raymonduniformestimates.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 01, (67–81). March 2010

Uniform Spectral Estimates for Families of Schrödinger

Operators with Magnetic Field of Constant

Intensity and Applications

Nicolas Raymond

Laboratoire de Mathématiques,

Université Paris-Sud 11, Bâtiment 425, F-91405

email : nicolas.raymond@math.u-psud.fr

ABSTRACT

The aim of this paper is to establish uniform estimates of the bottom of the spectrum of the

Neumann realization of (i∇ + qA)2 on a bounded open set Ω with smooth boundary when
|∇ × A| = 1 and q → +∞. This problem was motivated by a question occurring in the theory
of liquid crystals and appears also in superconductivity questions in large domains.

RESUMEN

El objetivo de este artículo es establecer estimativas uniformes del espectro inferior de la

realización de Neumann de (i∇ + qA)2 sobre un conjunto Ω abierto acotado con frontera
suave cuando |∇ × A| = 1 y q → +∞. Este problema fue motivado por una cuestión que
ocurre en teoría de cristales líquidos y aparece también en cuestiones de supercontudividad en

dominios grandes.

Key words and phrases: Spectral theory, semiclassical analysis, Neumann Laplacian, magnetic

field, liquid crystals.

Math. Subj. Class.: 35P, 35J10, 35Q, 81Q20, 82D30.



68 Nicolas Raymond CUBO
12, 1 (2010)

1 Introduction

Let Ω ⊂ R3 be an open bounded set with C3 boundary and A ∈ C2(Ω).
We will study the quadratic form qA defined by:

qA(u) =

∫

Ω

|(i∇ + qA)u|2 dx, ∀u ∈ H1(Ω)

and consider the associated selfadjoint operator, i.e the Neumann realization of (i∇ + qA)2 on Ω.
We denote µΩ(q, A) or µ(q, A) the lowest eigenvalue of the previous operator. Our purpose is to

study the behavior of this eigenvalue as q tends to infinity and to control the uniformity of the

estimates with respect to the magnetic field.

We let

A = {A ∈ C3(Ω) : |B| = 1 where B = ∇ × A}. (1.1)

The main estimates obtained in this paper are summarized in the two following theorems:

Theorem 1.1 (Uniform lower bound). For all ǫ ∈]0, 1
2
[, there exists

C = C(Ω,ǫ) > 0 and q0 > 0, such that, for all q ≥ q0 and for all A ∈ A,

µ(q, A) ≥ Θ0q − C
(
q1−2ǫ + (1 + |∇B|∞)q1/2+2ǫ

)
.

Theorem 1.2 (Uniform upper bound). For all δ ∈]0, 1/2[, there exists C = C(Ω,δ) > 0 and
q0 = q0(Ω; δ) > 0, such that for all q ≥ q0 and all A ∈ A:

µ(q, A) ≤ Θ0q + C(q2δ + |B|2C1q2−4δ + |B|C1q1−δ + |B|C2q3/2−3δ + |B|2C2q2−6δ),

where |B|2C1 = |B|∞ + |∇B|2∞ and |B|2C2 = |B|2C1 + |∇2B|2∞.

Remark 1.3.

In those theorems, ǫ and δ are left undefined because they will play a role in the application to

families of vector potentials where the semi-norms of B could become large. If the magnetic field

is fixed and if we are not interested in uniformity, we take ǫ = 1
8

and δ = 1
3

to have the optimal

estimates (relatively to the method), leading to the remainder O(q3/4) in the first case (in fact, one

can hope a remainder O(q2/3)) and O(q2/3) in the second case (cf. [HM04]). Let us also mention

that, in dimension 2, the remainder is O(q1/2) (see [FH08]).

�

Let us briefly recall the motivation of those estimates.



