Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains

Volume 2010, Article ID 853717,15pages doi:10.1155/2010/853717

Research Article

Two-Scale Convergence of Stekloff Eigenvalue

Problems in Perforated Domains

Hermann Douanla

Department of Mathematical Sciences, Chalmers University of Technology, 41296 Gothenburg, Sweden

Correspondence should be addressed to Hermann Douanla,[email protected]

Received 31 July 2010; Accepted 11 November 2010

Academic Editor: Gary Lieberman

Copyrightq2010 Hermann Douanla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By means of the two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloffeigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural local problem is not an eigenvalue problem.

1. Introduction

We are interested in the spectral asymptoticsas ε → 0 of the linear elliptic eigenvalue problem

−N

i,j1 ∂ ∂xi

aij

x,x ε

∂uε

∂xj

0, inΩε_,

N

i,j1 aij

x,x ε

∂uε

∂xjνi

λεuε, on∂Tε,

uε0, on∂Ω,

ε

Sε|uε| 2

dσεx 1,

1.1

where Ω is a bounded open set in N_x the numerical space of variables x x1, . . . , xN,

the symmetry conditionaji aij, the periodicity hypothesis: for eachx ∈ Ωand for every

k∈

N_{one hasa}

ijx, yk aijx, yalmost everywhere iny∈ Ny , and finally the ellipticity

condition: there existsα >0 such that for anyx∈Ω

Re

N

i,j1 aij

x, y ξjξi≥α|ξ|2 1.2

for allξ∈

N _{and for almost ally∈} N

y , where|ξ|2 |ξ1|2· · ·|ξN|2.

The setΩε _{ε >0is a domain perforated as follows. LetT ⊂Y 0,1}N_{be a compact}

subset in N

y with smooth boundary∂T≡Sand nonempty interior. Forε >0, we define

tεk∈

N_:εkT⊂Ω,

Tε

k∈tε

εkT,

Ωε _Ω\Tε_.

1.3

In this setup,Tis the reference hole, whereasεkTis a hole of sizeεandTε_{is the collection}

of the holes of the perforated domainΩε_{. The familyT}ε_{is made up with a finite number of}

holes sinceΩis bounded. Finally,ν νidenotes the outer unit normal vector to∂Tε≡Sε

with respect toΩε_.

The asymptotics of eigenvalue problems has been widely explored. Homogenization of eigenvalue problems in a fixed domain goes back to Kesavan1,2 . In perforated domains it was first considered by Rauch3 and Rauch and Taylor4 , but the first homogenization results on this topic pertains to Vanninathan5 , where he considered eigenvalue problems for the laplace operator aij δij Kronecker symbol in perforated domains, and

combined asymptotic expansion with Tartar’s energy method to prove homogenization results. Concerning homogenization of eigenvalue problems in perforated domains, we also mention the work of Conca et al.6 , Douanla and Svanstedt7 , Kaizu 8 , Ozawa and Roppongi 9 , Roppongi 10, and Pastukhova 11 and the references therein. In this paper, we deal with the spectral asymptotics of Stekloffeigenvalue problems for an elliptic linear differential operator of order two in divergence form with variable coefficients depending on the macroscopic variable and one microscopic variable. We obtain a very accurate, precise, and concise homogenization resultTheorem3.7by using the two-scale convergence method12–16 introduced by Nguetseng15 and further developed by Allaire

12 . A convergence result for gradients of eigenfunctions is provided, which improves the understanding of the asymptotic behavior of eigenfunctions. We also justify that the natural local problem is not an eigenvalue problem.

Unless otherwise specified, vector spaces throughout are considered over the complex field,, and scalar functions are assumed to take complex values. Let us recall some basic notations. LetY 0,1N, and letF N_{be a given function space. We denote byF}

perYthe space of functions inFloc Nthat areY-periodic and byF#Ythe space of those functions u ∈FperYwith

Yuydy 0. Finally, the letterEdenotes throughout a family of strictly

positive real numbers0< ε≤1admitting 0 as accumulation point. The numerical space N

The rest of the paper is organized as follows. In Section2, we recall some results about the two-scale convergence method, and the homogenization process is consider in Section3.

