Contribution to peroidic homogenization of a spectral problem and of the wave equation

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problem and of the wave equation

Thi Trang Nguyen

To cite this version:

Thi Trang Nguyen.

Contribution to peroidic homogenization of a spectral problem and of the

wave equation. Mechanics [physics.med-ph]. Université de Franche-Comté, 2014. English. �NNT :

2014BESA2027�. �tel-01140982�

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é c o l e d o c t o r a l e

_{s c i e n c e s p o u r l ’ i n g é n i e u r e t m i c r o t e c h n i q u e s}

Contribution à l’homog én éisation

p ´eriodique d’un probl `eme spectral

et de l’ ´equation d’onde

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é c o l e d o c t o r a l e

_{s c i e n c e s p o u r l ’ i n g é n i e u r e t m i c r o t e c h n i q u e s}

TH `

ESE pr ´esent ´ee par

T

HI

T

RANG

NGUYEN

pour obtenir le

Grade de Docteur de

l’Universit ´e de Franche-Comt ´e

Sp ´ecialit ´e :

Sciences pour l’ing ´enieur

Contribution à l’homog én éisation p ériodique d’un

probl `eme spectral et de l’ ´equation d’onde

Unit ´e de Recherche :

FEMTO-ST, D épartement Temps-Fr équence, Universit é de Franche-Comt é

Soutenue publiquement le 3 D ´ecembre 2014 devant le Jury compos ´e de :

M

ORVAN

OUISSE

Pr ´esident

Professeur, ENSMM, Besanc¸on

C

ARLOS

CONCA

Rapporteur

Professeur, Universit ´e du Chili, Chili

S ´

EBASTIEN

GUENNEAU

Rapporteur

Directeur

de

recherche

CNRS,

Institut

Fresnel,

Universit ´e

d’Aix

Marseille

J

UAN

CASADO-D´IAZ

Rapporteur

Professeur, Universit ´e de S ´eville,

Espagne

M

ICHEL

LENCZNER

Directeur de Th `ese

Professeur,

Universit ´e

de

Technologie Belfort-Montb ´eliard

M

ATTHIEU

BRASSART

Co-Directeur de Th `ese

Maˆıtre de conf ´erences, Universit ´e de

N

◦

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D´

epartement Temps-Fr´

equence

Ecole doctorale SPIM Besan¸

´

con

UFR ST Sciences et Techniques

Contributions to periodic

homogenization of a spectral problem

and of the wave equation

Dissertation

Submitted in Partial Fulfillment of the Requirements

for the Degree of Doctor of Ph.D. in Engineering Sciences

3 December 2014

Doctoral School of University of Franche-Comt´

e,

By

Thi Trang NGUYEN

Doctoral Committee:

President :

_{Professor Morvan OUISSE}

Reviewers :

_{Professor Carlos CONCA}

Director of research S´

ebastien GUENNEAU

Professor Juan CASADO-D´IAZ

Examiners :

_{Professor Michel LENCZNER}

Assistant professor Matthieu BRASSART

Professor Morvan OUISSE

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In this dissertation, we present the periodi homogenization of a spe tral

prob-lem and the wave equation with periodi rapidly varying oe ients in a bounded

domain. The asymptoti behavior isaddressed based ona method of Blo h wave

ho-mogenization. It allows modeling both the low and high frequen y waves. The low

frequen y part iswell-known and itis not anew pointhere. Inthe opposite, the high

frequen ypartofthe model,whi hrepresentsos illationso urringatthemi ros opi

andma ros opi s ales,wasnot wellunderstood. Espe ially,the boundary onditions

ofthehigh-frequen yma ros opi equationestablishedin[36℄werenotknown priorto

the ommen ementofthesis. Thelatterbringsthreemain ontributions. Thersttwo

ontributions, are about the asymptoti behavior of the periodi homogenization of

the spe tral problemand wave equation in one-dimension. They are derived starting

from a system of rst order equation as in [36℄ but also from the usual se ond order

equation. The two-s ale models are only for high frequen y waves in the ase of the

spe tral problem and for both high and low frequen ies for the wave equation. The

high frequen y models in lude a mi ros opi and a ma ros opi part, both in luding

boundary onditions, whi h for the latter is a novelty. Numeri al simulation results

are provided to orroborate the theory. The third ontribution onsists in an

exten-sion of the model for the spe tral problem to a thin two-dimensional bounded strip

Ω = (0, α) × (0, ε) ⊂ R

2

. The homogenization result in ludes boundary layer ee ts

o urring in the boundary onditions of the high-frequen y ma ros opi equation.

Keywords: Homogenization,Blo hwaves, Blo hwavede omposition,Spe tral

prob-lem, Wave equation, Two-s ale transform,Two-s ale onvergen e, Unfolding method,

Boundary layers, Boundary layer two-s ale transform, Ma ros opi equation,

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Dans ette thèse, nous présentons des résultats d'homogénéisation périodique d'un

problème spe tral et de l'équation d'ondes ave des oe ients périodiques variant

rapidement dans un domaineborné. Le omportement asymptotiqueest étudié en se

basant sur une méthode d'homogénéisation par ondes de Blo h. Il permet de

mod-éliser les ondes à basse et haute fréquen es. La partie du modèle à basse fréquen e

est bien onnu et n'est pas don abordée dans e travail. A ontrario, la partie à

hautefréquen edu modèle,quireprésentedes os illationsauxé hellesmi ros opiques

et ma ros opiques, est un problème laissé ouvert. En parti ulier, les onditions aux

limites de l'équation ma ros opique à hautes fréquen es établies dans [36℄ n'étaient

pas onnues avant ledébut de la thèse. Ce derniertravail apporte trois ontributions

prin ipales. Les deux premières ontributions, portent sur le omportement

asympto-tiqueduproblèmed'homogénéisationpériodiqueduproblèmespe traletde l'équation

des ondes en une dimension. Ellessont dérivées soità partir d'un système d'équation

du premier ordre omme dans [36℄, soit à partir de l'équation du se ond ordre. Les

modèlesàdeux é hellessontobtenuspour desondesàhautefréquen eseulementpour

le problème spe tral et pour les basses et hautes fréquen es pour l'équation des

on-des. Les modèles à haute fréquen e omprennent à la fois une partie mi ros opique

et une partie ma ros opique, ette dernière in luant des onditions au bord, e qui

est une nouveauté. Des résultats de simulations numériques orroborent la théorie.

La troisième ontribution onsiste en une extension du modèle du problème spe tral

posédans unebandemin ebidimensionnelleetbornée. Lerésultatd'homogénéisation

omprenddes eets de ou he limitequise produisentdans les onditions auxlimites

de l'équation ma ros opiqueà haute fréquen e.