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Uniform Spectral Estimates for Families ... 69

Liquid crystals

The first one occurs in the theory of liquid crystals. The asymptotic properties of the Landau-

de Gennes functional (cf. [BCLP02, dG95, HP07, Pan03, Pan06]) lead to the analysis of the

minimizers (or local minimizers) of the reduced functional:

F(ψ, n) =
∫

Ω

|(i∇ + qn)ψ|2 + r|ψ|2 + g
2
|ψ|4dx,

where r > 0, ψ ∈ H1(Ω, C) and n ∈ C(τ), with C(τ) defined for τ > 0 by:

C(τ) = {Qnτ Qt,Q ∈ SO3}, (1.2)

where SO3 denotes the set of the rotations in R
3 and

nτ = (cos(τx3), sin(τx3), 0). (1.3)

Let us notice that, if n ∈ C(τ),
∇ × n + τn = 0

and consequently

|∇ × n|∞ = τ.

Then, the analysis of the positivity of the Hessian of the functional at ψ = 0 leads to a spectral

problem and we are led to study the asymptotic properties of

µ∗(q,τ) = inf
n∈C(τ )

µ(q, n), (1.4)

as qτ → +∞.
In this context, X-B. Pan has given estimates (cf. [Pan06]) as:

qτ → +∞ and τ → 0

and Helffer and Pan give some extensions in [HP07] including the case:

qτ → +∞ and τ bounded.

In this paper, we treat the case where:

qτ → +∞ and τ → +∞.

Superconductivity

The second one occurs in the theory of superconductivity in large domains. In [Alm02, Alm08],

Y. Almog has analyzed properties of minimizers of Ginzburg-Landau’s functional when the size of

Ω tends to infinity; Theorems 1.1 and 1.2 permit to treat another regime (for the linear problem);

in fact, q will be allowed to tend to infinity.



70 Nicolas Raymond CUBO
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Organization of the paper

The paper is organized as follows. First, we prove Theorem 1.1 in Section 2 and Theorem 1.2 in

Section 3, then we will prove an Agmon estimate in Section 4 in order to study the localization

of first eigenfunctions in the considered asymptotic regime. Finally, Section 5 will present the

applications to the theory of liquid crystals and to the superconductivity in large domains. In each

case we will show that the eigenfunctions become localized at the boundary. This corresponds to

what is called surface smecticity in the first case and surface superconductivity in the second case

(see [Pan04]).

2 Lower Bound

In this section, we give the proof of Theorem 1.1. It is based on a localization technique through

a partition of unity and the analysis of simplified models.

2.1 Partition of unity

For each r > 0, we consider a partition of unity (cf. [HM04]) with the property that there exists

C = C(Ω) > 0 such that:

∑

j

|χrj |2 = 1 on Ω ; (2.5)

∑

j

|∇χrj|2 ≤
C

r2
on Ω. (2.6)

Each χrj is a C∞-cutoff function with support in the ball of center xj and radius r (denoted by
Bj ). We will choose r later for optimizing the error. We will use the IMS formula (cf. [CFKS86]):

Lemma 2.1.

qA(u) =
∑

j

qA(χju) −
∑

j

‖|∇χrj|u‖2, ∀u ∈ H1(Ω). (2.7)

So, in order to minimize qA(u), we will be reduced to the minimization of qA(v), with v

supported in some Bj , the price to pay being an error of order
C
r2

.

2.2 Approximation by the constant magnetic field in a ball or a semi-ball

We want to have estimates depending only on the magnetic field B = ∇ × A, that’s why we look
for a canonical choice of A depending only on B; it is the aim of the following lemmas. Let B a

ball (or semi-ball) of center 0 and radius r > 0.



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Uniform Spectral Estimates for Families ... 71

Lemma 2.2. Let F ∈ C2(B, R3).
We assume the existence a constant C > 0 such that:

|∇ × F| ≤ C|x|,

for x ∈ B. Then, there exists u ∈ C3(B) and α > 0 a constant such that:

|F(x) − ∇u(x)| ≤ αC|x|2,

for all x ∈ B.

Proof.

The proof is similar to the one of Poincaré’s theorem.

Let us define, for all x ∈ B:

u(x) =

∫ 1

0

F(tx) · xdt.

