On some connected non linear problems

L’explication que nous devons juger satisfaisante est celle qui adh`ere `a son objet : point de vide entre eux, pas d’interstice o`u une autre explication puisse aussi bien se loger ; elle ne convient qu’`a lui, il ne se prˆete qu’`a elle.

La pens´ee et le mouvant, Bergson

In this chapter we present two problems related to the non linear Schr¨odinger equation: (i) the semiclassical limit for the p-eigenvalues of the magnetic Laplacian,

(ii) the dimensional reduction of the time dependent non linear Schr¨odinger equation. 1. Non linear magnetic eigenvalues

1.1. Definition of the non linear eigenvalue. Let Ω be a bounded simply con-nected open set of _R2. We introduce the following “nonlinear eigenvalue” (or optimal magnetic Sobolev constant):

(6.1.1) λ(Ω,A, p, h) = inf ψ∈H1 0(Ω),ψ6=0 Q_h,A(ψ) R Ω|ψ|pdx 2 p = inf ψ∈H1 0(Ω), kψk_Lp_(Ω)=1 Qh,A(ψ),

where the magnetic quadratic form is defined by

∀ψ ∈H¹₀(Ω), Qh,A(ψ) =

Z

Ω

|(−ih∇+A)ψ|2dx.

Lemma 6.1. The infimum in (6.1.1) is attained.

Proof. Consider a minimizing sequence (ψ_j) that is normalized in L^p-norm. Then, by a H¨older inequality and using that Ω has bounded measure, (ψ_j) is bounded in L². Since A ∈ L^∞(Ω), we conclude that (ψ_j) is bounded in H1

0(Ω). By the Banach-Alaoglu Theorem there exists a subsequence (still denoted by (ψ_j)) and ψ∞ ∈ H1

0(Ω) such that

ψ_j * ψ∞ weakly inH1

0(Ω) andψ_j →ψ∞ in Lq(Ω) for all q ∈[2,+∞). This is enough to

conclude.

Lemma 6.2. The minimizers (which belong to H¹₀(Ω)) of the L^p-normalized version of (6.1.1) satisfy the following equation in the sense of distributions:

In particular (by Sobolev embedding), the minimizers belong to the domain of Lh,A.

1.2. A result by Esteban and Lions. By using the famous concentration-compactness method, Esteban and Lions proved the following proposition in [66].

Proposition 6.3. Let A ∈ L(_R^d,_R^d) such that B 6= 0 and p ∈ (2,2^∗), with 2^∗ = _d²₋^d₂. We let (6.1.3) S = inf ψ∈Dom(Q_A),ψ6=0 Q_A(ψ) kψk2 Lp(_Rd) .

Then, the infimum in (6.1.3) is attained.

Note that S >0. Indeed, if (φ_j)_j≥1 is a minimizing sequence, normalized in Lp, such thatQ_A(φ_j)→0, we deduce that, by diamagnetism, |φ_j| →0 inH1(_Rd). By the Sobolev embedding H1(_Rd) ⊂Lp(_Rd), we get that (φ_j)_j≥1 goes to zero in Lp(_Rd). We prove this proposition in Chapter 12, Section 1 by using an alternative method to the concentration-compactness principle. Moreover, it is possible to prove that the minimizers of (6.1.3) have an exponential decay.

Proposition 6.4. There exists α >0 such that, for any minimizerψ of (6.1.3), we have eα|x|ψ ∈L2(_Rd).

We focus on the two dimensional case. Definition 6.5. For p∈(2,+∞), we define

(6.1.4) λ^[0](p) =λ(_R²,A^[0], p,1) = inf ψ∈Dom(Q A^[0]),ψ6=0 Q_A[0](ψ) kψk2 Lp , where A^[0](x, y) = (0,−x). Here Q_A[0](ψ) = Z R² |(−i∇+A^[0])ψ|2dx, with domain Dom(Q_A[0]) = n ψ ∈L²(_R²) : (−i∇+A^[0])ψ ∈L²(_R²) o .

