Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface

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Homogenization and localization for a 1-D eigenvalue

problem in a periodic medium with an interface

Grégoire Allaire, Yves Capdeboscq

To cite this version:

Grégoire Allaire, Yves Capdeboscq. Homogenization and localization for a 1-D eigenvalue problem

in a periodic medium with an interface. Annali di Matematica Pura ed Applicata, Springer Verlag,

2002, 181 (3), pp.247–282. �10.1007/s102310100040�. �hal-02083808�

a periodi medium with an interfa e GregoireAllaire  Yves Capdebos q y Mar h 6, 2002 Abstra t

Inonespa e dimensionwe addressthehomogenizationof thespe tralproblemfor asingularly

perturbeddiusion equationinaperiodi medium. Denotingbythe period,thediusion

oeÆ- ient is s aledas 

2

. Thedomainismade oftwopurelyperiodi mediaseparatedbyaninterfa e.

Dependingonthe onne tionbetweenthetwo ellspe tralequations,threedierentsituationsarise

whengoestozero. First,thereisaglobalhomogenizedproblemasinthe asewithoutinterfa e.

Se ond,thelimitismadeoftwohomogenizedproblemswithaDiri hletboundary onditiononthe

interfa e. Third,thereisanexponentiallo alizationneartheinterfa eofthersteigenfun tion.

1 Introdu tion

Thispaperisdevotedtothehomogenizationoftheeigenvalueproblemforasingularlyperturbeddiusion

equation in a periodi medium. Although this problem is of interest in higher spa e dimensions, we

restri t ourselves to the one-dimensional ase be ause of the diÆ ulty of the analysis. In parti ular,

oneof ourkey tool is the theory of Hill's ordinary dierential equation [Eas73℄ for whi h there is no

equivalentin higherdimensions. Denotingby theperiod, thediusion oeÆ ientisassumedto beof

theorder of

2

. Thus,we onsiderthefollowingmodel

8 > > < > > :  2 d dx  a  x; x   d dx    +  x; x     =    x; x     in;   =0on; (1.1) where  ; 

isaneigenvalueandeigenfun tion(throughoutthispaper,theeigenfun tionsarenormalized

byk  k L 2 ()

=1). In(1.1)the oeÆ ientsareperiodi ofperiod1withrespe ttothefastvariablex=.

Thegeneralstudy ofthehomogenizationof(1.1)isfar frombeing omplete. Whenthe oeÆ ientsare

notrapidlyos illating(i.e.,theydependontheslowvariablexbutnotonx=),itisaproblemofsingular



Centre de Mathematiques Apliquees, E ole Polyte hnique, 91128 Palaiseau, Fran e |

allaire mapx.polyte h niqu e.f r

y

e.g.,[Pia98℄). When the oeÆ ientsarepurelyperiodi fun tions (i.e., theydepend solelyon x=),the

homogenizationof(1.1)(andsimilarmodelsin higherdimension) hasbeena hievedin [AB99℄, [AC00℄,

[AM97℄. Inthe aseofsmooth oeÆ ientswitha on entrationhypothesis,partialresultshavere ently

beenobtainedin[AP02℄(againinanyspa edimension). Herewefo usonthedierent ase(ofpra ti al

aswellastheoreti al importan e)where the oeÆ ientsaredis ontinuous. Morepre isely,wefo uson

the simplest possible model in this ontext, assuming that the domain is omposed of two periodi al

mediaseparatedbyaninterfa e.

Thedomainisoftheform( l;L),wherelandLarestri tlypositive onstants,andweintrodu ethe

twosub-domains

1

=( l;0)and

2

=(0;L)separatedbyaninterfa elo atedatthepoint0. Denoting

by 

i

(x) the hara teristi fun tion of

i (satisfying  1 + 2 =1and  1  2 =0in ), the oeÆ ients

areassumedtobegivenas

8 < : a(x;y)= 1 (x)a 1 (y)+ 2 (x)a 2 (y); (x;y)= 1 (x) 1 (y)+ 2 (x) 2 (y); (x;y)= 1 (x) 1 (y)+ 2 (x) 2 (y): (1.2) All fun tions a 1 ;a 2 ; 1 ; 2 ; 1 and  2

are assumed to be measurable, 1-periodi , bounded from above

andbelowbypositive onstants. Underthese assumptions,itis wellknownthatequation (1.1)admits

a ountable innitenumberof non-trivialsolutions(

 m ;  m ) m1

. Bystandardregularityresults,ea h

eigenfun tion   m belongs to H 1 0 ()\C 0;s

(), with s > 0, and by the Krein-Rutman theorem the

rsteigenvalue is simpleand the orrespondingeigenfun tion anbe hosenpositive. Be ause of this

property, the rst eigenpair has a spe ial physi al signi ation, and we are mostly interested in its

behavior,althoughthe aseof higherleveleigenpairsisalsotreatedinsomeo asions.

Themotivationforstudyingthismodel omesfromseveralappli ations. First,it anbeseenasa

semi- lassi allimitproblemforaS hrodinger-typeequationwithperiodi potential,aswellasperiodi metri

(thisis the so- alledground-state asymptoti problem, see, e.g., [KP93℄, [Pia98℄). Se ond, it plays an

importantrole in the uniform ontrollability of thewaveequation (see, e.g., [CZ℄). Third, and this is

ourmainmotivation,itisasimplemodelfor omputingthepowerdistributioninanu learrea tor ore.

Thisis theso- alled riti alityproblem for theone-groupneutron diusionequation (formoredetails,

wereferto[AC00℄,[Cap99℄andreferen estherein). Inalltheseappli ations,theassumptionofapurely

periodi medium(i.e.,nodependen eonxofthe oeÆ ients)ismu htoostrong. Ontheotherhandthe

oeÆ ientsarenotsmoothlyvarying butexhibit jumps at materialinterfa es. This makesmodel(1.1)

withassumptions(1.2)physi allyrelevant.