Mots- lés: Homogénéisation, Ondes de Blo h, Dé omposition en ondes de Blo h,

Problème spe tral, Equation des ondes, Transformée à deux-é helles, Convergen e à

deux é helles, Méthode d'é latement périodique, Cou hes limites, Transformation à

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First of all, I would like to deeply thank to my advisor, Prof. Mi hel LENCZNER,

for his kindheartedly advising. With honor, I would like to say Thank you, Prof.

Mi hel LENCZNER, for all of your help, guidan e, dedi ation, enthusiasm, patien e

and instru tions. It wasreallylu ky forme towork with you. Moreover, I owe a big

thank tomy o-advisor,Prof. MatthieuBRASSART, whowas alwaysready toguide,

help and en ourage me throughout this thesis. I do appre iate all of your support,

enthusiasm, advi e, and allmathemati aldis ussions in this thesis.

I would like to send my appre iation to all the members of Jury, Prof. Carlos

CONCA, Prof. Sébastien GUENNEAU, Prof. Juan CASADO-DÍAZ and Prof.

Mor-van OUISSE for reviewingand examining my thesis.

I hereby express mygratitude toespe iallythank toall stamembers atInstitute

of FEMTO-ST and all my friends at University of Fran hé-Comté for providing me

with alot of kindassistan es during the time when I have worked atFEMTO-ST. In

parti ular, I would like to sin erely thank to Dr. Philip Lutz, my PhD dire tor, and

Dr. Vin ent GIORDANO for allof their generous support. I amvery grateful to get

the aid and kindness of Prof. Ni olas RATIER, Prof. EmmanuelBIGLER and Prof.

BernardDULMET. I amvery thankfultothe se retaries Mrs. FabienneCORNUand

Mrs. Sarah DJAOUTI from Time frequen y department, Mrs. Isabelle GABET and

Mrs. Sandrine FRANCHI of our Lab for helping me in work ontra t issues. I am

alsothankful to the se retaries of do toral s hool of University of Fran hé-Comté for

helpingmewith registrationand preparation of my defense.

I also express my gratitude to thank Prof. Du Trong DANG, Prof. Minh Du

DUONGandProf. Pas alOMESfortea hingmeforthepastseveralyears,helpingme

to develop my ba kground knowledge in mathemati s and en ouragingme to pursue

my resear h areer.

A huge thank is given to my olleagues, Hui HUI, Bin YANG, Raj Narayan

DHARA, Youssef YAKOUBI, Huu Quan DO, Duy Du NGUYEN, and Mohamed

ABAIDI for their kindly help and instru tions. A lot of thanks are sent to all of my

friends living in Vietnam and Fran e for their kindness and sharing in my work and

my life.

Lastbutnotleast,agreatthankgoestomyparentswhoalwayswishmeallthebest

inlife,tomy youngerbrotherandtwoyoungersisterswho always en ourageand help

mewhenever I need them. A spe ial thank is given to my husbandfor his invaluable

support, love, andunderstanding. He isalways by my side to heer meup and stands

by me through the good times and bad. Many respe tful thanks are expressed to my

parents-in-law, un le Duy Chinh LE, aunt Thi Minh Thanh NGUYEN, ousin Anh

Tuan LE and allof my family.

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my younger sisters

my younger brother

my husband

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List of Figures ix

List of Tables xi

Introdu tion 1

Chapter 1 Notations, assumptions and elementary properties 7

1.1 Notations . . . 7

1.2 Blo hwaves and two-s ale transform . . . 8

1.2.1 In one dimension . . . 8

1.2.2 In atwo-dimensionalstrip . . . 12

1.3 Assumption of sequen e

ε

. . . 15

Chapter 2 Homogenization of the spe tral problem in one-dimension 17 2.1 Introdu tion . . . 17

2.2 Statement of the problem . . . 19

2.3 Homogenizationof the high-frequen yeigenvalue problem . . . 19

2.3.1 Mainresult . . . 20

2.3.2 Modalde omposition onthe Blo hmodes . . . 23

2.3.3 Derivation of the high-frequen yma ros opi equation . . . 24

2.3.4 Analyti solutions. . . 28

2.3.5 Neumannboundary onditions. . . 30

2.4 Homogenizationbased ona rst order formulation . . . 31

2.4.1 Reformulationof the spe tral problemand the main result . . . 31

2.4.2 Modelderivation . . . 33

2.5 Numeri alsimulations . . . 36

2.5.1 Simulationmethods and onditions . . . 36

2.5.2 Approximation of physi al modes by two-s ale modes . . . 37

2.5.3 The modelingproblem . . . 41

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Chapter 3 Homogenization of the one-dimensional wave equation 43

3.1 Introdu tion . . . 44

3.2 Statement of the results for the wave equation . . . 45

3.2.1 Assumptions. . . 46

3.2.2 The model . . . 47

3.2.3 Approximation result . . . 49

3.2.4 Analyti solutionsfor the homogeneous equation(

f

ε

_{= 0}

). . . . 51

3.3 Model derivation . . . 54

3.3.1 Preliminaryhomogenization results and their proofs . . . 56

3.3.2 Proof of main Theorem . . . 71

3.4 Other ases . . . 72

3.4.1 Neumannboundary onditions. . . 72

3.4.2 Generalizationof the wave equation . . . 73

3.5 Homogenizationbased ona rst order formulation . . . 81

3.5.1 Reformulationof the wave equation underthe rst order formulation 82 3.5.2 Homogenizedresults and proofs . . . 82

3.6 Numeri alexamples. . . 86

Chapter 4 Homogenization of the spe tral problem in a two dimensional strip 91 4.1 Introdu tion . . . 91

4.2 Statement of the results . . . 93

4.2.1 Assumptions. . . 94

4.2.2 The model . . . 94

4.2.3 Two-s ale asymptoti behaviour . . . 95

4.3 Model derivation . . . 97

4.3.1 Derivation of the HF-mi ros opi equation . . . 97

4.3.2 Derivation of the boundary layerequation . . . 99

4.3.3 Derivation of the ma ros opi equation . . . 102

4.3.4 Proof of Theorem 46 . . . 111

Chapter 5 Con lusions and perspe tives 113

Appendix 115

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1 Composite materialhas ama ros opi shape and a mi rostru ture. . . 6

