CUBO A Mathematical Journal
Vol.11, No¯ 05, (1–22). December 2009

On Some Spectral Problems and Asymptotic Limits

Occuring in the Analysis of Liquid Crystals

Bernard Helffer

Département de Mathématiques, Univ Paris-Sud and CNRS,

91405 Orsay cedex, France

email: Bernard.Helffer@math.u-psud.fr

and

Xing-Bin Pan

Department of Mathematics, East China Normal University,

Shanghai 200062, P.R. China

email : xbpan@math.ecnu.edu.cn

ABSTRACT

On the basis of de Gennes’ theory of analogy between liquid crystals and superconductivity, the

second author introduced the critical wave number Qc3 of liquid crystals, which is an analog

of the upper critical field Hc3 for superconductors, and he predicted the existence of a surface

smectic state, which was supposed to be an analog of the surface superconducting state. In

this article we study this problem and our study relies on the Landau-de Gennes functional

of liquid crystals in connection with a simpler functional called the reduced Ginzburg-Landau

functional which appears to be relevant when some of the elastic constants are large. We

discuss the behavior of the minimizers of these functionals. We describe briefly some results

obtained by Bauman-Carme Calderer-Liu-Phillips, and present more recent results on the

reduced Ginzburg-Landau functional obtained by the authors. This paper is partially extracted

of lectures given by the first author in Recife and Serrambi in August 2008.



2 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

RESUMEN

Sobre la base de teoria de Gennes, la analogía entre cristales liquidos y superconductividad,

el segundo autor introdujo el número de onda crítico Qc3 de cristales liquidos, el cual es un

análogo del campo crítico superior Hc3 para superconductores, él predijo la existencia de una

superficie “smetic state”, la qual fué supuesta ser un análogo de la superficie de estado de

supercondutividad. En este artículo estudiamos este problema y nuestro estudio se basa en

el funcional de Landau-de Gennes de cristales liquidos en conexión con un simple funcional

llamado el funcional de Ginzburg-Landau reduzido que resulta ser relevante cuando algunas

de las constantes elasticas son grandes. Nosotros discutimos el comportamiento de los min-

imizadores de esos funcionales. Describimos brevemente algunos resultados obtenidos por

Bauman-Carme Calderer-Liu-Phillips, y presentamos resultados mas recientes sobre el fun-

cional de Ginzburg-Landau reduzido obtenidos por los autores. Este artículo es parcialmente

extraido de las conferencias dadas por el primier autor en Recife y Serrambi en Agosto 2008.

Key words and phrases: Liquid crystals, surface smectic state, Landau-de Gennes functional,

reduced Ginzburg-Landau functional, critical wave number, critical elastic coefficients.

Math. Subj. Class.: 82D30, 82D55, 35J55, 35Q55.

1 Introduction

In [P1], based on the de Gennes analogy between liquid crystals and superconductivity [dG1, dGP],

one of the authors (X. Pan) introduced the critical wave number Qc3 (which is an analog of the

upper critical field Hc3 for superconductors) and predicted the existence of a surface smectic state,

which was supposed to be an analog of the surface superconducting state. The phase transition

between the nematic state and smectic state of liquid crystals as the wave number varies around

Qc3 was studied in [P1]. Bauman, Calderer, Liu and Phillips [BCLP] studied the phase transition

of liquid crystals for a different set of parameters1. In this article (which is partially extracted

of lectures given by one of the authors (B. Helffer) in Recife and Serrambi in August 2008), we

study this problem which relies on the Landau-de Gennes functional (modeling the properties

of liquid crystals) in connection with a simpler functional called the reduced Ginzburg-Landau

functional which appears to be relevant when some of the elastic constants are large. We discuss the

behavior of the minimizers. We describe mainly some results obtained by Bauman-Carme Calderer-

Liu-Phillips [BCLP], Pan [P1, P4, P5] and more recent results on the reduced Ginzburg-Landau

functional obtained in [HP2, HP3]. We also add a few new results.

All these results suggest that a liquid crystal with large Ginzburg-Landau parameter κ will

be in the surface smectic state if the number qτ lies asymptotically between κ2 and κ2/Θ0 with

κ → ∞, where Θ0 ∈ (0, 1) is the lowest eigenvalue of the Schrödinger operator with a unit magnetic

field in the half plane.

1In particular [P1] considers the case where K2 + K4 = 0 and in [BCLP] it is assumed that c0 ≤ K2 + K4 ≤ c1
where c0 and c1 are fixed positive constants.



CUBO
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On Some Spectral Problems and Asymptotic ... 3

This is also a natural extension of what was done by Fournais and Helffer in superconductivity

in continuation of previous results of Lu-Pan, Bernoff-Sternberg, Helffer-Morame and many others

(See [FH4] and references therein).

We will only present some of the results, emphasize on some points, recall some basic material

and refer to the original papers or works in progress for more details and proofs.

