0
0
(T
1
)j
Z
T
1
0
dt
v
2
0
�
3
4
; 8 2 [�R;�
0
℄;
and
(3.10) U
(t) 2 [�
0
; 0); 8T � T
1
; 8 2 [�R;�
0
℄:
By (2.13) there exists �
1
> 0 su h that
v
0
= U
0
� �
1
; on [0; T
1
℄; 8 2 [�R;�
0
℄:
Therefore, we dedu e from Poin ar�e inequality that there exists �
2
= �
2
(T
1
) > 0 su h that,
(3.11)
Z
T
1
0
v
2
0
�
�
(v=v
0
)
0
�
�
2
� �
2
Z
T
1
0
v
2
; 8 2 [�R;�
0
℄:
By (3.6),(3.9) and (3.11) it follows that,
(3.12)
Z
T
1
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
�
�
2
4
Z
T
1
0
v
2
:
But by (3.3) and (3.10) we have learly,
(3.13)
Z
T
T
1
(v
0
)
2
+
1
2
F
00
(U
)v
2
�
�
1
2
Z
T
T
1
v
2
; 8T � T
1
:
14
From (3.12){(3.13) we infer that,
(3.14)
Z
T
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
� �
3
Z
T
0
v
2
; 8T � T
1
;
with �
3
= min(�
2
=4; �
1
=2).
On the other hand, we learly have,
(3.15)
Z
T
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
�
Z
T
0
(v
0
)
2
�
B
2
Z
T
0
v
2
;
where B is any positive number satisfying �B � min
[�R;R℄
F
00
(t). Combining (3.14) with
(3.15) we obtain,
�
1 +
2�
3
B
�
Z
T
0
(v
0
)
2
+
1
2
F
00
(U
)v
2
�
�
2�
3
B
�
Z
T
0
(v
0
)
2
;
and the result of the lemma follows in this ase too (i.e. for 2 [�R;�
0
℄) with �
1
=
2�
3
B+2�
3
. �
Lemma 3.2. There exists T
2
> 0 su h that for every T � T
2
and 2 [�R;R℄, the unique
solution to the problem,
u
00
=
1
2
F
0
(u) on [0; T ℄;(3.16)
u(0) = ; u
0
(T ) = U
0
(T );(3.17)
is u = U
.
Proof. We shall assume without loss of generality that 2 [�R; 0℄. The proof onsists of
several steps.
Step 1: u
0
(t) > 0 on [0; T ℄ if < 0; u � 0 if = 0:
Proof of Step 1: We onsider �rst the ase < 0. If u
0
(0) < 0 then it is easy to see from
(3.16) that u
0
& on [0; T ℄ and this is in ontradi tion with the boundary ondition u
0
(T ) =
U
0
(T ) > 0. If u
0
(0) = 0 we get from u
00
(0) =
1
2
F
0
(u(0)) < 0 that u
0
is negative and de reasing
in a right neighborhood of 0, hen e on all of (0; T ℄, and we get a ontradi tion as before.
Therefore u
0
(0) > 0. Assuming by negation that the assertion u
0
(t) > 0; 8t 2 [0; T ℄ is false,
there must be a minimal s 2 (0; T ℄ su h that u
0
(t) > 0 for t 2 [0; s) while u
0
(s) = 0. Sin e
u
00
(s) =
1
2
F
0
(u(s)) � 0 it follows that u(s) � 0. The possibility u(s) = 0 is ruled out sin e it
would lead to u � 0. We are left with the possibility u(s) < 0. But then u
00
(s) =
1
2
F
0
(u(s)) < 0
and it follows that u
0
is negative and de reasing in a right neighborhood of s, hen e on all of
(s; T ℄ and we get a ontradi tion as above.
In the ase = 0 we annot have u
0
(0) < 0 as above. u
0
(0) > 0 is impossible sin e the same
argument as in the ase < 0 would lead to u
0
(T ) > 0, ontradi ting the boundary ondition
u
0
(T ) = 0 (re all that U
0
� 0). We are left with the possibility u
0
(0) = 0. But it is lear that
15
the only solution to (3.16) satisfying u(0) = u
0
(0) = 0 is u � 0. We therefore onsider in the
sequel only the ase 2 [�R; 0).
Step 2: There exists a fun tion f(s) with lim
s!1
f(s) = 0 su h that u(T ) � f(T ) for every
solution of (3.16){(3.17) with 2 [�R; 0).
Proof of Step 2: Arguing as in the proof of Lemma 2.3, we see that there exists a onstant
su h that,
(3.18) (u
0
)
2
= F (u) + ; 8t 2 [0; T ℄:
If � 0 then by (3.18) we must have F (u(t)) 6= 0 for all t, so F (u(t)) < 0 on [0; T ℄ and we
an take f � 0. If > 0 then again by (3.18), F (u(T )) < (u
0
(T ))
2
= (U
0
(T ))
2
, and sin e
lim
T!1
U
0
(T ) = 0 (uniformly in 2 [�R; 0)) we get the on lusion in this ase too.
Step 3: 8� > 0; 9K > 0 su h that ju(t)j � � for any solution u of (3.16){(3.17) with 2
[�R; 0);8T > 2K;8t 2 [K;T ℄:
Proof of Step 3: By Step 2 there exists K
1
> 0 su h that,
(3.19) f(s) � �; 8s � 2K
1
:
From Step 1 we then infer that,
(3.20) � u(t) � � on [0; T ℄; 8T > 2K
1
:
The result is lear then in the ase � � j j (by taking K = K
1
). It is enough to onsider the
ase � < j j. By the assumption (1.3) there exists � > 0 su h that,
(3.21) �
1
2
F
0
(s) � �(s� ); 8s 2 [ ;��℄;8 2 [�R;��):
The fun tion �(t) = os(�t=2) is an eigenfun tion orresponding to the prin ipal eigenvalue
�
1
= �
2
=4 of the operator Lv = �v
00
on (�1; 1) with zero boundary ondition. We laim that
the on lusion of Step 3 holds for any K > 0 satisfying
(3.22) K � max(
p
�
1
=�;K
1
) (see (3.19)):
We �x any t
0
2 [K;T �K℄. For every � 2 (0;�� � ) we set �
�
(t) = + ��(
t�t
0
K
): By (3.21)
and (3.22) we have,
��
00
�
=
�
�
1
�
K
2
�
�
�
t� t
0
K
�
=
�
1
K
2
(�
�
� ) � �(�
�
� ) � �
F
0
(�
�
)
2
:
Sin e by (3.20) u � , ea h �
�
is a subsolution for the problem,
8
<
:
� v
00
= �
1
2
F
0
(v) on [t
0
�K; t
0
+K℄;
v(t
0
�K) = u(t
0
�K):
For small enough � > 0 we have u � �
�
on [t
0
�K; t
0
+K℄. Set
�
0
= supf0 < � � �� � : u � �
�
on [t
0
�K; t
0
+K℄g:
16
By the maximum prin iple we have �
0
= ��� , so in parti ular u(t
0
) � + (��� ) = ��.
We proved that u(t
0
) � �� for any t
0
2 [K;T �K℄, but by the monotoni ity of u (see Step 1)
the inequality holds for t
0
2 [K;T ℄. Combining it with (3.20) we obtain the desired on lusion.
Step 4: ( on lusion) For T large enough the solution to (3.16){(3.17) is unique.
Proof of Step 4: Arguing by ontradi tion, suppose that there exist sequen es f
n
g � [�R; 0)
and fT
n
g, T
n
! 1, su h that for every n there is a solution u
n
6= U
n
of (3.16){(3.17) with
=
n
and T = T
n
. For ea h n, let t
n
2 [0; T
n
℄ be su h that
j(U
n
� u
n
)(t
n
)j = max
[0;T
n
℄
jU
n
� u
n
j:
Passing to a subsequen e if ne essary we may assume that the sign of (U
n
� u
n
)(t
n
) is the
same for all n. Without loss of generality we suppose that (U
n
� u
n
)(t
n
) > 0 for all n.
Take � =
0
(de�ned in (3.3)) and let K > 0 orrespond to it by Step 3. We may assume
that T
n
> 2K for all n. Next we laim that,
(3.23) t
n
2 [0;K℄; 8n:
Indeed, assume �rst that t
n
2 (K;T
n
): Then,
(U
n
� u
n
)
00
(t
n
) =
1
2
�
F
0
(U
n
(t
n
))� F
0
(u
n
(t
n
))
�
=
1
2
F
00
(z
n
)(U
n
(t
n
)� u
n
(t
n
));
for some z
n
2 (u
n
(t
n
); U
n
(t
n
)) � [��; �℄: Thus F
00
(z
n
) > 0, and we get that (U
n
�u
n
)
00
(t
n
) >
0; whi h is impossible for a maximum. Assume next that t
n
= T
n
. The same argument as
above shows that (U
n
� u
n
)
00
(t) > 0 on the interval [K;T
n
). Hen e (U
n
� u
n
)
0
is in reasing
on that interval. But by (3.17), (U
n
� u
n
)
0
(T
n
) = 0, so that (U
n
� u
n
)
0
< 0 on [K;T
n
),
ontradi ting our assumption that T
n
is a maximum point and ompleting the proof of our
laim (3.23).
Passing to a diagonal subsequen e, we may assume that for ea h T > 0 we have u
n
! �u in
C
2
[0; T ℄, where �u is a solution of,
8
<
:
�u
00
=
1
2
F
0
(�u) on [0;1);
�u(0) = � 2 [�R; 0℄;
with � = lim
n!1
n
. Moreover, by Step 3 we have also lim
t!1
�u(t) = 0. Therefore by
Lemma 2.3 we must have �u = U
�
and we on lude, using (3.23), that,
(3.24) lim
n!1
jju
n
� U
�
jj
L
1
(0;1)
= 0:
Next we de�ne,
v
n
(t) =
(U
�
� u
n
)(t)
(U
�
� u
n
)(t
n
)
for t 2 [0; T
n
℄:
Clearly jv
n
j � 1 on [0; T
n
℄, v
n
(0) = 0 and max
[0;K℄
v
n
= 1. The equation satis�ed by v
n
is,
v
00
n
=
1
2
F
0
(U
�
)� F
0
(u
n
)
(U
�
� u
n
)(t
n
)
=
1
2
F
00
(h
n
(t)) v
n
;
17
with h
n
(t) an intermediate point between u
n
(t) and U
�
(t). Passing to a diagonal subsequen e,
using (3.24), we may assume that v
n
! V in C
1
[0; T ℄, 8T > 0, where V satis�es,
V
00
=
1
2
F
00
(U
�
)V on (0;1);(3.25)
max
[0;K℄
V = 1; V (0) = 0; and kV k
L
1
(0;1)
� 1:(3.26)
We are going to get a ontradi tion by showing that there is no solution to (3.25){(3.26).
Assume �rst that � < 0. Then, v
0
= U
0
�
is a solution of (3.25) whi h is positive on [0;1).
For small enough � > 0 we have,
(3.27) v
0
� �V on [0;K℄:
Set �
0
= supf� > 0; � satis�es (3.27)g. Let t
0
2 [0;K℄ be su h that v
0
(t
0
) = �
0
V (t
0
): By our
hoi e of K, for t � K we have F
00
(U
�
(t)) > 0. So applying the maximum prin iple to the
(bounded) fun tion v
0
� �
0
V , whi h satis�es,
8
<
:
(v
0
� �
0
V )
00
=
1
2
F
00
(U
�
)(v
0
� �
0
V );
(v
0
� �
0
V )(K) � 0;
we obtain that v
0
� �
0
V � 0 on [K;1), hen e also on [0;1). Sin e �
0
V (0) = 0 < v
0
(0) the
strong maximum prin iple gives v
0
> �
0
V in (0;1), ontradi ting (v
0
� �
0
V )(t
0
) = 0.
In the ase � = 0 we have U
�
� 0. Then, we apply the above argument using v
0
(t) =
exp(�
p
F
00
(0)=2t), whi h is a positive solution of v
00
0
=
1
2
F
00
(0)v
0
.
�
Proof of Proposition 3.2. We �rst noti e two simple onsequen es of (1.1){(1.3) and (3.1).
There exist m
0
> 0 su h that
(3.28) F (a+ b)� F (a)� bF
0
(a) � �m
0
; 8a 2 [�R;R℄; 8b 2 (�1;1):
and �
0
> 0 su h that,
(3.29) F (a+ b)� F (a)� bF
0
(a) � 0; 8a 2 [��
0
; �
0
℄; 8b 2 (�1;1):
We next �x T
0
su h that
(3.30) T
0
� max(T
1
; T
2
) and jU
(t)j � �
0
; 8t � T
0
; 8 2 [�R;R℄:
Re all that T
2
is given in Lemma 3.2 and T
1
is given in Lemma 3.1. Sin e for any T > T
0
we
have by (3.29),(3.30),
Z
T
T
0
�
(v
0
)
2
+ F (U
+ v)� F (U
)� F
0
(U
)v
�
�
Z
T
T
0
(v
0
)
2
;
it is enough to prove that for some �
0
> 0 we have,
(3.31)
Z
T
0
0
�
(v
0
)
2
+ F (U
+ v)� F (U
)� F
0
(U
)v
�
� �
0
Z
T
0
0
(v
0
)
2
; 8v 2 C
1
[0; T
0
℄ with v(0) = 0:
18
Looking for a ontradi tion, we assume that for some sequen es f
n
g � [�R;R℄ and fv
n
g 2
C
1
[0; T
0
℄; v
n
(0) = 0; 8n; we have,
(3.32) lim
n!1
R
T
0
0
(v
0
n
)
2
+ F (U
n
+ v
n
)� F (U
n
)� F
0
(U
n
)v
n
R
T
0
0
(v
0
n
)
2
� 0:
Passing to a subsequen e, we need to onsider ea h of the three possibilities:
(i) lim
n!1
R
T
0
0
(v
0
n
)
2
=1;
(ii) lim
n!1
R
T
0
0
(v
0
n
)
2
= 0;
(iii) lim
n!1
R
T
0
0
(v
0
n
)
2
= > 0:
We �rst onsider ase (i). By (3.28) we have,
lim
n!1
R
T
0
0
(v
0
n
)
2
+ F (U
n
+ v
n
)� F (U
n
)� F
0
(U
n
)v
n
R
T
0
0
(v
0
n
)
2
� 1;
ontradi ting (3.32).
In ase (ii) we have v
n
! 0 in C[0; T
0
℄. Then by Poin ar�e inequality, for some sequen e
r
n
! 0 we have
�
�
Z
T
0
0
F (U
n
+ v
n
)� F (U
n
)� F
0
(U
n
)v
n
�
Z
T
0
0
1
2
F
00
(U
n
)v
2
n
�
�
� r
n
Z
T
0
0
(v
0
n
)
2
;
whi h together with Lemma 3.1 yields a ontradi tion to (3.32).