2.1 Firstten eigenvalues of the Blo hwave spe tral problem. . . 38

2.2 (a)Errors for

p = 85

and

k ∈ L

∗+

125

. (b) Errors for asele tion of

k

. . . . 38

2.3 (a)Blo hwave solution

φ

k

n

. (b) Ma ros opi solutions

u

k

n,ℓ

and

u

−k

n,ℓ

. . . 39

2.4 (a)Physi aleigenmode

w

ε

p

. (b) Relative error between

w

ε

p

and

ψ

ε,k

n,ℓ

. . . 39

2.5 (a)Errors for

p

varying in

J

ε

0

. (b) Ma ros opi eigenvalues. . . 40

2.6 (a)Error of approximation for

∆

k

= 3.0e − 3

. Ratios of errorredu tion. 40 2.7 (a)Two-s ale eigenmode

ψ

ε,k

n,ℓ

. (b) Relative error ve tor. . . 41

2.8 (a)

λ

1,ℓ

with respe t to

n

. (b)

γ

k

n,ℓ

with respe t to

n

. . . 42

3.1 (a)Initial ondition

u

ε

0

. (b) Initial onditions of HF-ma ros opi equation. 87 3.2 HF-ma ros opi solutions

u

k

n

and

u

−k

n

at

t = 0.466

and

x = 0.699

. . . . 88

3.3 (a)Physi alsolution at

x = 0.699

. (b) Relativeerror ve tor. . . 88

t = 0.466

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2.1 Errorsfor

∆

k

= 8.e − 3

and

3e − 3

. . . 40

2.2 Resultsfor the modelingproblem. . . 41

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Thehomogenizationtheory wasintrodu edinordertodes ribethe behaviourof

om-posite materials. Composite materials are hara terized by both a mi ros opi and

ma ros opi s alesdes ribing heterogeneitiesand the globalbehaviourofthe

ompos-iterespe tively,seeFigure1asanexample. Theaimof homogenizationispre isely to

give ma ros opi properties of the omposite by taking intoa ountproperties of the

mi ros opi stru ture. Thenamehomogenizationwasintrodu edin1974byBabuska

in[14℄ and it be ame animportantsubje t in Mathemati s. In the mathemati al

lit-erature, the homogenization of physi al systems with a periodi mi rostru ture or

periodi media is alled "periodi homogenization". Avastliterature exists whereone

distinguishes between sto hasti and deterministi homogenization orresponding to

sto hasti and deterministi mi ro-stru tures, the later being mostly on erned with

periodi homogenization. Nevertheless, there are also resear h works on non-periodi

deterministi mi ro-stru tures as in [91℄, [93℄, [92℄, [34℄. I re ommend the

introdu -tory book by D. Cioranes u and P. Donato [44℄ whi h is a good start to study the

homogenization theory of partial dierential equations. I also re ommend the books

[99℄, [19℄, [63℄, [109℄ to understand, not only the homogenization theory, but also its

motivation,histori aldevelopment and largerviewovervarious methods, see alsotwo

thesis works [108℄ and [55℄ for abrief history.

This thesis falls within the area of deterministi homogenization and its aim is to

study the periodi homogenization of a spe tral problem, at high frequen y, and of

the wave equation, simultaneously at high and low frequen ies, in an open bounded

domain

Ω ⊂ R

N

withtime-independentperiodi oe ients. Our omplete resultsare

presented for a one-dimensional geometry and also for a thin two-dimensional strip,

however a signi ant part of our results extend trivially to multi-dimensional ases.

The modelderivation method isbased onthe modulated-two-s aletransform and the

Blo hwave de omposition. I re allthat the two-s aletransformorperiodi unfolding

operator [77℄, [79℄, [78℄, [76℄, [45℄, [37℄ or[46℄, transforms a fun tionof the variablein

the physi al spa e into a fun tion of two variables, namely the ma ros opi variable

and the mi ros opi variables. This ishowthe on ept oftwos ale- onvergen e turns

out to bea usual notion of onvergen e of fun tions that an be weak orstrong. The

modulated-two-s aletransform was dened in [36℄ by multiplyingthe usual two-s ale

transform by a family of os illating exponential fun tions whi h ee t is to yield a

orresponding family of two-s ale limitswith all possible periodi ities also refered as

quasi-periodi ities.Wealsouse its ounterpartdenedfromthetwo-s ale onvergen e

issued from[89℄,[90℄, [1℄, [2℄, [81℄.

The Blo h wave de omposition, alsoknown as Floquet de omposition, was

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A Blo h wave de omposition of a fun tion, onsists in an expansion over a family of

the eigenfun tions solutionto the spe tral problem

div

y

a∇

y

φ

k

= −λ

k

φ

k

posedinthereferen e ell

Y ⊂ R

N

equippedwith

k

-quasi-periodi boundary onditions

forsome

k ∈ [−1/2, 1/2)

N

. Wereferto[104℄,[99℄,[111℄foranintrodu tiontotheBlo h

waves in spe tral analysis. Su h a de omposition is used in the so- alled Blo h wave

homogenization method for spe tral problems [7℄, [51℄, [8℄ and for ellipti problems

[48℄, [50℄. We noti e that we all our approa h with the same name "Blo h wave

homogenization"even if the te hniques dier insome aspe ts but we thinkthat they

yieldsimilar results.

For a two-dimensional strip, a boundary orre tor is required so that the

asymp-toti solution satises the nominal boundary ondition. It is solution to a boundary

layer problem posed in

R

+

_{× (0, 1)}

. Its solution might de reases exponentially with

respe t to the rst variable. The derivation of this part of model is a hieved by a

two-s ale transformdedi ated toboundary layers that an berelated tothe two-s ale

onvergen e for boundary layers as in[9℄.

Inallthiswork,thehomogenizationpro essstartswithaveryweakformulationofthe

spe tralorwave equation. Applying our method,providestwo-s alemodels in luding

theexpe tedhighfrequen y partsbutalsoalowfrequen ypartforthewaveequation.

Thelatteriswellknownsin eithasbeenfoundbyvariousauthors,soourworkfo uses

mainlyonthe high frequen y part. It omprises so- alled high-frequen ymi ros opi

and ma ros opi equations,the rst being ase ond orderpartial dierentialequation

and the se ond a system of rst order partialdierentialequations. In the strip ase,

the boundary layer problemisa se ond order partialdierentialequation.

The thesis in ludes three main ontributions. In the rst one, we onsider the

solution(

w

ε

_{, λ}

ε

)

of the spe tral problem

−∂

x

(a

ε

∂

x

w

ε

) = λ

ε

ρ

ε

w

ε

,

(1)

posed in a one-dimensional open bounded domain

Ω ⊂ R

, with Diri hlet or

Neu-mann boundary onditions. An asymptoti analysis of this problem is arried out

where

ε > 0

is aparameter tendingtozero andthe oe ientsare

ε

-periodi , namely

a

ε

_{= a}

x

ε

and

ρ

ε

_{= ρ}

x

ε

,

a (y)

and

ρ (y)

being

1

-periodi in

R

. Homogenization

of spe tral problems has been studied in various works providing the asymptoti

be-haviour of eigenvalues and eigenve tors. The low frequen y part of the spe trum has

been investigated in [69℄, [70℄, [110℄. Then, many ongurations have been analyzed,

as[52℄and[49℄forauid-stru tureintera tion,[21℄,[5℄forneutrontransport,[86℄,[98℄

for

ρ

whi h hanges sign or [6℄for the rst high frequen y eigenvalue and eigenve tor

for a one-dimensional non-self-adjoint problem with Neumann boundary onditions.