Thelimitbehaviorof(1.1)ismainlygovernedbythersteigenpair(

i ;

i

)intheunit ellof

i ,i=1;2, solutionof ( d dy  a i (y) d dy i  + i (y) i = i  i (y) i in[0;1℄; y! i

(y) 1 periodi andpositive:

(1.3)

Beforeweexplainourmainresults,letusre allwhatwasalreadyprovedin[AM97℄inthepurelyperiodi

ase,namelywhena

1 =a 2 , 1 = 2 ,and 1 = 2

. Asymptoti ally,thema ros opi trendof



isgiven

by anhomogeneouseigenvalueproblem, whereasits os illatorybehaviorisgovernedby

1 (

x



)(we all

Theorem1.1. Assuming that a 2 = a 1 ,  2 =  1 , and  2 =  1 , the m eigenpair  m ; m of (1.1) satises   m (x)=u  m (x) 1 ( x  )and  m = 1 + 2  m +o  2
;

where, uptoasubsequen e, thesequen eu

 m onvergesweakly inH 1 0 () tou m ,and( m ;u m )isthem th

eigenvalueandeigenve torfor the homogenizedproblem

 D d 2 dx 2 u m = m u m in; u m =0 on: (1.4)

Thehomogenized oeÆ ientsaregiven by

D= Z 1 0 a 1 (y) 2 1 (y)  1+ d dy (y)  dy and= Z 1 0  1 (y) 2 1 (y)dy; (1.5)

where the fun tion isthe solutionof

( d dy  a 1 (y) 2 1 (y)  d dy +1  =0 in[0;1℄;

pny!(y) 1 periodi :

(1.6)

Letus summarizeourresultsinthe aseofequalrsteigenvaluein the ells,

1

=

2

. Inthesequelwe

hoosetonormalizetherstperiodi eigenfun tionsasfollows

1

(0)=

2

(0)=1: (1.7)

Weintrodu easo- alleddis ontinuity onstantdenedby

=a 1 (0) d 1 dy (0) a 2 (0) d 2 dy (0): (1.8) Notethata i d i dy belongstoH 1 ( i )whi hisembeddedin C( i

)(in 1-D)andthereforeiswelldened

asthe tra eofa ontinuousfun tion at theorigin. Threedierentsituations arepossiblea ordingto

thesignof.

If=0, then thetwoperiodi media aresaid to bewell- onne ted. Inparti ular, thefun tion equal

to a i (d i )=(dx) in i

is ontinuous through the interfa e (as well as

i

be ause of the normalization

ondition(1.7)). Therefore,Theorem1.1extendseasilytothis ase,andthedis ontinuityattheinterfa e

isnot seenin thelimit. Introdu ing afun tion (x=)=

1 (x) 1 (x=)+ 2 (x) 2 (x=),the eigenpairs (  m ;  m ) m1 satisfy   m = 1 + 2  m +o( 2 )and  m (x)=u  m (x)  x   ; (1.9) whereu  m onvergesweaklytou m ,and( m ;u m ) m1

aretheeigenpairsofthehomogenizedproblem(see

Theorem3.1andFigure3.1)

8 > > < > > : d dx   1 (x)D 1 + 2 (x)D 2
du dx  =( 1 (x) 1 + 2 (x) 2 )u in; u=0 on:

vergen eresult(1.9)stillholdstrue,butthehomogenizedproblemhasanadditionalDiri hletboundary

onditionatx=0. Morepre isely,thelimithomogenizedproblemis(seeTheorem3.1 andFigure3.2)

8 > > > > > > > < > > > > > > > : D 1 d 2 dx 2 u= 1 u in 1 ; D 2 d 2 dx 2 u= 2 u in 2 ; u=0 on 1 [ 2 :

If  < 0, the situation is ompletely dierent sin e the rst eigenfun tion on entrates exponentially

fastattheinterfa e. Inthislatter ase,thereisnofa torizationprin ipleasinTheorem1.1, butrather

alo alization prin ipleat the dis ontinuity(see Theorem 3.5 and Figure 3.3). The rsteigenvalue

 1 onvergesto alimit 0< 1 < 1 = 2 , and0<  1  1

<Cexp( =),whereasthe rstnormalized

eigenve torsatises d dx   1 (x) 1 p  d dx   x   L 2 () +   1 (x) 1 p  ( x  ) L 2 () Cexp     :

Thelimitfun tion 2H

1

(R ) de reasesexponentiallyawayfromtheinterfa e,sin eitisgivenby

(x)=  1;1 (x) forx<0; 2;2 (x) forx>0; with 1 = 1 ( 1 )=  2 ( 2

), and ea h of the eigenpairs (

i (

i );

i;i

) beingthe rst eigen oupleof the

followingspe tral ellproblem

8 > > < > > : d dx  a i (x) d i; i dx  + i (x) i;i = i ( i ) i (x) i;i in[0;1℄; x! i; i (x)e ix 1-periodi . (1.10)

Therequiredproperties ofthe-parameterizedfamilyof spe tral ellproblems(1.10) aregiven in

se -tion2.

Wenowturn to the ase

1

6=

2

,and withno lossof generalityweassume

1 >

2

. Inthis ase too,

thespe tral ellproblems (1.10)governthelimit behaviorof(1.1). Weintrodu eapositiveparameter

 0 >0,su h that 1 ( 0 )= 2

,andanotherdis ontinuity onstant(seeLemma3.9)

( 0 )=a 1 (0) d 1;0 dy (0) a 2 (0) d 2 dy (0):

Thesignofthisnewdis ontinuity onstantdeterminestheasymptoti behaviorof(1.1).

If(

0

)>0,theeigenfun tions



m

on entrateinthesub-domain

2

wheretherstperiodi eigenvalue

isthe smallest(seeTheorem 3.8 in thesimpler asewhen 0,and Theorem 3.11when(

0 )>0).