Finally we onsider ase (iii). Sin e fv
n
g is bounded in H
1
[0; T
0
℄, passing to a subsequen e,
we an assume that v
n
! v weakly in H
1
[0; T
0
℄, strongly in C[0; T
0
℄ and also that
n
! � . If
v � 0, we get a ontradi tion as in ase (ii). We assume then that v 6� 0. It follows that
I := inf
w2H
1
[0;T
0
℄;w(0)=0
Z
T
0
0
(w
0
)
2
+ F (U
�
+ w)� F (U
�
)� wF
0
(U
�
) � 0:
Note that I > �1 by (3.28). Consider a minimizing sequen e fw
n
g for I. If lim
n!1
R
T
0
0
(w
0
n
)
2
=
1, we get a ontradi tion as in ase (i). So we may assume that (w
n
) is bounded in H
1
[0; T
0
℄.
Passing to a subsequen e we obtain that w
n
! w weakly in H
1
[0; T
0
℄ and strongly in C[0; T
0
℄.
If w � 0 we get a ontradi tion as in ase (ii), so we may assume that w 6� 0. But then w is
a minimizer for I and thus u = U
�
+w is a solution of (3.16){(3.17), with T = T
0
and = � ,
whi h is di�erent from U
�
, ontradi ting Lemma 3.2. �
Now we are in position to present the proof of Proposition 3.1. It uses Proposition 3.2 and
the approximation
~
d
"
for d
0;"
.
Proof of Proposition 3.1. Using Lemma 2.4 we �x K > 0 su h that jd
0;"
(x)j � �
0
on �
K"
(�
0
is de�ned by (3.29)). It is lear then that for any T > K" we have,
Z
�
K"
jrvj
2
+
1
"
2
(F (d
0;"
+ v)� F (d
0;"
)� F
0
(d
0;"
)v) �
Z
�
K"
jrvj
2
:
19
Therefore, it is enough to show that for some
0
> 0 we have,
(3.33)
Z
G
K"
jrvj
2
+
1
"
2
(F (d
0;"
+ v)� F (d
0;"
)� F
0
(d
0;"
)v) �
0
Z
G
K"
jrvj
2
:
Using (3.1) we �x A > 0 su h that,
(3.34) F
0
(�t) < min
[�R;R℄
F
0
< max
[�R;R℄
F
0
< F
0
(t); 8t � A:
SettingM := A+R we dedu e from (3.34) that the fun tion g
a
(b) := F (a+b)�F (a)�F
0
(a)b
satis�es,
(3.35) g
a
& for b � �M and g
a
% for b �M; 8a 2 [�R;R℄:
We de�ne a trun ation fun tion (s) by,
(s) =
8
>
>
<
>
>
:
�M s 2 (�1;�M);
s s 2 [�M;M ℄;
M s 2 (M;1):
The fun tion �v := (v) satis�es: �v 2 H
1
0
(G), jr�vj � jrvj a.e. , k�vk
L
1
� M , and thanks to
(3.35) we have,
F (d
0;"
+ �v)� F (d
0;"
)� F
0
(d
0;"
)�v � F (d
0;"
+ v)� F (d
0;"
)� F
0
(d
0;"
)v a.e. in G:
Therefore it suÆ es to prove (3.33) for �v, or equivalently, we may assume a priori that,
(3.36) kvk
L
1
�M :
Next we note that by Taylor formula we have,
(3.37) F (a+ b)� F (a)� F
0
(a)b =
Z
b
0
F
00
(a+ s)(b� s) ds:
Hen e, there exists a ontinuous in reasing fun tion h : [0;1) ! [0;1) with h(0) = 0 su h
that,
(3.38)
�
�
F (a+ b)� F (a)� F
0
(a)b�
�
F (~a+ b)� F (~a)� F
0
(~a)b
�
�
�
� h(ja� ~aj) � b
2
;
8a; ~a 2 [�R;R℄; 8b 2 [�M;M ℄:
By (3.38),(2.20),(2.2) and the one dimensional Poin ar�e inequality,
Z
K"
0
w
2
dt � (2=�)
2
(K")
2
Z
K"
0
(w
0
)
2
dt; for w satisfying w(0) = 0;
we infer that,
(3.39)
1
"
2
Z
G
K"
�
�
(F (d
0;"
+ v)� F (d
0;"
)�F
0
(d
0;"
)v)� (F (
~
d
"
+ v)� F (
~
d
"
)� F
0
(
~
d
"
)v)
�
�
�
h(C
1
")
"
2
Z
G
K"
jvj
2
� C
2
h(C
1
")
Z
�G
Z
K"
0
�
�
�v
�Æ
�
�
2
dÆ d�;
for some onstant C
2
> 0.
20
Next, by (2.3) and (2.2) we obtain (passing to the variables �; Æ),
(3.40)
�
�
Z
G
K"
�
�
�v
�Æ
�
�
2
�
Z
�G
Z
K"
0
�
�
�v
�Æ
�
�
2
dÆ d�
�
�
� C
3
"
Z
�G
Z
K"
0
�
�
�v
�Æ
�
�
2
dÆ d� :
Similarly, denoting B = max
[�M�R;M+R℄
jF
00
j we have,
1
"
2
�
�
Z
G
K"
(F (
~
d
"
+ v)� F (
~
d
"
)� F
0
(
~
d
"
)v) �
Z
�G
Z
K"
0
(F (
~
d
"
+ v)� F (
~
d
"
)� F
0
(
~
d
"
)v) dÆ d�
�
�
�
1
"
2
Z
�G
Z
K"
0
jF (
~
d
"
+ v)� F (
~
d
"
)� F
0
(
~
d
"
)vj Æ dÆ d�
�
B
2"
2
Z
�G
Z
K"
0
v
2
Æ dÆ d� � C
4
"
Z
�G
Z
K"
0
�
�
�v
�Æ
�
�
2
dÆ d� ;
(3.41)
for some C
4
> 0, where in the last inequality we used again the one dimensional Poin ar�e
inequality.
Finally, ombining (3.39){(3.41) and applying Proposition 3.2 yields for " � "
0
,
Z
G
K"
jrvj
2
+
1
"
2
(F (d
0;"
+ v)� F (d
0;"
)� F
0
(d
0;"
)v) �
Z
G
K"
�
�
�v
��
�
�
2
+
Z
�G
Z
K"
0
�
�
�v
�Æ
�
�
2
+
1
"
2
(F (
~
d
"
+ v)� F (
~
d
"
)� F
0
(
~
d
"
)v) dÆ d�
�
�
C
2
h(C
1
") + C
3
"+ C
4
"
�
Z
�G
Z
K"
0
�
�
�v
�Æ
�
�
2
dÆ d�
�
Z
G
K"
�
�
�v
��
�
�
2
+ (�
0
=2)
Z
�G
Z
K"
0
�
�
�v
�Æ
�
�
2
dÆ d� � min
�
1;
�
0
=2
1 + C
3
"
0
�
Z
G
K"
jrvj
2
;
(3.42)
and the result follows. �
We lose this se tion with Proposition 3.3 whi h is a spa ial version of Proposition 3.1. It
extends the one dimensional result Lemma 3.1 to in dimension 2 (just as Proposition 3.1
extends Proposition 3.2). A tually, we do not need the result in this paper but we bring it as
it may be of independent interest. The proof is similar to the one of Proposition 3.1. In fa t, it
is even onsiderably simpler, sin e it relies only on Lemma 3.1, and will be therefore omitted.
We remark that although stated in dimension 2, both Proposition 3.3 and Proposition 3.1 are
valid for arbitrary dimension (with the same proof). Proposition 3.3 improves upon a result
of Berger and Fraenkel [5, Lemma 4.2℄ whi h treated the ase of a Ginzburg-Landau energy.
Proposition 3.3. Let the fun tion F satisfy onditions (1.1){(1.3). Then, there exist on-
stants "
1
;K
1
; ~
0
> 0 su h that for every " � "
1
; T � K
1
" and v 2 H
1
0
(G) we have,
Z
G
T
jrvj
2
+
F
00
(d
0;"
)v
2"
2
� ~
0
Z
G
T
jrvj
2
:
21
4. The lower bound: I. Preliminaries
The next two se tions are devoted to the proof of the lower bound for E
"
(u
"
) whi h in
onjun tion with Proposition 2.2 will omplete the proof of Theorem 1. The proof onsists of
two main steps. In the �rst, arried out in this se tion (see Proposition 4.1 below), we prove
a lower bound of the form,
E
"
(u
"
) � E
"
(d
0;"
) +
�
E
"
(u
"
) + o(1);
where
�
E
"
(u) is a new energy, de�ned for any u 2 H
1
(�
"
a
;R
2
) as follows. We �rst de�ne for
ea h � < b
0
a C
1
-map g
�
: ��
�
! R
2
by,
g
�
(~�) = s(g(�(~�))); 8~� 2 ��
�
;
where we denoted as usual by �(~�) the nearest point proje tion of ~� on �G. The energy
�
E
"
(u)
will be de�ned, for properly hosen onstants ~
1
> 0 and a 2 (0; 1), by
(4.1)
�
E
"
(u) = ~
1
Z
��
"
a
ju(~�)� g
"
a
(~�)j
2
"
a
d~� +
Z
�
"
a
jruj
2
+
F (d(u))
"
2
:
The se ond main step, the analysis of the minimizers for
�
E
"
, or rather those of the slightly
di�erent energy
e
E
"
, is the subje t of the next se tion.
We start with some notations. For ea h " > 0 we de�ne,
(4.2) A
"
= fx 2 G; u
"
(x) =2 Sg :
For x 2 A
"
we set v
"
(x) = s(u
"
(x)), where, we re all, s is the nearest point proje tion on
�. We shall denote for short d
"
(x) = d(u
"
(x)), i.e. d
"
(x) is the signed distan e of u
"
(x) to
�. We start with a simple lemma. It generalizes for arbitrary the elementary formula,
orresponding to the ase = B(0; 1), jruj
2
= �
2
jrvj
2
+ jr�j
2
, where � = juj and u = �v
whenever u 6= 0.
Lemma 4.1. On the set A
"
we have,
(4.3) jru
"
(x)j
2
= (1� �
"
d
"
)
2
jrv
"
(x)j
2
+ jrd
"
(x)j
2
;
where �
"
:= �(v
"
(x)) is the urvature of � at v
"
(x).
Proof. Let � and � denote respe tively the unit tangent ve tor and inward unit normal ve tor
along �. On A
"
we write u
"
= v
"
+ d
"
�
"
where �
"
(x) := �(v
"
(x)). Sin e by Frenet formulas,
d�
ds
= ��� ; along �;
we on lude that,
(4.4) r�
"
= ��
"
rv
"
on A
"
:
Moreover, sin e � ? � and � ?
�
� we obtain that
(4.5) �
"
? rv
"
and �
"
? r�
"
on A
"
:
22
From (4.4){(4.5) we infer that, for any x 2 A
"
we have,
jru
"
j
2
= jrv
"
+ d
"
r�
"
+rd
"
� �
"
j
2
= jrv
"
j
2
+ d
2
"
jr�
"
j
2
+ jrd
"
j
2
+ 2d
"
rv
"
� r�
"
= (1� �
"
d
"
)
2
jrv
"
(x)j
2
+ jrd
"
(x)j
2
:
�
We expe t the term of the order
1
"
in (1.9) to be a ontribution of the energy of d
"
in a
boundary layer. A �rst step in this dire tion is made in the next lemma. We de�ne,
d
1;"
= d
"
� d
0;"
;
noting that d
1;"
= 0 on �G.
Lemma 4.2. For ea h � 2 ("
9=10
; b
0
) we have,
(4.6) E
"
(d
"
; G
�
) = E
"
(d
0;"
; G
�
)+
Z
G
�
jrd
1;"
j
2
+
1
"
2
Z
G
�
�
F (d
"
)�F (d
0;"
)�F
0
(d
0;"
)d
1;"
�
+o(1);
where o(1) denotes a quantity whi h goes to 0 as "! 0, uniformly in � 2 ("
9=10
; b
0
).
Proof. From Green formula and (2.9) we infer that,
E
"
(d
"
; G
�
) =
Z
G
�
jrd
0;"
j
2
+ 2rd
0;"
� rd
1;"
+ jrd
1;"
j
2
+
1
"
2
Z
G
�
F (d
"
)
=
Z
G
�
jrd
0;"
j
2
+
Z
G
�
jrd
1;"
j
2
+
1
"
2
�
F (d
"
)� F
0
(d
0;"
)d
1;"
�
+ 2
Z
��
�
�d
0;"
�n
d
1;"
= E
"
(d
0;"
; G
�
) +
Z
G
�
jrd
1;"
j
2
+
1
"
2
Z
G
�
�
F (d
"
)� F (d
0;"
)� F
0
(d
0;"
)d
1;"
�
+ o(1);
where in the last equality we used (2.32). �
Our next result establishes a lower bound for the energy of d
"
in a boundary layer, of the
order
1
"
. Moreover, we get even an extra positive term that will help us in the analysis of the
term of order j log "j of E
"
(u
"
).
Lemma 4.3. For ea h � 2 ("
9=10
; b
0
) we have,
(4.7) E
"
(d
"
; G
�
) � E
"
(d
0;"
; G
�
) +
0
Z
G
�
jrd
1;"
j
2
+ o(1):
Proof. We simply apply Proposition 3.1 to the r.h.s. of (4.6) with v = d
1;"
. Note that there
is no loss of generality in assuming the additional hypothesis (3.1) sin e only the values of F
on [�R;R℄ are relevant in this ase. �
In the next proposition, whi h is the main result of this se tion, we take into a ount also
the ontribution of the term
R
G
jru
"
j
2
� jrd
"
j
2
.
23
Proposition 4.1. There exist onstants ~
1
; "
0
> 0 su h that for every � 2 ("
9=10
; b
0
) and
" 2 (0; "
0
℄ we have,
(4.8) E
"
(u
"
)�E
"
(d
0;"
) � ~
1
Z
��
�
ju
"
(~�)� g
�
(~�)j
2
�
d~� +
Z
�
�
jru
"
j
2
+
F (d
"
)
"
2
+ o(1) :
Proof. In the sequel we shall identify a point x 2 G
b
0
with the pair (�(x); Æ(x)). For any point
z 2 R
2
n S and � > 0 small enough we de�ne a \re tangle" by,
R(z; �) =
n
y 2 R
2
; js(y)� s(z)j < �; d(y) 2
8
<
:
(��; d(z) + �) if d(z) � 0
(d(z) � �; �) if d(z) < 0
o
:
It is lear that for � small enough R(z; �) is well de�ned, so that, R(z; �) � R
2
n S. By our
assumption (1.7) it is lear that we an �x a value of �
0
> 0 su h that,
(4.9)
[
�2�G
R(g(�); �
0
) �� R
2
n S:
We then de�ne,
(4.10) 0 < r
0
:= min
�
1� �(s(y))d(y) ; y 2
[
�2�G
R(g(�); �
0
)
:
Thanks to (4.7) we have,
E
"
(u
"
)�E
"
(d
0;"
) �
Z
G
�
�
0
jrd
1;"
j
2
+ jru
"
j
2
� jrd
"
j
2
�
+
Z
�
�
jru
"
j
2
+
F (d
"
)
"
2
+ o(1):
(4.11)
For ea h �
0
2 �G we denote by ~�
0
the point in ��
�
with �(~�
0
) = �
0
. We de�ne the \interval"
I
�
0
:= fx 2 G : �(x) = �
0
; 0 < Æ(x) < �g;
whi h joins �
0
to ~�
0
. Next we laim that there exists a onstant �
1
> 0 su h that, for all
�
0
2 �G,
(4.12)
Z
I
�
0
�
0
jrd
1;"
j
2
+ jru
"
j
2
� jrd
"
j
2
�
� �
1
ju
"
(~�
0
)� g
�
(~�
0
)j
2
�
+ o(1):
Fix any �
0
2 �G. We distinguish two ases.