Higher order of asymptoti of the eigenvalueshave been studied in[106℄ and [101℄. A

survey on re ent spe tral problems en ountered in mathemati al physi s is available

in[71℄. In animportant ontribution[8℄, G.Allaireand C. Con a studiedthe

asymp-toti behaviour of both the low and high frequen y spe trum. In order to analyze

theasymptoti behaviourofthe highfrequen y eigenvalues,they used theBlo hwave

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eigenvalues

ε

2 _λ

ε

isthe unionof theBlo hspe trumand theboundary layerspe trum,

when

ε

goes to

0

. However, the asymptoti behaviourof the orresponding

eigenve -tors wasnot addressed for abounded domain

Ω

. Thisis the goalofthis ontribution.

We only fo uses on the Blo h spe trum of the high frequen y part. By applying the

Blo h wave homogenization method, the two-s ale model is derived in luding both

mi ros opi and ma ros opi eigenmodes with boundary onditions. We derive the

homogenization model from both the se ond order equation (2.1) and an equivalent

rst order system of equations. We observe that the two models are equivalent. The

asymptoti behaviour of the eigenvalue

λ

ε

and orresponding eigenve tor

w

ε

are

pro-vided.

In the se ond ontribution, we establisha homogenized model for the wave

equa-tion,

ρ

ε

_∂

tt

u

ε

− ∂

x

(a

ε

∂

x

u

ε

) = f

ε

,

u

ε

_{(t = 0, x) = u}

ε

0

and

∂

t

u

ε

_{(t = 0, x) = v}

ε

0 ,

(2)

posedinanite timeinterval

I ⊂ R

+

and inaone-dimensionalopen boundeddomain

Ω ⊂ R

withDiri hletboundary onditions. The asymptoti analysis is arriedout

un-der thesame assumptionsasfor thespe tral problemregarding

ε

andthe oe ients.

The homogenization of the wave equation has been studied in various works. The

onstru tionof homogenization and orre tor results for the lowfrequen y waves has

beenpublishedin[33℄,[60℄. Theseworkswerenottakingintoa ountfasttime

os illa-tions,sothe models ree tonlyapartof the physi alsolution. Similarsolutionshave

been derived for the ase where the oe ients depend onthe time variable

t

in [47℄,

[38℄. In [35℄ and [36℄, an asymptoti analysis of the solution

u

ε

_{(t, x)}

, that onserves

time and spa e os illationso urring both at low and high frequen ies in a bounded

domain, has been introdu ed. It is derived from a formulation of the wave equation

as a rst order system and uses a de omposition over Blo h modes. It extends the

thesis work [65℄ a hieved inone-dimension. Byusing the Blo hwave homogenization

method, the resulting asymptoti model in ludes separated parts for low and high

frequen ywavesrespe tively. Thelatteris omprisedwithami ros opi equationand

with a rst order ma ros opi equation whi h boundary onditions are missing. A

similarresult has been obtained in [39℄, based onthe se ond order formulation of the

wave equation, whi h homogenized solution is periodi in spa e be ause it does not

in lude a de omposition on Blo h modes. In the present ontribution, we synthesize

these ideas in a model, based on the se ond order formulation of the wave equation,

using the Blo h wave de omposition of the solution and more importantly in luding

boundary onditions. The main resultof this ontributionisthe boundary onditions

of the high frequen y ma ros opi model. However, the high frequen y ma ros opi

model is also new sin e it diers from this in [36℄ derived from a rst order system

only. In addition,the proofhasbeensimplied. Moreover, forthe sakeof omparison,

the homogenization is also presented under the rst order formulation as in [35℄ and

[36℄,thenboundary onditionsfortheone-dimensionalmodeloftheseworkshavebeen

announ ed. In on lusion, the physi alsolution

u

ε

is approximated by asum of alow

frequen y term, the usual orre tor in ellipti problems, using the solutionof the ell

problem, and a sum of Blo h waves being the orre tor for the high frequen y part.

Thesame resultis alsoestablished forthe Neumannboundary onditions andalsofor

ageneralizationof the wave equationtaking intoa ount azero order term aswellas

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We quote that in both ontributions, the models and proofs have been written

in one-dimension but they extend trivially to multi-dimensional ases, ex ept what

refers to the high frequen y ma ros opi boundary onditions whi h remainsanopen

question in higher dimension. Hen e, to do a step towards the possibility of taking

into a ount a multi-dimensionalgeometry, we address the ase of a two-dimensional

bounded strip. This yieldsthe third ontribution. Duetotimelimitation,onlyresults

on the spe tral problem are reported, but we expe t that they extend to the wave

equation. We study the periodi homogenizationof the spe tralproblem

−div (a

ε

∇w

ε

) = λ

ε

ρ

ε

w

ε

posed inanopen bounded strip

Ω = ω

1 × (0, ε) ⊂ R

2

with

ω

1 = (0, α) ⊂ R

+

,with the boundary onditions

w

ε

_{= 0}

on

∂ω

1 × (0, ε)

and

a

ε

_∇

x

w

ε

.n

x

= 0

on

ω

1 × {0, ε} ,

withthesameassumptionsregarding

ε

andthe oe ientsex eptedthatthereferen e

ell

Y ⊂ R

2 _.

The results of this part are an extension of those obtained in the rst

one, and the main remaining di ulty onsists in establishing the boundary

ondi-tions of the ma ros opi equation. The modelderivationmethodis stillbased on the

Blo h wave homogenization method using the modulated-two-s ale transform,

how-ever this tool is not enough. So, a boundary orre tor is added, it is solution to a

boundarylayerproblemwhi hisanHelmholtzequationposed in

R

+

_{× (0, 1)}

andwith

a non-homogeneous boundary ondition at left. Its solution is expe ted to de rease

exponentiallywith respe ttothe rstvariable. Thederivationofthis partof modelis

a hieved by atwo-s aletransformdedi ated toboundarylayers that anbe relatedto

the two-s ale onvergen e for boundarylayers as in[9℄. The omplete asymptoti

be-haviouroftheeigenvalue

λ

ε

and orrespondingeigenve tors

w

ε

in ludingtheboundary

layeree ts are provided. We observe that a similarproblem wasalsoinvestigated in

[53℄ but posed inthe unbounded domain

Ω = R

2

and for

k ∈

0,

1 ₂

only. The

deriva-tion uses the asymptoti expansionte hnique and the ma ros opi equation arises as

a ompatibility ondition. Higherorder equationsarealsoderived. Relatedworks[26℄

and [27℄ fo us onthe homogenizationin avi inity of agap edge of the Blo hspe tra.