Morepre isely, thelimitof  m

vanishesin thesub-domain

1

. Introdu ing thefa torization 

m (x)= u  m (x) 2 (x=)in 2

,thehomogenizedproblem forthelimitofu



m

issimply(seeFigure3.4)

8 > < > : D 2 d 2 dx 2 u= 2 u in 2 ; u=0 on 2 : The ase( 0

)=0 orrespondstothelimitbetweenlo alizationat theinterfa eand on entrationin

2

. Thelimitoftheeigenfun tion



m

stillvanishesin

1

,butinthehomogenizedproblemtheDiri hlet

boundary onditionatx=0isrepla edbyaNeumannboundary ondition(seeTheorem3.12)

8 > > > < > > > : D 2 d 2 dx 2 u= 2 u in 2 ; u(L)=0and du dx (0)=0: Finally, when ( 0

)< 0, alo alization phenomenon appears, and therst eigenfun tion on entrates

exponentially fast at the interfa e. The result is then similar to the oneobtained when 

1

= 

2 and

<0(seeTheorem3.10).

Ourmainresultsarestatedinse tion3when

1

isequalornotto

2

. Previously,inse tion2wegivea

fewte hni alresultsonthespe tral ellproblemsthatare ru ialnotonlyfortheproof,butalsoforthe

statementofourmainresults. Se tion4 ontainstheproofswhenthedis ontinuity onstantispositive,

  0, while se tion 5 fo us on the lo alization phenomena, namely  < 0 or(

0

) < 0. Se tion 6

ontainstheproofsinthespe ialsituationwhen<0but nolo alization o urs((

0 )0),asit an happen when 1 isnotequalto  2

. Se tion 7 ontainsthe proofofa ru ialte hni alresultaboutthe

Hillequationin onedimension.

2 Cell problems

Inordertostatepre iselyour onvergen eresults,theknowledgeofthespe tral ellproblem(1.3)isnot

enough. Asin[Cap98℄, weneedtointrodu eaparameterizedfamilyofspe tral ellproblems. Theyare

reminis entoftheso- alledBlo hwavede omposition(seee.g. [CPV95℄, [RS78℄),but theyinvolvereal

exponentialsinstead of omplexones. All the resultsin this se tionare proved under the assumption

thatthe periodi oeÆ ientsa

i

;

i ;

i

are positive, bounded, measurablefun tions, ex eptProposition

2.2whi h asksformoresmoothnessorpie ewise onstant oeÆ ients.

Lemma2.1. For ea h2R thereexistsauniquerst eigen ouple (

i; ; i ()), of theproblem 8 < : d dx  a i (x) d i; dx  + i (x) i; = i () i (x) i; in [0;1℄; x! i; (x)e x

1 periodi andpositive ;

(2.1)

whi h isnormalizedby

i;

i  2  i (0)  i ()C 2 ;

where C and arepositive onstants,independent of .

A further property of the rst eigenfun tion

i;

; is given in the next Proposition. Its proof is quite

deli ateandrelies onpurely 1-Darguments(wepostponeittose tion7). Wegivetwodierentproofs:

rstinthe aseofC

2

oeÆ ients,whi hallowstoperformaLiouvilletransformationandtouse lassi al

resultsonthe 1-DHillequation,se ond inthe aseof pie ewise onstant oeÆ ients, whi h permitsto

doexpli it omputations.

Proposition2.2. Assumingthatthe oeÆ ients areC

2

or pie ewise onstant, forea h2R the rst

eigenve tor i;

ofproblem(2.1) withthe normalization

i; (0)=1satises lim ! 1 d i; dx (0)= 1and lim !+1 d i; dx (0)=+1:

Proof ofLemma2.1. Byintrodu ingthe hangeofvariable

 i; (x)= i; (x)e x ; equation(2.1)isequivalentto 8 < : d dx  a i d i; dx    d dx (a i  i; )+a i d i; dx  +  i a i  2
 i; = i () i  i; in[0;1℄; x! i;

(x)1 periodi andpositive;

(2.3)

withthesamenormalization ondition

 i;

(0)=1:

Theexisten eofauniquerstpositiveeigen oupleforproblem(2.3)isknown,seee.g. [GT83,Theorem

8.38℄,andwehave i; 2H 1 # ([0;1℄)\C 0;s

([0;1℄),withs>0. Inparti ular,thisimplythatC>

i; (x)>

>0in [0;1℄. Itis provedin [Cap98℄ thatthefun tion !

i

()issmooth,stri tly on aveonallR,

andrea hesitsmaximumat=0.

Toobtainthegrowth onditionon

i

(),weperformthefollowing hangeofunknown

u  (x)= i; (x) i;0 (x)

whi hisli itbyvirtueofProposition4.1. Then,u



is solutionofthefollowingproblem

8 < : d dx  b(x) du  dx  =()s(x)u  in [0;1℄; x!u  (x)e x 1 periodi ; (2.4) withb(x)=a i (x) 2 i;0 (x), s(x)= i (x) 2 i;0 (x),and()= i ()  i

(0). These oeÆ ientsarebounded,

onstantM>m>0. Forea h 2R the rsteigenvalue ()of problem (2.4)satises m M  2  () M m  2 :

Proof. Wealreadyknowthat()<0forall6=0. We anassumethat>0sin e hangingthesignof

in (2.4)is equivalentto onsideritsadjointequationwhi hhasthesamersteigenvalue. Be ausewe

areworkingin onespa edimension,(2.4) anbewritten asasystemofordinarydierentialequations.

Namely,denotingby0 thex-derivation,

Y 0 (x)=A(x)Y(x)andA= 2 4 0 b 1 ()s 0 3 5 and Y = 0  Y 1 =u  Y 2 =bu 0  1 A : (2.5)

By enfor ing the normalization u



(0) = Y

1

(0) = 1, the Krein-Rutman Theorem implies that Y

1 is

positive,andthus Y

2 isin reasing. Sin eY 2 (n)=e n Y 2

(0),and>0,thisimpliesthat Y

2

(0)>0,and

thusY

2

(x)>0forx0. Thisin turngives,bytherstequation, thatY

1 isin reasingthus Y 1 1for x0. Be auseY 1 andY 2

arepositivefun tionsonR

+ ; we anwrite A Y Y 0 A + Y withA + = 2 4 0 m 1 ()M 0 3 5 ; andA = 2 4 0 M 1 ()m 0 3 5 :

