Strong diamagnetism for general

domains in R3 and applications

to superconductivity

Bernard HelfferMathematiques -

Univ Paris Sud- UMR CNRS 8628(After S. Fournais and B. Helffer)

Conference in St-Petersburg

June 2007

– Typeset by FoilTEX –

Main goals

We consider the Neumann Laplacian with constantmagnetic field on a regular domain. Let B be thestrength of the magnetic field, and let λ1(B) be thefirst eigenvalue of the magnetic Neumann Laplacianon the domain. It is proved that B 7→ λ1(B) ismonotone increasing for large B.

This result was proved by Fournais-Helffer in the caseof dimension 2 (first under a generic assumption,one year later in full generality). Our purpose (thisis again a common work with S. Fournais) is toshow here how one can prove the same result indimension 3 (but under generic assumptions). Theproof depends heavily on the two term asymptoticsof λ1(B) obtained by Pan and Helffer-Morame in2002.

If time permits,

we will discuss also applications of this monotonicity

for the identification of the critical fields insuperconductivity

and

present similar questions in the context of the theoryof liquid crystals.

Three models with parameters.

Model 1The spectral analysis is based in particular on theanalysis of the family

H(ξ) = D2t + (t+ ξ)2 , (1)

on the half-line (Neumann at 0) whose lowesteigenvalue µ(ξ) admits a unique minimum at ξ0 < 0.

We have to keep in mind two universal constantsattached to the problem on R

+.

The first one is

Θ0 = µ(ξ0) . (2)

It corresponds to the bottom of the spectrum of theNeumann realization in R2

+ (with B = 1).

Note thatΘ0 ∈]0, 1[ .

The second constant is

δ0 =1

2µ′′(ξ0) , (3)

Model 2The second model is quite specific of the problem indimension 3. We look in x1 > 0 to

L(ϑ,−i∂t) = −∂2x1−∂2

x2+(−i∂t+cosϑx1+sinϑx2)

2 .

By Partial Fourier transform, we arrive to :

L(ϑ, τ) = −∂2x1

− ∂2x2

+ (τ + cosϑx1 + sinϑx2)2 ,

in x1 > 0 and with Neumann condition on x1 = 0.

It is enough to consider the variation with respect toϑ ∈ [0, π

2 ].

The bottom the spectrum is given by :

ς(ϑ) := inf Spec (L(ϑ,−i∂t)) = infτ

(inf Spec (L(ϑ, τ))) .

We first observe the following lemma :

Lemma a.If ϑ ∈]0, π

2 ], then Spec (L(ϑ, τ)) is independent of τ .

This is trivial by translation in the x2 variable.

One can then show that the function ϑ 7→ ς(ϑ) iscontinuous on ]0, π

2 [ .This is based on the analysis of the essential spectrumof

L(ϑ) := D2x1

+D2x2

+ (x1 cosϑ+ x2 sinϑ)2 .

and that the bottom of the spectrum of this operatorcorresponds to an eigenvalue.We then show easily that

ς(0) = Θ0 < 1 .

andς(π

2) = 1 .

Finally, one shows that ϑ 7→ ς(ϑ) is monotonicallyincreasing and that

ς(ϑ) = Θ0 + α1|ϑ| + O(ϑ2) , (4)

with

α1 =

√µ′′(ξ0)

2. (5)

Model 3 : Montgomery’s model.

When the assumptions are not satisfied, and thatthe magnetic field B vanishes. Other models shouldbe considered. An interesting case is the case whenB vanishes along a line. This model was proposedby Montgomery in connection with subriemanniangeometry but this model appears also in the analysisof the dimension 3 case.

More precisely, we meet the following family(depending on ρ) of quartic oscillators :

D2t + (t2 − ρ)2 . (6)

Denoting by ν(ρ) the lowest eigenvalue, Kwek-Panhave shown that there exists a unique minimum ofν(ρ) leading to a new universal constant

ν0 = infρ∈R

ν(ρ) . (7)

Main results

Here we will describe the results of Helffer-Morame,Lu-Pan, Pan and the recent paper of Fournais-Helffer.

Let Ω ⊂ R3 be a bounded open set with smoothboundary, let β ∈ R3 be a unit vector, and define F

to be a vector field such that

curl F = β, in Ω, F ·N = 0 on ∂Ω,(8)

where N(x) is the unit interior normal vector to ∂Ω.Define QB to be the closed quadratic form

W 1,2(Ω) ∋ u 7→ QB(u) :=

∫

Ω

|(−i∇ +BF)u(x)|2 dx.

(9)

Let H(B) be the self-adjoint operator associated toQB.

