Moving Domain Walls in Magnetic Nanowires

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www.elsevier.com/locate/anihpc

Moving domain walls in magnetic nanowires

Katharina Kühn

^∗

Max-Planck-Institute for Mathematics in the Science, Inselstr. 22, 04103 Leipzig, Germany Received 18 March 2008; received in revised form 18 October 2008; accepted 21 November 2008

Available online 24 December 2008

Abstract

This paper investigates the reversal of magnetic nanowires via a perturbation argument from the static case. We consider the gradient flow equation of the micromagnetic energy including the nonlocal stray field energy. For thin wires and weak external magnetic fields we show the existence of travelling wave solutions. These travelling waves are almost constant on the cross section and can thus be seen as moving domain walls of a type called transverse wall.

Résumé

Cet article présente une étude du renversement des nano-fils magnétiques par une méthode de perturbation du cas statique. On considère l’équation du flot-gradient associé à l’énergie micromagnétique en incluant l’énergie de la perturbation non locale du champ magnétique. Pour des fils fins et des champs magnétiques externes de faible amplitude, on montre que les solutions prennent la forme d’ondes progressives. Ces ondes progressives ont une amplitude pratiquement homogène sur l’ensemble de la section, et peuvent donc être assimilées à des parois de domaines transverses.

MSC:82D40; 47J35

Keywords:Micromagnetism; Domain walls; Nonlinear evolution equations

1. Introduction

Because of possible technical applications [1,10] in the recent years there has been a growing interest in magnetic nanowires and especially in their reversal modes. It is known that the reversal of the magnetisation starts at one end of the wire and then a domain wall separating the already reversed part from the not yet reversed part is propagating through the wire.

In the micromagnetic model, the evolution of the magnetisation is described by the Landau–Lifshitz–Gilbert (LLG) equation. We simplify this equation taking the overdamped limit, that is, we consider the gradient flow equation of the micromagnetic energy. Viewing static domain walls as travelling waves with speed 0, we show the existence of travelling wave solutions for thin wires and weak external magnetic fields via a perturbation argument. This argument relies crucially on the fact that the wires are thin, since we need strong regularity of the static domain wall. We have

* Tel.: 089/72301607; fax: 0341/9959 633.

E-mail address:Katharina.Kuehn@gmx.de.

doi:10.1016/j.anihpc.2008.11.004

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proved strong regularity in the case of thin wires [7], and we cannot expect it for thick wires where the examples of low energy configurations are vortex walls which have a singularity and are not even continuous [6].

For thin wires, static domain walls are almost constant on the cross section [6]. Thus, after perturbing the equation with a weak external field, the moving domain walls are still almost constant on the cross section. Such a reversal mode has been observed in numerical simulations [4,5,11] and is called transverse mode.

Various models for the transverse mode have been analysed previously. Thiaville and Nakatani [10] study a one- dimensional model for the transverse mode and compare it with numerical simulations. Carbou and Labbé [3] consider a similar model. They prove that one-dimensional domain walls are asymptotically stable. Sanchez [9] considers the limit of the Landau–Lifshitz equation when the diameter of the domain and the exchange coefficient in the equation simultaneously tend to zero and performs an asymptotic expansion.

The final goal in understanding the transverse mode is to find solutions to the full Landau–Lifshitz–Gilbert equation, to describe their properties, and to rigorously derive a reduced theory. This paper is a step towards that goal which, contrary to the other approaches, takes into account the full three-dimensional structure of the problem. We expect that the methods developed in this paper can be applied to find solutions for the full Landau–Lifshitz–Gilbert equation.