We re ently have been aware of thepaper[40℄ whi h provides theboundary ondition

for the high frequen y ma ros opi equation forthe periodi ase (

k = 0

).

In our viewpoint, the asymptoti s of eigenvalues, eigenve tors and wave

propa-gation are important problems. They have been widely studied in transport theory,

rea tion-diusion equations and uid dynami s, so we present more bibliographi al

referen es. For general results on the spe tral problem, we refer to [18℄, [24℄, [41℄,

[66℄, [69℄, [70℄, [84℄, [85℄, [86℄, [110℄ and the referen es therein. In a xed domain, the

homogenization of spe tral problems with point-wise positive density fun tion goes

ba kto[69℄,[70℄. Inperforateddomains,therst homogenization resultisreferred to

[110℄. Furthermore, many other authors have addressed similar problems onne ted

with the homogenization of the wave equation for long-termapproximationbased on

onvergen e methodsor asymptoti expansions as [99℄, [32℄, [31℄, [42℄, [56℄, [57℄, [43℄,

[58℄,[20℄,[11℄,[13℄totakeintoa ountrapidspatialu tuations. Theyarevalidinthe

lowfrequen y rangeonly. Mostofthese resultsinvolvemore thanthetwousualterms

inthe asymptoti expansion, sothey involvehigherorderpartialdierentialequations

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onsidered as for instan e in [59℄ and the bibliography herein. Similar problems have

beenaddressed withthe perspe tiveofee tive oe ientderivationbasedonvarious

approa hes, see for instan e the re ent works [87℄ or [12℄. They involve Blo h-mode

analysis but also referto along history of works as the self- onsistents hemes in

dy-nami homogenization by [103℄,[67℄and [68℄ to ite onlyfew. Other relatedproblems

havebeenstudiedontheasymptoti regimeofthe singularlyperturbedwaveequation

forpropagationinaperiodi mediumwith volume mass

ε

2 _ρ

ε

asin[10℄orwith alarge

potentialasin[3℄. Anotherworkin[73℄studiedtheverylongtimebehaviourofwaves

in a strongly heterogeneous medium. In addition, other asymptoti results for the

wave equation an be found in[105℄, [75℄, [97℄ and [61℄. Another point of view refers

tothe midfrequen y approa hbuilt uponthe notionsof ee tiveenergy density. One

of su h method was initiated in [22℄, [23℄ and has been pursued in the re ent years

by several authors in luding[64℄,[74℄ and [30℄. Othernumeri alte hniques have been

developed in the re ent years as [72℄ orthose in the review paper[54℄.

We on ludethisintrodu tionbygivingafewreferen estorelatedworkson

bound-ary layers in homogenization. Alsorelated to the homogenization of eigenvalue

prob-lem, boundary layer has been studied in many works su h as [106℄, [82℄, [83℄, [96℄,

[101℄. In these publi ations, boundary layer equations are ellipti equation posed in

thema ros opi domain

Ω

,andthey yields orre torsforthelowfrequen ypart. In[9℄

and[8℄, the boundaryspe traisstudied, itinvolved aspe tralproblemwhi hsolution

is lo alized along the boundary. Moreover, we refer to [88℄, [62℄, [100℄ for studies of

boundarylayers for homogenizationof highlyos illatingsolutionofellipti equations.

These boundary layers are orre tors to the formalthe two-s ale expansion. In

addi-tion, we referto [19℄, [106℄, [28℄, [109℄ for other works and referen es about boundary

layers.

This dissertation isorganized asfollows. Chapter 1introdu esthe notations,

de-nitionsandpropertieswhi hare usedthroughoutthe thesis. InChapter 2,wepresent

thehomogenizationofthe spe tralprobleminonedimension. This orrespondstothe

published paper [95℄. Chapter3 addresses the homogenizationof the one-dimensional

waveequation. Arstpartisbasedonthese ondorderformulation,whi h

orrespond-ingpaperisinpreparation. Itsse ondpart isbasedontherst orderformulationand

is to appear in the pro eeding of the onferen e ENUMATH 2013 held in Lausanne.

The results for the strip are presented in Chapter 4. We draw our on lusions in

Chapter5withsomeremarksonfutureresear hwork. Somemathemati alproofsand

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Figure 1: Composite material has a ma ros opi shape and a mi rostru ture. The

(26)

Notations, assumptions and

elementary properties

Contents

1.1 Notations . . . 7

1.2 Blo h waves and two-s ale transform. . . 8

1.2.1 Inone dimension . . . 8

1.2.2 Inatwo-dimensional strip . . . 12

1.3 Assumptionof sequen e

ε

. . . 15

This hapter introdu e the notations, denitions, elementary properties and

as-sumptions whi h are used throughout the thesis.

1.1 Notations

For

N ∈ N

∗

and an open bounded domain

Ω ⊂ R

N

, the fun tional spa e

L

2 _(Ω)

of

square integrable fun tions is over

C

. For

m

-dimensional omplex-valued fun tions

u = (u

i

)

_i

and

v = (v

i

)

i

of

L

2 _(Ω)

m

, the dot produ t is denoted by

u.v :=

P

i

u

i

v

i

and

the hermitianinner produ t by

Z

Ω

u · v dx =

Z

Ω

u(x).v(x) dx.

(1.1)

The notation

O (ε)

refers to numbers or fun tions tending to zero when

ε → 0

in a

sensemade pre iseinea h ase,

∂

x

u =

∂u

∂x

isthe

x−

derivativeof the fun tion

u

inone

dimensionand

[u]

z=α

2 z=α

1

istheintegrationofafun tion

u

ontheboundary

∂X = {α

1 , α

2 }

of an interval

X = (α

1 , α

2 ) ⊂ R

. The ve tors

n

x

and

n

y

are the outer unit normals

to the boundaries

∂Ω

and

∂Y

of

Ω

and

Y

. For the sake of onvenien e, we shall use

the abbreviation "LF" and "HF" to refer to "low frequen y" and "high frequen y"

respe tively. Moreover, we introdu e a hara teristi fun tion

χ

0 (k) = 1

if

k = 0

and

= 0

otherwise.