In other words, H(B) is the differential operator(−i∇ +BF)2 with domain

u ∈W 2,2(Ω) : N · ∇u|∂Ω = 0 .

The operator H(B) has compact resolvent and weintroduce

λ1(B) := inf Spec H(B) . (10)

We will prove (under a generic assumption onthe domain Ω) that the mapping B 7→ λ1(B) ismonotonically increasing for sufficiently large valuesof B.

We will work under the following geometricassumption.

“Generic” Assumptions = G-Ass.We assume that the set of boundary points where βis tangent to ∂Ω, i.e.

Γ := x ∈ ∂Ω∣∣β ·N(x) = 0, (11)

is a regular submanifold of ∂Ω :

grad ′(β ·N)(x) 6= 0 , ∀x ∈ Γ . (12)

We finally assume that the set of points where β istangent to Γ is finite.

These assumptions are rather generic and for instancesatisfied for ellipsoids.

We will need the known two-term asymptotics of thegroundstate energy of H(B). The following resultwas proved by Helffer-Morame (the correspondingupper bound was also given by Pan).

Theorem 1If Ω and β satisfy G-Assumptions, then asB → +∞

λ1(B) = Θ0B + γ0B23 + O(B

23−η), (13)

for some η > 0.Here γ0 is defined by

γ0 := infx∈Γ

γ0(x), (14)

where

γ0(x) := 2−2/3ν0δ1/30 |kn(x)|2/3

(δ0+(1−δ0)|T (x)·β|2

)1/3

.

(15)Here T (x) is the oriented, unit tangent vector to Γat the point x and

kn(x) = | grad ′(β ·N)(x)| .

The new result obtained in collaboration withS. Fournais is the

Theorem 2Let Ω ⊂ R3 and β satisfying G- Assumptions

Let Γ1, . . . ,Γn be the collection of disjoint smoothcurves making up Γ. We assume that, for all j thereexists xj ∈ Γj such that γ0(xj) > γ0.

Then the directional derivativesλ′1,± := limt→0±

λ1(B+t)−λ(B)t ,

exist.

Moreover

limB→∞

λ′1,+(B) = limB→∞

λ′1,−(B) = Θ0. (16)

Localization estimatesWe start by recalling the decay of a groundstate inthe direction normal to the boundary. We will oftenuse the notation

t(x) := dist (x, ∂Ω). (17)

Now, if φ ∈ C∞0 (Ω), i.e. has support away from the

boundary, a simple integration by parts implies that

QB(φ) ≥ B‖φ‖22. (18)

It is a consequence of this elementary inequality(and the fact that Θ0 < 1) that groundstates areexponentially localized near the boundary.

Theorem 3There exist constants C, a1 > 0, B0 > 0 such that

∫Ωe2a1B1/2t(x)

(|ψB(x)|2

+B−1|(−i∇ +BF)ψB(x)|2)dx

≤ C ‖ψB‖22,

(19)

for all B ≥ B0, and all groundstates ψB of theoperator H(B).

We will mainly use this localization result in thefollowing form.

Corollary 4For all n ∈ N, there exists Cn > 0 and Bn ≥ 0 suchthat, ∀B ≥ Bn,

∫t(x)n|ψB(x)|2 dx ≤ Cn B

−n/2‖ψB‖22 .

We work in tubular neighborhoods of the boundaryas follows. For ǫ > 0, define

B(∂Ω, ǫ) = x ∈ Ω : t(x) ≤ ǫ. (20)

For sufficiently small ǫ0 we have that, for all x ∈B(∂Ω, 2ǫ0), there exists a unique point y(x) ∈ ∂Ωsuch that t(x) = dist (x, y(x)).

Define, for y ∈ ∂Ω, the function ϑ(y) ∈ [−π/2, π/2]by

sinϑ(y) := −β ·N(y). (21)

We extend ϑ to the tubular neighborhood B(∂Ω, 2ǫ0)by ϑ(x) := ϑ(y(x)).

In order to obtain localization estimates in thevariable normal to Γ, we use the following operatorinequality (due to Helffer-Morame).

Theorem 5Let B0 be chosen such that B

−3/80 = ǫ0 and define,

for B ≥ B0, C > 0 and x ∈ Ω,

WB(x) :=

B − CB1/4, t(x) ≥ 2B−3/8,

Bς(ϑ(x)) − CB1/4, t(x) < 2B−3/8.

(22)

Then, for C large enough

H(B) ≥WB, (23)

(in the sense of quadratic forms) for all B ≥ B0.

We use this energy estimate to prove Agmon typeestimates on the boundary.