1.1. Static domain walls

We work in the framework of micromagnetism. This is a mesoscopic continuum theory that assigns a nonlocal nonconvex energy to each magnetisationmfrom the domainΣ⊂R³to the sphereS²⊂R³. Experimentally observed ground states correspond to minimisers of the micromagnetic energy functional. When appropriately rescaled, for a soft magnetic material with an external field of strengthhin direction ofe_xthis energy is

Eh(m)=

Σ

|∇m|²+hex·m +

R³

H (m)². (1)

HereH (m):R³→R³is the projection ofmon gradient fields, i.e.,

H (m)= ∇u withu=divminR³. (2)

We consider magnetisations where the domainΣ_R=R×D_Ris an infinite cylinder with radiusRand set M(R):=

m:Σ_R→S²E₀(m) <∞

. (3)

To specify the conditions at±∞we need to define a smooth functionχ:R→R³with limx→±∞χ (x)= ±e_x. Our choice is

χ:R→R³, x→tanh(x)e_x. (4)

In [6] we have shown that for m:ΣR→S²the conditionE0(m) <∞is equivalent to the statement that one of the four mapsm± ex,m±χ is inH¹(ΣR). Thus, to single out the magnetisations that correspond to a 180 degree domain wall we define

Ml(R):=

m:Σ_R→S²m−χe_x∈H¹(Σ_R)

. (5)

For everyR >0 there exist energy minimising 180 degree domain walls, i.e., minimisers ofE₀inMl(R)[6]. For R→0 the energy minimisation problemΓ-converges to a reduced, one-dimensional problem whose minimiser can be calculated explicitly to be

m^red:R→S², x→

tanh x

√2 , 1 cosh(x/√

2),0 . (6)

In [7] we have shown that the minimisers converge tom^rednot only in a topology implied by the energy estimates but also in stronger norms.

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Theorem 1.Letm^Rbe a minimiser ofE₀inMl(R).

(i) ForRsmall enough,m^R∈H²(Σ_R)+χ∩C¹(Σ_R).

(ii) We have

Rlim→0

1

Rm^R−m^red

H¹(Σ_R)=0,

Rlim→0

m^R−m^red

C¹(ΣR)=0.

1.2. The dynamic model

We assume that the evolution of the magnetisation can be described by gradient flow of the energy under the condition|m| ≡1 with Neumann boundary conditions, that is,

∂_tm= −δ_mE_h(m)+

δ_mE_h(m)·m

m inΣ_R, ∂_νm=0 on∂Σ_R, (7)

where

δ_mE_h(m)= −2m+2H (m)−he_x. (8)

This equation is the overdamped limit of the Landau–Lifshitz–Gilbert equation. We are interested in travelling wave solutions. Because of the rotational symmetry of the cylinder we have to take into account that the solutions may rotate around the axis of the cylinder. We set

Q_φ:=

1 0 0 0 cos(φ) sin(φ) 0 −sin(φ) cos(φ)

, Q˜_φ:=

0 0 0 0 cos(φ) sin(φ) 0 −sin(φ) cos(φ)

, (9)

and note that∂_tQ_ωt=ωQ˜_ωt₊^π

2. Rotating travelling waves with speedcand angular velocityωsatisfy m(t, x, y)=Qωtm

0, Q₋ωt(x−ct, y) . Defining

Φ(m):=

0

−m_y₂ m_y₁

+

0 0 0 0 ∂_y₁m_y₁ ∂_y₂m_y₁ 0 ∂_y₁m_y₂ ∂_y₂m_y₂

0 y₂

−y₁

, (10)

we have

∂_tm(t, x, y)=ωQ˜_ωt₊^π

2m

0, Q₋ωt(x−ct, y)

−cQ_ωt∂_xm

0, Q₋ωt(x−ct, y)

−Q_ωt∇ym

0, Q₋ωt(x−ct, y)

ωQ˜₋_ωt₊^π

2y

= −c∂_xm(t, x, y)+ωQ˜^π

2m(t, x, y)−ω∇ym(t, x, y)Q˜^π

2y

= −c∂xm(t, x, y)−ωΦ

m(t, x, y) .