Sin ethematri esA

+

andA have onstant oeÆ ients,itisstraightforwardtoobtainthesolutionsof

theinitial valueproblems

Z 0 = (A ) T Z ; Z(0)=Z 0 ; andX 0 = (A + ) T X; X(0)=X 0 :

Inparti ular,the hoi e Z

0 =X 0 =  1;( ()mM) 1=2 

leadstothepositivesolutions

Z(x)=Z 0 exp x r ()m M ! andX(x)=X 0 exp x r ()M m ! :

We an ompute that (Y
Z)

0 = Y 0
Z +Y
Z 0 = ( Y 0

A Y)
Z  0 sin e Z is positive. Thus

Y
Z Y(0)
Z(0)forallx0,and hoosingx=n2N leadsto

Y(n)
Z(n)=exp n  r ( ())m M !! Y(0)
Z(0)Y(0)
Z(0); and therefore   q ( ())m M . Similarly, we have (Y
X) 0 =(Y 0 A + Y)
X 0sin e X is positive,

whi hgivesinturn foralln,

Y(n)
X(n)=exp n  r ( ())M m !! Y(0)
X(0)Y(0)
X(0); andtherefore q ( ())M m .

Itisprovedin[Cap02℄thatin general!()isastri tly on avefun tion,i.e.,thatonanybounded

subset K  R

N

(with N the spa e dimension) the Hessian matrix H =

  2  ij  1i;jN is negative

denite andHx
x  C(K)x
x with C(K)>0. The fun tion () a hievesits maximumin 0 and

lim jj!1

()= 1.

3 Main results

Inthespirit of the method of proofof Theorem 1.1 (see [AM97℄), weintrodu ein (1.1)the hange of

unknown u  (x)=   (x) x; x 
;

withafun tion (x;y)denedby

(x;y)= 1 (x) 1 (y)+ 2 (x) 2 (y); (3.1) where( 1 ; 1 )and( 2 ; 2

)arethersteigen ouplesinea hperiodi ellof(1.3). Byournormalization

ondition(1.7),thefun tion (x;x=)is ontinuousattheinterfa ex=0. Onthe ontrary,thefun tion

a(x;x=)(d (x;x=))=(dx)isnotne essarily ontinuousanditsjump attheinterfa eismeasuredbythe

dis ontinuity onstantintrodu edin (1.8).

Therstresult on ernsthespe ial asewhentherst elleigenvaluesof(1.3)areequal,

1

=

2

, and

thedis ontinuity onstantisnon-negative,0. Under these assumptions,weobtainageneralization

ofTheorem1.1. Theorem3.1. Let   m and   m

be the m-th eigenvalue andnormalized eigenve tors of (1.1). Assume

thatthe dis ontinuity onstantdenedin(1.8)isnon negative0, andthat

1 = 2 . Then   m (x)=u  m (x)  x; x   and   m = 1 + 2  m +o  2
; uptoasub-sequen e, u  m onvergesweakly in H 1 0 () towards u m ,and( m ;u m )isthe m-th eigen ouple

ofthe homogenizedproblem, whi h,if =0, is

8 > > < > > : d dx   1 (x)D 1 + 2 (x)D 2
du dx  =( 1 (x) 1 + 2 (x) 2 )u in ; u=0 on ; (3.2) and,if >0, is 8 > > > > > > > < > > > > > > > : D 1 d 2 dx 2 u= 1 u in 1 ; D 2 d 2 dx 2 u= 2 u in 2 ; u=0 on 1 [ 2 : (3.3)

−20

−10

0

10

20

0

0.2

0.4

0.6

0.8

1

Figure3.1: Firsteigenfun tionforproblem(1.1)inthe aseoftwowell- onne tedmedia,i.e.,=0.

−50

−40

−30

−20

−10

0

10

20

30

40

50

0

0.2

0.4

0.6

0.8

1

Figure3.2: Firsteigenfun tionforproblem(1.1)inthe aseofnonwell- onne tedmediawithapositive

dis ontinuity onstant>0.

AsanillustrationofTheorem3.1, wepresentsomedire t omputationsof thersteigenfun tion



1 of

problem (1.1). The ase 

1

=

2

and =0is shownon Figure 3.1. (The domain is omposed of an

homogeneousmediumontheleftandanheterogeneousoneontheright). The ase

1

=

2

and>0is

shownonFigure3.2(thedomainis omposedoftwoheterogeneousmediawiththesame ell oeÆ ients

but witha onstantphaseshiftbetween therightand theleft). Thedata usedfor the omputation is

presentedinRemark3.7.

Remark3.2. Of ourse, sin e the homogenized oeÆ ients are onstant in ea h sub-domain we an

omputeexpli itlytheeigenvaluesofthehomogenizedproblemsinTheorem3.1.

Remark3.3. There is asimple suÆ ient onditionfor havingwell- onne tedmedia, i.e.,  =0. If all

oeÆ ientssatisfy a entral symmetry ondition, i.e., are symmetri with respe t to the enter of the

unit ell [0;1℄, then it is easy to he kthat

i

satises a Neumannboundary onditionat x =0 and

x =1, and therefore = 0. A tually, Theorem 3.1 wasalready proved by Malige [Mal96℄ under this

assumption. Thesymmetrywasusedforthe onstru tionoftheexampleshowninFigure3.1: in

2

,the

periodi oeÆ ientsarepie ewise onstanton(0:3;0:7)and(0:1;0:3)[(0:7;1:0); in

1 , a 1 1, 1 1 and 1 = 2 .

1 2

Inother words,therearetwode oupledhomogenizedproblems. Therefore,therealwaysexisttwo

non-negativeeigenfun tionswithdisjointsupports,u

1 (x)=sin(  l x) 1

(x) orrespondingtotheeigenvalue

 1 =  2 D 1 1l 2 and u 2 (x) = sin(  L x) 2

(x) orresponding to the eigenvalue 

2 =  2 D 2 2L 2 . If the rst

eigenvaluesin ea h sub-domain are distin t, e.g., L

2  2 D 1 > l 2  1 D 2

, the rstfa torized eigenfun tion

u 

1

will tend to u

2

, i.e., will on entrate in the sub-domain that has the smallest rst eigenvalue and

onvergetozeroin theotherone. Intheother asewhere thersteigenvaluesin

1

and

2

areequal,

thersteigen-subspa eisofdimension2,spanbyu

1

andu

2

andtheuniquenessofthelimitof



1 islost

(onFigure3.2thelimitseemstobealinear ombinationofthersteigenfun tionsonea hsub-domain).