Theorem 6Suppose that Ω ⊂ R3 and β satisfy G-Assumptions.Define for x ∈ ∂Ω,

dΓ(x) := dist (x,Γ),

and extend dΓ to a tubular neighborhood of theboundary by dΓ(x) := dΓ(y(x)).Then there exist constants C, a2 > 0, B0 ≥ 0, suchthat

∫

B(∂Ω,ǫ0)

e2a2B1/2dΓ(x)3/2|ψB(x)|2 dx ≤ C‖ψB‖

22,

(24)

for all B ≥ B0 and all groundstates ψB of H(B).

We have the following easy consequence.

Corollary 7Suppose that Ω ⊂ R

3 satisfies G-Assumptionsrelatively to β. Then for all n ∈ N there existsCn > 0 such that

∫

B(∂Ω,ǫ0)

dΓ(x)n|ψB(x)|2 dx ≤ CnB−n/3‖ψB‖

22,

(25)

for all B > 0 and all groundstates ψB of H(B).

Consider now the set MΓ ⊂ Γ where the functionγ0 is minimal,

MΓ := x ∈ Γ : γ0 = γ0. (26)

For simplicity, we asume that Γ is connected.

Theorem 8Suppose that Ω ⊂ R3 satisfies G-Assumptionsrelatively to β and let δ > 0. Then for all N > 0there exists CN such that if ψB is a groundstate ofH(B), then

∫

x∈Ω : dist (x,MΓ)≥δ

|ψB(x)|2 dx ≤ CNB−N ,

(27)

for all B > 0.

Proposition 9Let dΓ be the function defined in Theorem 6 . Lets0 ∈ Γ and define, for ǫ > 0,

Ω(ǫ, s0)= x ∈ Ω : dist (x,Γ) < ǫ and dist (x, s0) > ǫ.

Then, if ǫ is sufficiently small, there exists a functionφ ∈ C∞(Ω) such that

A := F + ∇φ ,

and satisfies

|A(x)| ≤ C(t(x) + dΓ(x)2

),

for all x ∈ Ω(ǫ, s0).

[Proof of Prop. 9 ]We use adapted coordinates (r, s, t) near Γ (N = 1).Γ is parametrized by arc-length as

|Γ|

2πS

1 ∋ s 7→ Γ(s) ∈ ∂Ω.

Given x ∈ Ω, close to Γ, there is a uniquepoint Γ(s(x)) ∈ Γ such that dist ∂Ω(y(x),Γ) =dist ∂Ω(y(x),Γ(s(x))), where dist ∂Ω denotes thegeodesic distance on the boundary. The coordinates(r, s, t) associated to the point x now satisfy

|r| = dist ∂Ω(y(x),Γ), s = s(x), t = dist (x, ∂Ω).

Notice that dΓ(x) ∼ |r(x)|, so we may replace dΓ

by r in the proposition.

Let A1dr + A2ds+ A3dt be the magnetic one-formωA = A · dx pulled-back (or pushed forward) to thenew coordinates (r, s, t). Also write

dωA = B12dr ∧ ds+ B13dr ∧ dt+ B23ds ∧ dt.

Clearly, Bij = ∂iAj − ∂jAi, for i < j. Here weidentify (1, 2, 3) with (r, s, t) for the derivatives.

The magnetic field β corresponds to the magnetictwo-form via the Hodge-map. In particular, since βis tangent to ∂Ω at Γ we get that

B12(0, s, 0) = 0. (28)

We now find a particular solution A such thatcurl A = B .

We make the Ansatz

A1 = −

∫ t

0

B13(r, s, τ) dτ, (29)

A2 = −

∫ t

0

B23(r, s, τ) dτ +

∫ r

0

B12(ρ, s, 0) dρ,

(30)

A3 = 0. (31)

One verifies by inspection that with these choices

|A| ≤ C(r2 + t). (32)

Transporting this A back to the original coordinatesgives an A with

curl A = 1, |A(x)| ≤ C(t(x) + dΓ(x)2).

Since Ω(ǫ, s0) is simply connected (for sufficiently

small ǫ) A is gauge equivalent to F and theproposition is proved.

Monotonicity

We now prove how one can derive the monotonicityresult from the known asymptotics of the groundstateenergy and localization estimates for the groundstateitself.

Based on these estimates the proof of Theorem isvery similar to the two-dimensional case.

Proof of Theorem 2For simplicity, we assume that Γ is connected.Applying analytic perturbation theory to H(B) weget the first part.

Let s0 ∈ Γ be a point with γ(s0) > γ0. Let A bethe vector potential defined in Proposition 9.

Let QB the quadratic form

W 1,2(Ω) ∋ u 7→ QB(u) =

∫

Ω

| − i∇u+BAu|2dx ,

and H(B) be the associated operator.