Let us now state the main theorem of this section (the proof is given in Chapter 12, Section 2).

Theorem 6.6. Let p≥2. Let us assume that A is smooth on Ω, that B=∇ ×A does

not vanish on Ω and that its minimum b₀ is attained in Ω. Then there exist C > 0 and

h₀ >0 such that, for all h∈(0, h₀),

(1−Ch¹8)λ^[0](p)b 2 p 0h²h⁻²p ≤λ(Ω,A, p, h)≤(1 +Ch^1/2)λ^[0](p)b 2 p 0h²h⁻²p. 2. Non linear dynamics in waveguides

Let us now discuss another non linear problem.

With the same formalism, we will consider the case of unbounded curves and the case of closed curves. Consider a smooth, simple curve Γ in _R² defined by its normal parametrization γ :x₁ 7→γ(x₁). For ε >0 we introduce the map

(6.2.1) Φ_ε:S =_M×(−1,1)3(x₁, x₂)7→γ(x₁) +εx₂ν(x₁) =x,

whereν(x₁) denotes the unit normal vector at the pointγ(x₁) such that det(γ⁰(x₁), ν(x₁)) = 1 and where

M=

(

R for an unbounded curve,

T=_R/(2π_Z) for a closed curve.

We recall that the curvature at the pointγ(x1), denoted by κ(x1), is defined by

γ⁰⁰(x₁) = κ(x₁)ν(x₁).

The waveguide is Ω_ε = Φ_ε(S) and we will work under the following assumption which states that waveguide does not overlap itself and that Φ_ε is a smooth diffeomorphism.

Assumption 6.7. We assume that the function κ is bounded, as well as its derivatives

κ⁰ andκ⁰⁰. Moreover, we assume that there existsε₀ ∈(0,_kκk¹

L^∞) such that, forε∈(0, ε₀), Φ_ε is injective.

We will denote by −∆^Dir_Ωε the Dirichlet Laplacian on Ω_ε. We are interested in the following equation:

(6.2.2) i∂_tψ^ε=−∆^Dir_Ωεψ^ε+λε^α|ψ^ε|2ψ^ε

on Ω_ε with a Cauchy conditionψε(0;·) = ψε

0 and whereα ≥1 andλ∈_Rare parameters. By using the diffeomorphism Φ_ε, we may rewrite (6.2.2) in the space coordinates (x₁, x₂) given by (6.2.1). For that purpose, let us introduce m_ε(x₁, x₂) = 1−εx₂κ(x₁) and consider the function ψε transported by Φ_ε,

U_εψ^ε(t;x₁, x₂) = φ^ε(t;x₁, x₂) =ε^1/2m_ε(x₁, x₂)^1/2ψ^ε(t; Φ_ε(x₁, x₂)).

Note thatU_εis unitary fromL²(Ω_ε, dx) toL²(S, dx₁dx₂) and mapsH¹₀(Ω_ε) (resp. H²(Ω_ε)) to H¹₀(S) (resp. to H²(S)). Moreover, the operator −∆^Dir_Ωε is unitarily equivalent to the self-adjoint operator onL2(S, dx₁dx₂),

Uε(−∆^Dir_Ωε)U_ε⁻¹ =Hε+Vε, with Hε=P_ε,²₁+P_ε,²₂,

where

P_ε,1 =m⁻_ε^1/2D_x1m⁻_ε^1/2, P_ε,2 =ε⁻¹D_x2

and where the effective electricV_ε potential is defined by

Vε(x1, x2) = − ^κ(x1)

2

4(1−εx₂κ(x₁))2 .