In particular, rotating travelling waves that are a solution of (7) satisfy the stationary equation

−δ_mE_h(m)+

δ_mE_h(m)·m

m+c∂_xm+ωΦ(m)=0 inΣ_R,

∂_νm=0 on∂Σ_R. (11)

To find solutions of (11) we consider first the caseh=0 and then use a perturbation argument. For this we have to work in a function space that is large enough to contain the solutions and small enough that the left-hand side of (11) is differentiable in this function space. As we will see,H²(Σ_R,R³)+χ is a good choice. In this space we have to restrict the search to solutions with|m| ≡1. We have to include further conditions in the set of admissible solutions to

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break the translation invariance and the rotation invariance of the problem. Forc=0,ω=0,h=0, Eq. (11) simplifies to

0= −δ_mE₀(m)+

δ_mE₀(m)·m

m inΣ_R, ∂_νm=0 on∂Σ_R. (12)

This is the Euler Lagrange equation for the energyE₀under the condition|m| =1. Thus, Theorem 1 implies that, for R >0 small enough, minimisersm^Rof the energyE₀are solutions of (12) inH²(Σ_R,R³)+χ.

We proceed a follows.

1. Depending on m^R we define the set of admissible functions S and show that S is a Banach submanifold of H²(ΣR,R³)+χ.

2. We find a continuously differentiable function N:S×L²(Σ_R,R)×R³→L²

Σ_R,R³

×R

such that(m, c, ω, h)is a solution of (11) if and only if there existsα∈L²(ΣR,R)that satisfiesN (m, α, c, ω, h)= (0, h).

3. We show that the derivativeDNofN in(m^R,0,0,0,0)is invertible.

4. Then, according to the inverse function theorem [12, Theorem 73.B, p. 552], there exists a neighbourhoodUof (m^R,0,0,0,0)and a neighbourhoodV of(0,0)such thatN|U→V is bijective. In particular, there existsh₀>0, such that for all|h|< h₀there arem_h, α_h, c_h, ω_h withN (m_h, α_h, c_h, ω_h, h)=0. In other words, for all|h|< h₀ there exists a solution of (11).

In Section 2 we go through the steps 1–4 to show the existence of travelling wave solutions for small radii and small external magnetic field. The arguments of Section 2 use the invertibility of an operator representing the “interesting”

part ofDN (m^R,0,0,0,0). This invertibility is shown in Section 3 and relies on the fact thatm^Ris close tom^red. 1.3. Definitions and notation

The letterp denotes a point inR³and has the componentsp=(x, y1, y2)=(x, y). A mapf with values inR³ has the componentsf =(fx, fy₁, fy₂). We writefy for(0, fy₁, fy₂), i.e., we viewfy as a map to{0} ×R². For a setA⊂L²(Rⁿ), we denote the closure ofAinL²(Rⁿ)byA_L2 and the characteristic function by1A. Fora, b∈Rⁿ, n∈Nwe denote the scalar product bya·b. ForΩ⊂R³andf, g:Ω→Rⁿ,n∈N, we set

f, gΩ:=

Ω

f (p)·g(p) dp,

whenever the integral on the right-hand side is defined. Moreover we set D_R(p):=

q∈R²: |p−q|< R

, D_R:=D_R(0), Σ_R:=R×D_R.

The definitions ofχin (4), ofMl in (5), and ofΦin (10) remain valid. Withm^redas in (6) we define

m^red_R :ΣR→S², (x, y)→m^red(x). (13)

Form:Ω⊂R³→R³letH (m):R³→R³be the projection ofmon gradient fields as in (2). The micromagnetic energy without external magnetic field is denoted by E(m) and the micromagnetic energy including the external magnetic field is denoted byE_h(m).

Finally, letm^R:Σ_R→S²always be a minimiser ofEinMl(R). To break the translation and rotation invariance we additionally require

m^R−m^red_R

L²(ΣR)v−m^red_R

L²(ΣR) for all other minimisersv∈Ml(R).

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2. The perturbation argument

As described above, the first step in the perturbation argument is to show that we are working on a sufficiently smooth manifold. Set

S^R:=

f ∈H²

Σ_R,R³ +χ

|f| ≡1, ∂_νf =0 on∂Σ_R,

∂_xm^R, f

Σ_R=0, Φ

m^R , f

Σ_R=0,

,

TS^R:=

f ∈H²

Σ_R,R³f ·m^R≡0, ∂νf =0 on∂ΣR,

∂_xm^R, f

Σ_R=0, Φ

m^R , f

Σ_R=0

.