2 Some Questions in the Theory of Liquid Crystals and First

Answers

2.1 The model

The model in liquid crystals can be described2 by the functional

(ψ, n) 7→ EK[ψ, n] =

∫

Ω

{
|∇qnψ|

2 − κ2|ψ|2 +
κ2

2
|ψ|4 + K1 |div n|

2

+ K2 |n · curl n + τ|
2

+ K3 |n × curl n|
2
}
dx,

where:

• Ω ⊂ R3 is the region occupied by the liquid crystal,

• ψ is a complex-valued function called the order parameter,

• n is a real vector field of unit length called director field,

• q is a real number called wave number,

• ∇qn is the magnetic gradient: ∇qn = ∇ − iqn ,

• τ is a real number measuring the chiral pitch,

• K = (K1,K2,K3) with K1 > 0, K2 > 0 and K3 > 0 is the triple of the elastic coefficients or

Frank coefficients,

• κ > 0 depends on the material and on temperature and is called the Ginzburg-Landau

parameter of the liquid crystal.

This functional is called the Landau-de Gennes functional. We are interested in minimizing

the functional over the pairs (ψ, n) ∈ H1(Ω, C)×V (Ω, S2), where H1(Ω, C) is the standard Sobolev

space for complex-valued functions, V (Ω, S2) consists of vector fields n such that div n ∈ L2(Ω),

curl n ∈ L2(Ω, R3) and |n(x)|2 = 1 almost everywhere. We refer to [C], [BCLP] and [P1, P4, P6]

2This is an already simplified model where boundary terms (see [BCLP, P1]) have been eliminated. With the
notations of these authors, we are considering as in [HP2] the case K2 + K4 = 0.



4 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

for a more complete discussion of the mathematical issues and a discussion of the contents of the

references to the physics literature [dG1, dG2, dGP]. We only mention here that the physical

interpretation is that n is the molecular director field and that, if we write

ψ(x) = ρ(x)eiφ(x) ,

where ρ(x) ≥ 0 and φ(x) is a real function, we recover the molecular mass density by

δ(x) = ρ0(x) + ρ(x) cos φ(x) ,

where ρ0(x) is some given reference density.

Observing that we have the lower bound

EK[ψ, n] ≥ −
κ2

2
|Ω|, (2.1)

it is not too difficult to show that this functional admits minimizers. But the main questions are

then:

• What is the minimum of the energy ?

• What is the nature of the minimizers ?

Of course the answer depends heavily on the various parameters and we will only be able to

give answers in some asymptotic regimes.

As in the theory of superconductivity, a special role will be played by some critical points of

the functional, the pairs (0, n) , where n is a minimizer of the so called Oseen-Frank functional:

n 7→ EKOF [n] :=

∫

Ω

{
K1 |div n|

2
+ K2 |n · curl n + τ|

2
+ K3 |n × curl n|

2
}
dx.

These special solutions are called “nematic phases” and one is naturally asking if they are minimizers

or local minimizers of the functional EK.

In any case, the minimizers of EK satisfy some Euler-Lagrange equation. We do not write the

complete system but note that the variation with respect to the order parameter leads to

−∇2qnψ − κ
2ψ + κ2|ψ|2ψ = 0 in Ω , (2.2)

together with the boundary condition

ν · ∇qnψ = 0 on ∂Ω , (2.3)

where ν denotes the normal to the boundary.

Using the maximum principle for u = |ψ|2 which can be seen as a solution of




−∆u + κ2u(1 − u) ≥ 0 in Ω ,

∂u

∂ν
= 0 on ∂Ω ,

one can show that

‖ψ‖L∞(Ω) ≤ 1 . (2.4)



CUBO
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On Some Spectral Problems and Asymptotic ... 5

2.2 A universal upper bound

For τ > 0, let us consider C(τ) the set of the S2-valued vector fields satisfying:

curl n = −τn , div n = 0 .

It has been shown in [BCLP] that C(τ) consists of the vector fields NQτ such that, for some Q ∈

SO(3) ,

N
Q
τ (x) ≡ QNτ(Q

tx) , x ∈ Ω , (2.5)

where

Nτ (y1,y2,y3) = (cos(τy3), sin(τy3), 0) , y ∈ R
3 . (2.6)

This is also equivalent, as |n|2 = 1, to

div n = 0 , n · curl n + τ = 0 , n × curl n = 0 . (2.7)

So the last three terms in the functional EK vanish if and only if n ∈ C(τ) .

As a consequence, if we denote by

C(K1,K2,K3,κ,q,τ) = inf
(ψ,n)∈H1(Ω,C)×V (Ω,S2)

EK[ψ, n] , (2.8)

the infimum of the energy over the natural maximal form domain of the functional, then

C(K1,K2,K3,κ,q,τ) ≤ c(κ,q,τ) , (2.9)

where

c(κ,q,τ) = inf
n∈C(τ)

inf
ψ∈H1(Ω,C)

Gqn[ψ] , (2.10)

and Gqn[ψ] is the so called reduced Ginzburg-Landau functional which will be defined in the next

subsection.