2. Two-Scale Convergence on Periodic Surfaces

We first recall the definition and the main compactness theorems of the two-scale convergence method. LetΩbe an open bounded set in N_x integerN≥2andY 0,1N, the unit cube.

Definition 2.1. A sequenceuεε∈E ⊂ L2Ω is said to two-scale converge inL2Ωto

someu0 ∈L2Ω×Yif, asE ε → 0,

Ωuεxφ

x,x ε

dx−→

Ω×Y

u0

x, y φx, y dx dy 2.1

for allφ∈L2Ω;CperY.

Notation 1. We express this by writinguε−−→2s u0inL2Ω.

The following theorem is the backbone of the two-scale convergence method.

Theorem 2.2. Letuεε∈Ebe a bounded sequence inL2Ω. Then, a subsequenceEcan be extracted fromEsuch that, asE ε → 0, the sequenceuεε∈E two-scale converges inL2Ωto someu0 ∈ L2_Ω×Y.

Here follows the cornerstone of two-scale convergence.

Theorem 2.3. Letuεε∈Ebe a bounded sequence inH1Ω. Then, a subsequenceEcan be extracted fromEsuch that, asE ε → 0,

uε−→u0, in H1Ω-weak,

uε−→u0, inL2Ω,

∂uε

∂xj

2s

−−→ ∂u0 ∂xj

∂u1 ∂yj,

in L2Ω1≤j≤N ,

2.2

whereu0∈H1Ωandu1∈L2Ω;H_#1Y.

In the sequel, we denote bydσy y∈Y,dσεx x∈Ω, ε∈E, the surface measures

onSandSε_{, respectively. The surface measure ofSis denoted by|S|. The space of squared}

integrable functions, with respect to the previous measures onSandSε_{are denoted byL}2_S andL2_Sε_{, respectively. Since the volume ofS}ε _{grows proportionally to 1/εasε → 0, we}

endowL2_Sε_{with the scaled scalar product17}

u, vL2_Sεε

Sεu

xvxdσεx

u, v∈L2Sε

. 2.3

Definition 2.4. A sequenceuεε∈E⊂L2Sεis said to two-scale converge to someu0∈L2Ω× Sif as follows.E ε → 0,

ε

Sεuεxφ

x,x ε

dσεx−→

Ω×S

u0

x, y φx, y dx dσy 2.4

for allφ∈ CΩ;CperY.

The following result paves the way of the general version of Theorem2.2.

Lemma 2.5. Letφ∈ CΩ;CperY. Then, we have

ε Sε φ x,x ε

2dσεx≤Cφ2_∞ 2.5

for some constantCindependent ofεand, asE ε → 0,

ε Sε φ x,x ε

2dσεx−→

Ω×S

_φ

x, y 2dx dσy . 2.6

Proof. The first part is left to the reader. Letϕ∈ CΩandψ∈ CperY. We have

ε

Sε

ϕxψ

x ε

2dσεx ε

k∈tε

εkS

ϕxψ

x ε

2dσεx. 2.7

Using the second mean value theorem, for anyk∈tε_{, we have}

εkS

ϕxψ

x ε

2dσεx ϕxk2

εkS

ψ x ε

2dσεx 2.8

for somexk∈εkS⊂εkY. Hence,

ε

Sε

ϕxψ

x ε

2dσεx ε

k∈tε

εkS

ϕxψ

x ε

2dσεx

ε

k∈tε

ϕxk2

εkS

ψ x ε 2dσεx

ε

k∈tε

ϕxk2_εN−1

kS

_ψ

y 2dσy

S

_ψ

y 2dσy

k∈tε

εNϕxk2.

But, asE ε → 0,

k∈tε

εNϕ

xk2−→

Ω

_ϕx2

dx, 2.10

and the proof is completed due to the density ofCΩ⊗ CperYinCΩ;CperY.