Ourse ondresult ompletesthe ase

1

=

2

whenthedis ontinuity onstantisnegative,<0. Under

theseassumptions,weobtainalo alizationphenomena.

Theorem3.5. Let(  1 ;  1

)betherstnormalizedeigen oupleof(1.1). Assumethat

1

=

2

and<0.

Then,thereexistsaunique

1

>0and auniquepositive (x)2H

1 (R ) su hthat 0  1  1 Cexp     and d dx   1 (x) 1 p  d dx   x   L 2 () +   1 (x) 1 p  ( x  ) L 2 () Cexp     ;

where C and are positive onstant,independentof . The limiteigenvaluesatises

1 < 1 = 2 ,and

thelimit eigenfun tionisdenedby

(x)=  1; 1 (x) for x<0; 2; 2 (x) for x>0; with 1 >0and 2 <0and(  1 ; i; i

)isthe rsteigen ouple ofthe ellproblem (2.1), i.e.,

8 > > < > > : d dx  a i (x) d i;i dx  + i (x) i;i = 1  i (x) i;i in[0;1℄; x! i; i (x)e ix 1 periodi :

Remark3.6. Theorem 3.5 is illustrated by Figure 3.3: the rst eigenve torof system (1.1) onverges

exponentially fasttowardsa lo alized eigenfun tionnear the interfa ebetweenthe twodomains.

Fur-thermore, the orresponding eigenvalueis smallerthan 

1

=

2

, whi h is the limitobtainedin all the

other ases. In ontrast with Theorem 3.1, no fa torization, or limit homogenized problem appear in

thewordingofTheorem3.5. Thelimiteigenfun tion ontainsboththeperiodi alos illationsandthe

ma ros opi trend.

Remark3.7. The omputationsshownonFigure 3.2andFigure3.3 wereperformedwiththesametwo

media,but theirpositionsareswit hed withrespe ttotheinterfa ewhenpassingfromone asetothe

other. Wetake l=L=1with100periodi ity ells,whi hyields=0:02. Allthemoretheperiodi ell

oeÆ ientsforthetwomediaarethesameuptoaphaseshiftintheunit ell. Morepre isely,inFigure3.2

the oeÆ ients are a

1 (y) = a(y), a 2 (y) = a(y+ ),  1 (y) = (y),  2 (y) = (y+ ),  1 (y) = (y),  2

(y) = (y+ ), while in Figure 3.3 they are a

1 (y) = a(y+ ), a 2 (y) = a(y),  1 (y) = (y+ ),  2 (y)=(y),  1 (y)=(y+ ),  2

(y)=(y), where =0:6 isa onstantphaseshift, anda;and 

−50

−40

−30

−20

−10

0

10

20

30

40

50

0

0.2

0.4

0.6

0.8

1

Figure3.3: Firsteigenfun tionforproblem(1.1)inthe aseofnonwell- onne tedmediawithanegative

dis ontinuity onstant<0.

spe iedorderasfollows

(a;;)= 8 > > < > > : (a I ; I ; I ) if0<y<0:1 (a II ; II ; II ) if0:1<y<0:5 (a III ; III ; III ) if0:5<y<0:8 (a I ; I ; I ) if0:8<y<1 with ConstituentI a I =0:9666  I =2:1080  I =2:8283 ConstituentII a II =2:0086  II =2:3878  II =2:9451 ConstituentIII a III =2:0444  III =2:9945  III =1:1493

Notethat,by onstru tion,

1

=

2

1:3863. Theshapeofthersteigenve tor

 1 onFigure3.2(with eigenvalue  1

1:3899), orrespondstowhatisannoun edbyTheorem3.1: asymptoti ally,bothmedia

tendtoseparatewhen>0. Therefore,bysymmetryFigure3.3 orrespondstoasituationwhere<0:

thersteigenve tor on entratesexponentiallyat theinterfa ebetweenthetwomedia. Thenumeri al

al ulation onrms that the orresponding eigenvalue (



1

 1:3720) is below that of the periodi ity

ell. This phenomenonis explainedby Lemma 5.4whi h givesane essaryandsuÆ ient onditionfor

theexisten eofalo alizedeigensolution.

Wenowturntothegeneral ase

1

6=

2

. Inthesequel,weshallassume,withoutlossofgenerality,that

 1

> 2

:

Ifthedis ontinuity onstantis non-negative,i.e., 0,the eigenfun tions on entrateasymptoti ally

inthesub-domain

2

wheretherstperiodi eigenvalueisthesmallest.

Theorem3.8. Let   m and  m

bethe m-theigenvalue andnormalizedeigenfun tion of (1.1). Assume

that0and  1 > 2 . Then,   m (x)=u  m (x)  x; x   and   m = 2 + 2  m +o  2
;

−20

−10

0

10

20

0

0.2

0.4

0.6

0.8

1

Figure3.4: Firsteigenfun tionof(1.1)inthe aseoftwomediawith

1 > 2 and=0. where, upto asubsequen e, u  m onverges weakly inH 1 0 () tou m , with u m =0 in 1 and( m ;u m ) is

them-th eigenpair of the following homogenizedproblem

 D 2 d 2 dx 2 u m = m  2 u m in 2 ; u m =0 on 2 ; (3.4)

andthehomogenized oeÆ ientsarestill given by(1.5).

Figure3.4 illustratesTheorem 3.8. It displaysthersteigenfun tion 



1

in the aseoftwomedia with

symmetri periodi stru tures (so that =0), with

1

'1:58 and

2

'0:43and 20 periodi ellson

ea hsideoftheinterfa e.

When 

1 > 

2

, a lo alization phenomena an also o ur. Let us rst remark that, as an obvious

onsequen eofLemma2.1, wehavethefollowingresult.