Let us verify that u is suitable. As F ∈ C2(B, R3), we can extend F in a C2 function on R3, so by
computing, we have, for x ∈ B:

∂iu(x) = Fi(x) +

3∑

j=1,j 6=i

∫ 1

0

(∂iFj − ∂jFi) (tx)txjdt.

�

As a consequence, we immediately deduce the lemma which will give us uniformity in our further

estimates :

Lemma 2.3. There exists C > 0 such that, for all A ∈ C2(B), there exists φ ∈ C3(B) verifying:

|A(x) − Alin(x) − ∇φ(x)| ≤ C|∇B|∞|x|2,

for x ∈ B and where Alin is defined by:

A
lin(x) =

1

2
B(0) ∧ x.

2.3 Local estimates for the lower bound

We now distinguish two cases: the balls inside Ω and those which intersect the boundary. For the

balls inside Ω, we are reduced to the problem of Dirichlet with constant magnetic field, and for the

other balls, to the problem of Neumann on an half plane with constant magnetic field.



72 Nicolas Raymond CUBO
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2.3.1 Study inside Ω

Let j such that Bj does not intersect the boundary.

We recall the inequality (problem of Dirichlet with constant magnetic field), for all ψ ∈ C∞0 (Ω):
∫

Ω

|(i∇ + qAlin)ψ|2dx ≥ q
∫

Ω

|ψ|2dx. (2.8)

Using Lemma 2.3 we make the change of gauge v 7→ e−iφv and with the classical inequality:

|a + b|2 ≥ (1 − λ2)|a|2 − 1
λ2

|b|2,

for λ > 0, we get:
∫

Ω

|(i∇ + q(A − ∇φ))(χjue−iφ)|2dx ≥
(
(1 − λ2)

∫

Ω

|(i∇ + qAlin)(χjue−iφ)|2dx

−C2q2 |∇B|
2
∞

λ2
r4

) ∫

Ω

|χju|2dx,

where Alin is defined in Lemma 2.3. Thus, we find with (2.8) applied to ψ = χjue
−iφ:

∫

Ω

|(i∇ + qA)χju|2dx ≥
(

(1 − λ2)q − C2q2 |∇B|
2
∞

λ2
r4

) ∫

Ω

|χju|2dx.

2.3.2 Study near the boundary

We refer to [HM02, HM04], but we will control carefully the uniformity. We first recall some

properties of the harmonic oscillator on an half axis (see [DH93, HM01]).

Harmonic oscillator on R+

For ξ ∈ R, we consider the Neumann realization hN,ξ in L2(R+) associated with the operator

− d
2

dt2
+ (t + ξ)2, D(hN,ξ) = {u ∈ B2(R+) : u′(0) = 0}. (2.9)

One knows that it has compact resolvent and its lowest eigenvalue is denoted µ(ξ); the associated

L2-normalized and positive eigenstate is denoted uξ and is in the Schwartz class. The function

ξ 7→ µ(ξ) admits a unique minimum in ξ = ξ0 and we let: Θ0 = µ(ξ0).
We now introduce local coordinates near the boundary in order to compare with the harmonic

oscillator on R+:

Local coordinates near the boundary

Let’s assume that 0 ∈ ∂Ω. In a neighborhood V of 0, we take local coordinates (y1,y2) on ∂Ω
(via a C3 map φ). We denote N(φ(y1,y2)) the interior unit normal to the boundary at the point
φ(y1,y2) and define local coordinates in V :

Φ(y1,y2,y3) = φ(y1,y2) + y3N(φ(y1,y2)).



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Uniform Spectral Estimates for Families ... 73

More precisely, for a point x ∈ V , φ(y1,y2) is the projection of x on ∂Ω ∩ V and y3 = d(x,∂Ω).
Taking a convenient map φ, we can assume

Φ(0) = 0 and D0Φ = Id.