Remark 2.6. Even if often usedsee, e.g.,13,17 , this is the first time Lemma2.5is rigorously proved. It is worth noticing that because of a trace issue one cannot replace therein the spaceCΩ;CperY byL2Ω;CperY.

Theorem2.2generalizes as follows.

Theorem 2.7. Letuεε∈Ebe a sequence inL2Sεsuch that

ε

Sε

|uεx|2dσεx≤C, 2.11

whereCis a positive constant independent ofε. There exists a subsequenceEofEsuch thatuεε∈E two-scale converges to someu0∈L2Ω;L2Sin the sense of Definition2.4.

Proof. PutFεφ ε

Sεuεxφx,x/εdσεxforφ∈ CΩ;CperY. We have

_Fε

φ ≤C

ε

Sε

φ

x,x ε

2dσεx

1/2

≤Cφ_∞, 2.12

which allows us to viewFεas a continuous linear form onCΩ;CperY. Hence, there exists a bounded sequence of measuresμεε∈Esuch thatFεφ με, φ. Due to the separability of

CΩ;CperYthere exists a subsequenceEofEsuch that in the weak∗topology of the dual ofCΩ;CperYwe haveμε → μ0 asE ε → 0. A limit passageE ε → 0in2.12 yields

_{μ0, φ≤C}

Ω×S

_φ

x, y 2dx dσy 1/2

. 2.13

Butμ0is a continuous form onL2Ω;L2Sby density ofCΩ;CperYin the later space, and there existsu0 ∈L2Ω;L2Ssuch that

μ0, φ

Ω×S

u0

x, y φx, y dx dσy 2.14

for allφ∈ CΩ;CperY, which completes the proof.

In the case whenuεε∈Eis the sequence of traces onSεof functions inH1Ω, a link

Proposition 2.8. Letuεε∈E⊂H1Ωbe such that

uεL2_ΩεDuε_L2_ΩN ≤C, 2.15

whereCis a positive constant independent ofεandD denotes the usual gradient. The sequence of traces ofuεε∈EonSεsatisfies

ε

Sε|uεx| 2

dσεx≤C ε∈E, 2.16

and up to a subsequence E of E, it two-scale converges in the sense of Definition 2.4 to some

u0 ∈ L2Ω;L2Swhich is nothing but the trace on Sof the usual two-scale limit, a function in L2Ω;H_#1Y. More precisely, asE ε → 0,

ε

Sεuεxφ

x,x ε

dσεx−→

Ω×S

u0

x, y φx, y dx dσy ,

Ωuεxφ

x,x ε

dx dy−→

Ω×Y

u0

x, y φx, y dx dy

2.17

for allφ∈ CΩ;CperY.

3. Homogenization Procedure

We make use of the notations introduced earlier in Section1. Before we proceed we need a few details.

3.1. Preliminaries

We introduce the characteristic functionχGof

G Ny \Θ 3.1

with

Θ

k∈ N

kT. 3.2

It follows from the closeness ofT thatΘis closed in N_y so thatGis an open subset of N_y . Next, letε∈Ebe arbitrarily fixed, and define

Vε

u∈H1Ωε:u0 on∂Ω. 3.3

We equipVεwith theH1Ωε-norm which makes it a Hilbert space. We recall the following

Proposition 3.1. For eachε ∈ Ethere exists an operatorPε ofVε intoH₀1Ω with the following properties:

iPεsends continuously and linearlyVεintoH01Ω;

ii Pεv|Ωε vfor allv∈V_ε;

iiiDPεvL2_ΩN≤cDv_L2_ΩεNfor allv∈V_ε, wherecis a constant independent ofεand

Ddenotes the usual gradient operator.

It is also a well-known fact that, under the hypotheses mentioned earlier in the introduction, the spectral problem1.1has an increasing sequence of eigenvalues{λk

ε}∞k1,

0< λ1

ε≤λ2ε≤λ3ε≤ · · · ≤λnε,

λnε −→∞, asn−→∞.