Inthe following,we usethe notationsforBlo hwavede ompositiondened in[36℄

where the dual ell or rst Brillouin zone is

Y

∗

_{= [−1/2, 1/2)}

(27)

wave numbers used inthe modelis

L

∗

_K

=

{−

2K

K

, ..,

K

2K

−

1 K

} ⊂ L

if

K

iseven,

{−

K−1

2K

, ..,

K−1

2K

} ⊂ L

if

K

is odd, (1.2) for

K ∈ N

∗

. Notethat

L

∗

K

→ Y

∗

when

K → ∞.

Thesuper- ell

Y

K

= (0, K)×(0, 1)

N −1

is made of

K

ells translated from

Y = (0, 1)

N

. For

r ∈ {1, ..., N}

, the variable

x

is writtenas

x = (x

r

, e

x

r

)

with

x

e

r

= (x

1 , ..., x

r−1

, x

r+1

, ..., x

N

) .

Forany

k ∈ Y

∗

the spa e of square integrable

k−

quasi-periodi fun tions in

x

r

dire -tion is

L

2 _k

_{= {u ∈ L}

2 _loc

(R

N

_{) | u(x}

r

+ ℓ, e

x

r

) = u(x)e

2iπkℓ

a.e. in

R

N

for all

ℓ ∈ Z},

orequivalently

L

2 _k

_{= {u ∈ L}

2 _loc

(R

N

_{) | ∃v ∈ L}

2 _♯

su h that

u(x) = v(x)e

2iπkx

r

a.e. in

R

N

},

where

L

2 ♯

is the traditional notation for

L

2 k

in the periodi ase i.e. when

k = 0.

Likewise, we set

H

_k

2 := L

2 _k

_{∩ H}

_loc

2 R

N

bearing in mind that the subs ript

♯

would be more appropriate in the periodi ase

k = 0.

In addition, the operator

̟

k

: L

2 _{(Y ) → L}

2 k

denotes the

k−

quasi-periodi

extensionoperator. Finally,we denote

I

k

_{= {−k, k}}

if

k ∈ Y

∗

_\

0, −

1 ₂

and

I

k

= {k}

otherwise. (1.3)

1.2 Blo h waves and two-s ale transform

We distinguish between two ases: inone dimension and ina two-dimensionaltrip.

1.2.1 In one dimension

We onsider

Ω = (0, α) ⊂ R

+

aninterval,whi h boundary isdenoted by

∂Ω

, and two

fun tions

(a

ε

_{, ρ}

ε

₎

assumed toobey a pres ribed prole,

a

ε

:= a

x

ε

and

ρ

ε

_{:= ρ}

x

ε

,

where

ρ ∈ L

∞

_(R)

,

a ∈ W

1,∞

_(R)

are both

Y

-periodi where

Y = (0, 1)

. Moreover,

they are required tosatisfy the standard uniform positivity and ellipti ity onditions:

ρ

0 _{≤ ρ ≤ ρ}

1

and

a

0 ≤ a ≤ a

1 ,

for some given stri tly positive

ρ

0

,

ρ

1

,

a

0

and

a

1

.

Two-s ale operators Forthe sakeofnotationsimpli ity,wedenote

P

ε

_{= −∂}

x

(a

ε

∂

x

)

and

Q

ε

_{= ρ}

ε

_∂

tt

. Forafun tion

u (x, y)

dened in

Ω × R

and afun tion

v (t, τ )

dened

in

I × R

, we introdu e,

P

0 _{u = −∂}

x

(a∂

x

u) , P

1 u = −∂

x

(a∂

y

u) − ∂

y

(a∂

x

u)

and

P

2 u = −∂

y

(a∂

y

u) ,

(1.4)

Q

0 v = ρ∂

tt

v, Q

1 v = 2ρ∂

t

∂

τ

v

and

Q

2 _{v = ρ∂}

τ τ

v.

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Blo h wavesForagiven

k ∈ Y

∗

, the Blo heigenelements

(λ

k

n

, φ

k

n

)

indexed by

n ∈ N

are solution to

P(k) : −∂

y

a∂

y

φ

k

n

= λ

k

_n

ρφ

k

_n

in

Y

with

φ

k

n

∈ H

k

2 (Y )

and

φ

k

n

L

2 _{(Y )}

= 1,

(1.5)

where the eigenvalues

λ

k

n

onstitute a non-negative in reasing sequen e. The zero

eigenvalue only for

k = 0

is denoted by

λ

0

. We state some properties of the Blo h

eigenelements

λ

k

n

, φ

k

n

solution to (1.5) whi h are useful in studying the HF-waves.

For a given

k ∈ Y

∗

, the operator

P

2 k

:= −∂

y

(a∂

y

.) : D (P

k

2 ) ⊂ L

2 k

(Y )/ Ker(P

k

2 ) →

L

2 k

(Y )/ Ker(P

k

2 )

with dense domain is positive self-adjoint and with ompa t inverse,

soits spe trum is made with anin reasing sequen e of positivereal numbers tending

to innity. Moreover, the family

φ

k

n

onstitutes an orthonormal basis of the spa e

L

2 _{(Y )}

for the hermitianinner produ t. The only zero eigenvalue is

λ

0

orresponding

toa onstanteigenve tor, equaltoone by normalization. Therefore,

Ker (P

2 k

) = ∅

for

all

k ∈ Y

∗

ex ept for

k = 0

. This is the same for the ase of a two-dimensional strip

inSe tion 1.2.2. Notation 1 For

k 6= 0, n ∈ M

k

_,

the onjugate

φ

k

n

of

φ

k

n

is solution of

P(−k)

. We

hoose the numbering of the eigenve tors

φ

−k

n

so that

φ

−k

n

= φ

k

n

whi h implies that

λ

−k

_n

= λ

k

_n

.

Remark 2 For ea h

k ∈ Y

∗

,

n ∈ N

∗

, the se ond order dierential equation (1.5)

admits two independent solutions, whi h a ording to Notation 1, are

φ

k

n

and

φ

−k

n

when

k 6∈ {0, −

1

2 }

. So, the eigenvalues

λ

k

n

and

λ

−k

n

are both simplewhile in the other

ase theeigenve tors areor periodi or anti-periodi and theeigenvalues are or simple

or double.

The

L

2 ₋

orthogonalproje tor onto

φ

k

n

is denoted by

Π

k

n

and the asso iated time s ale

is

α

k

n

=

√

2π

_λ

k

n

, with

α

0

0 = ∞

. Denote by

M

k

the set of the indi es

n

of all Blo h

eigenelements,

M

k

= N

for

k = 0

and

M

k

_{= N}

∗

for

k 6= 0

. (1.6)

The spa e-modulated-two-s ale transform Let us assume fromnow on that the

domain

Ω

is the union of a nite number of entire ells of size

ε

or equivalently

that

ε

belongs to a subsequen e of

ε

n

=

α

n

for

n ∈ N

∗

. The set of all ells of

Ω

is

C := {ω

ε

= εl + εY | l ∈ Z, εl + εY ⊂ Ω}

.