Let j such that Bj ∩ ∂Ω 6= ∅. We can assume that xj ∈ ∂Ω and xj = 0 without loss of generality.
After a change of variables, we have:

∫

Ω

|(i∇ + qA)χju|2dx =
∫

y3>0

|(i∇y + qÃ)χ̃ju|2(DΦ)−1((DΦ)−1)t| det(DΦ)|dy,

where the tilde denotes the functions in the new coordinates and

à = DyΦ(A(Φ(y))).

There exists C > 0 (uniform in j) such that:
∫

y3>0

|(i∇y + qÃ)χ̃ju|2(DΦ)−1((DΦ)−1)t | det(DΦ)|dy ≥ (1 − Cr)
∫

y3>0

|(i∇y + qÃ)χ̃ju|2dy.

We use again the approximation by the constant magnetic field (for semi-balls) on the support of

χ̃j .

More precisely, there exists α > 0 uniform in j, such that

supp(χ̃j ) ⊂ B(xj,αr).

Then, we change our partition of unity: we replace the balls which intersect the boundary by

Φ(B(xj,αr)). There exists C > 0 such that for all j, there exists Ã
lin

(defined in Lemma 2.3)

satisfying:
∫

y3>0

|(i∇y + qÃ)χ̃ju|2dy ≥ (1 − λ2)
∫

y3>0

|(i∇y + qÃ
lin

)χ̃ju|2dy

−C
2q2

λ2
|∇B̃|2∞r4

∫

Ω

|χ̃ju|2dy,

where B̃ = ∇y × Ã. In order to express B̃ as a function of B, we need the following lemma:

Lemma 2.4. With the previous notations, we have:

B̃ = det(DΦ)((DΦ)−1)tB.

Proof.

The result is standard. Let us recall the proof for completness. Let us introduce the 1-form ω:

ω = A1dx1 + A2dx2 + A3dx3.

In the new coordinates x = Φ(y), we have, with the previous notations:

ω = Ã1dy1 + Ã2dy2 + Ã3dy3.



74 Nicolas Raymond CUBO
12, 1 (2010)

Then, it remains to write:

dω = (∇ × A)1dx2 ∧ dx3 + (∇ × A)2dx1 ∧ dx3 + (∇ × A)3dx1 ∧ dx2,

and to express dxi as a function of (dyj ).

The comatrix formula gives the conclusion.

�

We are reduced to the case of constant magnetic field (of intensity q) on R3+ = {y3 > 0} (see
[HM02, HM04, LP00]) and we get:

∫

y3>0

|(i∇y + qÃ)χ̃ju|2dy ≥ Θ0q
∫

y3>0

|χ̃ju|2dy.

Thus, we find:
∫

y3>0

|(i∇y + qÃ)χ̃ju|2dy ≥
(

(1 − λ2)qΘ0 −
C2q2

λ2
(1 + |∇B|2∞)r4

) ∫

y3>0

|χ̃ju|2dy.

2.3.3 End of the proof

We take, for 0 < ǫ <
1

2
, r = 1

q1/2−ǫ
. We divide by q and we choose λ such that:

λ2 =
q

λ2
(1 + |∇B|2∞)r4.

Then, the previous estimates lead to the existence of C > 0 and q0 > 0 depending only on Ω such

that for all q ≥ q0:
qA(u)

q
≥

(
qΘ0 − C

(
r + λ2 +

1

qr2

)) ∫

Ω

|u|2dx.

We finally find, by the minimax principle:

µ(q, A)

q
≥ Θ0 − C

(
1

q1/2−2ǫ
+

1

q2ǫ
+

|∇B|∞
q1/2−2ǫ

)
.

3 Upper Bound

In this section, we give a proof of Theorem 1.2. We refer to [HM04] and, in the next lines, we

emphasize the crucial points where uniformity is concerned. In the case of the constant magnetic

field on R3+, we know (cf. [HM02, LP00]) that the bottom of the spectrum is minimal when the

magnetic field is tangent to the boundary. So, we will look for a quasimode localized near a point

where the magnetic field is tangent. Then, we will take as trial function some truncation of uξ0 .