2.3 Reduced Ginzburg-Landau functional

Given a vector field A, the reduced Ginzburg-Landau functional GA with magnetic potential A is

defined on H1(Ω, C) by

ψ 7→ GA[ψ] =

∫

Ω

{|∇Aψ|
2 − κ2|ψ|2 +

κ2

2
|ψ|4} dx. (2.11)

The standard Ginzburg-Landau functional with external vector field σHe (where He = curl F is a

divergence free vector field on Ω) (see [FH4] and references therein) takes the form

EGL[ψ, A] = GκσA[ψ] + κ
2σ2

∫

R3

|curl A − He|2 dx (2.12)

where



6 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

• Ω is a bounded and simply connected domain,

• (ψ, A) ∈ H1(Ω, C) × Ḣ1
div,F(R

3, R3),

• Ḣ1
div,F(R

3, R3) = {A | div A = 0 , A − F ∈ Ḣ1(R3, R3) } ,

• Ḣ1(R3) denotes the homogeneous Sobolev space, i.e. the closure of C∞0 (R
3
) under the norm

u 7→ ‖∇u‖L2(R3) , and Ḣ
1
(R

3, R3) denotes the corresponding space of vector fields.

So the Oseen-Frank energy in liquid crystals theory replaces the magnetic energy measuring

the square of the L2 norm of curl A − He in R3 in Ginzburg-Landau theory.

For convenience, we also write GA[ψ] as G[ψ, A]. So we have

c(κ,q,τ) = inf
n∈C(τ),ψ∈H1(Ω,C)

G[ψ,qn] ≤ 0 , (2.13)

and if n ∈ C(τ) , then the following equality holds

EK[ψ, n] = G[ψ,qn] . (2.14)

3 A Limiting Case: The Case of Large Frank Constants

We have seen that in full generality (2.9) holds. Conversely, it can be shown (see [BCLP, P1, HP2])

that when the elastic parameters tend to +∞, the converse is asymptotically true.

Proposition 3.1.

lim
K1,K2,K3→+∞

C(K1,K2,K3,κ,q,τ) = c(κ,q,τ) . (3.1)

So c(κ,q,τ) is a good approximation for the minimal value of EK for large Kj’s.

Of course, a basic initial remark for the proof is that if (ψ, n) is a minimizer EK then we

always have

EKOF [n] ≤
κ2|Ω|

2
. (3.2)

Remark 3.2.

In [BCLP], the authors used instead the bound

EKOF [n] ≤ C(Ω)q
2τ2 , (3.3)

which is obtained from the universal upperbound

C(K1,K2,K3,κ,q,τ) ≤ C(Ω)q
2τ2 −

κ2|Ω|

2
. (3.4)

This upper bound is obtained (see Lemma 1 in [BCLP]) by computing the energy of the pair (ψ, n) =

(eiqx·Nτ (x), Nτ(x)), with Nτ defined in (2.6).



CUBO
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On Some Spectral Problems and Asymptotic ... 7

This gives the two following controls

‖div n‖2L2(Ω) ≤
κ2|Ω|

2K1
, (3.5)

and

‖curl n − τn‖2L2(Ω) ≤
κ2|Ω|

2 min{K2,K3}
. (3.6)

We also have

‖∇qnψ‖
2
L2(Ω) ≤

κ2|Ω|

2
. (3.7)

Remark 3.3.

1. This limiting case where all the elastic coefficients tend to +∞ appears naturally in the

transition from smectic-C to nematic phase (see [dG2]).

2. As observed by D. Phillips in a conference in Ryukoku University in Japan, it would be also

interesting to have the result with fixed K1 > 0, in the limit when K2 and K3 tend to +∞. A

modification of the proof in [HP2] leads indeed (see Lemma 3.4) to this result. For parameters

in different regime (in particular c0 < K2 + K4 < c1), the same conclusion was obtained in

[BCLP].

3. An interesting open problem is to control the rate of convergence in (3.1) (see [Ray3]).

Proof of Proposition 3.1

To illustrate the last point of the remark and to complete the proof of the proposition, we

need the following result 3

Lemma 3.4.

Let τ0 > 0 and C0 > 0. Then for any ǫ > 0, there exists α > 0 such that if n ∈ V (Ω, S
2
),

τ ∈ (0,τ0], and

‖curl n + τn‖L2(Ω) ≤ α,

‖div n‖L2(Ω) ≤ C0τ ,

then there exists Q ∈ SO(3) such that

‖n − NQτ ‖L4(Ω) ≤ ǫ . (3.8)

Proof. We give the proof in the case of a fixed τ > 0.

If it were not true, we will find ǫ0 > 0 and a bounded sequence nj ∈ V (Ω, S
2
) such that

‖div nj‖L2(Ω) is bounded,

lim
j→+∞

‖curl nj + τnj‖L2(Ω) = 0 ,

3The same conclusion was proved in [BCLP] (Lemma 4) in the case where K2 + K4 ≥ c0 > 0 and hence
‖∇n‖2

L2(Ω)
can be controlled by the energy.



8 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

and

inf
Q∈SO(3)

‖nj − N
Q
τ ‖L4(Ω) ≥ ǫ0 .