Denition 3 For any

k ∈ Y

∗

, the modulated-two-s ale transform

S

ε

k

: L

2 (Ω) →

L

2 _{(Ω × Y )}

of a fun tion

u ∈ L

2 _(Ω)

is dened by

S

ε

k

u (x, y) =

X

ω

ε

∈C

ε

u (εl

ω

ε

+ εy) χ

ω

ε

(x) e

−2iπkl

ωε

_,

(1.7) where

εl

ω

ε

standsfortheunique nodein

εL

of

ω

ε

and

χ

ω

ε

isthe hara teristi fun tion

of

ω

ε

.

FromDenition3ofthemodulated-two-s aletransform,thethreefollowingproperties

an be he ked by using (1.7) and are admitted. For

k ∈ Y

∗

(29)

u, v ∈ L

2 _(Ω)

kS

k

ε

uk

2 L

2 _{(Ω×Y )}

=

Z

Ω×Y

|S

ε

k

u|

2 dxdy =

X

ω

ε

∈C

Z

ω

ε

|u|

2 dx = kuk

2 _L2(Ω)

, (1.8)

S

_k

ε

(uv) = S

₀

ε

(u)S

_k

ε

(v)

, and

S

ε

k

(∂

x

u) (x, y) =

1 ε

∂

y

S

ε

k

u (x, y)

for

u ∈ H

1 _(Ω)

. (1.9) The adjoint

S

ε∗

k

: L

2 (Ω × Y ) → L

2 (Ω)

of

S

ε

k

is dened by

Z

Ω

(S

_k

ε∗

_{v) (x) · w (x) dx =}

Z

Ω×Y

v (x, y) · (S

ε

k

w) (x, y) dxdy,

(1.10) forall

w ∈ L

2 _(Ω)

and

v ∈ L

2 _{(Ω × Y )}

. A dire t omputation,see [95℄, shows that the

expli it expression of

S

ε∗

k

v

is

(S

_k

ε∗

v) (x) =

X

ω

ε

∈C

ε

−1

Z

ω

ε

v

z,

x − εl

ω

ε

dzχ

ω

ε

(x)e

2iπkl

ωε

_,

(1.11)

itmaps regular fun tions in

Ω × Y

to a pie ewise- onstantfun tions in

Ω

.

Remark 4 Let

k ∈ Y

∗

anda bounded sequen e

u

ε

in

L

2 _(Ω)

su h that

S

ε

k

u

ε

onverges to

u

k

in

L

2 _{(Ω × Y )}

weakly when

ε → 0

, then

S

ε

−k

u

ε

onverges to some

u

−k

in

L

2 _{(Ω ×}

Y )

weakly. Moreover, sin e

S

ε

k

u

ε

and

S

ε

−k

u

ε

are onjugate then

u

k

and

u

−k

are also onjugate. A ording to (1.11),

S

ε∗

k

v

is not a regular fun tion. For various reasons, we need a

regularapproximationof

S

ε∗

k

v

thatwilldenoteby

R

k

_v

. Theexpressionof

R

k

_v

depends

on the regularity of

v

with respe t to its rst variable. Prior to dening

R

k

_v

, it is

required to extend

v (x, y)

to

y ∈ R

by

k−

quasi-periodi ity. Hen e, we denote by

R

k

the operator operatingon fun tions

v(x, y)

dened in

Ω × R

and

k−

quasi-periodi in

y

,

(R

k

v)(x) = v(x,

x

ε

).

(1.12)

Thenext lemmashows that

R

k

is anapproximationof

S

ε∗

k

for

k−

quasi-periodi

fun -tions.

Lemma 5 Let

v ∈ C

1 _{(Ω × Y )}

be a

k−

quasi-periodi fun tion in

y

then

S

_k

ε∗

v = R

k

v + O (ε)

in the

L

2 _(Ω)

weak sense

.

(1.13)

Moreover, for

v ∈ C

2 _{(Ω × Y )}

a

k−

y

then

R

k

_{v = S}

ε∗

k

v + ε

y −

1 ₂

∂

x

v

+ εO (ε)

in the

L

2 (I × Ω)

weak sense. (1.14)

We refer toLemma 3 in[95℄and to[80℄ for the proof,see alsothe proof of

forth om-ing Lemma 8 in Appendix when the time variables are dismissed. In the proof, we

(30)

Corollary 6 Let

v ∈ C

1 _{(Ω × Y )}

and

k−

quasi-periodi in

y

, forany boundedsequen e

u

ε

in

L

2 _(Ω)

su h that

S

ε

k

u

ε

onverges to

u

in

L

2 _{(Ω × Y )}

weakly when

ε → 0

then

Z

Ω

u

ε

_{· R}

k

_{v dx →}

Z

Ω×Y

u · v dxdy

when

ε → 0.

Notethat for

k = 0

, this orresponds tothe denition of two-s ale onvergen e in [1℄

and [89℄.

The time-two-s ale transform A two-s ale transform is then introdu ed for the

time variable,let

Z

beas a anoni al latti e and

Λ = (0, 1)

asa time unit ell, we set

D := {θ

ε

= εl + εΛ | l ∈ Z, εl + εΛ ⊂ I}

the family of all

εΛ−

ells ontained in

I

.

Denition 7 The time two-s ale transform

T

ε

_{: L}

2 _{(I) → L}

2 _{(I × Λ)}

of the fun tion

u ∈ L

2 _(I)

is dened by

T

ε

_{u (t, τ ) :=}

X

θ

ε

∈D

u (εl

θ

ε

+ ετ ) χ

θ

ε

(t)

(1.15) where

εl

θ

ε

∈ εZ

stands fortheleftendpointof

θ

ε

and

χ

θ

ε

isthe hara teristi fun tion

of

θ

ε

.

Similarly,for

u ∈ L

2 _(I)

and

v ∈ H

1 _{(I) ,}

the two following properties an be he ked

by using (1.15),

kT

ε

uk

2 L

2 _(I×Λ)

=

Z

I×Λ

|T

ε

u|

2 dtdτ =

X

θ

ε

∈D

Z

θ

ε

|u|

2 dt = kuk

2 L

2 _(I)

(1.16)

and

T

ε

_(∂

t

v) (t, τ ) =

1 ε

∂

τ

(T

ε

_{v) (t, τ )}

. (1.17) The adjoint

T

ε∗

_{: L}

2 _{(I × Λ) → L}

2 _(I)

of

T

ε

isdened by

Z

I

(T

ε∗

_{v) (t) · w (t) dt =}

Z

I×Λ

v (t, τ ) · (T

ε

_{w) (t, τ ) dtdτ ,}

(1.18) for all

w (t) ∈ L

2 _(I)

and

v (t, τ ) ∈ L

2 _{(I × Λ)}

. The expli it expression of

T

ε∗

_v

is

(T

ε∗

v) (t) =

X

θ

ε

∈D

ε

−1

Z

θ

ε

v

z,

t − εl

θ

ε

dzχ

_θ

_ε

(t),

(1.19)

itmaps regular fun tions in

I × Λ

to apie ewise- onstant fun tions in

I.