Lemma3.9. For all

1 >

2

thereexistsaunique

0 >0su hthat  1 ( 0 )= 2 .

Indeed, Lemma 3.9 is obvious by remarking that 

1

(), dened in Lemma 2.1, is a on ave fun tion

with quadrati growth at innity and rea hing its maximum at  = 0, 

1 (0) =  1 >  2 . The rst eigenve tors orrespondingto  2 and 1 ( 0 )aredenoted by 2 and 1; 0

. Theyare ontinuousat the

interfa e,i.e.,

2

(0)=

1; 0

(0)=1,andweintrodu eanewdis ontinuity onstantthatwill hara terize

thelo alization phenomena

( 0 )=a 1 (0) d 1;0 dy (0) a 2 (0) d 2 dy (0): Theorem3.10. Let(  1 ;  1

)betherstnormalizedeigen oupleof(1.1). Assumethat(

0

)<0. Then,

thereexistsaunique

1

>0andauniquepositive 2H

1 (R ) su hthat 0  1  1 Cexp     and d dx   1 (x) 1 p  d dx   x   L 2 () +   1 (x) 1 p  ( x  ) L 2 () Cexp     ; (3.5)

1 2 1

thelimit eigenfun tionisdenedby

(x)=  1;1 (x) for x<0; 2;2 (x) for x>0; with 1 >0and 2 <0and( 1 ; i;i

)isthe rsteigen ouple ofthe ellproblem (2.1), i.e.,

8 > > < > > : d dx  a i (x) d i;i dx  + i (x) i; i = 1  i (x) i; i in[0;1℄; x! i;i (x)e  i x 1 periodi :

Finally, in the remaining ase  < 0 and (

0

)  0, there is no lo alization, and the eigenfun tions

still on entrateasymptoti allyinthesub-domain

2

wheretherstperiodi eigenvalueisthesmallest.

When (

0

) > 0, the limit problem has Diri hlet boundary onditions. When (

0

) = 0, the limit

problemhasaNeumannboundary onditionat theinterfa e.

Theorem3.11. Let  m and  m

bethem-th eigenvalueandnormalizedeigenfun tionof(1.1). Assume

that 1 > 2 ,<0and( 0 )>0. Then,   m = 2 + 2  m +o  2
;   m (x)!0in L 2 ( 1 ) and   m (x)=u  m (x) 2  x   ; where, uptoa subsequen e, u  m onverges weakly in H 1 ( 2 )tou m ,and ( m ;u m )is the m-th eigenpair

ofthe followinghomogenized problem

 D 2 d 2 dx 2 u m = m  2 u m in 2 ; u m =0 on 2 ; (3.6)

andthehomogenized oeÆ ientsarestill given by(1.5).

Theorem3.12. Let  m and  m

bethem-th eigenvalueandnormalizedeigenfun tion of(1.1). Assume

that 1 > 2 ,<0and( 0 )=0. Then,   m = 2 + 2  m +o  2
;   m (x)!0in L 2 ( 1 ) and   m (x)=u  m (x) 2  x   ; where, uptoa subsequen e, u  m onverges weakly in H 1 ( 2 )tou m ,and ( m ;u m )is the m-th eigenpair

ofthe followinghomogenized problem

 D 2 d 2 dx 2 u m = m  2 u m in 2 ; u m (L)=0; and du m dx (0)=0; (3.7)

andthehomogenized oeÆ ientsarestill given by(1.5).

Remark3.13. NotethatthehomogenizedproblemsofTheorems3.11and3.12( orrespondingto(

0 )>

0 and (

0

) = 0, respe tively) are similar ex ept the boundary ondition at x = 0. Then a simple

omputationshowsthat thersthomogenizedeigenvalue

1

is fourtimessmallerwhen(

0

)=0than

when(

0 )>0.

fa torizedeigensolutionsu 

m

havebounded gradientsin all of , but simplywithin

2

. It is therefore

diÆ ult(at least for us)to study the possible o urren eof boundary layersin

1

. In thelimit ase

of Theorem 3.12, be ause of the homogenized Neumann boundary ondition at x = 0, we expe t a

nontrivialboundarylayerin

1 .

Remark3.15. Thegeneralizationoftheresultsofthisse tiontohigherspa edimensionsisnotobvious

foratleasttworeasons. First,Theorems3.5 and3.10reliesonProposition2.2whi hisprovedonlyin

onedimension (by usingo.d.e. te hniques). Se ond, evenTheorems 3.1and 3.8 (whi h donotdepend

onProposition 2.2) arenotstraightforwardin higher dimensionsbe ause we annot assumeaperfe t

transmission ondition(1.7) at theinterfa e. Of ourse, ifit happens by han e that, for adimension

N>1,wehave 1;0 (0;y 0 )= 2 (0;y 0 ) (3.8)

foralmost everyy

0 2[0;1℄ N 1 ,and 0(y 0 )=a 1 (0;y 0 ) d 1;0 dy (0;y 0 ) a 2 (0;y 0 ) d 2 dy (0;y 0 )M <+1; (3.9)

thenTheorems3.1and3.8extendeasilysin eattheinterfa etheproblemisessentiallyone-dimensional

(see [Cap99℄). Of ourse, these onditions are very stri t and almost never satised in pra ti e. In

general,webelievethatboundarylayersat theinterfa emustbetakenintoa ount.

Remark3.16. Throughoutthispaperweassumethat, afterres alingby,theperiodi ityonbothsides

of the interfa e is exa tly one. The fa t that the period is the samein

1

and

2

is not important,

andthisispurelyby onvenien ethatwemadethis hoi e. All ourresultsapplyifthetwoperiods are

dierent,providedthat thedis ontinuity onstantsand(

0

)areproperlydened.

4 Proofs in the ase   0

Inorderto proveTheorem 3.1and 3.8,werstneedto justifythefa torization

 (x)=u  (x) x; x 
.

ThisisthegoalofthenextPropositionwhi hisageneralizationofapreviousresultof[AM97℄(seealso

[AC00℄).