Notice that, for all ε < ε₀, we havem_ε≥1−ε₀kκk_L∞ >0. The problem (6.2.2) becomes (6.2.3) i∂_tφ^ε =H_εφ^ε+V_εφ^ε+λε^α−1m⁻_ε¹|φ^ε|2φ^ε

with Dirichlet boundary conditionsφε(t;x₁,±1) = 0 and the Cauchy condition φε(·; 0) =

φ^ε₀ =Uεψ₀^ε. We notice that the domains of Hε and Hε+Vε coincide with H²(S)∩H¹₀(S). In order to study (6.2.8), it is natural to conjugate the equation by the unitary group

e^itHε so that the problem (6.2.8) becomes (6.2.4) i∂_tϕ_e^ε=e^itHε(V_ε−ε⁻²µ₁)e^−itHε

e

ϕ^ε+λW_ε(t;ϕ_e^ε), ϕ_e^ε(0;·) = φ^ε₀,

where

(6.2.5) W_ε(t;ϕ) = e^itHεm⁻_ε¹|e^−itHεϕ|²e^−itHεϕ

and where ϕ_eε =eitH_εϕε which satisfies ϕ_eε(t;x₁,±1) = 0.

We will analyze the critical case α = 1 where the nonlinear term is of the same order as the parallel kinetic energy associated to D²_x1. It is well-known that (6.2.2) (thus (6.2.3) also) has two conserved quantities: the L² norm and the nonlinear energy. Let us introduce the first eigenvalue µ₁ = ^π₄² of D²_x2 on (−1,1) with Dirichlet boundary conditions, associated to the eigenfunction e₁(x₂) = cos ^π₂x₂ and define the energy functional (6.2.6) E_ε(φ) = ¹ 2 Z S |P_ε,1φ|2dx₁dx₂+ ¹ 2 Z S |P_ε,2φ|2dx₁dx₂+¹ 2 Z S V_ε− ^µ1 ε2 |φ|2dx₁dx₂ +^λ 4 Z S m⁻_ε¹|φ|4dx₁dx₂.

Notice that we have substracted the conserved quantity ^µ1

2ε2kφk2

L2 to the usual nonlinear energy, in order to deal with bounded energies. Indeed, we will consider initial conditions with bounded mass and energy, which means more precisely the following assumption. Assumption 6.8. There exists two constants M₀ >0 and M₁ > 0 such that the initial data φε

0 satisfies, for all ε∈(0, ε₀),

kφ^ε₀k_L2 ≤M0 and Eε(φ^ε₀)≤M1.

Let us define the projection Π₁one₁by letting Π₁u=hu, e₁i_L2((−1,1))e₁. A consequence of Assumption 6.8 is that φ^ε₀ has a boundedH¹ norm and is close to its projection Π1φ^ε₀. Indeed, we will prove the following lemma (see Chapter 23, Section 2).

Lemma 6.9. Assume thatφ^ε₀ satisfies Assumption 6.8. Then there existsε1(M0)∈(0, ε0)

and a constant C > 0 independent of ε such that, for all ε ∈(0, ε1(M0)),

(6.2.7) kφ^ε₀k_H1(S)≤C and kφ^ε₀−Π₁φ^ε₀k_L2(_M,H1(−1,1)) ≤Cε .

It will be convenient to consider the following change of temporal gaugeφε(t;x₁, x₂) =

e⁻iµ1ε⁻2tϕε(t;x₁, x₂). This leads to the equation

(6.2.8) i∂_tϕ^ε =H_εϕ^ε+ (V_ε−ε⁻²µ₁)ϕ^ε+λm⁻_ε¹|ϕ^ε|2ϕ^ε

with conditions ϕε(t;x₁,±1) = 0, ϕε(0;·) = φε

0.

We can now state the main theorem of this section (see Chapter 23, Section 2.2). Theorem 6.10. Assume that φε

0 ∈H2∩H1

0(S) and that there existM₀ >0, M₂ >0such

that, for all ε ∈(0, ε₀),

(6.2.9) kφ^ε₀k_L2 ≤M₀, (H_ε−^µ1 ε2)φ^ε₀ L2 ≤M₂. Then φε

0 satisfies Assumption 6.8 and, for all ε ∈ (0, ε₁(M₀)), (6.2.8) admits a unique

solution ϕε ∈ C(_R+;H2∩H1

0(S))∩ C1(_R+;L2(S)). Moreover, there exists C >0 such that,

for all ε∈(0, ε₁(M₀)) and t≥0, we have

Part 2