Lemma 2.There existsR₀>0such that for allRR₀the setS^Ris a submanifold ofH²(Σ_R,R³)+χ. The tangent space ofS^R inm^RisTS^R.

Proof. We show the lemma in two steps. We define W^R:=

m∈H²

ΣR,R³ ∂νm|∂Σ_R=0,

m, ∂xm^R

Σ_R =0, m, Φ

m^R

Σ_R =0 .

First, since∂xm^R, Φ(m^R)∈L²(ΣR)and since the trace of a function inH²(ΣR)is inH¹(∂ΣR), the setW^R+χis a closed affine subspace ofH²(Σ_R,R³)+χ.

Second, we show thatS^Ris a submanifold ofW^R+χ. Set φ:W^R+χ→

f ∈H²(ΣR,R): ∂νf|∂Σ_R=0

, m→ |m| −1,

thenS^R=φ⁻¹(0). On{m∈W^R: |φ(m)|<1}the functionφis continuously differentiable and the derivative inmis Dφ(m):W^R→

f ∈H²(ΣR,R): ∂νf|∂Σ_R=0

, g→ g·m

|m| . (14)

IfRis small enough, for everym∈S^Rthe differentialDφ(m)is surjective: Indeed the equality∂_xm^red_R ·Φ(m^red_R )= 0 implies

det

∂_xm^red_R , ∂_xm^red_R Σ_R ∂_xm^R, Φ(m^red_R )Σ_R

∂_xm^red_R , Φ(m^red_R )ΣR Φ(m^red_R ), Φ(m^red_R )ΣR

=π R²∂_xm^red_R ²

L²(R)+Φ m^red_R

L²(R)

,

so with Theorem 1(ii) there existsR0such that for allRR0we have det

∂_xm^R, ∂_xm^RΣR ∂_xm^R, Φ(m^R)ΣR

∂_xm^R, Φ(m^R)ΣR Φ(m^R), Φ(m^R)ΣR

>0.

Therefore, for everyf ∈H²(Σ_R,R)with∂_νf|∂ΣR=0 we can find unique numbersb₁,b₂such that f m+b1∂xm^R+b2Φ

m^R , ∂xm^R

ΣR=0, f m+b₁∂_xm^R+b₂Φ

m^R , Φ

m^R

Σ_R=0,

andf m+b1∂xm^R+b2Φ(m^R)is a pre-image off inW^R. Moreover, since in a Hilbert space every subspace splits, in particularDφ⁻¹(0)splits. Thus 0 is a regular value ofφand we can apply [12, Thm. 73C, p. 556] to conclude that S^Ris a submanifold ofW^R+χ. Because of (14) the spaceTS^Ris the tangent space ofS^Rinm^R. 2

We consider the map s:S^R→L²

Σ_R,R³

, m→ −δ_mE_h(m)+

δ_mE_h(m)·m m, that is, with (8),

s(m)=2

m−(m·m)m−H (m)+

H (m)·m m

s1

+hex−(hex·m)m

s₂

.

The spaceH²(Σ_R,R)+χembeds intoC⁰(Σ_R,R), and functionsm→m, andm→H (m)are continuous linear maps fromS^RtoL²(Σ_R,R³). For the last statement see [8, Lemma 2.6]. Thuss₁:S^R→L²(Σ_R,R³)is well defined and continuously differentiable.

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Moreover, we have

he_x−(he_x·m)m=h1−m²_x

e_x+m_xm_y2h|m_y|,

sos₂:S^R→L²(Σ_R,R³)is well defined and continuously differentiable, too.

Thus we can define the continuously differentiable map N^R:S^R×L²(ΣR,R)×R³→L²

ΣR,R³

×R, (m, α, c, ω, h)→

−δmEh(m)+

δmEh(m)·m

m+c∂xm+ωΦ(m)+αm, h .