Case I:
R
I
�
0
jrd
1;"
j
2
�
�
2
0
4�
:
In this ase we have learly,
(4.13)
Z
I
�
0
�
0
jrd
1;"
j
2
+ jru
"
j
2
� jrd
"
j
2
�
�
0
�
2
0
4�
�
0
�
2
0
16R
2
ju
"
(~�
0
)� g
�
(~�
0
)j
2
�
;
sin e ju
"
(~�
0
)� g
�
(~�
0
)j � 2R.
Case II:
R
I
�
0
jrd
1;"
j
2
<
�
2
0
4�
:
24
For small " we have jI
�
0
j � 2�. Therefore, by Cau hy-S hwarz inequality we have for every
Æ 2 (0; �),
(4.14)
jd
1;"
(�
0
; Æ)j = jd
1;"
(�
0
; Æ) � d
1;"
(�
0
; 0)j � jI
�
0
j
1=2
�
Z
I
�
0
jrd
1;"
j
2
�
1=2
� (2�)
1=2
�
�
0
2�
1=2
<
3�
0
4
:
From (4.14) and Proposition 2.1 it follows that for all " 2 (0; "
0
) and Æ 2 (0; �) we have,
(4.15) d
"
(�
0
; Æ) 2
8
<
:
(��
0
; d(g(�
0
)) + �
0
) if d(g(�
0
)) � 0;
(d(g(�
0
))� �
0
; �
0
) if d(g(�
0
)) < 0:
By (4.15) we have two possibilities. The �rst is,
(i) u
"
(�
0
; Æ) 2 R(g(�
0
); �
0
); 8Æ 2 (0; �):
Then by (4.3), (4.10) and Cau hy-S hwarz inequality we have, again for " small,
Z
I
�
0
�
0
jrd
1;"
j
2
+ jru
"
j
2
� jrd
"
j
2
�
�
0
d
2
1;"
(~�
0
)
2�
+ r
2
0
Z
I
�
0
jrv
"
j
2
�
0
d
2
1;"
(~�
0
)
2�
+ r
2
0
jv
"
(~�
0
)� g
�
(~�
0
)j
2
2�
:
(4.16)
By Proposition 2.1, d
0;"
(~�
0
)=� = o(1), that is,
d
2
1;"
(~�
0
)
�
=
d
2
"
(~�
0
)
�
+ o(1):
Therefore, from (4.16) we infer, using the elementary inequality jv
1
+ v
2
j
2
� 2jv
1
j
2
+ 2jv
2
j
2
,
that
Z
I
�
0
�
0
jrd
1;"
j
2
+ jru
"
j
2
� jrd
"
j
2
�
�
0
ju
"
(~�
0
)� v
"
(~�
0
)j
2
2�
+ r
2
0
jv
"
(~�
0
)� g
�
(~�
0
)j
2
2�
+ o(1)
�
1
4
min(
0
; r
2
0
)
ju
"
(~�
0
)� g
�
(~�
0
)j
2
�
+ o(1) :
(4.17)
In view of (4.15) the remaining possibility is,
(ii) There exists Æ
0
2 (0; �) su h that
u
"
(�
0
; Æ) 2 R(g(�
0
); �
0
); 8Æ 2 (0; Æ
0
) and js(u
"
(�
0
; Æ
0
))� s(g(�
0
))j = �
0
:
Then by (4.3) and (4.10) we get, as in (4.16), that
Z
f�=�
0
;0<Æ<Æ
0
g
�
0
jrd
1;"
j
2
+ jru
"
j
2
� jrd
"
j
2
�
� r
2
0
jv
"
(�
0
; Æ
0
)� s(g(�
0
))j
2
2Æ
0
= r
2
0
�
2
0
2Æ
0
� r
2
0
�
2
0
2�
� r
2
0
�
2
0
8R
2
ju
"
(~�
0
)� g
�
(~�
0
)j
2
�
:
(4.18)
25
Combining (4.13),(4.17) and (4.18) we get that (4.12) holds with,
�
1
= min
�
0
�
2
0
16R
2
;
1
4
min(
0
; r
2
0
); r
2
0
�
2
0
8R
2
�
:
Finally, integrating (4.12) over � 2 �G and using (2.2){(2.3) we obtain (4.8) with ~
1
=
�
1
2
. �
5. The lower bound II: Con lusion
In view of Proposition 4.1, the proof of the lower bound of Theorem 1 will be ompleted
on e we prove that,
(5.1)
�
E
"
(u
"
) �
l
2
2�
Dj log "j+O(1);
where
�
E
"
is de�ned in (4.1), for some a 2 (0; 9=10). A tually, we will allow a to be "-dependent,
i.e. a = a
"
, and we will require from it to satisfy,
(5.2) a 2 (
7
8
;
9
10
):
For te hni al reasons it will be onvenient to work with a slightly di�erent energy
e
E
"
whi h
is de�ned below. We �rst introdu e some notations. We denote
~
Æ(x) = dist(x; ��
"
a
). The
fun tion p
"
(x) is de�ned on �
"
a
by,
(5.3) p
"
(x) =
8
<
:
(
~
Æ(x)="
1
2
)"+ (1�
~
Æ(x)="
1
2
)"
a
if
~
Æ(x) � "
1
2
;
" if
~
Æ(x) > "
1
2
:
We also denote,
~g(~�) = g
"
a
(~�) = s(g(�(~�))) for ~� 2 ��
"
a
;
where, as above, �(~�) denotes the nearest point proje tion of ~� on �G. Finally we de�ne,
(5.4)
e
E
"
(u) = ~
1
Z
��
"
a
ju(~�)� ~g(~�)j
2
p
"
(~�)
d~� +
Z
�
"
a
jruj
2
+
F (d(u))
p
2
"
(x)
; 8u 2 H
1
(�
"
a
;R
2
):
Note that p
"
(x) = "
a
on ��
"
a
and p
"
(x) � " in �
"
a
. Therefore,
e
E
"
(u) �
�
E
"
(u) and the
following proposition implies (5.1), and onsequently, as explained above, the lower bound of
Theorem 1.
Proposition 5.1. We have,
(5.5)
e
E
"
(u) �
l
2
2�
Dj log "j+O(1); 8u 2 H
1
(�
"
a
;R
2
); 8" 2 (0; 1); 8a 2 (7=8; 9=10):
We denote for ea h " > 0 by w
"
a minimizer for
e
E
"
over H
1
(�
"
a
;R
2
). The proof of
Proposition 5.1 is quite involved and onsists of several lemmas in whi h the asymptoti
behavior of fw
"
g as " ! 0 is studied. The analysis is similar to the one of the Ginzburg-
Landau energy as in [7, 16℄. The di�eren e is that in the minimization problem for (5.4)
there is no boundary ondition. The minimizer w
"
is for ed to take boundary values loser
26
and loser to eg as " be omes smaller and smaller, be ause of the explosion of the boundary
integral, as "! 0.
We start with some basi properties of the minimizers fw
"
g. First, using the same argument
as in Lemma 2.1 we get,
(5.6) kw
"
k
L
1
(�
"
a)
� R; 8":
Sin e by Proposition 4.1,
E
"
(u
"
)�E
"
(d
0;"
) �
e
E
"
(u
"
) + o(1) �
e
E
"
(w
"
) + o(1);
we infer from (2.37) and (2.31) that,
(5.7)
e
E
"
(w
"
) �
l
2
2�
Dj log "j+O(1):
Proposition 5.1 will follow from the lower bound that we are going to prove, namely,
(5.8)
e
E
"
(w
"
) �
l
2
2�
Dj log "j+O(1):
The Euler-Lagrange equation satis�ed by w
"
is,
(5.9)
8
>
>
<
>
>
:
�w
"
=
1
2p
2
"
(x)
F
0
(d(w
"
)) � (rd)(w
"
) in �
"
a
;
�w
"
�n
= ~
1
~g � w
"
p
"
on ��
"
a
:
An easy onsequen e of (5.9) is the following gradient bound.
Lemma 5.1. There exists a onstant ~ su h that,
(5.10) jrw
"
(x)j �
~
p
"
(x)
; 8x 2 �
"
a
Proof. We shall use the following property of the fun tion p
"
. There exists "
1
> 0 su h that,
for every " 2 (0; "
1
℄ we have,
(5.11) x; y 2 �
"
a
and jy � xj � p
"
(x) =)
1
2
p
"
(x) � p
"
(y) � 2p
"
(x):
Indeed, �x any x; y 2 �
"
a
with jy � xj � p
"
(x). With a ertain abuse of notations we view p
"
also as a fun tion of
~
Æ. By (5.2){(5.3),
j log p
"
(y)� log p
"
(x)j �
�
�
�
Z
~
Æ(y)
~
Æ(x)
�p
"
=�
~
Æ
p
"
d
~
Æ
�
�
�
� p
"
(x) sup
�
�
�
�p
"
=�
~
Æ
p
"
�
�
�
� p
"
(x) �
"
a�1=2
� "
1=2
"
� 2"
2a�3=2
! 0 as "! 0;
and (5.11) follows.
Consider any x 2 �
"
a
. We shall distinguish two ases:
(i)
~
Æ(x) > p
"
(x)=4,
(ii)
~
Æ(x) � p
"
(x)=4.
27
In ase (i), we de�ne on B(0; 1) by res aling the fun tion W
"
(y) = w
"
(x + p
"
(x)y=4). It
satis�es,
(5.12) �W
"
=
p
2
"
(x)
32p
2
"
(x+ p
"
(x)y=4)
F
0
(d(W
"
)) � (rd)(W
"
):
By (5.11) the r.h.s. of (5.12) is bounded in L
1
(B(0; 1)). Standard ellipti estimates yield
a bound for jrW
"
j on B(0; 1=2), and res aling ba k we obtain (5.10) in this ase.
In the se ond ase (ii) we dedu e from (5.11) that
1
2
p
"
(x) � p
"
(~�(x)) � 2p
"
(x), where ~�(x)
denotes the nearest point proje tion of x on ��
"
a
. Set K
"
= p
"
(~�(x)), so that
~
Æ(x) �
K
"
2
. Let
C
"
= fy 2 B(0; 1); ~�(x) +K
"
y 2 �
"
a
g, and note that C
"
tends to a half disk B
+
(0; 1) when
" ! 0. Similarly to ase (i) we de�ne res aled fun tions on C
"
by W
"
(y) = w
"
(~�(x) +K
"
y):
Then, W
"
satis�es,
(5.13)
8
>
>
<
>
>
:
�W
"
=
K
2
"
2p
2
"
(~�(x) +K
"
y)
F
0
(d(W
"
)) � (rd)(W
"
) on C
"
;
�W
"
�n
=
~
1
K
"
p
"
(~�(x) +K
"
y)
�
~g(~�(x) +K
"
y)�W
"
�
on �C
"
n �B(0; 1):
The r.h.s in both equalities in (5.13) are bounded in L
1
(C
"
) thanks again to (5.11). By
ellipti estimates, as those of [10, Th. 4.1℄, we get that for every 2 (0; 1) the family fW
"
g is
bounded in C
(C
"
\ B(0; 3=4)). Plugging this bound in (5.13), and taking into a ount the
estimate krp
"
k
L
1
� "
a�1=2
! 0 as "! 0, we an now infer from the ellipti estimates of [10,
Th. 4.1℄ that fW
"
g is bounded in C
1;
(C
"
\B(0; 2=3)). Res aling ba k we obtain the desired
result. �
An immediate onsequen e of Lemma 5.1 is
Lemma 5.2. For any m > 0 there exists a onstant � > 0 su h that:
(i) If for some x 2 �
"
a
we have jd(w
"
(x))j � m, then jd(w
"
(y))j �
m
2
for every y 2 �
"
a
satisfying jy � xj � �p
"
(x):
(ii) If for some x 2 ��
"
a
we have jw
"
(x) � ~g(x)j � m, then jw
"
(y) � ~g(y)j �
m
2
for every
y 2 ��
"
a
satisfying jy � xj � �p
"
(x):
Proof. Consider ase (i), i.e. of x 2 �
"
a
with jd(w
"
(x))j � m. For 0 < � < 1=2, if jy �
xj < �p
"
(x) then (for " small) there exists a Lips hitz urve : [0; 1℄ ! �
"
a
su h that
(0) = x; (1) = y and j
0
(t)j < 2jy � xj; 8t. By (5.10) and (5.11) we obtain,
jw
"
(y)� w
"
(x)j � 2
�
Z
1
0
jrw
"
( (t))j dt
�
jy � xj � 2~
�
Z
1
0
dt
p
"
( (t))
�
jy � xj
�
4~
p
"
(x)
jy � xj � 4~ �:
It suÆ es thus to take � < min(1=2;m=(4~ )). Case (ii) is treated similarly. �
28
Let
(5.14) e� := max
y2�
j�(y)j;
and �x m satisfying,
(5.15) 0 < m < min
�
l
32
;
1
4
dist(image(g);S);
1
100e�
�
:
In the spirit of [7, 16℄ we de�ne a set of \bad points" by,
(5.16) S
"
= fx 2 �
"
a
; jd(w
"
(x))j � mg [ fx 2 ��
"
a
; jw
"
(x)� ~g(x)j � mg :
In other words, x 2 �
"
a
is a \bad point" if either x 2 �
"
a
and w
"
(x) is \far away" from �,
or x 2 ��
"
a
and w
"
(x) is \far away" from ~g(x). Our next laim is that the set S
"
is \small",
in the sense that it an be overed by a �nite number of \bad" dis s and \half dis s". For a
point ~�
0
2 ��
"
a
and r > 0 we denote B
+
(~�
0
; r) := B(~�
0
; r) \ �
"
a
and all it a \half dis ".
Proposition 5.2. There exist a onstant k > 0 and an integer N su h that, for every " 2
(0; 1), there is a olle tion fB(z
"
j
; �
"
p
"
(z
"
j
)) \ �
"
a
g
N(")
j=1
of mutually disjoint dis s and \half
dis s" whi h satis�es,
(5.17) S
"
�
N(")
[
j=1
B(z
"
j
; �
"
p
"
(z
"
j
)) \ �
"
a
;
with �
"
� k and N(") � N .