3.4

It is to be noted that if the coefficients aε_ij are real valued then the first eigenvalue λε₁ is isolated. Moreover, to each eigenvalue,λk

ε is attached to an eigenvectorukε ∈Vεand{ukε}∞k1 is an orthonormal basis in L2Sε. In the sequel, the couple λkε, ukε will be referred to as

eigencouple without further ado.

We finally recall the Courant-Fisher minimax principle which gives a usefulas will be seen latercharacterization of the eigenvalues to problem1.1. To this end, we introduce the Rayleigh quotient defined, for eachv∈Vε\ {0}, by

Rεv

ΩεAεDv, Dvdx

Sε|v|2dσεx

, 3.5

whereAε_{is theN}2_{-square matrixa}ε

ij1≤i,j≤NandDdenotes the usual gradient. Denoting by

Ek_{k≥0the collection of all subspaces of dimensionkofV}

ε, the minimax principle is stated

as follows: for anyk≥1, thekth eigenvalue to1.1is given by

λkε min W∈Ek

max

v∈W\{0}R

ε_v

max

W∈Ek−1

min

v∈W⊥_\{0}R

ε_v

. 3.6

In particular, the first eigenvalue satisfies

λ1ε min v∈Vε\{0}

Rεv, 3.7

and every minimum in3.6is an eigenvector associated withλ1

ε.

Now, letQε _{Ω\εΘ. This is an open set in} N_{, andΩ}ε_\Qε_{is the intersection of}

Ωwith the collection of the holes crossing the boundary∂Ω. We have the following result which implies, as will be seen later, that the holes crossing the boundary∂Ωare of no effects as regards the homogenization process since they are in arbitrary narrow stripe along the boundary.

Lemma 3.2see19 . LetK⊂Ωbe a compact set independent ofε. There is someε0 >0such that

Ωε_\Qε_{⊂Ω\Kfor any0< ε≤ε}

Next, we introduce the space

1

0 H01Ω×L2

Ω;H_#1Y. 3.8

Endowed with the following norm

v 1 0

_Dxv0Dyv1

L2_Ω×Y

v v0, v1∈

1 0

, 3.9

1

0 is a Hilbert space admittingF0∞DΩ×DΩ⊗ C∞# Y as a dense subspace. This being so, for

u,v∈

1 0×

1 0, let

aΩu,v

N

i,j1

Ω×Y∗

aij

x, y

∂u0 ∂xj

∂u1 ∂yj

∂v0 ∂xj

∂v1 ∂yj

dx dy. 3.10

This defines a hermitian, continuous sesquilinear form on

1 0×

1

0. We will need the following results.

Lemma 3.3. FixΦ ψ0, ψ1∈F0∞, and defineΦε:Ω → (ε >0) by

Φεx ψ0x εψ1

x,x ε

x∈Ω. 3.11

Ifuεε∈E⊂H01Ωis such that

∂uε

∂xi

2s

−−→ ∂u0 ∂xi

∂u1 ∂yi,

inL2Ω 1≤i≤N 3.12

asE ε → 0, whereu u0, u1∈

1 0, then

aεuε,Φε−→aΩu,Φ 3.13

asE ε → 0, where

aεuε,Φε N

i,j1

Ωεa

ε ij

∂uε

∂xj

∂Φε

∂xi dx. 3.14

Proof. Forε >0,Φε∈ DΩand all the functionsΦεε >0have their supports contained in a

fixed compact setK⊂Ω. Thanks to Lemma3.3, there is someε0>0 such that

Using the decomposition Ωε _Qε_∪Ωε_\Qε _{and the equalityQ}ε _{Ω∩εG, we get for}

E ε≤ε0

aεuε,Φε N

i,j1

Ωε aij x,x ε ∂uε ∂xj

∂Φε

∂xidx

N

i,j1

Qεaij

x,x ε ∂uε ∂xj

∂Φε

∂xidx

N

i,j1

Ω∩εG

aij x,x ε ∂uε ∂xj

∂Φε

∂xidx

N

i,j1

Ωaij

x,x ε

χεGx∂uε

∂xj

∂Φε

∂xi dx

N

i,j1

Ωaij

x,x ε χG x ε ∂uε ∂xj

∂Φε

∂xidx.