Proposition4.1. Let (x;y)bethe fun tion denedby (3.1). Then, the linear operator T denedby

T :H 1 0 () ! H 1 0 () (x) ! (x) x; x 

isbounded,invertible andbi ontinuous.

Proof. Thankstothenormalization ondition(1.7)thefun tion (x;x=)is ontinuousonR. Byvirtue

ofLemma2.1weknowthatthereexisttwopositive onstantsC> >0su hthatC

1 (y);

2

(y)

forally2[0;1℄,andtheseboundsalsoholdsfor . Therefore,forall2H

1 0 (),ifwedeneu=T() , wehave C 1 kk L 2 () kuk L 2 ()  1 kk L 2 () ; (4.1)

andT isanhomeomorphism onL (). Ontheotherhand, Z a d dx d dx = Z 1 a 1 2 1 du dx du dx + Z 2 a 2 2 2 du dx du dx + Z 1 a 1 d 1 dx d(u 2 1 ) dx + Z 2 a 2 d 2 dx d(u 2 2 ) dx : (4.2) Equation(1.3)dening 1 (x=)tested againstu 2 (x) 1 (x=)writes Z 1 a 1 d 1 dx d(u 2 1 ) dx 1  a 1 (0) d 1 dy (0)u 2 (0)= 1  2   1 Z 1  1 2 1 u 2 Z 1  1 2 1 u 2  ; (4.3)

andsimilarlywehave

Z 2 a 2 d 2 dx d(u 2 2 ) dx + 1  a 2 (0) d 2 dy (0)u 2 (0)= 1  2   2 Z 2  2 2 2 u 2 Z 2  2 2 2 u 2  : (4.4)

Whenwerepla e(4.3)and(4.4)into (4.2)weobtain

Z a  x; x   d dx d dx dx+ 1  2 Z   x; x    2 dx = 2 X i=1 Z i a i  x   2 i  x   du dx du dx dx + 1  2 2 X i=1 Z i  i  i  x   2 i  x   u 2 dx + 1  u 2 (0): (4.5)

Where is the dis ontinuity onstant given by (1.8). If  0, all the left hand side terms are

non-negativein(4.5). Sin e a 1 ;a 2 ; 1 and 2

arebounded belowbypositive onstants,we andedu ethat

du dx 2 L 2 () +kuk 2 L 2 () C() d dx 2 L 2 () +kk 2 L 2 () ! : (4.6) Conversely,wehave 0u 2 (0)C Z  du dx  2 dx; (4.7)

thereforewealsoobtainfrom(4.5)that

d dx 2 L 2 () +kk 2 L 2 () C() du dx 2 L 2 () +kuk 2 L 2 () ! (4.8)

andthis on ludestheproofofthepropositionfor0. If0,notethatthankstothenormalization

ondition(1.7),u

2

(0)=

2

(0). Consequently,identity(4.5)isalso

Z a  x; x    d dx  2 dx+ 1  2 Z   x; x    2 dx 1   2 (0) = 2 X i=1 Z i a i  x   2 i  x    du dx  2 dx + 1  2 2 X i=1 Z i  i  i  x   2 i  x   u 2 dx:

Ifwepro eedto the hangeofunknownu 

=T(



),problem(1.1)istransformedintoaneweigenvalue

problem,wherethesingularperturbationinfrontofthedivergen etermhasdisappeared. Proposition4.2

givestheformofthisnewproblemafter somesimplealgebra.

Proposition4.2. Introdu ing u  (x) =   (x)= x; x 

, (1.1) is equivalent to the following eigenvalue

problem 8 > > > > > > > < > > > > > > > : d dx  D  x; x   du  dx  +  1  2  2  1 (x)B  x; x   u  + 1 u  (0)Æ(x)=  B  x; x   u  u  =0on in (4.9)

where Æ(x) istheDira fun tion,the (positive)diusion oeÆ ient isdenedby

D  x; x   = 1 (x) 2 1  x   a 1  x   + 2 (x) 2 2  x   a 2  x   ;

the(positive) oeÆ ient B by

B  x; x   = 1 (x) 2 1  x    1  x   + 2 (x) 2 2  x    2  x   ;

andthenew eigenvalueby

  =    2  2 :

Remark4.3. WeprovedProposition4.1regardlessofthesignof,thereforeProposition4.2isalsovalid

when<0. Weshallusethisequivalentformof(1.1)in Se tion6.

Following astrategy already used in [AC00℄, [AC98℄, the asymptoti study of the eigenvalue problem

(4.9)relies onthedetailedhomogenization,astendtozero,ofthefollowingproblem

8 > > > > > > < > > > > > > : d dx  D  x; x   du  dx  +  1  2  2  1 (x)B  x; x   u  + 1 u  (0)Æ(x)=f  u  =0on; in (4.10)

witharighthandsidef



whi hisaboundedsequen eofL

2

(),weakly onvergingtoalimitf 2L

2 ().

Werstobtainaprioriestimates

Proposition4.4. If0,the solution u

 of equation(4.10) satises ku  k H 1 0 () +  1  2  ku  k L 2 (1) + r   ku  (0)kCkf  k L 2 () (4.11)

where C is a onstant independent of . Therefore, up toasubsequen e, u



onverges weakly toa limit

uinH

1

0

 if 1

> 2

,the limituvanishes in

1

andthusbelongstoH

0 ( 2 ),  if  1 =  2

and  > 0, the limit satises u(0) = 0 and thus an be written u = u

1 +u 2 with u 1 2H 1 0 ( 1 )andu 2 2H 1 0 ( 2 ).

Proof. Ifwetest variationallyequation(4.10)dening u

 againstu  ,weobtain Z D  x; x    du  dx  2 dx+  1  2  2 Z 1 B  x; x   (u  ) 2 dx+ 1  (u  ) 2 (0)= Z f  u  dx:

Sin eD andBareboundedbelowbyapositive onstant,andsin eweassumethat

1  2 and0, weobtain du  dx 2 L 2 () +  1  2  2 ku  k 2 L 2 ( 1 ) +   ku  (0)k 2 Ckf  k L 2 () ku  k L 2 () ;

whi hyieldsthedesiredresultthankstoPoin areinequality.