The proof of Proposition 5.2, whi h uses a variant of the method of [7, 8, 16℄, requires some
preliminary results. We �rst introdu e some notations. The two interse tion points of the
ir le �B(~�
0
; r) with ��
"
a
(for r small enough there are indeed exa tly two su h points) are
denoted by ~�
�r
0
and ~�
+r
0
. More pre isely, if we run over �B
+
(~�
0
; r) in the positive sense, then
the �rst end point of the \interval" �B
+
(~�
0
; r) \ ��
"
a
is de�ned to be ~�
�r
0
. Analogously to
[16℄ we de�ne,
f
"
(x
0
; r) =
r
2
Z
�B(x
0
;r)
jrw
"
j
2
+
F (d(w
"
))
p
2
"
for x
0
2 �
"
a
su h that B(x
0
; r) � �
"
a
;
f
"
(~�
0
; r) =
r
2
Z
�B(~�
0
;r)\�
"
a
jrw
"
j
2
+
F (d(w
"
))
p
2
"
+
r
2
~
1
p
"
(~�
�r
0
)
jw
"
(~�
�r
0
)� ~g(~�
�r
0
)j
2
+
r
2
~
1
p
"
(~�
+r
0
)
jw
"
(~�
+r
0
)� ~g(~�
+r
0
)j
2
for ~�
0
2 ��
"
a
:
(5.18)
The next two lemmas are ru ial for the proof of Proposition 5.2.
Lemma 5.3. Let b
1
2 (0;
4a
3
�
7
6
). There exists � > 0 su h that: for all b 2 (0; b
1
℄, for every
x
0
2 �
"
a
su h that
~
Æ(x
0
) � p
"
(x
0
)"
�b
and for every r � p
"
(x
0
)"
�b
there holds (for " � "
0
),
Z
B(x
0
;r)
F (d(w
"
))
p
2
"
� f
"
(x
0
; r) + "
�
:
29
Proof. Applying Pohozaev's identity, i.e. multiplying both sides of the �rst equation in (5.9)
by (x� x
0
) � rw
"
, integrating over B(x
0
; r) and applying Green formula we get,
Z
B(x
0
;r)
F (d(w
"
))
p
2
"
=
r
2
Z
�B(x
0
;r)
�
�
�
�w
"
��
�
�
2
�
�
�
�w
"
�n
�
�
2
+
F (d(w
"
))
p
2
"
�
�
1
2
Z
B(x
0
;r)
�
(x� x
0
) � r
�
1
p
2
"
�
�
F (d(w
"
)) ;
(5.19)
where
�
��
and
�
�n
denote respe tively tangential and normal (pointed outward) derivatives on
�B(x
0
; r). Sin e jF (d(w
"
))j � max
[�R;R℄
jF j we infer from (5.19) that,
Z
B(x
0
;r)
F (d(w
"
))
p
2
"
� f
"
(x
0
; r) + C
Z
B(x
0
;r)
�
jx� x
0
j
jrp
"
j
p
3
"
�
� f
"
(x
0
; r) + Cr
3
max
B(x
0
;r)
jrp
"
j
p
3
"
:
(5.20)
Sin e " � p
"
� "
a
and jrp
"
j � "
a�1=2
we obtain, using our assumptions on r and b, that
(5.21) r
3
max
�
"
a
jrp
"
j
p
3
"
� r
3
"
a�7=2
� "
3(a�b)+a�7=2
= "
4a�3b�7=2
� "
4a�3b
1
�7=2
:
The result follows from (5.20){(5.21) for any � 2 (0; 4a � 3b
1
� 7=2) (for " � "
0
). �
The next lemma is the analog of Lemma 5.3 for boundary points.
Lemma 5.4. Let b
1
be as in Lemma 5.3. Then, there exists � > 0 su h that: for all b 2 (0; b
1
℄,
for every ~�
0
2 ��
"
a
and every r � p
"
(~�
0
)"
�b
= "
a�b
there holds (for " < "
0
),
Z
B
+
(~�
0
;r)
F (d(w
"
))
p
2
"
+
~
1
2
Z
�B
+
(~�
0
;r)\��
"
a
jw
"
(~�)� ~g(~�)j
2
p
"
d~� � f
"
(~�
0
; r) + "
�
:
Proof. The proof uses again Pohozaev's identity, but the omputations are more involved than
in Lemma 5.3. After multiplying the �rst equation in (5.9) by (x � ~�
0
)rw
"
and integrating
over B
+
(~�
0
; r) we get for the r.h.s.,
Z
B
+
(~�
0
;r)
1
2p
2
"
r
x
�
F (d(w
"
))
�
(x� ~�
0
)
= �
Z
B
+
(~�
0
;r)
�
F (d(w
"
))
p
2
"
+ (x� ~�
0
) � r
�
1
2p
2
"
�
F (d(w
"
))
�
+
Z
�B
+
(~�
0
;r)
F (d(w
"
))
2p
2
"
(x� ~�
0
) � n
= �
Z
B
+
(~�
0
;r)
F (d(w
"
))
p
2
"
�
Z
B
+
(~�
0
;r)
(x� ~�
0
) � r
�
1
2p
2
"
�
F (d(w
"
))
+ r
Z
C
r
F (d(w
"
))
2p
2
"
+
Z
L
r
F (d(w
"
))
2p
2
"
(x� ~�
0
) � n := I
1
+ I
2
+ I
3
+ I
4
;
30
where C
r
:= �B
+
(~�
0
; r) \ �
"
a
, L
r
:= ��
"
a
\B(~�
0
; r) and n denotes the unit normal, pointed
outward, on �B
+
(~�
0
; r). As in the previous lemma, I
2
= O("
�
1
) for some �
1
> 0. Sin e �
"
a
is a C
2
domain, we have for x 2 L
r
,
j(x� ~�
0
) � nj � Cjx� ~�
0
j
2
� Cr
2
=) jI
4
j � C
r
3
"
2a
� C"
a�3b
:
Therefore for �
2
= min(�
1
; a� 3b
1
) we have,
Z
B
+
(~�
0
;r)
1
2p
2
"
r
x
�
F (d(w
"
))
�
(x� ~�
0
)
= �
Z
B
+
(~�
0
;r)
F (d(w
"
))
p
2
"
+ r
Z
C
r
F (d(w
"
))
2p
2
"
+O("
�
2
):
(5.22)
Next we study the analogous term obtained for the l.h.s. of (5.9). We get,
Z
B
+
(~�
0
;r)
�w
"
(x� ~�
0
)rw
"
=
Z
�B
+
(~�
0
;r)
�
1
2
((x� ~�
0
) � n)jrw
"
j
2
+
�w
"
�n
((x� ~�
0
)rw
"
)
=
r
2
Z
C
r
�
�
�w
"
�n
�
�
2
�
�
�
�w
"
��
�
�
2
�
Z
L
r
1
2
((x� ~�
0
) � n)jrw
"
j
2
+
Z
L
r
�w
"
�n
((x� ~�
0
)rw
"
)
=
r
2
Z
C
r
�
�
�w
"
�n
�
�
2
�
�
�
�w
"
��
�
�
2
�
Z
L
r
1
2
((x� ~�
0
) � n)jrw
"
j
2
+
Z
L
r
((x� ~�
0
) � n)
�
�
�w
"
�n
�
�
2
+
Z
L
r
((x� ~�
0
) � � )
�
�w
"
�n
�
�w
"
��
�
:= I
5
+ I
6
+ I
7
+ I
8
;
(5.23)
where � is the unit tangent ve tor along L
r
, in the positive dire tion. As above, j(x�~�
0
) �nj �
Cr
2
on L
r
and by (5.10), jrw
"
j � ~ "
�a
on L
r
. Therefore,
(5.24) jI
6
j+ jI
7
j � Cr
3
"
�2a
� C"
a�3b
;
as in the estimate for I
4
.
Next, using the boundary ondition in (5.9) we an write,
(5.25) I
8
= ~
1
Z
L
r
((x� ~�
0
) � � )
�
~g � w
"
"
a
�
�w
"
��
�
:
Let (�); � 2 [�
�
; �
+
℄; be an ar length paramaterization of L
r
, that we an hoose to satisfy
(0) = ~�
0
, so that (�
�
) = ~�
�r
0
and (�
+
) = ~�
+r
0
. Using this parameterization we rewrite
(5.25) as follows,
(5.26) I
8
=
~
1
"
a
Z
�
+
�
�
(( (�)� (0)) �
0
(�))
�
~g( (�)) � w
"
( (�))
�
�
d
d�
w
"
( (�)) d� :
31
Sin e
j( (�) � (0)) �
0
(�)� � j = j
�
(�)� (0) � �
0
(�)
�
�
0
(�)j
� j (�)� (0)� �
0
(�)j �
�
�
Z
�
0
j
0
(t)�
0
(�)j dt
�
�
� C�
2
;
we get from (5.26), using again (5.10), that,
(5.27)
�
�
I
8
�
~
1
"
a
Z
�
+
�
�
�
�
~g( (�))� w
"
( (�))
�
�
d
d�
w
"
( (�)) d�
�
�
� C"
�2a
r
3
� C"
a�3b
:
Next we noti e that,
~
1
"
a
Z
�
+
�
�
�
�
~g( (�)) � w
"
( (�))
�
�
d
d�
w
"
( (�)) d� =
~
1
"
a
Z
�
+
�
�
�
�
~g( (�)) � w
"
( (�))
��
~g( (�))
�
�
d�
�
~
1
"
a
Z
�
+
�
�
�
2
�
j~g( (�)) � w
"
( (�))j
2
�
�
d�
:= I
9
+ I
10
:
(5.28)
Sin e j
d~g( (�))
d�
j � C, we obtain
(5.29) jI
9
j � C"
�a
r
2
� C"
a�2b
:
Next, integration by parts yields,
I
10
=
~
1
�
�
2"
a
j~g(~�
�r
0
)�w
"
(~�
�r
0
)j
2
�
~
1
�
+
2"
a
j~g(~�
+r
0
)� w
"
(~�
+r
0
)j
2
+
~
1
2
Z
L
r
jw
"
(~�)� ~g(~�)j
2
p
"
d~�
= �
~
1
r
2"
a
�
j~g(~�
�r
0
)� w
"
(~�
�r
0
)j
2
+ j~g(~�
+r
0
)� w
"
(~�
+r
0
)j
2
�
+
~
1
2
Z
L
r
jw
"
(~�)� ~g(~�)j
2
p
"
d~� +O("
a�2b
) ;
(5.30)
where in the last step we used the estimates j�
+
� rj; j�
�
+ rj � Cr
2
. Finally, ombining
together (5.22),(5.23),(5.24),(5.27),(5.28), (5.29) and (5.30) we obtain,
Z
B
+
(~�
0
;r)
F (d(w
"
))
p
2
"
+
~
1
2
Z
L
r
jw
"
(~�)� ~g(~�)j
2
p
"
d~� =
r
2
Z
C
r
�
�
�w
"
��
�
�
2
�
�
�
�w
"
�n
�
�
2
+
F (d(w
"
))
p
2
"
+
~
1
r
2"
a
�
j~g(~�
�r
0
)� w
"
(~�
�r
0
)j
2
+ j~g(~�
+r
0
)� w
"
(~�
+r
0
)j
2
�
+O("
�
2
) ;
and the result follows for any 0 < � < �
2
. �
From now on we �x,
(5.31) b
1
=
2a
3
�
7
12
;
so that we an hoose � > 0 for whi h the on lusions of both Lemma 5.3 and Lemma 5.4 are
satis�ed.
32
Lemma 5.5. There exists a onstant C
0
> 0 su h that, for every x
0
2 �
"
a
we have,
(5.32)
Z
B(x
0
;p
"
(x
0
)"
�7b
1
=8
)\�
"
a
F (d(w
"
))
p
2
"
+
~
1
2
Z
B(x
0
;p
"
(x
0
)"
�7b
1
=8
)\��
"
a
jw
"
(~�)� ~g(~�)j
2
p
"
d~� � C
0
:
Proof. Assume �rst that
~
Æ(x
0
) � p
"
(x
0
)"
�8b
1
=9
. Using the upper bound (5.7) for
e
E
"
(w
"
) we
hoose r 2 (p
"
(x
0
)"
�7b
1
=8
; p
"
(x
0
)"
�8b
1
=9
) su h that,
r
2
Z
�B(x
0
;r)
jrw
"
j
2
+
F (d(w
"
))
p
2
"
� C
1
;
for some onstant C
1
> 0. Thus f
"
(x
0
; r) � C
1
and (5.32) is a dire t onsequen e of
Lemma 5.3.
Assume next that
~
Æ(x
0
) < p
"
(x
0
)"
�8b
1
=9
. Set ~�
0
= ~�(x
0
), i.e. ~�
0
is the nearest point
proje tion of x
0
on ��
"
a
. We have,
B(x
0
; p
"
(x
0
)"
�7b
1
=8
)\�
"
a
� B(x
0
; p
"
(x
0
)"
�8b
1
=9
)\�
"
a
� B
+
(~�
0
; 2p
"
(x
0
)"
�8b
1
=9
) � B(~�
0
; p
"
(~�
0
)"
�9b
1
=10
):
Using (5.7) we hoose r 2 (p
"
(~�
0
)"
�9b
1
=10
; p
"
(~�
0
)"
�10b
1
=11
) su h that f
"
(~�
0
; r) � C
2
, and the
result follows, this time from Lemma 5.4. �
Next we prove,
Lemma 5.6. For every 0 < � < � < 1 there exists a onstant C(�; �) > 0 su h that,
(5.33)
e
E
"
(w
"
; B(x
0
; p
"
(x
0
)"
��b
1
) \ �
"
a
) � C(�; �)j log "j;
if either,
(i)
~
Æ(x
0
) � p
"
(x
0
)"
��b
1
and B(x
0
; p
"
(x
0
)"
��b
1
) \ S
"
6= ;,
or,
(ii) x
0
2 ��
"
a
and B
+
(x
0
; p
"
(x
0
)"
��b
1
) \ S
"
6= ;.
Proof. By Lemma 5.2 there exists a onstant l
0
> 0 su h that,
(5.34)
y
0
2 S
"
=)
Z
B(y
0
;p
"
(y
0
))\�
"
a
F (d(w
"
))
p
2
"
+
~
1
2
Z
B(y
0
;p
"
(y
0
))\��
"
a
jw
"
(~�)� ~g(~�)j
2
p
"
d~� � l
0
:
Consider now ase (i). Let y
0
2 B(x
0
; p
"
(x
0
)"
��b
1
) satisfy jd(w
"
(y
0
))j � m. We laim that
(5.35) B(y
0
; p
"
(y
0
)) � B(x
0
; p
"
(x
0
)"
�(�+�)b
1
=2
) :
Indeed, if jz� y
0
j � p
"
(y
0
) then jz�x
0
j � jz� y
0
j+ jy
0
�x
0
j � p
"
(y
0
)+ p
"
(x
0
)"
��b
1
. We need
to show that
(5.36) p
"
(y
0
) � p
"
(x
0
)
�
"
�(�+�)b
1
=2
� "
��b
1
�
:
33
In fa t, by (5.3),
log p
"
(y
0
)� log p
"
(x
0
) �
�
�
�
Z
~
Æ(y
0
)
~
Æ(x
0
)
�p
"
=�
~
Æ
p
"
d
~
Æ
�
�
�
� j
~
Æ(y
0
)�
~
Æ(x
0
)j
"
a�1=2
� "
1=2
"
� jy
0
� x
0
j"
a�3=2
� p
"
(x
0
)"
��b
1
+a�3=2
� "
2a�3=2�b
1
= "
2a�3=2�2a=3+7=12
= "
4a=3�11=12
! 0:
This learly implies (5.36) and our laim (5.35) follows. From (5.34) and (5.35) we dedu e
that,
Z
B(x
0
;p
"
(x
0
)"
�(�+�)b
1
=2
)
F (d(w
"
))
p
2
"
�
Z
B(y
0
;p
"
(y
0
))
F (d(w
"
))
p
2
"
� l
0
:
It follows from Lemma 5.3 that for some C
3
> 0 we have,
(5.37) f
"
(x
0
; r) � C
3
; 8r 2 (p
"
(x
0
)"
�(�+�)b
1
=2
; p
"
(x
0
)"
��b
1
):
Integration of (5.37) yields,
e
E
"
(w
"
; B(x
0
; p
"
(x
0
)"
��b
1
)) � 2
Z
p
"
(x
0
)"
��b
1
p
"
(x
0
)"
�(�+�)b
1
=2
f
"
(x
0
; r)
r
dr � C
3
(� � �)b
1
j log "j;
and (5.33) follows in this ase. In ase (ii) the argument is almost identi al, with the only
di�eren e that we use Lemma 5.4 instead of Lemma 5.3. �
We are now ready to present the proof of Proposition 5.2.