From the assumptions, it is clear that the sequence is bounded in V (Ω, S2), and hence bounded

in H1
loc

(Ω, R3) (see [P4], Lemma 2.3), hence we can extract a subsequence, still denoted by nj,

and find n∞ such that nj tends weakly to n∞ in H
1
loc

(Ω, R3). One can also show that |n∞|
2

= 1

a.e. in Ω and that curl n∞ + τn∞ = 0. So n∞ belongs to C(τ) and there exists Q ∈ SO(3) such

that n∞ = N
Q
τ . Now by compactness of the injection of H

1
loc

(Ω, R3) in L4
loc

(Ω, R3), we get that nj

tends to n∞ in L
4
loc

(Ω, R3). Let D be a compact subset of Ω such that |Ω \ D| < ǫ0/48. For large

j we have ∫

D

|nj − n∞|
4dx <

ǫ40
3
.

Then ∫

Ω

|nj − n∞|
4 dx ≤

∫

D

|nj − n∞|
4 dx + 16|Ω \ D| <

2ǫ40
3
,

which leads to a contradiction.

The control of the rate of convergence in Proposition 3.1 should pass through a good knowl-

edge of α(ǫ).

The second step in the proof of Proposition 3.1 consists in observing that, if (ψ, n) is a

minimizer, we can, for any Q ∈ SO(3) , get the lower bound

G[ψ,qn] = ‖∇qnψ‖
2
L2(Ω) − κ

2‖ψ‖2L2(Ω) +
κ2

2
‖ψ‖4L4(Ω)

≥ (1 − η)‖∇
qN

Q
τ
ψ‖2L2(Ω) − κ

2‖ψ‖2L2(Ω) +
κ2

2
‖ψ‖4L4(Ω)

−
q2

η
|Ω|1/2‖n − NQτ ‖

2
L4(Ω) .

(3.9)

Here we have used Cauchy-Schwarz inequality and (2.4). This we can rewrite, using (3.7), in the

form

G[ψ,qn] ≥ G[ψ,qNQτ ] −
η

2
κ2|Ω| −

q2

η
|Ω|1/2‖n − NQτ ‖

2
L4(Ω) . (3.10)

This gives

G[ψ,qn] ≥ c(κ,q,τ) −
η

2
κ2|Ω| −

q2

η
|Ω|1/2‖n − NQτ ‖

2
L4(Ω) . (3.11)

Putting together the estimates, we obtain

EKOF (n) ≤
η

2
κ2|Ω| +

q2

η
|Ω|1/2‖n − NQτ ‖

2
L4(Ω) , (3.12)



CUBO
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On Some Spectral Problems and Asymptotic ... 9

and
c(κ,q,τ) ≥ C(K1,K2,K3,κ,q,τ)

≥ c(κ,q,τ) −
η

2
κ2|Ω| −

q2

η
|Ω|1/2‖n − NQτ ‖

2
L4(Ω) .

(3.13)

These estimates are valid for any η > 0 and any Q ∈ SO(3) . The remainder on the right hand

side can be chosen arbitrarily small by choosing first η small. Then we can use (3.5) and (3.6) to

have ‖curl n + τn‖L2(Ω) and ‖div n‖L2(Ω) small and using Lemma 3.4 (and a good choice of Q) to

have ‖n − NQτ ‖L4(Ω) small.

4 Analysis of the Reduced Ginzburg-Landau Functional

We now analyze the non-triviality of the minimizers realizing c(κ,q,τ). As for the Ginzburg-

Landau functional in superconductivity, this question is closely related to the analysis of the lowest

eigenvalue λN1 (qn) of the Neumann realization of the magnetic Schrödinger operator −∇
2
qn in Ω

that we met already when describing the Euler-Lagrange equation associated to the functional.

Namely λN1 = λ
N
1 (qn) is the lowest eigenvalue of the following problem

{
−∇2qnφ = λ

N
1 φ in Ω ,

ν · ∇qnφ = 0 on ∂Ω ,
(4.1)

where ν is the unit outer normal of ∂Ω.

But the new point is that we will minimize λN1 (qn) over n ∈ C(τ). So we shall actually meet

the quantity:

µ∗(q,τ) = inf
n∈C(τ)

λN1 (qn) . (4.2)

We preliminarily observe the

Lemma 4.1.

If (ψ,qn) is a nontrivial minimizer of G, then G[ψ,qn] < 0 .

The proof is simple. ψ is a solution of the (Euler-Lagrange) equation (2.2) with Neumann

condition (2.3). Multiplying (2.2) by ψ and integrating over Ω, we have, after an integration by

parts and taking account of the boundary condition (2.3),

∫

Ω

{|∇qnψ|
2 − κ2(1 − |ψ|2)|ψ|2} dx = 0 , (4.3)

and hence

c(κ,q,τ) = Gqn[ψ] = −
κ2

2

∫

Ω

|ψ|4 dx < 0 . (4.4)

This gives:
1

2
(µ∗(q,τ) − κ

2
)‖ψ‖2L2(Ω) ≤

1

2
(λN1 (qn) − κ

2
)‖ψ‖2L2(Ω) = c(κ,q,τ) . (4.5)



10 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

Hence, when c(κ,q,τ) < 0, we should have µ∗(q,τ) < κ
2. Pushing forward, one has the main

comparison statement (analogous to a statement in Fournais-Helffer [FH3] for surface supercon-

ductivity) in [HP2]:

Proposition 4.2.