Since(−δ_mE_h(m)+(δ_mE_h(m)·m)m+c∂_xm+ωΦ(m))⊥mfor allm∈S^Rwe haveN^R(m, α, c, ω, h)=(0, h)if and only ifmis a solution of (11) andα=0.

The differential ofN^Rin(m^R,0_L2(Σ_R,R),0_R3)is DN^R

m^R,0,0

:TS^R×L²

ΣR, R³

×R³→L²

ΣR,R³

×R, (g, α, c, ω, h)→

−L^R(g)+c∂xm^R+ωΦ(m0)+αm^R, h , where

L^R:H²

Σ_R,R³

→L²

Σ_R,R³ , g→δ_mE(g)−

δ_mE(g)·m^R m^R−

δ_mE m^R

·g m^R−

δ_mE m^R

·m^R

g. (15)

With (8) we have the following explicit formula forL^R(g):

L^R(g)= −2g+2H (g)+2

g·m^R

m^R−2

H (g)·m^R

m^R+2

m^R·g m^R

−2 H

m^R

·g

m^R+2

m^R·m^R g−2

H m^R

·m^R

g. (16)

We will consider the restrictions ofL^Rto different subspaces ofH²(ΣR,R³). We will call these restrictionsL^R as well, but name always the domain and the range.

Lemma 3.For allR >0and allg, f ∈TS^Rwe have L^R(g)=δmE(g)−

δmE(g)·m^R m^R−

δmE m^R

·m^R

g, (17)

L^R(g)·f =δmE(g)·f− δmE

m^R

·m^R

g·f . (18)

MoreoverL^R(TS^R)⊆(TS^R)_L2and the operatorL^R:TS^R→(TS^R)_L2 is symmetric.

Proof. Sincem^Ris a solution of (12),δmE(m^R)is pointwise parallel tom^R. The elements ofTS^Rare pointwise or- thogonal tom^R. This implies (17) and (18). By definition the elements ofTS^Rsatisfy Neumann boundary conditions, so for allg, f ∈TS^Rwe haveL^Rf, gΣ_R= f, L^RgΣ_R.

It remains to show thatL^R(TS^R)⊆(TS^R)_L2. We have TS^R

L² :=

f∈L²

Σ_R,R³f ·m^R≡0,

∂_xm^R, f

Σ_R=0, Φ

m^R , f

Σ_R=0

.

Looking at (17), we see that L^R(g)⊥m^R. Set v(t, x, y):=m^R(x+t, y). Then v(t,·)satisfies for all t∈R the equation

0=δ_mE v(t,·)

− δ_mE

v(t,·)

·v(t,·) v(t,·), therefore we have for allg∈TS^R

0=∂t

δmE v(t,·)

− δmE

v(t,·)

·v(t,·)

v(t,·), g

ΣR

t=0

= L

∂_xm^R , g

Σ_R=

L(g), ∂_xm^R

Σ_R.

Analogously, withQ_φas in (9) we have forw(φ, x, y):=Q_φ(m^R(Q₋_φ(x, y)))the equation 0=δ_mE

w(φ,·)

− δ_mE

w(φ,·)

·v(φ,·) w(φ,·)

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and thus for allg∈TS^R 0=∂_φ

δ_mE w(φ,·)

− δ_mE

w(φ,·)

·v(φ,·)

w(φ,·), g

ΣR

φ=0

= L

Φ m^R

, g

ΣR=

L(g), Φ m^R

ΣR. 2 Note thatDN^R(m^R,0,0)is bijective if and only if (a) ∂_xm^RandΦ(m^R)are linearly independent, (b) L^R:TS^R→(TS^R)_L2 bijective.

Since limR→0m^R−m^red_R C¹(Σ_R)=0 and since ∂_xm^red_R andΦ(m^red_R )are linearly independent, (a) is satisfied ifR is small enough. In Section 3 we will show that (b) is satisfied for smallR, too. Altogether, we have the following theorem.

Theorem 4.(m, c, ω)is a solution of (11)if and only if there existsα∈L²(ΣR,R)such thatN^R(m, α, c, ω, h)= (0, h).