Proof of Proposition 5.2. We apply a similar argument to the one used in [2, Lemma 3.5℄,
whi h is a variant of the method of Struwe [16℄. We denote,
S
1
"
= S
"
\ f
~
Æ(x) � "
a�4b
1
=5
g and S
2
"
= S
"
\ f
~
Æ(x) < "
a�4b
1
=5
g :
First we laim that there exists a olle tionA of mutually disjoint dis s fB(x
"
j
; p
"
(x
"
j
)"
�3b
1
=4
)g
N(")
j=1
with x
"
j
2 S
1
"
for all j su h that,
(5.38)
[
x2S
1
"
B(x; p
"
(x)"
�3b
1
=4
) �
N(")
[
j=1
B(x
"
j
; 5p
"
(x
"
j
)"
�3b
1
=4
) ;
with N(") � N
1
uniformly in ". We onstru t this olle tion as follows. First we hoose
x
"
1
2 S
1
"
and initialize A by setting A = fB(x
"
1
; p
"
(x
"
1
)"
�3b
1
=4
)g. If
S
x2S
1
"
B(x; p
"
(x)"
�3b
1
=4
) �
B(x
"
1
; 5p
"
(x
"
1
)"
�3b
1
=4
) we are done. Otherwise, there exists some y 2
S
x2S
1
"
B(x; p
"
(x)"
�3b
1
=4
) n
B(x
"
1
; 5p
"
(x
"
1
)"
�3b
1
=4
). Then we hoose x
"
2
2 S
1
"
satisfying jx
"
2
� yj < p
"
(x
"
2
)"
�3b
1
=4
and add
B(x
"
2
; p
"
(x
"
2
)"
�3b
1
=4
) to our olle tion A. We need to he k that,
(5.39) B(x
"
2
; p
"
(x
"
2
)"
�3b
1
=4
) \B(x
"
1
; p
"
(x
"
1
)"
�3b
1
=4
) = ;:
Assume that (5.39) is false. Then,
5p
"
(x
"
1
)"
�3b
1
=4
� p
"
(x
"
2
)"
�3b
1
=4
� jx
"
1
� yj � jx
"
2
� yj � jx
"
2
� x
"
1
j � (p
"
(x
"
1
) + p
"
(x
"
2
))"
�3b
1
=4
:
34
Thus,
(5.40) p
"
(x
"
2
) � 2p
"
(x
"
1
) :
Writing for short
~
Æ
1
=
~
Æ(x
"
1
);
~
Æ
2
=
~
Æ(x
"
2
), we get using (5.31) that
(5.41) j
~
Æ
2
�
~
Æ
1
j � jx
"
2
� x
"
1
j � 2"
a�3b
1
=4
= 2"
a=2+7=16
:
On the other hand, by (5.3) and (5.41) we have, as in the proof of (5.36),
log p
"
(x
"
2
)� log p
"
(x
"
1
) =
Z
~
Æ
2
~
Æ
1
�
�p
"
=�
~
Æ
p
"
�
d
~
Æ � j
~
Æ
1
�
~
Æ
2
j
"
a�1=2
"
= j
~
Æ
1
�
~
Æ
2
j"
a�3=2
� 2"
3a=2�3=2+7=16
! 0; as "! 0;
ontradi ting (5.40) (for " small enough) and thus establishing (5.39). Continuing re ursively,
it is lear that after a �nite number of steps we will end up with a olle tion A of mutually
disjoint dis s satisfying (5.38). By Lemma 5.6,
e
E
"
(w
"
; B(x
"
i
; p
"
(x
"
i
)"
�3b
1
=4
)) � C(1=2; 3=4)j log "j; 8i;
whi h together with the upper bound (5.7) implies that N(") � N
1
for some N
1
independent
of ".
By a similar onstru tion to the above, we onstru t a olle tion of mutually disjoint dis s,
B = fB(y
"
j
; p
"
(y
"
j
))g
M(")
j=1
, with y
"
j
2 S
1
"
, su h that
(5.42) S
1
"
�
M(")
[
j=1
B(y
"
j
; 5p
"
(y
"
j
)) :
It follows from (5.38) that,
M(")
[
j=1
B(y
"
j
; p
"
(y
"
j
)) �
N(")
[
j=1
B(x
"
j
; 5p
"
(x
"
j
)"
�3b
1
=4
) :
Sin e N(") � N
1
, and by Lemma 5.5,
Z
B(x
"
j
;5p
"
(x
"
j
)"
�3b
1
=4
)
F (d(w
"
))
p
2
"
� C
0
; 8j 2 f1; : : : ; N(")g;
while by (5.34),
Z
B(y
"
j
;p
"
(y
"
j
))
F (d(w
"
))
p
2
"
� l
0
; 8j 2 f1; : : : ;M(")g;
we infer that M(") � N
2
for some integer N
2
, independent of ". After a �nite number m
1
of
iterations, m
1
� N
2
, ea h onsisting of multiplying all radii by 5, and deleting some dis s, we
end up with a olle tion fB(~y
"
j
; 5
m
1
p
"
(~y
"
j
))g
M
1
(")
j=1
of mutually disjoint dis s whi h over S
1
"
.
35
Next we apply a very similar argument in order to show that S
2
"
too an be overed by a �nite
number (bounded uniformly in ") of dis s, or \half dis s", with radii of the order p
"
of their en-
ters. The �rst step is to �nd a olle tion of mutually disjoint dis s fB(x
"
j
; p
"
(x
"
j
)"
�5b
1
=6
)g
N(")
j=1
with x
"
j
2 S
2
"
for all j, su h that,
(5.43)
[
x2S
2
"
B(x; p
"
(x)"
�5b
1
=6
) \ �
"
a
�
N(")
[
j=1
B(x
"
j
; 5p
"
(x
"
j
)"
�5b
1
=6
) \ �
"
a
:
As in the ase of S
1
"
we an show that N(") � N
1
, and then �nd a olle tion of mutually
disjoint fB(y
"
j
; p
"
(y
"
j
)) \ �
"
a
g
M(")
j=1
, with y
"
j
2 S
2
"
, su h that
S
2
"
�
M(")
[
j=1
B(y
"
j
; 5p
"
(y
"
j
)) \ �
"
a
:
As above we an show that M(") � N
2
for some N
2
. Finally we apply a �nite number
of iterations whi h give at the end a olle tion of mutually disjoint dis s and \half dis s",
fB(z
"
j
; 5
m
2
p
"
(z
"
j
)) \ �
"
a
g
M
2
(")
j=1
, su h that for ea h j either z
"
j
2 �
"
a
with
~
Æ(z
"
j
) � 5
m
2
p
"
(z
"
j
) or
z
"
j
2 ��
"
a
. Ea h iteration onsists as above of multiplying all radii by 5 and deleting some
dis s, but we add also a step in whi h we repla e a dis of enter y
"
j
whi h interse ts ��
"
a
by
a \half dis " entered at ~�(y
"
j
) with radius multiplied by 5 (and then multiply also all other
radii by 5).
To on lude the proof we regroup together the olle tion of dis s and \half dis s" whi h
over S
1
"
and S
2
"
to a single olle tion. We may still need to apply another round of iterations
of multiplying radii by 5 and deleting some dis s in order to get �nally a olle tion of mutually
disjoint dis s and half dis s as required. �
Consider the olle tion of dis s and \half dis s" given by Proposition 5.2. Passing to a
subsequen e "
n
! 0, we may assume that N("
n
) = N and �
"
n
= � are independent of "
n
and that z
"
n
j
! z
j
2
�
G; 8j. We denote by a
1
; : : : ; a
N
1
the distin t limits of the sequen es
fz
"
n
j
g; j = 1; : : : ; N , and de�ne,
(5.44) �
k
= fj 2 f1; : : : ; Ng : z
"
n
j
! a
k
g; k = 1; : : : ; N
1
:
We deal �rst with the ase a
k
2 G. For every j 2 �
k
we have p
"
n
(z
"
n
j
) = "
n
for n large.
From (5.10) it follows that d
"
n
j
:= deg(w
"
n
; �B(z
"
n
j
; �"
n
)) is bounded uniformly in n, so passing
to a subsequen e we may assume that d
"
n
j
= d
j
; 8n. In parti ular, it follows that for every r
satisfying,
(5.45) 0 < r <
1
2
min
�
Æ(a
k
);min(ja
j
� a
k
j : j 6= k)
�
;
the degree K
k
:= deg(w
"
n
; �B(a
k
; r)) is independent of n. Next we prove a lower bound for
e
E
"
near a
k
.
36
Lemma 5.7. If a
k
2 G then for any r satisfying (5.45) we have,
(5.46)
e
E
"
n
(w
"
n
; B(a
k
; r)) � jK
k
j
l
2
2�
log
r
"
n
� C:
Proof. The proof is almost identi al to the one in [7, 16℄. The only di�eren e here, is that
we have to repla e [9, Th. 4℄ and [16, Prop. 3.4'℄ by the following lemma, whose proof is the
same of that of [9, Th. 4℄ and is therefore omitted.
Lemma 5.8. Let A = B(0; R)n
M
S
j=1
B(p
j
; R
0
) with R
0
�
1
4
R, p
j
2 B(0;
1
4
R); 8j; and jp
j
�p
k
j �
4R
0
; 8j 6= k. Suppose that u 2 C
1
(A;R
2
) satis�es
jd(u)j � m on A;
and
1
R
2
0
Z
A
F (d(u)) � K;
for some onstant K and let d
j
= deg(u; �B(p
j
; R
0
)); 8j. Consider the \referen e map"
u
0
(z) = �
�
M
Y
j=1
�
z � p
j
jz � p
j
j
�
d
j
�
;
where � is de�ned in (1.10). Then, for some onstant C > 0, depending only on K and m,
we have,
(5.47)
Z
A
jruj
2
�
Z
A
jru
0
j
2
� CM
2
�
M
X
j=1
jd
j
j
�
2
:
�
Next we turn to the ase where a
k
2 �G. We need a notion of \degree" for ertain maps
whi h are de�ned on a \half ir le" C
+
(~�
0
; r) := �B(~�
0
; r) \ �
"
a
n
with ~�
0
2 ��
"
a
n
and small
enough r > 0. We de�ne a positive sense on C
+
(~�
0
; r) as the one whi h is indu ed by taking
the positive sense on �B
+
(~�
0
; r). We denote, as before, by ~�
�r
0
and ~�
+r
0
the end points of
C
+
(~�
0
; r). Thus C
+
(~�
0
; r) starts at the point ~�
+r
0
and ends at ~�
�r
0
. There exist r
0
; "
0
> 0,
depending only g;G and , su h that,
(5.48) j~g(�
�r
0
)� ~g(�
+r
0
)j � m and jC
+
(~�
0
; r)j � (
100
99
)�r; 8r � r
0
; 8" � "
0
;
Consider a ontinuous map w : C
+
(~�
0
; r)! R
2
satisfying,
(5.49) jw(~�
+r
0
)� ~g(�
+r
0
)j; jw(~�
�r
0
)� ~g(�
�r
0
)j � m and jd(w(t))j � m; 8t 2 C
+
(~�
0
; r):
Clearly there exists a ontinuous extension ~w : �B
+
(~�
0
; r)! R
2
su h that ~w = w on C
+
(~�
0
; r)
and j ~w� ~gj � 2m on �B
+
(~�
0
; r) nC
+
(~�
0
; r). Then we de�ne the degree of w on C
+
(~�
0
; r) to
be d
0
= deg( ~w; �B
+
(~�
0
; r)). Using (5.15) it is easy to verify that d
0
does not depend on the
37
extension ~w and so we get a well de�ned notion. In the ase of a C
1
-map w we an get an
expli it formula as follows. We denote by �(s(w(x))) the normal to � at the point s(w(x)).
The quantity
Z
C
+
(~�
0
;r)
�(s(w)) ^
�s(w)
�t
dt
represents the displa ement of s(w(x)) along � as x varies over C
+
(~�
0
; r). For r � r
0
and
"
n
� "
0
it follows by (5.48){(5.49) and (5.15) that d
0
is hara terized as the unique integer
satisfying,
(5.50)
�
�
�
Z
C
+
(~�
0
;r)
�(s(w)) ^
�s(w)
�t
dt� d
0
l
�
�
�
� 3m < l=10:
In parti ular, for z
"
n
j
2 ��
"
a
n
we denote by d
"
n
j
the degree of w
"
n
on C
+
(z
"
n
j
; �"
a
n
). By (5.3)
and (5.10) we obtain that p
"
n
� "
a
n
and jrw
"
n
j � C"
�a
n
on C
+
(z
"
n
j
; �"
a
n
). It follows that fd
"
n
j
g
remains bounded as "
n
! 0, and we may assume, by passing to a further subsequen e, that
d
"
n
j
= d
j
; 8n. Next onsider a point ~�
0
2 ��
"
a
n
(for "
n
small) and r � r
0
su h that C
+
(~�
0
; r)
does not interse t any of the bad dis s/half dis s fB(z
"
n
j
; �p
"
n
(z
"
n
j
))\�
"
a
n
g
N
j=1
. Sin e (5.49) is
satis�ed for w = w
"
n
, the degree d
0
of w
"
n
on C
+
(~�
0
; r) is well de�ned. Moreover, this degree
does not depend on n. This follows from the formula,
d
0
=
X
�
d
j
: B(z
"
n
j
; �p
"
n
(z
"
n
j
)) \ �
"
a
n
� B
+
(~�
0
; r)
;
whi h is a dire t onsequen e of the additive property of the \usual degree" and the degree
that we de�ned above. We shall use the following simple lower bound for the energy on an
\half annulus".
Lemma 5.9. Let ~�
0
2 ��
"
a
n
, and 0 < r
1
< r
2
< r
0
be given su h that the losure of
A
+
:= B
+
(~�
0
; r
2
) n B
+
(~�
0
; r
1
) does not interse t any of the bad dis s/half dis s. Then,
(5.51)
Z
A
+
jrw
"
n
j
2
� 1:5
�
l
2
2�
�
d
2
0
log
r
2
r
1
;
where d
0
is the degree of w
"
n
on C
+
(~�
0
; r
1
) ( learly the degree on C
+
(~�
0
; r) is the same for
any r 2 [r
1
; r
2
℄).