−
κ2|Ω|

2
[1 − κ−2µ∗(q,τ)]

2 ≤ c(κ,q,τ) (4.6)

and

c(κ,q,τ) ≤ −
κ2

2
[1 − κ−2µ∗(q,τ)]

2
+ sup

n∈C(τ)

sup
φ∈Sp(qn)

(
∫
Ω
|φ|2 dx)2∫

Ω
|φ|4 dx

, (4.7)

where Sp(qn) is the eigenspace associated to λN1 (qn), and [ · ]+ denotes the positive part of the

enclosed quantity.

For the upper bound, we can minimize G[ψ,qn] over the pairs (ψ, n) with ψ = tψqn, where

ψqn is an eigenfunction of −∇
2
qn, t ∈ C, and n ∈ C(τ).

For the lower bound, we have just to use the Hölder inequality and (4.4) and (4.5).

This completes the (sketch of the) proof that c(κ,q,τ) is strictly negative if and only if

µ∗(κ,τ) < κ
2.

5 Main Questions

As a consequence of Proposition 4.2, we obtain that the transition from a nematic phase to a

smectic phase is strongly related to the analysis of the solution of

1 − κ−2µ∗(q,τ) = 0 . (5.1)

This is a pure spectral problem concerning a family indexed by n ∈ C(τ) of Schrödinger operators

with magnetic field −∇2qn.

Remark 5.1.

In the analysis of (5.1), the monotonicity of q 7→ µ∗(q,τ) is an interesting open question (see

Fournais-Helffer [FH3] for the phase transition from normal state to surface superconducting state

of type II superconductors).

This will permit indeed to find a unique solution of (5.1) permitting a natural definition of the

critical value Qc3(κ,τ).

We hope to answer this question in [HP3] in the case of a strictly convex domain.

We have proved in [HP2] that if τ stays in a bounded interval, then Qc3(κ,τ) and µ∗(q,τ) can

be controlled in two regimes



CUBO
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On Some Spectral Problems and Asymptotic ... 11

• σ → +∞ ,

• σ → 0 ,

where

σ = qτ ,

which, as it appears already in [BCLP], is in some sense the leading parameter in the theory. We

mention that if we examine the magnetic Schrodinger operator −∇2qn, the parameter σ corresponds

indeed to the intensity of the magnetic field corresponding to the magnetic potential qn, with

n ∈ C(τ). This will be detailed in Sections 7 and 8.

6 A Simpler Question

A simpler question, which was first introduced in [P1], and partially solved in [P5] with the help

of [P3, HM4], but can be completed under the additional assumption below by a careful control

(see [HP2, HP3]) of the uniformity in the proof of [HM4], can be stated as follows:

Question 6.1.

Given a strictly convex open set Ω ⊂ R3, find the direction h of the constant magnetic field giving

asymptotically as σ → +∞ the lowest energy for the Neumann realization in Ω of the Schrödinger

operator with magnetic field σh.

Let us present shortly the answer to this question. We assume that

Assumption 6.2.

At each point of ∂Ω the curvature tensor has two strictly positive eigenvalues κ1(x) and κ2(x), so

that

0 < κ1(x) ≤ κ2(x) .

Under this assumption, the set Γh of boundary points where h is tangent to ∂Ω, i.e.

Γh := {x ∈ ∂Ω
∣∣ h · ν(x) = 0}, (6.1)

is a regular submanifold of ∂Ω:

dT (h · ν)(x) 6= 0 , ∀x ∈ Γh , (6.2)

where dT denotes the tangential gradient along Γh. The basic example where this assumption is

satisfied is the ellipsoid.

For any given h, let Fh be the magnetic potential such that

curl Fh = h and div Fh = 0 in Ω, Fh · ν(x) = 0 on ∂Ω .

We have the following two-term asymptotics of λN1 (σFh) of the Neumann Laplacian −∇
2
σFh

, (due

to Helffer-Morame-Pan [HM4, P3]).



12 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

Theorem 6.3.

If Ω and h are as above, then, as σ → +∞,

λN1 (σFh) = Θ0σ + γ̂hσ
2/3

+ O(σ2/3−η) , (6.3)

for some η > 0.

Moreover η is independent of h and the control of the remainder is uniform with respect to h.

Here Θ0 ∈ (0, 1), δ0 ∈ (0, 1) and ν̂0 > 0 are spectral quantities (see in the Appendix), and γ̂h

is defined by

γ̂h := inf
x∈Γh

γ̃h(x) , (6.4)

where

γ̃h(x) := 2
−2/3ν̂0δ

1/3
0 |kn(x)|

2/3
(
1 − (1 − δ0)|Th(x) · h|

2
)1/3

. (6.5)

Here Th(x) is the oriented, unit tangent vector to Γh at the point x and

kn(x) = |d
T
(h · ν)(x)| .

Note that the constant Θ0 has been denoted by β0 in [LuP1, LuP2, LuP3] and in [P1, P3, P4] etc.

Here is now the answer to the question 6.1. We have just to determine infh∈S2 γ̂h or equiva-

lently

inf
h∈S2

inf
x∈Γh

|kn(x)|
2/3

(
1 − (1 − δ0)|Th(x) · h|

2
)1/3

.

So everything is reduced to the analysis of the map

Γh ∋ x 7→ kn(x)
2
(
1 − (1 − δ0)|Th(x) · h|

2
)
.