So, we fix x0 ∈ ∂Ω such that: B(x0) · ν = 0. Such a x0 exists; indeed, noticing that div(B) = 0,
the Stokes formula gives: ∫

∂Ω

B · νdσ =
∫

Ω

div(B)dx = 0.



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Uniform Spectral Estimates for Families ... 75

We take δ ∈]0, 1/2[ and we suppose that u is such that

supp(u) ⊂ B(x0,αr),

with r = 1
qδ

.

We assume that the support of u is small enough and after a change of coordinates, we can use

the same arguments as in Lemma 2.3 and take a gauge in which A satisfies:

|A(y) − A0(y)| ≤ C|B|C2|y|3,

where A0 = Alin + R, with R = (R1,R2,R3) and Rj homogeneous polynomial of order 2 and

where ∇2B denotes the hessian matrix of B.
We find:

qA(u) ≤ (1 + Cr)(qA0 (u) + C(|B|2C2r6q2‖u‖2 + qr3|B|C2‖u‖qA0 (u)1/2)).
We let:

u(y) = q1/4+δe−iξ0y2q
1/2

uξ0 (q
1/2y3)χ(4q

δy3)χ(4q
δ(y21 + y

2
2)

1/2).

We have to compare: qA0 (u) and qAlin (u). We get:

qA0 (u) ≤ qAlin (u) + Cq2r4(1 + |∇B|2∞)‖u‖2

+2ℜ
{∫

|y|≤αr,y3>0
(i∇ + qAlin)u · (qA0 − qAlin)udy

}
.

We have to estimate the double product (cf. [HM04, section 6, p. 120]):
∣∣∣∣∣ℜ

{∫

|y|≤αr,y3>0
(i∇ + qAlin)u · (qA0 − qAlin)udy

}∣∣∣∣∣ ≤ C(1 + |∇B|∞)q
1−δ.

Moreover, using the exponential decrease of uξ0 , we have:

q
A

lin (u) ≤ Θ0q + Cq2δ.

4 Agmon’s Estimates

In order to estimate the asymptotic localization of the first eigenfunctions in the applications, we

will need Agmon’s estimates to have some exponential decrease inside Ω; that is the aim of this

section. Let us introduce some notations (see [Agm82, Alm08, FH08]). For γ > 0 small enough,

let ηγ be a smooth cutoff function such that:

ηγ =

{
1 if d(y) = d(y,∂Ω) ≥ γ
0 if y /∈ Ω

,

with

|∇ηγ| ≤
C

γ
.



76 Nicolas Raymond CUBO
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We let: Ωγ = {y ∈ Ω : d(y,∂Ω) ≥ γ}. For α > 0, we let

ξ(y) = ηγe
αd(y).

We finally denote µ0(q, A) the lowest eigenvalue of the Dirichlet’s realization of (i∇ + qA)2 on Ω.
We have the following localization property:

Proposition 4.1. There exists C > 0 and γ0 > 0, depending only on Ω such that for all 0 < ǫ ≤ 1
and α verifying:

0 < α <

(
1

1 + ǫ

)1/2
(µ0 − µ)1/2,

and for all 0 < γ ≤ γ0, if u is a normalized mode associated with the lowest eigenvalue µ = µ(q, A)
of the Neumann realization of (i∇ + qA)2, then:

‖ηγeαd(y)|u|‖H1(Ω) ≤
C

√
ǫγ

(
µ0 + 1

µ0 − µ − (1 + ǫ)α2
)1/2

eαγ.

Proof.

We consider the equation verified by u:

(i∇ + qA)2u = µu.

One multiplies by ξ2u and integrate by parts (using (i∇ + qA)u · ν = 0 on ∂Ω) to get:

|(i∇ + qA)(ξu)|22 = µ|ξu|22 + |(∇ξ)u|22.

We have, for all ǫ > 0:

|(∇ξ)u|2 ≤ (1 + 1
ǫ
)

∫

Ω\Ωγ
|∇η|2e2αd(y)|u|2 + (1 + ǫ)α2

∫

Ω

|ξu|2.