3.16

Bear in mind that, asE ε → 0, we havesee, e.g.,19, Lemma 2.4

N

i,j1 ∂uε

∂xj

∂Φε

∂xi

2s

−−→ N

i,j1

∂u0 ∂xj ∂u1 ∂yj ∂ψ0 ∂xj ∂ψ1 ∂yj

, inL2Ω. 3.17

We also recall that aijx, yχGy ∈ CΩ;L2perY1 ≤ i, j ≤ Nand that Property2.1 in Definition2.1still holds forf inCΩ;L2

perYinstead ofL2Ω;CperYwhenever the two-scale convergence therein is ensuredsee, e.g.,14, Theorem 15 . Thus, asE ε → 0,

aεuε,Φε N

i,j1

Ωaij

x,x ε χG x ε ∂uε ∂xj

∂Φε

∂xidx

−→N

i,j1

Ω×Y

aij

x, y χG

y ∂u0 ∂xj ∂u1 ∂yj ∂ψ0 ∂xj ∂ψ1 ∂yj dx dy N

i,j1

Ω×Y∗

aij x, y ∂u0 ∂xj ∂u1 ∂yj ∂ψ0 ∂xj ∂ψ1 ∂yj dx dy

aΩu,Φ,

3.18

We now construct and point out the main properties of the so-called homogenized coefficients. Let 1≤j≤N, and fixx∈Ω. Put

ax;u, v

N

i,j1

Y∗

aij

x, y ∂u ∂yj

∂v ∂yidy,

ljx, v N

k1

Y∗akj

x, y ∂v ∂ykdy

3.19

foru, v∈H_#1Y. Equipped with the seminorm

Nu Dyu_L2_Y∗N

u∈H#1Y

, 3.20

H_#1Y is a pre-Hilbert space that is nonseparate and noncomplete. Let H_#1Y∗ be its separated completion with respect to the seminorm N· and i the canonical mapping of

H1

#YintoH#1Y∗. We recall that

iH1

#Y∗is a Hilbert space;

iiiis linear;

iiiiH1

#Yis dense inH#1Y∗;

iviuH1

#Y∗Nufor everyuinH 1 #Y;

vifFis a Banach space andla continuous linear mapping ofH_#1YintoF, then there exists a unique continuous linear mappingL:H1

#Y∗ → Fsuch thatlL◦i.

Proposition 3.4. Letj 1, . . . , N, and fixxinΩ. The noncoercive local variational problem

u∈H#1Y, ax;u, v ljx, v, ∀v∈H#1Y 3.21

admits at least one solution. Moreover, ifχj_xandθj_{xare two solutions,}

Dyχjx Dyθjx a.e.inY∗. 3.22

Proof. Proceeding as in the proof of19, Lemma 2.5 , we can prove that there exists a unique hermitian, coercive, continuous sesquilinear form Ax;·,· on H1

#Y∗×H#1Y∗ such that Ax;iu,iv ax;u, vfor allu, v∈H1

#Y. Based onvabove, we consider the antilinear formljx,·onH_#1Y∗such thatljx,iu ljx, ufor anyu∈H_#1Y. Then,χjx∈H_#1Y

satisfies3.21if and only ifiχj_xsatisfies

iχjx∈H_#1Y∗, Ax;iχjx, Vljx, V, ∀V ∈H#1Y∗. 3.23

Butiχj_{xis uniquely determined by3.23 see, e.g.,20, page 216 . We deduce that3.21}

which means thatχj_xandθj_{xhave the same neighborhoods inH}1

#Yor equivalently Nχj_x−θj_{x 0. Hence,3.22.}

Corollary 3.5. Let1≤ i, j ≤N, andxfixed inΩ. Letχj_x∈H1

#Ybe a solution to3.21. The

following homogenized coefficients

qijx

Y∗

aij

x, y dy−

N

l1

Y∗

ail

x, y ∂χ

j

∂yl

x, y dy 3.24

are well defined in the sense that they do not depend on the solution to3.21.

