266 M. Dutour – B. Helffer
Proof. Let ψ
n
be a sequence of C
∞
functions such that • 0 ≤ ψ
n
≤ 1; • ψ
n
= 0 in a neighborhood of ; • ψ
n
x → 1, ∀x 6∈ ;
We observe that Z
✂
2
1 − ψ
n
H
e 2
d x −→
Z
H
2 e
d x . 34
We can consequently choose n such that: κ λ
· ||
2 R
✂
2
1 − ψ
n
H
e 2
d x
1 2
, 35
We now try to find A
n
such that • curl A
n
= ψ
n
H
e
; • supp A
n
∩ = ∅. We have already shown how to proceed when is starshaped. In the general case, we first choose
e A
n
such that: curl e A
n
= ψ
n
H
e
, without the condition of support see 2 for the argument. We now observe that curl e
A
n
= 0 in . Using the simple connexity, we can find φ
n
in C
∞
such that e
A
n
= ∇φ
n
. We can now extend φ
n
outside as a compactly supported C
∞
function in
2
e φ
n
. We then take A
n
= e A
n
− ∇e φ
n
. It remains to compute the energy of the pair 1, A
n
which is strictly negative in order to achieve the proof of the proposition.
R
EMARK
6. In the case when is not simply connected, Proposition 8 remains true, if we replace by e
, where e
is the smallest simply connected open set containing .
R
EMARK
7. It would be interesting to see how one can use the techniques of [1] for ana- lyzing the properties of the zeros of the minimizers, when they are not normal solutions. The
link between the two papers is given by the relation λ = κd
2
. In conclusion, we have obtained, the following theorem:
T
HEOREM
2. Under condition 15, there exists α 0, such that:
|| 2
R
e
H
2 e
d x
1 2
≤ λ
opt
κ κ
≤ inf α
, λ
1
κ .
36
4. Localization of pairs with small energy, in the case κ large
When κ is large and λ − λ
1
is small enough, we will show as in [7] that all the solutions of non positive energy of the GL-systems are in a suitable neighborhood of 0, A
e
independent of κ
≥ κ 0. This suggests that in this limiting regime these solutions of the GL-equations if
there exist and if they appear as local minima will furnish global minimizers. Let us show this
On bifurcations from normal solutions 267
localization statement. The proof is quite similar to the proof of Theorem 1. We recall that we have 17-23. Now we add the condition that, for some η 0,
λ ≤ λ
1
+ η . 37
Note that we have already solved the problem when λ ≤ λ
1
−
C κ
, so we are mainly interested in the λ’s in an interval of the form
λ
1
−
C κ
, λ
1
+ η .
The second assumption is that we consider only pairs u, A ∈ H
1
× H
1 loc
2
such that G
λ
u, A ≤ 0 .
38 We improve 23 into
k∇ − i Auk
2 L
2
≥ 1
− ǫ − C
ǫ kak
2 L
4
+
λ
1
− C
ǫ kak
2 L
4
kuk
2 L
2
. 39
Taking ǫ =
1 κ
, we get, using also 22, the existence of κ and C such that, for λ
∈ 0, λ
1
+ η and for κ
≥ κ ,
k∇ − i Auk
2
≥ 1
− C
κ λ
1
− C
κ kuk
2
, 40
for any u, A such that G
λ,κ
u, A ≤ 0.
Coming back to 1, and, using again the negativity of the energy G
λ,κ
u, A of the pair u, A, we get
λ Z
|u|
4
d x ≤
η +
C κ
kuk
2 L
2
. 41
But by Cauchy-Schwarz, we have Z
|u|
2
d x ≤ ||
1 2
Z
|u|
4
d x
1 2
. 42
So we get kuk
L
2
≤ ||
λ
1 2
η +
C κ
1 2
43 We see that this becomes small with η and
1 κ
. It is then also easy to control the norm of u in H
1
. We can indeed use successively 25, 26, 38 and the trivial inequality:
k∇ − i Auk
2 L
2
≤ λkuk
2 L
2
+ G
λ
u, A . 44
The control of A − A
e
in the suitable choice of gauge is also easy through 17 and 20. Note also that if λ λ
1
, we obtain the better kuk
L
2
≤ C
κλ
1 2
. 45
So we have shown in this section the following theorem:
268 M. Dutour – B. Helffer
T
HEOREM
3. There exists η 0 such that, for 0 η η
and for λ ≤ λ
1
+ η, then there exists κ
such that, for κ ≥ κ
, all the pairs u, A with negative energy are in a suitable neighborhood
η,
1 κ
of the normal solution in H
1
,
✁
× H
1
,
2
whose size tends to 0 with η and
1 κ
. R
EMARK
8. Using the same techniques as in [7], one can also show that there are no solu- tions of the Ginzburg-Landau equations outside this neighborhood. This is discussed in Section
5.
5. A priori localization for solutions of Ginzburg-Landau equations