We use that |u|2 = 1 to find:

|ξu|2 ≤ C
γ

(1 +
1

ǫ
)

e2αγ

µ0 − µ − (1 + ǫ)α2
.

Moreover, the diamagnetic inequality gives:

|∇|ξu||2 ≤ |(i∇ + qA)(ξu)|2.

It follows that:

||ξ|u|||2H1(Ω) ≤
C

γ
(1 +

1

ǫ
)e2αγ

µ0 + 1

µ0 − µ − (1 + ǫ)α2
.

�

5 Applications

We now describe two applications of our main results.



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Uniform Spectral Estimates for Families ... 77

5.1 Application to an helical vector field

In this section, we study µ∗(q,τ) defined in (1.4) as qτ → +∞ and τ → +∞. Due to the definition
(1.4), we will use the uniform analysis of µ(q, n) with n ∈ C(τ).

5.1.1 Estimate of the first eigenvalue

The main theorem in this section is the following:

Theorem 5.1. Let c0 > 0 and 0 ≤ x < 12 . There exists C > 0 and q0 > 0 depending only on Ω,
c0 and x such that, if (q,τ) verifies qτ ≥ q0 and

τ ≤ c0(qτ)x, (5.10)
then:

Θ0 −
C

(qτ)1/4−x/2
≤ µ

∗(q,τ)

qτ
≤ Θ0 +

C

(qτ)1/3−2x/3
. (5.11)

Remark 5.2.

This statement was obtained for x = 0 in [HP07] and rough estimates where given in [BCLP02] as
τ

q
→ 0.

�

Proof.

Let us notice that:

µ(qτ,
Qnτ Q

t

τ
) = µ(q,Qnτ Q

t).

Moreover, there exists C > 0 such that for all τ > 0 and n ∈ C(τ), if A = n
τ

and B = ∇× A, then:

|B|∞ = 1,
|∇B|∞ ≤ Cτ,
|∇2B|∞ ≤ Cτ2.

For the lower bound, we apply Theorem 1.1 to the subfamily C(τ) of A and, using (5.10), we get:
µ∗(q,τ)

qτ
≥ Θ0 − C

(
1

(qτ)1/2−2ǫ
+

1

(qτ)2ǫ
+ c0

(qτ)x

(qτ)1/2−2ǫ

)
.

We choose ǫ such that 1
2
− 2ǫ − x = 2ǫ, i.e: 2ǫ = 1

4
− x

2
.

For the upper bound, we apply Theorem 1.2 with A =
nτ

τ
. Then, we get:

µ∗(q,τ) ≤ Θ0qτ + C((qτ)2δ + (qτ)2−4δ+2x + (qτ)1−δ+x + (qτ)3/2−3δ+2x + (qτ)2−6δ+4x).
We choose δ such that:

2δ = 1 − δ + x.
Thus, we take δ = 1+x

3
and the upper bound follows.

�



78 Nicolas Raymond CUBO
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5.1.2 Localization of the ground state near the boundary as τ → + ∞

We first state a proposition:

Proposition 5.3. For A ∈ A (cf. (1.1)), we denote µ0(q, A) the bottom of the spectrum of the
Dirichlet realization of (i∇ + qA)2.
Then, for all ǫ ∈]0, 1/2[, there exists C = C(Ω,ǫ) > 0 and q0 = q0(Ω,ǫ) s.t. if q ≥ q0, for all
A ∈ A:

µ0(q, A)

q
≥ 1 − C

(
1

q2ǫ
+

|∇B|∞
q1/2−2ǫ

)
.

Proof.

We again use partition (2.5), formula (2.7) and the proof is the same as for Theorem 1.1.

�

We deduce:

Corollary 5.4. Let c0 > 0. For all x ∈ [0, 1/2[, there exists C = C(Ω,x,c0) > 0, such that, for
all n ∈ C(τ) and (q,τ) such that τ ≤ c0(qτ)x:

µ0(qτ,
n

τ
)

qτ
≥ 1 − C

(
1

qτ

)1/4−x/2
.