As observed in the appendix of [HM3], where the comparison is done between the results of [P3]

and [HM4], this last expression can be written in the form

Γh ∋ x 7→ κ1(x)
2
cos

2 φ(x) + κ2(x)
2
sin

2 φ(x)

− (1 − δ0)(κ1(x) − κ2(x))
2
sin

2 φ(x) cos2 φ(x) ,

where, for x ∈ ∂Ω, φ(x) is defined by writing

h = cos φ(x)u1(x) + sin φ(x)u2(x) ,

with (u1(x), u2(x)) being the orthonormal basis of the curvature tensor at x, associated to the

eigenvalues κ1(x) and κ2(x).

We easily observe that, is 0 < κ1 ≤ κ2 the function

[0, 1] ∋ t 7→ κ21t + κ
2
2(1 − t) − (1 − δ0)(κ1 − κ2)

2t(1 − t) ,

admits a minimum at t = 1. Hence, when minimizing over h and x ∈ Γh, it is rather easy to show

that the infimum is obtained by first choosing a point x0 of ∂Ω where κ1(x) is minimum and then

taking h = u1(x0). This leads to the following proposition which was conjectured and partially

proved by X. Pan [P5] and then completed in [HP2].



CUBO
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On Some Spectral Problems and Asymptotic ... 13

Proposition 6.4.

Under Assumption 6.2, we have

inf
h∈S2

γ̂h = inf
x∈∂Ω

(κ1(x))
2/3 , (6.6)

hence

inf
h∈S2

λN1 (σFh) = Θ0σ + inf
x∈∂Ω

(κ1(x))
2/3σ2/3 + O(σ2/3−η) . (6.7)

This answers Question 6.1.

7 Semi-Classical Case: qτ Large

When looking at the general problem, various questions occur. The magnetic field −qτn (corre-

sponding when n ∈ C(τ) to the magnetic potential qn) is no more constant, so one should extend

the analysis to this case. A first analysis [HM4, P3, HP2] (semi-classical in spirit) gives:

Theorem 7.1.

As σ = qτ → +∞,

µ∗(q,τ) = Θ0 qτ + O((qτ)
2/3

) , (7.1)

where the remainder is controlled uniformly for 4 τ ∈ (0,τ0] .

This is a consequence of

λN1 (qn) = Θ0 qτ + O((qτ)
2/3

) , (7.2)

with O((qτ)2/3) uniform with respect to n ∈ C(τ) and τ ∈ (0,τ0] .

The reader could be astonished to have this uniformity. The first thing is to observe that,

when Ω has no holes, it is not the magnetic potential A = qn which is important in the analysis

of the Neumann groundstate energy of −∇2
A

but the magnetic field which is −qτn. We observe

that the magnetic field is of constant strength qτ and that its variation is controlled if τ ∈ (0,τ0].

The analysis of [HM4] which was devoted to the constant magnetic field case can go through (see

[HP2]).

This leads (assuming the monotonicity of µ∗ with respect to q), to obtain for the solution

Qc3(κ,τ) of (5.1) the expansion

τ Qc3(κ,τ) =
κ2

Θ0
+ O(κ4/3) . (7.3)

We refer to [HP2] and to the last section for a discussion of this critical wave number.

We hope to give in [HP3] a two-term asymptotic of µ∗(q,τ) and consequently of Qc3(κ,τ) for

large κ (with τ ∈ (0,τ0]).

4This condition can be relaxed [Ray2] at the price of a worse remainder.



14 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

8 Perturbative Case: qτ Small

A second analysis (perturbative in spirit) gives (see [HP2])

Theorem 8.1.

As σ = qτ → 0,

µ∗(q,τ) = Θ(τ)(qτ)
2

+ O((qτ)4) , (8.1)

where the remainder is controlled uniformly for τ ∈ (0,τ0], and τ 7→ Θ(τ) is a continuous function

on [0,τ0] such that

Θ(0) = inf
h∈S2

1

|Ω|

∫

Ω

|Fh|
2 dx. (8.2)

One can also give an asymptotic of c(κ,q,τ), see [HP2].

Theorem 8.2.

Let Ω be a bounded smooth domain in R3 and τ0 > 0. Then, there exist positive constants C1(τ0),

C2(τ0) and σ1(τ0), such that, for any q,τ,κ satisfying

τ ∈ (0,τ0], σ = qτ ∈ (0,σ1], κ ≥ C1(τ0)σ , (8.3)

we have ∣∣∣∣c(q,τ,κ) +
κ2

2
|Ω| − σ2|Ω|Θ(τ)

∣∣∣∣ ≤ C2(τ0)(1 + κ
−2

)σ4 . (8.4)

As a corollary and using (2.1), we immediately obtain under the same assumptions
∣∣∣∣ inf
(ψ,n)∈H1(Ω,C)×V (Ω,S2)

EK[ψ, n] +
κ2

2
|Ω|

∣∣∣∣ ≤ C(τ0)(qτ)
2
(1 + κ−2) . (8.5)

This estimate is independent of the elastic parameters.

Remark 8.3.

It would be good to have a second term in this last expansion.