Then H(B) and H(B) are unitarily equivalent:

H(B) = eiBφH(B)e−iBφ, for some φ independentof B.With ψ+

1 ( · ;β) being a suitable choice of normalizedgroundstate, we get (by analytic perturbation theoryapplied to H(B) and the explicit relation between

H(B) and H(B),

λ′1,+(B)

= 〈Aψ+1 ( · ;B) , pB bA

ψ+1 ( · ;B)〉

+〈pB bAψ+

1 ( · ;B) , Aψ+1 ( · ;B)〉 .

(33)

We now obtain for any b > 0,

λ′1,+(B) =QB+b(ψ

+1 ( · ;B)) − QB(ψ+

1 ( · ;B))

b(34)

− b

∫

Ω

|A|2 |ψ+1 (x;B)|2 dx

≥λ1(B + b) − λ1(B)

b− b

∫

Ω

|A|2 |ψ+1 (x;B)|2 dx .

(35)

We choose b := MB23−η, with η from (13) and

M > 0 (to be taken arbitrarily large in the end).Then, using (13), (34) becomes

λ′1,+(B) ≥ Θ0 + γ0B−1/3(1+b/B)2/3−1

b/B

−CM−1 − b∫Ω|A|2 |ψ+

1 (x;B)|2 dx ,(36)

for some constant C independent of M,B.

If we can prove that

B23

∫

Ω

|A|2 |ψ+1 (x;B)|2 dx ≤ C, (37)

for some constant C independent of B, then we cantake the limit B → ∞ in (36) and obtain

lim infB→∞

λ′1,+(B) ≥ Θ0 − CM−1. (38)

Since M was arbitrary this implies the lower boundfor λ′1,+(B). Applying the same argument to thederivative from the left, λ′1,−(B), we get (theinequality gets turned since b < 0)

lim supB→∞

λ′1,−(B) ≤ Θ0. (39)

Since, by perturbation theory, λ′1,+(B) ≤ λ′1,−(B)for all B, we get (16).

Thus it remains to prove (37).

By Proposition 9 we can estimate

∫Ω|A|2 |ψ+

1 (x;B)|2 dx≤ C

∫Ω(ǫ,s0

)(t2 + r4)|ψ+1 (x;B)|2 dx

+‖A‖2∞

∫Ω\Ω

(ǫ,s0

) |ψ+1 (x;B)|2 dx .

Combining Corollaries 4 and 7 and Theorem 8,we therefore find the existence of a constant C > 0such that :

∫

Ω

|A|2 |ψ+1 (x;B)|2 dx ≤ C B−1 , (40)

which is stronger than the needed estimate (37).

Ginzburg-Landau functional

The Ginzburg-Landau functional is given by

Eκ,H[ψ,A] =∫Ω

|∇κHAψ|

2 − κ2|ψ|2 + κ2

2 |ψ|4

+κ2H2| curl A − β|2dx ,

withΩ simply connected,(ψ,A) ∈W 1,2(Ω; C) ×W 1,2(Ω; R3),β = (0, 0, 1)and where∇A = (∇ + iA).

We fix the choice of gauge by imposing that

div A = 0 in Ω , A · ν = 0 on ∂Ω .

Terminology for the minimizers

The pair (0,F) is called the Normal State.

A minimizer (ψ,A) for which ψ never vanishes willbe called SuperConducting State.

In the other cases, one will speak about Mixed State.

The general question is to determine the topology ofthe subset in R

+ × R+ of the (κ,H) corresponding

to minimizers belonging to each of these threesituations.

Theorem 10 (Lu-Pan-Fournais-Helffer)

There exists κ0 such that, ∀κ ≥ κ0, (0,F) is a globalminimizer of Eκ,H iff λ1(κH) < κ2 .

Remark.

This makes our analysis of the monotonicity of λ1

(for B large), which implies, for κ large, the existenceof a unique H such that

λ1(κH) = κ2 .

particularly interesting.

Same questions in the theory of Liquid

crystals

A similar arises in the theory of Liquid crystals. Thesimples question is to consider the case when themagnetic vector field is only of constant module !Typically one is interested in the case when

β(x) = B (cos τx3, sin τx3, 0) ,

as B → +∞. There are partial results for thismodel obtained by Almog and Pan.

The energy for this model can be written as

E [ψ,n] =

∫

Ω

|∇qnψ|

2 − k2|ψ|2 +

k2

2|ψ|4

+K1 | div n|2 + K2 |n · curl n + τ |2

+K3 |n× curl n|2

dx ,

where :

• Ω 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,

• τ is a real number measuring the chiral pitch insome liquid crystal materials,

• K1, K2 and K3 are positive constants called theelastic coefficients,

and

• k is a positive constant which depends on thematerial and on the temperature.

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