The functionN^Ris continuously differentiable and, ifRis small enough,DN^R(m^R,0,0)is bijective.

If N^R is continuously differentiable and DN^R(m^R) is invertible, according to the inverse function theorem [12, Theorem 73.B, p. 552] there exists a neighbourhood U of (m^R,0_L2(Σ_R,R),0_R3)and a neighbourhood V of (0_L2(ΣR,R³),0_R)such thatN^R|U→V is bijective. So for everyhsmall enough, we can findm_h, α_h, c_h, ω_hsuch that N^R(mh, αh, ch, ωh, h)=0. That is, we have proved our main theorem.

Theorem 5.For allR >0small enough there existsh_R>0such that for allhwithh < h_Rthere is exists a solution (m_h, c_h, ω_h)of(11).

3. Invertibility ofL^R

The goal of this section is to prove the following theorem.

Theorem 6.ForRsmall enough, the operatorL^R:TS^R→(TS^R)_L2, as defined in(15), is invertible, and its inverse is continuous.

We proceed in two steps. First, we define a mapL^R_∗ and show that for functionsmin a certain spaceTS₀^Rwe have L^R_∗(m), mΣ_R ¹₄m²_L2(ΣR). Then we prove that, for smallR, the operator L^R on the spaceTS^R is in a certain sense similar toL^R_∗ onTS₀^R.

In analogy to (1) and (15) we set E^R_∗ :M(R)→R, m→

Σ_R

|∂_xm|²+1

2|m_y|²+20R²|∇ym|², (19)

L^R_∗:H²

ΣR,R³

→L²

ΣR,R³ , g→δ_mE^R_∗(g)−

δ_mE_∗^R(g)·m^red_R m^red_R −

δ_mE_∗^R m^red_R

·g

m^red_R − δ_mE_∗^R

m^red_R

·m^red_R

g, (20)

where

δ_mE^R_∗(m)= −2∂xxm+(0, my₁, m_y₂)−40R²_ym.

Moreover we define TS₀^R:=

f ∈H²

Σ_R,R³ f ·m^red_R ≡0, ∂νf =0 on∂ΣR,

∂_xm^red_R , f

=0, Φ

m^red_R , f

=0

.

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Lemma 7.The minimiser ofE_∗^RinMl(R)is unique up to translation and rotation. It is given by m^red_R :Σ_R→S², (x, y)→

tanh

x

√2 , 1 cosh(x/√

2),0 , and we have

∂_xm^red_R (x, y)= 1

√2 m^red_R

y(x, y)

∂_xm^red_R (x, y)

|∂_xm^red_R (x, y)|=

1 cosh(x/√

2),−tanh x

√2 ,0 , Φ

m^red_R (x, y)

=

0,0, 1

cosh(x/√ 2) . Proof. The function

m^red:R→S², x→

tanh x

√2 , 1 cosh(x/√

2),0 is the only minimiser of

R|∂_xm|²+¹₂|m_y|², up to translation and rotation [8, Lemma 2.26]. Thus the functionm^red_R is the only minimiser ofE_∗^RinMl(R), up to translation and rotation.

A direct calculation yields the results for∂_xm^red_R andΦ(m^red_R ). 2 Lemma 8.For allR >0and allg, f ∈TS₀^Rwe have

L^R_∗(g)=δ_mE_∗^R(g)−

δ_mE_∗^R(g)·m^red_R m^red_R −

δ_mE_∗^R m^red_R

·m^red_R

g, (21)

L^R_∗(g)·f =δ_mE^R_∗(g)·f− δ_mE_∗^R

m^red_R

·m^red_R

g·f . (22)

Moreover,L^R_∗(TS₀^R)⊆(TS₀^R)_L2, and the operatorL^R_∗:TS₀^R→(TS₀^R)_L2 is symmetric.

Proof. We can argue exactly as in Lemma 3. 2 Theorem 9.For allR >0and allm∈TS₀^Rwe have

L^R_∗(m), m

ΣR1

4m²_L2(Σ_R).