Proof. If d
0
= 0 the result is obvious. We thus assume in the sequel that d
0
6= 0. As
in the proof of Lemma 4.1 we an write on A
+
, w
"
n
= v
"
n
+ d
"
n
�
"
n
where v
"
n
= s(w
"
n
),
�
"
n
(x) = �(w
"
n
(x)) and d
"
n
= d(w
"
n
). Denoting by �
"
n
= �(v
"
n
(x)) the urvature of � at
38
v
"
n
(x) we have for every r 2 (r
1
; r
2
), using (4.3), (5.15),(5.48) and (5.50),
Z
C
+
(~�
0
;r)
jrw
"
n
j
2
�
Z
C
+
(~�
0
;r)
(1� �
"
n
d
"
n
)
2
�
�
�v
"
n
�t
�
�
2
dt �
(1� e�m)
2
jC
+
(~�
0
; r)j
�
Z
C
+
(~�
0
;r)
�
�
�v
"
n
�t
�
�
dt
�
2
�
�
99
3
�r100
3
��
Z
C
+
(~�
0
;r)
�
"
n
^
�v
"
n
�t
dt
�
2
�
�
99
3
�r100
3
�
(jd
0
j �
1
100
)
2
l
2
� 1:5 d
2
0
�
l
2
2�r
�
:
(5.52)
Integrating (5.52) over (r
1
; r
2
) yields (5.51). �
Next we prove an analog of Lemma 5.7 for the ase a
k
2 �G.
Lemma 5.10. If a
k
2 �G then for any
0 < r < min
�
1; r
0
;
1
2
min(ja
j
� a
k
j : j 6= k)
�
;
we have,
(5.53)
e
E
"
n
(w
"
n
; B(a
k
; r) \G) � jK
k
j
l
2
2�
j log "
n
j+O(1);
where K
k
is the degree of w
"
n
on �B(a
k
; r) \ �
"
a
n
(or more pre isely, on C
+
(~a
k
; r) where ~a
k
is the point on ��
"
a
n
su h that �(~a
k
) = a
k
).
Proof. The argument is similar to the one used in [2℄. Consider the olle tion of dis s/half dis s
fB(z
"
n
j
; "
a
n
)\�
"
a
n
g
j2�
k
. Sin e we enlarged the radii of the original dis s/half dis s, interse tions
may o ur. After a �nite number,
~
l
"
n
1
, bounded uniformly in n, of iterations, ea h onsisting
of multiplying all radii by 9, deleting some dis s and repla ing, if ne essary, z
"
n
j
by ~�(z
"
n
j
),
we get a new olle tion of dis s/half dis s fB(~z
"
n
j
; �
1
) \ �
"
a
n
g
~m
1
j=1
with �
1
= �
"
n
1
= 9
~
l
"
n
1
"
a
n
su h
that,
(i) Ea h B(z
"
n
i
; �p
"
n
(z
"
n
i
)) is ontained in some B(~z
"
n
j
; �
1
),
(ii) Ea h ~z
"
n
j
equals z
"
n
i
or ~�(z
"
n
i
) for some i,
(iii) j~z
"
n
j
1
� ~z
"
n
j
2
j � 4�
1
; 8j
1
6= j
2
,
(iv) For ea h j, either ~z
"
n
j
2 ��
"
a
n
or
~
Æ(~z
"
n
j
) � 2�
1
.
Passing to a subsequen e we may assume that
~
l
"
n
1
is independent of n. We may also assume
that
~
d
j
, the degree of w
"
n
on either �B(~z
"
n
j
; �
1
), for a dis , or C
+
(~z
"
n
j
; �
1
), for a half-dis , is
independent of n. Clearly
~
d
j
equals the sum of all the d
i
's su h that B(z
"
n
i
; �p
"
(z
"
n
i
))\�
"
a
n
�
B(~z
"
n
j
; �
1
) \ �
"
a
n
.
Consider next the olle tion of dis s/half dis s fB(~z
"
n
j
; "
1=2
n
) \ �
"
a
n
g
j2�
k
. As above, we
apply a number l
1
(whi h an be assumed independent of n) of iterations, ea h onsisting of
multiplying all radii by 9, deleting some dis s and repla ing, if ne essary, ~z
"
n
j
by ~�(~z
"
n
j
), until
39
we end up with a new olle tion of dis s/half dis s fB(z
"
n
j;1
; r
1
)\�
"
a
n
g
m
1
j=1
with r
1
= r
"
n
1
= 9
l
1
"
1=2
n
su h that,
(i) Ea h B(~z
"
n
i
; �
"
n
1
) is ontained in some B(z
"
n
j;1
; r
1
) \ �
"
a
n
,
(ii) Ea h z
"
n
j;1
equals ~z
"
n
i
or ~�(~z
"
n
i
) for some i,
(iii) jz
"
n
j
1
;1
� z
"
n
j
2
;1
j � 4r
1
; 8j
1
6= j
2
,
(iv) For ea h j, either z
"
n
j;1
2 ��
"
a
n
or
~
Æ(z
"
n
j;1
) � 2r
1
.
We denote by d
j;1
the degree of w
"
n
on either �B(z
"
n
j;1
; r
1
), for a dis , or C
+
(z
"
n
j;1
; r
1
), for a
half-dis . As above, we may assume this degree is independent of n. In the sequel we shall use
freely without further details this kind of argument leading to independen e of n of numbers
of iteration and degrees.
On ea h dis B(z
"
n
j;1
; r
1
) we have by (5.3) that p
"
n
= "
n
. Hen e, by the argument of
Lemma 5.7 it follows that,
(5.54)
Z
B(z
"
n
j;1
;r
1
)
jrw
"
n
j
2
� jd
j;1
j
l
2
2�
log
r
1
"
n
� C; 8j:
Next onsider a half-dis B
+
(z
"
n
j;1
; r
1
). Put,
I
j
= fi : B(~z
"
n
i
; �
1
) \ �
"
a
n
� B
+
(z
"
n
j;1
; r
1
)g:
Clearly d
j;1
=
P
i2I
j
~
d
i
. We denote
(j)
= B
+
(z
"
n
j;1
; r
1
) n
[
i2I
j
B(~z
"
n
i
; �
1
);
and laim that
(5.55)
Z
(j)
jrw
"
n
j
2
� jd
j;1
j
l
2
2�
log
r
1
�
1
�C:
The proof of (5.55) follows from the argument of [16, Prop 3.3℄, using the estimates (5.51) for
a half-annulus, and (5.47) for an annulus; the details are left to the reader.
Set
R
1
= R
"
n
1
= 1=2min
�
f
~
Æ(z
"
n
j;1
) : z
"
n
j;1
2 �
"
a
n
g; fjz
"
n
j
1
;1
� z
"
n
j
2
;1
j : j
1
6= j
2
g
�
:
After a �nite number l
2
of iterations, as above, we obtain a new olle tion of dis s/half-dis s
fB(z
"
n
j;2
; r
2
) \ �
"
a
n
g
m
2
j=1
with r
2
= r
"
n
2
= 9
l
2
R
1
su h that
(i) Ea h B(z
"
n
i;1
; r
1
) is ontained in some B(z
"
n
j;2
; r
2
),
(ii) Ea h z
"
n
j;2
equals z
"
n
i;1
or ~�(z
"
n
i;1
) for some i,
(iii) jz
"
n
j
1
;2
� z
"
n
j
2
;2
j � 4r
2
; 8j
1
6= j
2
,
(iv) For ea h j, either z
"
n
j;2
2 ��
"
a
n
or
~
Æ(z
"
n
j;2
) � 2r
2
.
40
For a dis B(z
"
n
i;1
; R
1
) we have
Z
B(z
"
n
i;1
;R
1
)nB(z
"
n
i;1
;r
1
)
jrw
"
n
j
2
�
l
2
2�
d
2
i;1
log
R
1
r
1
�O(1):
For a half dis B
+
(z
"
n
i;1
; R
1
) we have, by Lemma 5.9,
Z
B
+
(z
"
n
i;1
;R
1
)nB
+
(z
"
n
i;1
;r
1
)
jrw
"
n
j
2
� 1:5
l
2
2�
d
2
i;1
log
R
1
r
1
� 1:5
l
2
2�
jd
i;1
j log
R
1
r
1
:
We denote by d
j;2
the degree of w
"
n
on the boundary of the dis /half dis B(z
"
n
j;2
; r
2
) \ �
"
a
n
.
Continuing this way, we get a sequen e,
r
1
= 9
l
1
"
1=2
n
< R
1
< r
2
= 9
l
2
R
1
< � � � < r
M
= 9
l
M
R
M�1
;
with the orresponding dis s/half-dis s fB(z
"
n
i;h
; r
h
) \ �
"
a
n
g, h = 1; : : : ;M; i = 1; : : : ;m
h
, and
m
M
= 1, so that the half-dis B
+
(z
"
n
1;M
; r
M
) ontains all the previous dis s/half dis s. We
denote by d
i;h
the degree of w
"
n
on �B(z
"
n
i;h
; r
h
) \ �
"
a
n
. We put R
M
= r=2 and we note that
d
1;M
= K
k
.
We next laim that for ea h h and every half-dis B
+
(z
"
n
i;h
; R
h
), we have,
(5.56)
Z
B
+
(z
"
n
i;h
;R
h
)
jrw
"
n
j
2
� jd
i;h
j
l
2
2�
(1:5 logR
h
+ j log "
n
j)� C:
We prove (5.56) by indu tion. Consider �rst the ase h = 1. For a half-dis B
+
(z
"
n
i;1
; R
1
) we
have by Lemma 5.9,
(5.57)
Z
B
+
(z
"
n
i;1
;R
1
)nB
+
(z
"
n
i;1
;r
1
)
jrw
"
n
j
2
� jd
i;1
j
l
2
2�
1:5 log(R
1
=r
1
):
On the other hand, (5.55) yields,
(5.58)
Z
B
+
(z
"
n
i;1
;r
1
)
jrw
"
n
j
2
� jd
i;1
j
l
2
2�
log("
1=2
n
="
a
n
)� C = jd
i;1
j
l
2
2�
(a�
1
2
)j log "
n
j � C:
Adding (5.58) and (5.57) leads to,
Z
B
+
(z
"
n
i;1
;R
1
)
jrw
"
n
j
2
� jd
i;1
j
l
2
2�
h
�
3
4
+ a�
1
2
�
j log "
n
j+ 1:5 logR
1
i
� C
� jd
i;1
j
l
2
2�
(1:5 logR
1
+ j log "
n
j)� C;
sin e a >
3
4
.
Assume now that (5.56) holds for h and let us prove it for a ertain half-dis B
+
(z
"
n
j;h+1
; R
h+1
).
We denote,
J
(in)
j
= fi : z
"
n
i;h
2 B
+
(z
"
n
j;h+1
; R
h+1
)g and J
(bd)
j
= fi : z
"
n
i;h
2 �B
+
(z
"
n
j;h+1
; R
h+1
) \ ��
"
a
n
g :
41
By the same argument that led to (5.54) we have,
(5.59)
Z
B(z
"
n
i;h
;R
h
)
jrw
"
n
j
2
� jd
i;h
j
l
2
2�
log
R
h
"
n
� C; 8i 2 J
(in)
j
:
The indu tion assumption gives,
(5.60)
Z
B
+
(z
"
n
i;h
;R
h
)
jrw
"
n
j
2
� jd
i;h
j
l
2
2�
(1:5 logR
h
+ j log "
n
j)� C; 8i 2 J
(bd)
j
:
Set
D
I
=
X
i2J
(in)
j
d
i;h
and D
B
=
X
i2J
(bd)
j
d
i;h
= d
j;h+1
�D
I
:
By Lemma 5.9 we obtain,
Z
B
+
(z
"
n
i;h+1
;R
h+1
)nB
+
(z
"
n
i;h+1
;r
h+1
)
jrw
"
n
j
2
� jD
B
+D
I
j
l
2
2�
1:5 log
R
h+1
r
h+1
� C
� jD
B
+D
I
j
l
2
2�
1:5 log
R
h+1
R
h
� C:
(5.61)
Sin e R
h
� "
1=2
n
, we have j log "
n
j + 1:5 logR
h
� 0. Therefore, ombining (5.59){(5.61) we
infer that,
Z
B
+
(z
"
n
i;h+1
;R
h+1
)
jrw
"
n
j
2
�
l
2
2�
�
1:5jD
B
+D
I
j log
R
h+1
R
h
+ jD
B
j(j log "
n
j+ 1:5 logR
h
) + jD
I
j log
R
h
"
n
�
� C
�
l
2
2�
�
1:5jD
B
+D
I
j log
R
h+1
R
h
+ (jD
B
j+ jD
I
j)(j log "
n
j+ 1:5 logR
h
)
�
� C
�
l
2
2�
�
jD
B
+D
I
j(1:5 logR
h+1
+ j log "
n
j)
�
� C
= jd
j;h+1
j
l
2
2�
(1:5 logR
h+1
+ j log "
n
j)� C:
We thus proved (5.56). In parti ular, at the end, for h =M we get,
Z
B
+
(z
"
n
1;M
;r)
jrw
"
n
j
2
� jK
k
j
l
2
2�
(1:5 log r + j log "
n
j)� C;
whi h is the desired result. �
Proof of Proposition 5.1. It is enough to prove that for any sequen e "
n
! 0 we have
(5.62)
e
E
"
n
(w
"
n
) � D
l
2
2�
j log "
n
j �C:
Passing to a subsequen e, we may assume that a system of bad dis s/half dis s, with the same
notations as above, is asso iated with fw
"
n
g. Sin e D =
P
N
1
j=1
K
j
, we get from Lemma 5.7
42
and Lemma 5.10, for any 0 < r < (
1
2
)min
j 6=k
ja
j
� a
k
j that,
e
E
"
n
(w
"
n
) �
N
1
X
j=1
e
E
"
n
(w
"
n
; B(a
j
; r) \ �
"
a
n
) �
N
1
X
j=1
jK
j
j
l
2
2�
j log "
n
j � C � D
l
2
2�
j log "
n
j � C;
and (5.62) follows. �
Remark 5.1. From the proof of Lemma 5.10 it follows that if for some a
j
2 �G we have
K
j
6= 0, then
e
E
"
n
(w
"
n
)�D
l
2
2�
j log "
n
j ! 1 when "
n
! 0.
6. Convergen e of u
"
In this se tion we use the energy estimate of Theorem 1 in order to prove the onvergen e
result Theorem 2 for the minimizers fu
"
g. The next lemma provides us with some basi
estimates.