Lemma 3.6. The following assertions are true:

iqij∈ CΩ,

iiqjiq_ij,

iiithere exists a constantα0>0such that

Re

N

i,j1

qijxξjξi≥α0|ξ|2 3.25

for allx∈Ωand allξ∈

N_.

Proof. See for example,21 .

We are now in a position to state the main result of this paper.

3.2. Homogenization Result

Theorem 3.7. For eachk≥1and eachε∈E, letλk

ε, ukεbe the k’th eigencouple to1.1. Then, there exists a subsequenceEofEsuch that

1

ελ

k

ε −→λk0, in asE ε−→0, 3.26

Pεukε −→uk0, inH01Ω-weak asE ε−→0, 3.27

Pεukε −→uk0, inL2ΩasE ε−→0, 3.28

∂Pεukε

∂xj

2s

−−→ ∂uk0 ∂xj

∂uk₁ ∂yj,

inL2Ω, asE ε−→0 1≤j ≤N , 3.29

whereλk₀, uk₀∈ ×H

1

−N

i,j1 ∂ ∂xi

1

|S|qijx ∂u0 ∂xj

λ0u0, in Ω,

u00, on∂Ω,

Ω|u0|

2_dx 1

|S|,

3.30

whereuk₁ ∈L2Ω;H_#1Y. Moreover, for almost everyx∈Ω, the following hold true:

iuk

1xis a solution to the noncoercive variational problem

uk

1x∈H#1Y,

ax;uk₁x, v−

N

i,j1 ∂uk

0 ∂xj

Y∗aij

x, y ∂v ∂yidy,

∀v∈H1 #Y.

3.31

iiWe have

iuk₁x

N

j1 ∂uk

0 ∂xjxi

χjx, 3.32

whereχj_{is any function inH}1

#Ydefined by the cell problem3.21.

Proof. Let us first recall that, according to the properties of the coefficientsqijLemma3.6, the

spectral problem3.30admits a sequence of eigencouples with similar properties to those of problem1.1. However, this is also proved by our homogenization process.

Now, fixk≥1. There exists a constant 0< c1<∞independent ofεsuch that

0< λkε ≤c1μkε, 3.33

where

μkε min W∈Ek

max

v∈W\{0}

Ωε|Dv|2dx

Sε|uε|2dσεx

, 3.34

Ek _{still being the collection of subspaces of dimension k of V}

ε. But it is proved in

5, Proposition 12.1 that 0 < μk

ε < c2ε, c2 being a constant independent ofε. Hence the sequence1/ελk

εε∈Eis bounded in. Clearly, for fixedE ε >0,uk

εlies inVεand

N

i,j1

Ωεa

ε ij

∂ukε

∂xj

∂v ∂xidx

1

ελ

k ε

ε

Sεu

k

for anyv ∈ Vε. Bear in mind thatε

Sε|ukε|2dσεx 1, and chose v ukε in 3.35. The

boundedness of the sequence1/ελk

εε∈Eand the ellipticity assumption1.2implies at once

by means of Proposition3.1that the sequencePεukεε∈Eis bounded inH01Ω. Theorem2.3 and Proposition 2.8 apply simultaneously and give us uk _uk

0, uk1 ∈

1

0 such that for someλk₀ ∈ and some subsequenceE

_{⊂ Ewe have 3.26–3.29, where3.28is a direct} consequence of3.27by the Rellich-Kondrachov theorem. For fixedε ∈ E, letΦε be as in