As an immediate consequence of Proposition 4.1, we have the following theorem:

Theorem 5.5. For all x ∈ [0, 1/2[, there exists δ0 > 0, C > 0, c > 0 such that if (q,τ) verifies
qτ ≥ δ0 and τ ≤ c0(qτ)x, then for all n ∈ C(τ) and u a L2-normalized solution of

(i∇ + qn)2u = µ(q, n)u, in Ω
(i∇ + qn)u · ν = 0, on ∂Ω

we have:

‖η c√
qτ
e((1−Θ0)

1/2√qτ −r(qτ ))d(·,∂Ω)|u|‖H1(Ω) ≤ C,

where r(qτ) = (qτ)3/8+x/4.

Proof.

We apply Proposition 4.1 with ǫ = (qτ)−1/4+x/2, α = (1 − Θ0)1/2(
√
qτ − λ), γ = 1√

qτ
, λ =

(qτ)1/4+x1/2, with x < x1 <
1
2

and we notice, by taking a truncated gaussian, that

µ0(qτ,
n

τ
) ≤ Cqτ.

We finally take x1 =
1
2
( 1
2

+ x) and apply the estimate (5.11) of Theorem 5.1.

�



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Uniform Spectral Estimates for Families ... 79

5.2 Surface superconductivity in large domains

5.2.1 The problem

Let R > 0 and x0 ∈ Ω. We denote ΩR = {x0 + R(x − x0), x ∈ Ω}. The aim of this part is
to study the behaviour of µΩR (q, A), where A ∈ A (cf.(1.1)) as q → +∞ and R → +∞. In his
work [Alm08], Almog studies the regime q fixed and R → +∞ (see Theorem 1.1 in [Alm08]) and
then makes q to tend to infinity (see Section 3 of [Alm08]). In this section, we give a theorem

which treats another regime : q → +∞ and R with polynomial increase in q. We first observe the
following scaling invariance:

Lemma 5.6. Let R > 0. We have:

µΩR (q, A) =
1

R2
µΩ

(
qR2,

A(R·)
R

)
.

5.2.2 Estimate of the lowest eigenvalue

Theorem 5.7. Let c0 > 0 and y ≥ 0. There exists C>0, q0 > 0 and R0 > 0 depending only on Ω,
c0 and y such that, if (q,R) satisfies q ≥ q0, R ≥ R0 and R ≤ c0qy, then, for A ∈ A:

Θ0 −
C

(qR2)
1

4(1+2y)

≤ µΩR (q, A)
q

≤ Θ0 −
C

(qR2)
1

3(1+2y)

.

Proof.

We use Lemma 5.6. We let x =
y

1 + 2y
and notice that R ≤ c(y)(qR2)x with c(y) = c

1
1+2y

0 and

A(R·)
R

∈ A. Then by the Theorems 1.1 and 1.2, we have the wished conclusion by using the same
arguments as for Theorem 5.1.

�

5.2.3 Localization of the groundstate near the boundary in large domains

In the case of large domains, we prove a quite analogous theorem with Theorem 5.5:

Theorem 5.8. For all y ≥ 0, there exists δ0 > 0, δ1 > 0, C > 0, c > 0 such that if (q,R) verifies
q ≥ δ0, R ≥ δ1 and R ≤ c0qy, then for all A ∈ A and u a L2-normalized solution of

(i∇ + qA)2u = µΩR (q, A)u, in ΩR
(i∇ + qA)u · ν = 0, on ∂ΩR

we have,

‖η c√
q
e(1−Θ0)

1/2(
√

q−r(q,R))d(·,∂ΩR)|u|‖H1(ΩR) ≤ C,

where r(q,R) = q
1/2− 1

8(1+2y) R
− 1

4(1+2y) .



80 Nicolas Raymond CUBO
12, 1 (2010)

Proof.

After a rescaling, the proof is the same as the one of Theorem 5.5.

�

Received: October, 2008. Revised: October, 2009.

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