Lemma 6.1. There exists a onstant C
1
> 0 su h that for all a 2 (
8
9
;
9
10
) and " 2 (0; 1) there
holds,
(6.1)
Z
G
(2")
anG
"
a
jru
"
j
2
+
1
"
2
Z
�
"
a
F (d(u
"
)) +
Z
��
"
a
ju
"
(~�)� ~g(~�)j
2
"
a
d~� � C
1
:
Proof. By Proposition 4.1, (5.4), Proposition 2.2 and Corollary 2.1 we have,
e
E
"
(u
"
) � ~
1
Z
��
"
a
ju
"
(~�)� ~g(~�)j
2
"
a
d~� +
Z
�
"
a
jru
"
j
2
+
F (d(u
"
))
"
2
� E
"
(u
"
)�E
"
(d
0;"
) � D
l
2
2�
j log "j+ C:
(6.2)
Moreover, noting that the lower bound for
e
E
"
, established in the previous se tion, is valid if
we repla e ~
1
in (5.4) by any other positive onstant, say ~
1
=2, we infer that,
(6.3) D
l
2
2�
j log "j � C �
�
~
1
2
�
Z
��
"
a
ju
"
(~�)� ~g(~�)j
2
"
a
d~� +
Z
�
"
a
jru
"
j
2
+
F (d(u
"
))
"
2
:
Subtra ting (6.3) from (6.2) we get,
(6.4)
Z
��
"
a
ju
"
(~�)� ~g(~�)j
2
"
a
d~� � C; 8a 2 (
7
8
;
9
10
); 8" 2 (0; 1):
Applying Proposition 5.1 for 2" instead of " yields,
(6.5) ~
1
Z
��
(2")
a
ju
"
(~�)� ~g(~�)j
2
2
a
"
a
d~� +
Z
�
(2")
a
jru
"
j
2
+
F (d(u
"
))
4"
2
� D
l
2
2�
j log "j � C:
Put b
"
= a(1 +
log 2
log "
), so that "
b
"
= (2")
a
. Sin e lim
"!0
jb
"
� aj = 0, we may apply (6.4) with
a = b
"
to obtain that,
(6.6)
Z
��
(2")
a
ju
"
(~�)� ~g(~�)j
2
2
a
"
a
d~� � C; 8a 2 (
8
9
;
9
10
); 8" 2 (0; 1):
43
From (6.5) and (6.6) we infer that,
(6.7)
Z
�
(2")
a
jru
"
j
2
+
F (d(u
"
))
4"
2
� D
l
2
2�
j log "j � C; 8a 2 (
8
9
;
9
10
); 8" 2 (0; 1):
On the other hand, from (6.2) we get that,
(6.8)
Z
�
"
a
jru
"
j
2
+
F (d(u
"
))
"
2
� D
l
2
2�
j log "j+ C:
Subtra ting (6.7) from (6.8) yields,
(6.9)
Z
G
(2")
anG
"
a
jru
"
j
2
+
�
3
4
�
Z
�
(2")
a
F (d(u
"
))
"
2
� C; 8a 2 (
8
9
;
9
10
); 8" 2 (0; 1):
Combining (6.4) with (6.9) we are led to (6.1). �
Fix a 2 (
8
9
;
9
10
). Using (6.1) we an �nd t
"
2 ("
a
; 2
a
"
a
) su h that
(6.10)
Z
��
t
"
jru
"
j
2
�
C
"
a
:
Put ea = ea
"
= log
"
t
"
, i.e. t
"
= "
ea
. Note that ea 2
�
a(1+ log 2= log "); a
�
, so that jea� aj ! 0. In
the sequel we shall work on the domain �
"
ea
instead of �
"
�
, and modify our previous notations
a ordingly. In parti ular, we denote in the sequel B
+
(~�; r) = B(~�; r) \ �
"
ea
and C
+
(~�; r) =
�B(~�; r) \ �
"
ea
for any ~� 2 ��
"
ea
and any r > 0. We shall also denote ~g(~�) = s(g(�(~�))) for
~� 2 ��
"
ea
. Analogously to (5.16) we de�ne a set of \bad points" as follows,
(6.11)
e
S
"
= fx 2
�
�
"
ea
: jd(u
"
(x))j � mg [ fx 2 ��
"
ea
: ju
"
(x)� ~g(x)j � mg :=
e
S
";1
[
e
S
";2
:
Next we prove,
Lemma 6.2. There exist a onstant k
1
> 0 and an integer N
1
, both independent of ", and for
ea h ", a olle tion of mutually disjoint dis s/half dis s fB(x
"
i
; �
"
") \ �
"
ea
g
N
1
(")
i=1
that satis�es
(6.12)
e
S
";1
�
N
1
(")
[
i=1
B(x
"
i
; �
"
") \ �
"
ea
;
with �
"
� k
1
and N
1
(") � N
1
, su h that for ea h i we have: either
~
Æ(x
"
i
) � 2�
"
" or x
"
i
2 ��
"
ea
.
Moreover, there exist k
2
and N
2
(independent of "), and for ea h " > 0, a olle tion of
mutually disjoint half dis s fB
+
(y
"
j
;
"
"
ea
)g
N
2
(")
j=1
, with y
"
j
2 ��
"
ea
, that satis�es
(6.13)
e
S
";2
�
N
2
(")
[
j=1
B
+
(y
"
j
;
"
"
ea
);
with
"
� k
2
and N
2
(") � N
2
. Further, we an onstru t su h olle tions with the following
property: for ea h 1 � i � N
1
(") and 1 � j � N
2
(") there holds:
(6.14) either B(x
"
i
; �
"
") \B
+
(y
"
j
;
"
"
ea
) = ; or B(x
"
i
; �
"
") \ �
"
ea
� B
+
(y
"
j
;
"
"
ea
):
44
Proof. From the gradient estimate (2.6) we get immediately an analogous result to Lemma 5.2 (i),
namely: there exists �
1
> 0 su h that,
(6.15) x 2
e
S
";1
=) jd(u
"
(y))j � m=2; 8y 2 �
"
ea
with jy � xj � �
1
":
Using (6.15) we an repeat the pro edure of [7℄, see also the proof of Proposition 5.2, to
onstru t a olle tion fB(x
"
i
; �
"
")\�
"
ea
g
N
1
(")
i=1
satisfying (6.12). We have N
1
(") � N
1
for some
N
1
sin e
1
"
2
Z
�
"
~a
F (d(u
"
)) � C
1
(see (6.1)):
We laim that the following analog to Lemma 5.2 (ii) holds as well, i.e. there exists �
2
> 0
su h that,
(6.16) x 2
e
S
";2
=) ju
"
(y)� ~g(y)j � m=2; 8y 2 ��
"
ea
with jy � xj � �
2
"
ea
:
To prove (6.16) it is enough to noti e that by (6.10) and the Cau hy-S hwarz inequality we
have,
ju
"
(x)� u
"
(y)j
2
� C
�
Z
��
"
~a
jru
"
j
2
�
jx� yj �
�
C
"
ea
�
jx� yj; 8x; y 2 ��
"
ea
:
Using (6.16) we an apply the same pro edure as above to obtain a olle tion satisfying (6.13).
Finally, if property (6.14) is not satis�ed for some i; j, we multiply by 5 the radii of all the
\big half dis s", i.e. those of radii of order "
ea
. After a �nite number of su h iterations,
involving possibly deletion of some \big half dis s", we end up with olle tions of half dis s
and dis s satisfying all the requirements. The number N
2
(") of big half-dis s in this olle tion
is bounded, uniformly in ", thanks to (6.16) and the bound
R
��
"
ea
ju
"
(~�)�~g(~�)j
2
"
ea
� C
1
from (6.1).
�
We denote for ea h 1 � i � N
1
(") by d
"
i
the degree of u
"
on the boundary of B(x
"
i
; �
"
")\�
"
ea
(in ase of a half dis we use the de�nition of degree given after (5.47)). Similarly, for ea h
1 � j � N
2
(") we denote by
~
d
"
j
the degree of u
"
on C
+
(x
"
i
;
"
"
ea
). From the gradient bound
(2.6) it follows that for some
1
> 0 we have,
(6.17) jd
"
i
j �
1
; 8i = 1; : : : ; N
1
("); 8" 2 (0; 1):
We laim that the same is true for the \big half dis s", namely, there exists a onstant
2
su h
that,
(6.18) j
~
d
"
j
j �
2
; 8j = 1; : : : ; N
2
("); 8" 2 (0; 1):
Indeed, for any �xed j 2 f1; : : : ; N
2
(")g we set,
I
j
= f1 � i � N
1
(") : x
"
i
2 ��
"
ea
s.t. B
+
(x
"
i
;
"
"
ea
) � B
+
(y
"
j
;
"
"
ea
)g
and
J
j
= f1 � i � N
1
(") : B(x
"
i
; �
"
") � B
+
(y
"
j
;
"
"
ea
)g;
45
and then E
j
= B
+
(y
"
j
;
"
"
ea
) n
S
i2I
j
B
+
(x
"
i
; �
"
"). By (6.18) we have,
(6.19) jdeg(u
"
; �E
j
)j = j
X
i2J
j
d
"
i
j � C:
The estimate (6.19) is equivalent to
(6.20) j
Z
�E
j
�(s(u
"
)) ^
�s(u
"
)
�t
dtj � C:
Next, from (6.10) we infer that
(6.21) j
Z
�E
j
\��
"
ea
�(s(u
"
)) ^
�s(u
"
)
�t
dtj � C
Z
�E
j
\��
"
ea
jru
"
j � C"
ea=2
�
Z
��
"
ea
jru
"
j
2
�
1=2
� C;
while from (2.6) and the estimate N
1
(") � N
1
we have,
(6.22) j
Z
[
i2I
j
C
+
(x
"
i
;�
"
")
�(s(u
"
)) ^
�s(u
"
)
�t
dtj � N
1
� C:
Combining (6.20) with (6.21) and (6.22) we are led to
j
Z
C
+
(y
"
j
;
"
"
ea
)
�(s(u
"
)) ^
�s(u
"
)
�t
dtj � C;
and our laim (6.18) follows.
The next proposition provides a basi upper bound for the energy of the minimizers away
from the boundary, and a �nite number of points.
Proposition 6.1. For a subsequen e "
n
! 0, there exist D distin t points a
1
; : : : ; a
D
2 G
su h that for every ompa t subset E �� G n fa
1
; : : : ; a
D
g we have,
(6.23) E
"
n
(u
"
n
; E) � C(E):
Moreover, the degree of u
"
n
around ea h a
j
equals 1.
Proof. Passing to a subsequen e "
n
! 0 and using Lemma 6.2 and (6.17),(6.18) we an assume
that N
1
("
n
) = N
1
, N
2
("
n
) = N
2
and d
"
n
i
= d
i
,
~
d
"
n
j
=
~
d
j
for all i; j and all n. We an further
assume that lim
n!1
x
"
n
i
2
�
G exists for all i and denote. As in the proof of Proposition 5.2,
we denote by a
1
; : : : ; a
M
the distin t limits and put
�
k
= fj 2 f1; : : : ; N
1
g : x
"
n
j
! a
k
g; k = 1; : : : ;M :
For ea h r satisfying (5.45) we denote by K
k
the degree of u
"
n
on �B(a
k
; r), if a
k
2 G, or on
�B(a
k
; r) \ G, if a
k
2 �G. We have learly K
k
=
P
j2�
k
d
j
. Sin e the degree of ~g on ��
"
ea
equals the degree of g on �G, i.e. to D, we have,
(6.24) D =
M
X
k=1
K
k
:
46
Consider �rst a
k
2 G and r satisfying (5.45). By the argument of Lemma 5.7, we obtain that
(6.25) E
"
n
(u
"
n
; B(a
k
; r)) � jK
k
j
l
2
2�
log
r
"
n
� C:
A tually, here the proof is even easier sin e we have to our disposal the estimate,
1
"
2
Z
�
"
ea
F (d(u
"
)) � C
1
(see (6.1)):
Next, for ea h a
k
2 �G and r satisfying (5.45) we laim that
(6.26) E
"
n
(u
"
n
; B(a
k
; r) \ �
"
ea
n
) � jK
k
j
l
2
2�
j log "
n
j � C:
The proof of (6.26) follows by the same argument as the one used in the proof of (5.53).
Moreover, as in Remark 5.1 we get that when K
k
6= 0 there holds,
(6.27) lim
n!1
E
"
n
(u
"
n
; B(a
k
; r) \ �
"
ea
n
)� jK
k
j
l
2
2�
j log "
n
j =1:
Combining (6.25) with (6.26) we get that for any r > 0 satisfying (5.45),
(6.28) E
"
n
(u
"
n
;�
"
ea
n
\
M
[
j=1
B(a
j
; r)) �
�
M
X
j=1
jK
j
j
�
l
2
2�
j log "
n
j � C:
On the other hand, by Proposition 2.2 and Corollary 2.1 we infer that,
(6.29) E
"
n
(u
"
n
;�
"
ea
n
) � D
l
2
2�
j log "
n
j+O(1):
By (6.24) and (6.28){(6.29) we obtain that
(6.30) E
"
n
(u
"
n
;�
"
ea
n
n
M
[
j=1
B(a
j
; r)) � C(r);
and (6.23) follows.
Sin e D = j
P
M
j=1
K
j
j �
P
M
j=1
jK
j
j we infer from (6.28) and (6.29) that K
j
� 0 for all j.
Moreover, taking into a ount (6.27) we get that K
j
= 0 whenever a
j
2 �G. The arguments of
steps 1 and 2 in the proof of [7, Theorem VI.2℄ show that K
j
= 1 for ea h a
j
2 G. Relabeling
if ne essary, we may assume that a
j
2 G if and only if 1 � j � D, and therefore K
j
= 1 for
1 � j � D. We �nally remark that the argument whi h led to (6.30) now yields,
(6.31) E
"
n
(u
"
n
;�
"
ea
n
n
D
[
j=1
B(a
j
; r)) � C(r):
�
Now we are ready to omplete the proof of Theorem 2.