9 Coming Back to the Main Functional

We finish this survey by presenting the following results regarding the non-triviality of minimizers.

Similar but less accurate results have been obtained in [BCLP]. For the corresponding results for

superconductors see Giorgi and Phillips [GioP] and Lu and Pan [LuP1, LuP2, LuP3].

It is a consequence of (8.4) and of (2.9) that

Proposition 9.1.

Let Ω be a bounded smooth domain in R3, τ0 > 0 and κ0 > 0. Then, for any q,τ,κ satisfying

τ ∈ (0,τ0], σ = qτ ∈ (0,σ1], κ ≥ C1(τ0)σ , κ ≥ κ0 , (9.1)

the minimizers of EK are non trivial.



CUBO
11, 5 (2009)

On Some Spectral Problems and Asymptotic ... 15

We note that the constants involved in the previous statement are independent of the elastic

coefficients. The next proposition will show that, if σ/κ and the elastic constants are sufficiently

large, the minimizers are the nematic phases.

Proposition 9.2.

For any τ0 and any σ0, there exists a constant C > 0 such that, if

τ ∈ (0,τ0] , qτ ≥ σ0 , qτ ≥ C(1 + q
2
)(1 + κ2) ,

then the functional EK has no minimizer with ψ 6= 0.

Proof. Let (ψ, n) be a minimizer, with ψ not trivial. So

EK[ψ, n] < 0 .

We should keep in mind what we have obtained in (2.2). In particular, we deduce, like for getting

(4.3), that

‖∇qnψ‖
2
L2(Ω) ≤ κ

2‖ψ‖2L2(Ω) . (9.2)

As in the proof of Proposition 3.1, we can compare with some element in C(τ). For any Q ∈ SO(3) ,

we have

‖∇
qN

Q
τ
ψ‖2L2(Ω) ≤ 2‖∇qnψ‖

2
L2(Ω) + 2q

2‖n − NQτ ‖
2
L4(Ω)‖ψ‖

2
L4(Ω) . (9.3)

Then, one can use (9.2) and the so called diamagnetic inequality5 and this, together with Sobolev’s

injection of H1(Ω, C) in L4(Ω, C), leads to

‖ψ‖2L4(Ω) = ‖ |ψ| ‖
2
L4(Ω)

≤ C(Ω)(‖∇|ψ|‖2L2(Ω) + ‖ψ‖
2
L2(Ω))

≤ C(Ω)(‖∇qnψ‖
2
L2(Ω) + ‖ψ‖

2
L2(Ω)) .

(9.4)

Hence we obtain, using the characterization of the groundstate energy,

λN1 (qN
Q
τ ) ≤ 2(κ

2
+ C(Ω)(1 + κ2)q2‖n − NQτ ‖

2
L4(Ω)) . (9.5)

We now choose some Q and observe that, for any σ0 > 0, there exists Ĉ, such that, if qτ is larger

than σ0, we have from the proof of Theorem 7.1 (uniformly with respect to τ ∈ (0,τ0])

qτ

Ĉ
≤ λN1 (qN

Q
τ ) . (9.6)

This implies
qτ

Ĉ
≤ 2(1 + q2‖n − NQτ ‖

2
L4(Ω))(1 + κ

2
) . (9.7)

The observation –and this is again independent of the elastic coefficients– is that

qτ

Ĉ
≤ 2(1 + 4|Ω|1/2q2)(1 + κ2) . (9.8)

5We recall that diamagnetic inequality says that, for any u ∈ H1
loc

and any magnetic potential A in L2
loc

we have
|∇|u|| ≤ |∇Au| , almost everywhere.



16 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

The proof is finished by taking C = 4 max{1, 4|Ω|1/2Ĉ} .

Hence, this is only if we want to have a more precise information about the transition between

nematic phase and smectic phase that we will have to use that some of the elastic constants are

large. Let us give a statement, in this direction.

Proposition 9.3.

For any τ0 > 0, any k
0
1 > 0, any ǫ > 0, there exists κ0 and, for any q, C(ǫ,τ0,q) such that if

6

• κ ≥ κ0 ,

• τ ∈ (0,τ0] ,

• K1 ≥ k
0
1κ

2/τ,

• minj=2,3 Kj ≥ C(ǫ,τ0,q)κ
2 ,

• qτ ≥ (1 + ǫ)κ2/Θ0 ,

then the functional has no minimizers with ψ 6= 0.

Proof. We have to improve the argument starting from (9.3) which we replace for any η > 0

by

‖∇
qN

Q
τ
ψ‖2L2(Ω) ≤ (1 + η)‖∇qnψ‖

2
L2(Ω) + (1 +

1

η
)q2‖n − NQτ ‖

2
L4(Ω)‖ψ‖

2
L4(Ω) . (9.9)

This leads to replace (9.5) by

λN1 (qN
Q
τ ) ≤ (1 + η)κ

2
+ C(Ω,η)(1 + κ2)q2‖n − NQτ ‖

2
L4(Ω) , (9.10)

with two free parameters η and Q .

We first choose η = ǫ/2. We can now use Lemma 3.4 together with (3.5) and (3.6) in order to

get

C(Ω,η)q2‖n − NQτ ‖
2
L4(Ω) ≤

ǫ

2
.