Theoperator

B

k

n

,transformingtwo-s alefun tions

v(t, τ , x, y)

denedin

I × R×Ω × R

by fun tions of the physi alspa e-time variables,is then

(B

k

_n

v)(t, x) = v(t,

t

εα

k

n

, x,

x

ε

)

. (1.20)

NextLemma presentsthe relationbetween

B

k

n

v

and

T

εα

k

n

∗

S

ε∗

k

v

for afun tion

v

whi h

isperiodi in

τ

and

k−

quasi-periodi in

y

for any

n ∈ N

∗

.

(31)

Lemma 8 For

k ∈ Y

∗

and

n ∈ N

∗

, let

v ∈ C

1 _{(I × Λ × Ω × Y )}

be a periodi fun tion

in

τ

and

k−

y,

then

B

k

n

v = T

εα

k

n

∗

S

ε∗

k

v + O (ε)

in the

L

2 (I × Ω)

weak sense

.

(1.21) Moreover, if

v ∈ C

2 _{(I × Λ × Ω × Y )}

is a periodi fun tion in

τ

and

k−

quasi-periodi fun tionin

y,

then

B

k

n

v

an be approximated at the rst order by

B

k

n

v = T

εα

k

n

∗

S

ε∗

k

v + εα

k

_n

τ −

1 ₂

∂

t

v + ε

y −

1 ₂

∂

x

v

+ εO (ε)

(1.22) in the

L

2 _{(I × Ω)}

weak sense.

It would take long to present here the proof, based on Lemma 3 in [95℄ and [80℄, of

Lemma8indetails,thusitispostponedinAppendix. Moreover, forafun tion

u (x, y)

dened in

Ω × R

and a fun tion

v (t, τ )

dened in

I × R

, we observe that,

P

ε

R

k

_{u =}

2 X

n=0

ε

−n

R

k

_P

n

_u

,

P

ε

_B

k

n

u

= B

k

_n

P

0 u + ε

−1

P

1 u + ε

−2

P

2 u

,

(1.23) and

Q

ε

_B

k

n

v = B

k

n

Q

0 v + εα

k

_n

−1

Q

1 v + εα

k

_n

−2

Q

2 v

.

1.2.2 In a two-dimensional strip

We onsider an open bounded domain

Ω = ω

1 × ω

2

with

ω

1 = (0, α) ⊂ R

+

and

ω

2 = (0, ε)

withends

Γ

end

= ∂ω

1 × ω

2

and lateralboundary

Γ

lat

= ω

1 × ∂ω

2

. Asusual

inhomogenization papers,

ε > 0

denotes asmallparameter intended togo tozero. A

2 × 2

matrix

a

ε

and a real fun tion

ρ

ε

are assumed toobey a pres ribed prole,

a

ε

:= a

x

ε

and

ρ

ε

_{:= ρ}

x

ε

,

where

ρ ∈ L

∞

_(R

2 ₎

and

a ∈ W

1,∞

_(R

2 ₎

2×2

is symmetri . They are both

Y −

periodi

with respe t to the referen e ell

Y ⊂ R

2

. Moreover, they are required tosatisfy the

standard uniform positivity and ellipti ity onditions,

ρ

0 _{≤ ρ ≤ ρ}

1

and

a

0 ||ξ||

2 ≤ ξ

T

aξ ≤ a

1 ||ξ||

2

for all

ξ ∈ R

2

(1.24)

for some given stri tly positive numbers

ρ

0

,

ρ

1

,

a

0

and

a

1 _.

We note that variables

x

and

y

an be writtenas,

x = (x

1 , x

2 )

and

y = (y

1 , y

2 ) .

For

Y

1 = Y

2 = (0, 1)

, let us dene the unit ell

Y = Y

1 × Y

2 = (0, 1)

2

whi h, upon

res aling to size

ε

, be omes the period in

Ω.

The boundary

∂Y

is de omposed into

∂Y = γ

_end

_{∪ γ}

_lat

where

γ

end

= ∂Y

1 × Y

2

and

γ

lat

= Y

1 × ∂Y

2

. The ells

ω

j

1ε

j∈N

and

(ω

j

ε

)

_j∈N

of size

ε

,and their indi es

l

_ω

j

ε

j∈N

are de omposed a ordingly

ω

j

1ε

= ε (j + Y

1 )

,

ω

j

ε

= ω

j

1ε

× ω

2

and

l

ω

j

ε

= (j, 0)

(32)

Withthe same onvention, theboundarylayer ellisindire tion

y

1

and isdened as,

Y

_∞

+

= R

+

_{× Y}

2 .

Theboundary

∂Y

+

∞

of

Y

+

∞

isde omposedinto

γ

+

∞,end

= {0}×Y

2

and

γ

+

∞,lat

= R

+

×∂Y

2 .

Two-s aleoperatorsSimilarlytotheone-dimensional ase,wedenote

P

ε

_{= − div}

x

(a

ε

∇

x

.)

and

P

0 _{= −∂}

x

1 (a

11 ∂

x

1 .) , P

1 = −∂

x

1 (a

1. ∇

y

.) − div

y

(a

.1

∂

x

1 .) , P

2 = − div

y

(a∇

y

.) .

Blo h waves For

k ∈ Y

∗

, the Blo h eigenelements

(λ

k

n

, φ

k

n

)

indexed by

n ∈ N

are solutionto

P(k) : − div

y

a∇

y

φ

k

n

= λ

k

_n

ρφ

k

_n

in

Y

with

φ

k

n

∈ H

2 (Y ) ∩ L

2 H

k

2 (Y

1 ); Y

2

(1.25)

su h that

a∇

y

φ

k

n

.n

y

= 0

on

γ

lat

and

φ

k

n

L

2 _{(Y )}

= 1,

where the eigenvalues

λ

k

n

onstitute a non-negative in reasing sequen e. The zero

eigenvalue onlyfor

k = 0

is denoted by

λ

0

.

The modulated-two-s ale transform In the statement of the results, the

asymp-toti behaviour of the solution is expressed by using the following denition of the