Lemma3.3. Multiplying both sides of the first equality in1.1byΦεand integrating overΩ

leads us to the variationalε-problem

N

i,j1

Ωεa

ε ij

∂Pεukε

∂xj

∂Φε

∂xi dx

1 ελ k ε ε Sε

Pεukε

Φεdσεx. 3.36

Sendingε∈Eto 0, keeping3.26–3.29and Lemma3.3in mind, we obtain

N

i,j1

Ω×Y∗

aij ∂u0 ∂xj ∂u1 ∂yj ∂ψ0 ∂xj ∂ψ1 ∂yj

dx dyλk₀

Ω×S

uk₀ψ0dx dσ

y . 3.37

The right-hand side follows by means of Proposition2.8as explained:

ε

Sε

Pεukε

Φεdσεx ε

Sε

Pεukε

ψ0dσεx ε

ε

Sε

Pεukε

ψ1 x,x ε

dσεx

−→

Ω×S

uk0ψ0dx dσ

y 0, asE ε−→0.

3.38

Therefore,λk

0,uk∈ ×

1

0 solves the followingglobal homogenized spectral problem:

findλ,u∈ ×

1

0 such that

N

i,j1

Ω×Y∗aij

∂u0 ∂xj ∂u1 ∂yj ∂ψ0 ∂xi ∂ψ1 ∂yi

dx dyλ|S|

Ωu0ψ0dx, ∀Φ∈

1 0.

3.39

To provei, chooseΦ ψ0, ψ1in3.39such thatψ0 0 andψ1 ϕ⊗v1, whereϕ∈ DΩ andv1∈H_#1Yto get

Ωϕx

⎡

⎣N

i,j1

Y∗

aij

∂uk₀ ∂xj

∂uk₁ ∂yj

∂v1 ∂yidy

⎤

⎦_dx0. 3.40

Hence, by the arbitrariness ofϕ, we have a.e. inΩ

N

i,j1

Y∗aij

∂uk₀ ∂xj

∂uk₁ ∂yj

∂v1 ∂yidy

0 3.41

Regarding ii, pick any χj_{x solution to the cell problem 3.21, and put zx}

N

j1∂uk0/∂xjxχjx.

By multiplying both sides of3.21by−∂uk

0/∂xjxand then summing over 1≤j ≤ N, we see thatzxsatisfies3.31. Hence,izx iuk_{xby uniqueness of the solution}

to the coercive variational problem inH_#1Y∗corresponding to the noncoercive variational problem3.31 see the proof of Proposition3.4. Thus,3.32follows sinceiis linear.

Now, by consideringΦ ψ0, ψ1in3.39such thatψ10 andψ0∈ DΩ, we get

N

i,j1

Ω×Y∗

aij

∂uk

0 ∂xj

∂uk

1 ∂yj

∂ψ₀

∂xi dx dy|S|λ k

0

Ωu

k

0ψ0dx. 3.42

As3.32is equivalentsee the proof of Proposition3.4to

Dyuk₁x N

j1 ∂uk

0

∂xjxDyχ

j_{x, a.e.inY}∗_{, 3.43}

we arrive at

N

i,j1

Ω

Y∗

aijdy− N

l1

Y∗

ail∂χ j

∂yldy

∂uk₀ ∂xj

∂ψ₀

∂xi dx|S|λ k

0

Ωu

k

0ψ0dx, 3.44

that is,see3.24

N

i,j1

Ω

1

|S|qijx ∂uk

0 ∂xj

∂ψ₀ ∂xidxλ

k

0

Ωu

k

0ψ0dx. 3.45

Thanks to the arbitrariness ofψ0and the weak derivative formula, we conclude thatλk₀, uk₀ is thekth eigencouple to3.30and the whole sequence1/ελk

εε∈Econverges.

Finally, by using3.28and a similar line of reasoning as in the proof of Lemma2.5, we arrive at

lim

E _ε→0ε

Sε

PεukεPεulεdσεx |S|

Ω

uk₀ul₀dx. 3.46

The normalization condition in3.30follows thereby, and moreover{uk₀}∞k1is an orthogonal basis inL2_Ω.

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