This leads to our choice of Q. This time we have to use the uniform asymptotic established in (7.2).

Remark 9.4.

The results of this section have to be compared with similar, but less accurate, results of [BCLP].

6We can in the second item alternatively assume K1 ≥ k
0
1q

2τ .



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11, 5 (2009)

On Some Spectral Problems and Asymptotic ... 17

Application to the critical wave numbers

Proposition 9.3 admits a converse using what we know about c(κ,q,τ). This converse result

is independent of the Frank constants. More precisely, we can introduce (following [P1]):

For κ > 0, τ > 0, we define

Q
c3

(κ,τ) = inf{q > 0 : ψ = 0 is the minimizer of Gqn for all n ∈ C(τ)} ,

Qc3(κ,τ) = inf{q > 0 : ψ = 0 is the unique minimizer of Gq′n

for all q′ > q, n ∈ C(τ)}.

(9.11)

and
Q

K

c3(κ,τ) = inf{q > 0 : the (0, n) (n ∈ C(τ)) are minimizer of E
K },

QK
c3

(κ,τ) = inf{q > 0 : the (0, n) (n ∈ C(τ)) are the unique

minimizers of EK for all q′ > q,}.

(9.12)

Here we have explicated in the notation the dependence on K = (K1,K2,K3).

It results of course of the universal estimate (2.9) that

Q
K

c3(κ,τ) ≤ Qc3(κ,τ) and Q
K

c3
(κ,τ) ≤ Q

c3
(κ,τ) , (9.13)

for any elastic constants.

Proposition 9.3 implies that for large κ and sufficient large elastic constants the critical wave

numbers satisfy

τ lim
Kj →+∞

Q
K

c3(κ,τ) ∼
κ2

Θ0
and τ lim

Kj →+∞
QK
c3

(κ,τ) ∼
κ2

Θ0
. (9.14)

A The De Gennes Family

The family of operators H(ξ)

H(ξ) = D2t + (t − ξ)
2 (A.1)

on the half-line with Neumann boundary condition at 0 appears initially in the analysis of the

problem in the half plane R2+ := {x1 > 0} of the Neumann realization of the magnetic Schrodinger

operator with constant magnetic field D2x1 + (Dx2 − x1)
2. Here we write Dt =

1
i
∂
∂t

. The lowest

eigenvalue of the operator H(ξ),

ξ 7→ µ(ξ)

admits a unique minimum at ξ0 > 0. For analyzing the variation of µ(ξ), it is useful to combine

the following two formulas

• The Feynman-Hellmann formula:

µ′(ξ) = −2

∫ +∞

0

(t − ξ)uξ(t)
2 dt ,



18 Bernard Helffer and Xing-Bin Pan CUBO
11, 5 (2009)

where uξ is the normalized groundstate of H(ξ).

• The Bolley-Dauge-Helffer formula [DaH]:

µ′(ξ) = uξ(0)
2
(ξ2 − µ(ξ)) .

This permits to show that µ(ξ) has a unique minimum, which is attained at ξ0 > 0. Moreover

lim
ξ→+∞

µ(ξ) = 1 , lim
ξ→−∞

µ(ξ) = +∞ .

Graph of µ(ξ) and comparison with Dirichlet realization

Figure 1: De Gennes model, computed by V. Bonnaillie-Noël

Two constants have played a role in the main text. The first one is Θ0:

0 < Θ0 = µ(ξ0) = inf
ξ∈R

µ(ξ) < 1 . (A.2)

We have

ξ20 = Θ0 ∼ 0.59 . (A.3)

The second one is:

δ0 =
1

2
µ′′(ξ0) . (A.4)



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On Some Spectral Problems and Asymptotic ... 19

B Montgomery’s Model

We have also met the following family (depending on ρ) of quadratic oscillators:

D2t + (t
2 − ρ)2 . (B.1)

Denoting by ν(ρ) the lowest eigenvalue of this operator, Pan and Kwek [PK] (see also [Hel]) have

shown that there exists a unique minimum of ν(ρ) leading to a new universal constant

ν̂0 = inf
ρ∈R

ν(ρ) . (B.2)

One has (Feynman-Hellmann Formula)

ρmin = 2

∫
t2u2ρmin dt ,

where uρ denotes the normalized groundstate.

Numerical computations confirm that the minimum is attained for a positive value of ρ:

ρmin ∼ 0.35 ,

and that this minimum is

Θmin ∼ 0.5698 .

Numerical computations also suggest that the minimum is non degenerate. This result is proved

in [Hel].

Acknowledgements

The first author would like to thank F. Cardoso and A. Sá Barreto for their kind invitation at

this conference and the opportunity to give a mini-course the week before, and he also thank his

collaborators (particularly V. Bonnaillie-Noël, S. Fournais), his student N. Raymond (whose Master

thesis inspires partially this survey) and also P. Bauman and D. Phillips for useful discussions.

The second author was partly supported by the National Natural Science Foundation of China

grant no. 10471125, 10871071, the Science Foundation of the Ministry of Education of China grant

no. 20060269012, and the National Basic Research Program of China grant no. 2006CB805902.

Received: November, 2008. Revised: February, 2009.

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