Proof. The relations|∂_xm^red_R | =^√¹₂|(m^red_R )_y|and

∂_xxm^red_R ·m^red_R +∂_xm^red_R ²=∂_x

∂_xm^red_R ·m^red_R

=0 imply

δ_mE_∗^R m^red_R

·m^red_R = −2∂xxm^red_R ·m^red_R +m^red_R

y²=2m^red_R

y². Thus, with Lemma 8, for allg, h∈TS₀^Rwe have

L^R_∗(g)·h=δ_mE_∗^R(g)·h− δ_mE_∗^R

m^red_R

·m^red_R g·h

=

δ_mE₀(g)−2m^red_R

y²g

·h.

We define the vectore_sto be the unit vector in direction of∂_xm^red_R , i.e.,

e_s(x):= ∂_xm^red_R (x)

|∂_xm^red_R (x)|= m^red_R

y1(x),− m^red_R

x(x),0 , and introduce the sets

(9)

W1:=

m∈TS₀^R:

DR

m(x, y) dy≡0

, W2:=

m∈TS0^R: m(x, y)=α(x)e_y₂ for someα∈H²(R,R) , W3:=

m∈TS0^R: m(x, y)=α(x)es(x)for someα∈H²(R,R) .

ThenTS₀^Ris the direct sum ofW1,W2andW3, and we haveL^R_∗(Wi)⊂(Wi)_L2 fori∈ {1,2,3}.

Assumem∈W1. Using the Poincaré inequality, we have L^R_∗m, m

ΣR=40R²∇ym²_L2(Σ_R)+2∂xm²_L2(Σ_R)+ my²_L2(Σ_R)−2m^red_R

ym²

L²(ΣR)

40

16m²_L2(ΣR)−2m²_L2(ΣR)=1

2m²_L2(ΣR).

Assumem∈W2. Thenm(x, y)=α(x)1D_R(y)e_y₂ for someα∈H²(R,R), we have L^R_∗(m)

(x,y)=

−2∂xxα(x)+α(x)−2m^red_R

y(x)²α(x)

1D_R(y)ey₂, (23)

L^R_∗(m), m

ΣR=π R²

2∂_xα²_L2(R)+

R

1−2m^red_R

y²

α² (24)

and

1−2m^red_R

y(x)²1

4 for|x|1.6. (25)

SinceΦ(m^red_R )· e_y₂ is positive (Lemma 7), and sinceΦ(m^red_R ), m =0, the functionαhas to change sign.

First, assume thatαchanges sign in[−1.6,1.6]. We have

{f:[−1.6,1.6]→Rinf,f changes sign}

2∂_xf²_L2([−1.6,1.6])

f²_L2([−1.6,1.6])

=2π² 3.2²,

the infimum is attained and the minimisers are multiples ofx→sin(_3.2^π x). Thus we have 2∂_xα²_L2([−1.6,1.6])+

1.6

−1.6

1−2m^red_R

y²

α²2∂_xα²_L2([−1.6,1.6])− α²_L2([−1.6,1.6])

2π²

3.2²−1 α²_L2([−1.6,1.6]), and therefore, with (25) and (24),

L^R_∗(m), m

ΣRπ R² 2π²

3.2²−1 α²_L2([−1.6,1.6])+1

4α²_L2(R\[−1.6,1.6])

1

4m²_L2(ΣR).

Now assume thatαdoes not change sign in[−1.6,1.6]and letS₋⊂Rbe the set whereαhas the opposite sign as in[−1.6,1.6]. With Lemma 7 we see thatΦ(m^red_R (x, y))· e_y₂0.5 for|x|<1.6, y∈D_R, and sinceΦ(m^red_R )·

e_y₂, mΣR=0 we have

√1.6αL²([−1.6,1.6]) 1 π R²Φ

m^red_R

· e_y₂,|m|

[−1.6,1.6]×D_R

1

π R²Φ m^red_R

· e_y₂,|m|

S₋×DR

S₋

2e⁻^|

x|

√2|α|

S₋

√8e⁻^|

x|

√2|∂xα|