47
Proof of Theorem 2. By (6.31) we get (possibly after passing to a further subsequen e) that
(6.32) u
"
n
* u
�
weakly in H
1
lo
(G n fa
1
; : : : ; a
D
g):
for some u
�
2 H
1
(G n
D
S
j=1
B(a
j
; r); �), for all small r > 0. Fix any dis B(x
0
; r) ��
G n fa
1
; : : : ; a
D
g. By (6.23) we have E
"
n
(u
"
n
; B(x
0
; r)) � C. Using Fubini's theorem we an
�nd r
0
2 (r=2; r) su h that (after passing to a subsequen e if ne essary),
Z
�B(x
0
;r
0
)
jru
"
n
j
2
+
F (d(u
"
n
))
"
2
n
� C:
Applying the argument of [6℄ we get that,
u
"
n
! u
�
in H
1
(B(x
0
; r
0
)) and d(u
"
n
)! 0 uniformly on B(x
0
; r
0
):
In parti ular, for n large enough we on lude that F (d(u
"
n
)) is a C
2
fun tion on B(x
0
; r
0
)
and also that
(6.33) F
00
(d(u
"
n
)) > 0 on B(x
0
; r
0
):
Next we may apply the argument of Steps A.3 and A.4 from the proof of [6, Theorem 1℄ to
show that,
(6.34) fu
"
n
g is bounded in H
2
lo
(B(x
0
; r
0
)) and fru
"
n
g is bounded in L
1
lo
(B(x
0
; r
0
)):
Indeed, we drop for simpli ity the subs ript "
n
and set as in [6℄, A =
1
2
jruj
2
. Then,
(6.35) �A = jD
2
uj
2
+
X
i=1;2
u
x
i
�(u
x
i
);
and writing u = (u
1
; u
2
) we have also, for i = 1; 2,
u
x
i
�(u
x
i
) =
F
0
(d(u))
2"
2
n
�
d
u
1
u
1
(u
1
)
2
x
i
+ 2d
u
1
u
2
(u
1
)
x
i
(u
2
)
x
i
+ d
u
2
u
2
(u
2
)
2
x
i
�
+
F
00
(d(u))
2"
2
n
(d
u
1
(u
1
)
x
i
+ d
u
2
(u
2
)
x
i
)
2
:
(6.36)
Noting that by (2.7), j�uj =
jF
0
(d(u))j
2"
2
n
, we infer from (6.35){(6.36) and (6.33) that for some
> 0 we have,
(6.37) �A � jD
2
uj
2
� j�ujjruj
2
:
Sin e j�uj �
p
2jD
2
uj we get by (6.37) and Cau hy-S hwarz that,
(6.38) ��A+ jD
2
uj
2
�
1
2
jD
2
uj
2
+ 4
2
A
2
:
From the Bo hner-type inequality (6.38) we dedu e our laim (6.34), as in [7℄. In parti ular,
we obtain that,
(6.39) u
"
n
! u
�
in C
�
lo
(G n fa
1
; : : : ; a
D
g); 8� 2 (0; 1):
48
Next we assert that u
�
= s Æ g on �G (note that being an H
1
map in a neighborhood of
�G, u
�
has a well de�ned tra e in H
1=2
(�G)). We �rst laim that,
(6.40)
Z
��
�
ju
�
(~�)� g
�
(~�)j
2
�
d~� � C; uniformly for 0 < � < b
1
:= min(b
0
;dist(fa
j
g
D
j=1
; �G)):
Indeed, by (6.25) we infer that for ea h � < b
1
,
(6.41)
Z
�
�
jru
"
n
j
2
+
F (d
"
n
)
"
2
n
�
Dl
2
2�
j log "
n
j � C:
Combining (6.41) with (4.8), (2.37) and (2.31), passing to the limit n!1 and using (6.39),
we are led to (6.40). Next we �x a fun tion 2 C
1
(G) su h that � 1 on G
b
1
=2
and � 0
on �
b
1
and let v be de�ned on G as follows,
v(x) =
8
<
:
(x)s(g(�(x))) x 2 G
b
1
;
0 x 2 �
b
1
:
Clearly v 2 H
1
sÆg
(G;R
2
) and therefore u
�
= s Æ g on �G if and only if
(6.42) w := (u
�
� v) 2 H
1
0
(G;R
2
):
By (6.40) we have,
(6.43)
1
�
Z
��
�
jw(~�)j
2
d~� � C; uniformly for 0 < � < b
1
:
Choose a sequen e r
n
! 0 and de�ne a fun tion �
n
2 Lip(G) by,
�
n
(x) =
8
>
>
<
>
>
:
0 Æ(x) � r
2
n
;
log r
n
�log(Æ(x)=r
n
)
log r
n
r
2
n
< Æ(x) < r
n
;
1 Æ(x) � r
n
:
Then set w
n
= �
n
w. Clearly w
n
2 H
1
0
(G;R
2
), and it is enough to show that w
n
! w in H
1
in order to on lude that w 2 H
1
0
(G;R
2
): Sin e obviously w
n
! w in L
2
, it remains to verify
that rw
n
!rw in L
2
. Now,
Z
G
jr(w
n
�w)j
2
=
Z
G
r
2
n
jrwj
2
+
Z
G
r
n
nG
r
2
n
jr
�
(1� �
n
)w
�
j
2
�
Z
G
r
2
n
jrwj
2
+ 2
Z
G
r
n
nG
r
2
n
jrwj
2
+ 2
Z
G
r
n
nG
r
2
n
jwj
2
jr�
n
j
2
:
(6.44)
Next, by (2.2),(2.3) and (6.43) we obtain,
Z
G
r
n
nG
r
2
n
jwj
2
jr�
n
j
2
=
1
j log r
n
j
2
Z
G
r
n
nG
r
2
n
jwj
2
Æ
2
(x)
�
C
j log r
n
j
2
Z
r
n
r
2
n
�
Z
��
�
jwj
2
d~�
�
d�
�
2
�
C
j log r
n
j
2
Z
r
n
r
2
n
d�
�
=
C
j log r
n
j
! 0; as n!1:
(6.45)
49
Sin e learly
R
G
r
n
jrwj
2
! 0, we infer from (6.44){(6.45) that
R
G
jr(w
n
� w)j
2
! 0, and the
result follows.
By [7, Remark I.1℄, in order to prove that u
�
is indeed the anoni al harmoni �-valued
map asso iated with s Æ g and a = (a
1
; : : : ; a
D
), we only need to show that u
�
2 W
1;1
(G).
We shall explain now how the argument of Struwe [16℄ implies that u
�
2 W
1;p
(G) for all
p 2 [1; 2). It is enough to show that u
�
2 W
1;p
(B(a
j
; r)) for all 1 � j � D, all small r > 0
and all p 2 (1; 2). In fa t, the same argument whi h led to (6.25) (see [16, Proposition 3.3℄)
shows that,
E
"
n
�
u
"
n
; B(a
j
; r) n
[
i2�
j
B(x
"
n
j
; �)
�
� jK
j
j
l
2
2�
j log �j+ C =
l
2
2�
j log �j+ C;
for all � > 0. Applying the argument of [16℄ we infer that fu
"
n
g is bounded inW
1;p
(B(a
j
; r)),
for p < 2, hen e u
�
2W
1;p
(B(a
j
; r)) as laimed.
To on lude the proof of Theorem 2 we need to show that the on�guration a = (a
1
; : : : ; a
D
)
minimizes the renormalized energy. We re all from [7℄ that the renormalized energy W (
~
h;b)
asso iated with a smooth boundary ondition
~
h : �G ! S
1
of degree D and a on�guration
b = (b
1
; : : : ; b
D
) of D distin t points in G, satis�es,
(6.46) W (
~
h;b) = lim
�!0
+
1
2
Z
A
�
jrv
b
j
2
� �D log(1=�);
where A
�
= G n
D
S
j=1
B(b
j
; �) and v
b
is the anoni al S
1
-valued harmoni map asso iated with
b and h. Other expressions for W (h;b) are given in [7℄. If
~
h = �
�1
Æ h with h : �G! � of
degree D, then the anoni al �-valued harmoni map asso iated with b and h is u
b
= � Æv
b
(� is de�ned in (1.10)). From (6.46) we dedu e that,
(6.47)
l
2
4�
2
W (
~
h;b) = lim
�!0
+
1
2
Z
A
�
jru
b
j
2
�D
l
2
4�
log(1=�);
Analogously to [7, Chapter III℄ we de�ne for every "; � > 0,
(6.48) J("; �) = minfE
"
(u) : u 2 H
1
(B(0; �);R
2
); u(x) = �(x=jxj) on �B(0; �)g:
As in [7℄ we have by a simple res aling argument that J("; �) = J("=�; 1). De�ning J(t) =
J(t; 1) we get as in [7, Lemma III.1℄ that,
(6.49) J(t) +
l
2
2�
log t is nonde reasing for t > 0:
From the energy estimate (1.9) for the domain G = B(0; 1) and the �-valued boundary
ondition g(x) = �(x=jxj) we get in parti ular that J(") +
l
2
2�
log " � �C, for all " > 0
(a tually this ase requires only a simple adaptation of the results in [7℄). Combining it with
(6.49) we infer that the following �nite limit exists:
(6.50) e = lim
"!0
+
J(") +
l
2
2�
log ":
50
Our result that the on�guration a minimizes the renormalized energy will follow from the
next two laims.
Claim 1: Let b be any on�guration of D distin t points in G. Then, for any small � > 0 we
have,
(6.51) lim
n!1
E
"
n
(u
"
n
)�E
"
n
(d
0;"
n
)�DJ("
n
; �) �
l
2
2�
2
W (�
�1
ÆsÆg;b)+D
l
2
2�
log(1=�)+o
�
(1);
where o
�
(1) is a quantity whi h goes to 0 with �.
Claim 2: For the parti ular on�guration a we have,
(6.52) lim
n!1
E
"
n
(u
"
n
)�E
"
n
(d
0;"
n
)�DJ("
n
; �) �
l
2
2�
2
W (�
�1
ÆsÆg;b)+D
l
2
2�
log(1=�)+o
�
(1):
The proof of Claim 1 requires a re�nement of the upper bound onstru tion of Proposition 2.2.
We �x a on�guration b of D distin t points in G. For ea h small � > 0 we onstru t a
sequen e fw
"
n
g � H
1
g
(G;R
2
) as follows. On G
�
we de�ne (as in (2.39)),
(6.53) w
"
n
(x) =
8
<
:
s(g(�(x))) + d
0;"
n
(x)�(g(�(x))) for Æ(x) 2 [0; �=2);
s(g(�(x))) +
2
�
(� � Æ(x))d
0;"
n
(x)�(g(�(x))) for Æ(x) 2 [�=2; �℄:
By (2.20) and (2.32) it follows that,
(6.54) E
"
n
(w
"
n
; G
�
) = E
"
n
(d
0;"
n
) + o
�
(1):
Note that by de�nition w
"
n
= g
�
on ��
�
. On the domain A
�
:= �
�
n
D
S
j=1
B(b
j
; �) we let
w
"
n
= v
�
where v
�
is a minimizer for the Diri hlet energy
R
A
�
jrvj
2
for v 2 H
1
(A
�
; �)
satisfying the boundary ondition v = g
�
on ��
�
, and v(x) = �
�
(x� b
j
)=jx� b
j
j
�
on �B(b
j
; �)
for j = 1; : : : ;D. From the results of [7℄ we dedu e that,
(6.55)
Z
A
�
jrv
�
j
2
= D
l
2
2�
log(1=�) +
l
2
2�
2
W
�
�
(�
�1
Æ g
�
;b) + o
�
(1);
where here W
�
�
(�
�1
Æ g
�
;b) refers to the renormalized energy on the domain �
�
. Moreover,
it is easy to see that,
(6.56) W
�
�
(�
�1
Æ g
�
;b)�W (�
�1
Æ s Æ g;b) = o
�
(1):
Finally, on ea h dis B(b
j
; �); j = 1; : : : ;D, we de�ne w
"
n
as a minimizer for the energy E
"
n
with the boundary ondition �
�
(x� b
j
)=jx� b
j
j
�
on �B(b
j
; �). By de�nition,
(6.57) E
"
n
(w
"
n
; B(b
j
; �)) = J("
n
; �); j = 1; : : : ;D:
Combining (6.54){(6.57) we are led to
E
"
n
(w
"
n
) = E
"
n
(d
0;"
n
) +DJ("
n
; �) +
l
2
2�
2
W (�
�1
Æ s Æ g;b) +D
l
2
2�
log(1=�) + o
�
(1);
whi h learly implies (6.51) sin e E
"
n
(u
"
n
) � E
"
n
(w
"
n
).
51
Next we turn to the proof of Claim 2. We �rst note that by (4.7) we have,
(6.58) E
"
n
(u
"
n
; G
"
1=2
n
) � E
"
n
(d
"
n
; G
"
1=2
n
) � E
"
n
(d
0;"
n
; G
"
1=2
n
) + o
"
n
(1):
On A
�
:= �
�
n
D
S
j=1
B(a
j
; �) we have u
"
n
! u
�
in H
1
�norm. Therefore,
(6.59) lim
n!1
E
"
n
(u
"
n
; A
�
) =
Z
A
�
jru
�
j
2
=
l
2
2�
2
W (�
�1
Æ s Æ g;a) +D
l
2
2�
log(1=�) + o
�
(1):
Finally, applying the argument of the proof of [4, Lemma 5.4℄ we on lude that,
(6.60) lim
n!1
�
E
"
n
(u
"
n
; B(a
j
; �)) � J("
n
; �)
�
� o
�
(1); 8j:
Combining (6.58){(6.60) we are led to (6.52). �
Referen es
[1℄ S. B. Angenent, Uniqueness of the solution of a semilinear boundary value problem, Math. Ann. 272 (1985),
129{138.
[2℄ N. Andr�e and I. Shafrir, Minimization of a Ginzburg-Landau type fun tional with nonvanishing Diri hlet
boundary ondition, Cal . Var. Partial Di�erential Equations. 7 (1998), 191{217.
[3℄ N. Andr�e and I. Shafrir, On the minimizers of a Ginzburg-Landau energy when the boundary ondition has
zeros, preprint.
[4℄ N. Andr�e and I. Shafrir, On nemati s stabilized by a large external �eld, Reviews in Math. Phys. 11 (1999),
653{710.
[5℄ M. S. Berger and L. E. Fraenkel, On the asymptoti solution of a nonlinear Diri hlet problem, J. Math.
Me h. 19 (1970), 553{585.
[6℄ F. Bethuel, H. Brezis and F. H�elein, Asymptoti s for the minimizers of a Ginzburg-Landau fun tional, Cal .
Var. PDE 1 (1993), 123{148.
[7℄ F. Bethuel, H. Brezis and F. H�elein, Ginzburg-Landau Vorti es, Birkh�auser, 1994.
[8℄ F. Bethuel and T. Rivi�ere, Vorti es for a variational problem related to super ondu tivity, Ann. Inst. H.
Poi ar�e Anal. Non Lin�eaire 12 (1995), 243{303.
[9℄ H. Brezis and F. Merle and T. Rivi�ere, Quantization e�e ts for ��u = u(1� juj
2
) in R
2
, Ar h. Rational
Me h. Anal. 126 (1994), 35{58.
[10℄ M. Chipot, M. Chleb��k, M. Fila and I. Shafrir, Existen e of positive solutions of a semilinear ellipti
equation in R
n
+
with a nonlinear boundary ondition, J. Math. Anal. Apl. 223 (1998), 429{471.
[11℄ H. I. Choi, S. W. Choi and H. P. Moon, Mathemati al theory of Medial axis transform, Pa i� J. of
Mathemati s 181 (1997), 57{88.
[12℄ D. Gilbarg and N. Trudinger, Ellipti Partial Di�erential Equations of Se ond Order, Springer-Verlag,
Berlin and New York, 1983.
[13℄ L. Lassoued and P. Mirones u, Ginzburg-Landau type energy with dis ontinuous onstraint, J. Anal. Math.
77 (1999), 1{26.
[14℄ L. Modi a, The gradient theory of phase transitions and the minimal interfa e riterion, Ar h. Rational
Me h. Anal. 98 (1987), 123{142.
[15℄ J. Serrin, Nonlinear ellipti equations of se ond order, AMS Symposium in Partial Di�erential Equations,
Berkeley, 1971.
52
[16℄ M. Struwe, On the asymptoti behavior of minimizers of the Ginzburg-Landau model in 2 dimensions,
Di�erential Integral Equations 7 (1994), 1613-1624; erratum, lo . it. 8 (1995), 124.
53