Lemma4.5. Let S



bethe operator denedby

S  :L 2 ()! L 2 () f ! u  unique solutioninH 1 0 () ofequation (4.10) withr.h.s. f: (4.12) Forallxed>0, S 

isalinear ompa t operator inL

2 ().

This resultis a onsequen e of the apriori estimate (4.11) and of the ompa t in lusion of H

1

0

() in

L 2

(). Weshallshowthefollowingresult

Proposition4.6. Letf



beaweakly onvergingsequen etoalimitfinL

2 (). Thesequen eu  =S  (f  ) weakly onvergesinH 1 0 () towardsu 0 dened byu 0 =S(f). 1. If =0and 1 = 2

thenS isthe following ompa t operator

S: L 2 () !L 2 () f !uunique solutionof  d dx  1 (x)D 1 + 2 (x)D 2
d dx u(x)
=f in ; u=0 on : whereD 1 andD 2 aregiven by(1.5). 2. If 0and 1 > 2

thenS isthe following ompa t operator

S: L 2 () !L 2 () f !uuniquesolutionof  D 2 d 2 dx 2 u(x)=f in 2 ; u=0 on n 2 :

1 2 S: L 2 () !L 2 () f !uuniquesolutionof 8 < : D 1 d 2 dx 2 u(x)=f in 1 ; D 2 d 2 dx 2 u(x)=f in 2 ; u=0 on  2 [ 2 :

Proof. Theproofisquitestandardinhomogenizationtheory. Forexample,usingthenotionoftwo-s ale

onvergen e (see [All92℄, [Ngu89℄) it is an easyexer ise that we safely leaveto thereader (the details

an be found in [Cap99℄ if ne essary). Let us simply remark that, if 

1

= 

2

and  = 0, then the

homogenization of (4.10) is ompletely obvious. If 

1

=

2

and >0, then the apriori estimates of

Proposition4.4 showsthat u



(0) goesto zero,while, if

1 >

2

and0,theyimply thatu



goesto

zeroin

1 .

Wearenowableto on ludetheproofofTheorem 3.1andTheorem3.8.

Proof ofTheorem3.1 andTheorem 3.8. LetusrstremarkthatProposition4.6implythatthesequen e

ofoperatorsS



,denedby(4.12),uniformly onvergestothelimitoperatorS. Theasymptoti analysis

oftheeigenvalueproblem(4.9)istrulygivenbythat ofT

 givenby T  :L 2 () !L 2 () f !S  (B x; x 
f): Theeigenvaluesof T 

being theinverseof that of (4.9). Introdu ing (x) =

R 1

0

B(x;y)dy whi h is the

weaklimitofB(x;

x



);wedenethelimitoperatorT by

T :L 2 () !L 2 () f !S(f): The sequen e T 

does not uniformly onverge to T, but the sequen e T



is nevertheless sequentially

ompa t,in thesensethat

 8f 2L 2 () lim !0 kT  (f) T(f)k L 2 () =0; thesetfT  (f); kfk L 2 () 1;0gissequentially ompa t:

Theorems3.1and3.8arethen onsequen esofTheorem4.7(seealso hapter11in[JKO95℄).

Theorem4.7. (seee.g. [Ans71℄,[Cha83℄)LetT

n

beasequen eof ompa toperatorsthat onvergesto

T. Assumethat(T

n )

n1

is olle tively ompa t andT is ompa t. Let 2C be aneigenvalue ofT,of

multipli itym. Let beasmooth urveen losing inthe omplexplane andleavingoutsidethe restof

thespe trumof T. Then, for suÆ ientlylargevalues ofn , en losesalso exa tly meigenvaluesof T

n

andleavesoutside therestof the spe trumof T

n .

0

The goal of this se tion is to proveTheorems 3.5 and 3.10. To understand the asymptoti behavior

of problem (1.1) when the dis ontinuity onstant (

0

) is negative, we rst res ale the equations by

introdu ingthe hange ofvariables y=

x



. Then,problem(1.1)isequivalentto

8 > > < > > : d dy  a(y) d'  dy  +(y)'  =  (y)'  in  ; '  (  1 l)='  ( 1 L)=0; (5.1) with  =℄  1 l; 1 L[,'  (y)=  ( x  ),and

a(y)(resp: (y);(y))=

 a 1 (y)(resp: 1 (y); 1 (y)) ify<0; a 2 (y)(resp: 2 (y); 2 (y)) ify>0:

Asgoesto0,thedomain



onvergestoR, andformallythelimitproblemof(5.1)is

8 > > < > > : d dx  a(x) d dx  +(x) =(x) inR; 2H 1 (R) : (5.2)

Werstre all someproperties of thespe trum of (5.2). Weintrodu ethe Greenoperator S a tingin

L 2 (R) denedby S:L 2 (R) ! L 2 (R) f ! uuniquesolutioninH 1 (R) of d dx  a(x) du dx  +(x)u=(x)f inR: (5.3)

Theeigenvaluesof S arepre iselythe inverseof thoseof(5.2). Nevertheless,to simplify thedis ussion

weshallsaythat is aneigenvalueofS,or(5.2),ifitsinversebelongstothespe trumofS.

Proposition5.1. The operator S isself-adjoint and non- ompa t. Its spe trum an be de omposed in

itsdis reteandessential part, (S)=

dis

(S)[

ess

(S). The lower bound of the essential spe trumis

equaltothe smallest ell rsteigenvaluein(1.3), namely

min ess (S)=min(  1 ; 2 ):

If(; ) isan eigen ouple inthe dis rete spe trum,thenthere exist

1 >0and 2 <0su hthat (x)=  1 (x) ifx<0; 2 (x) ifx>0; and(; i )isaneigen oupleof 8 > > < > > : d dx  a i (x) d i dx  + i (x) i = i (x) i in[0;1℄; x! i (x)e  i x 1 periodi : (5.4)

tipli ity,whileitsessentialspe trumis hara terizedbyWeyl riterion, i.e.,82 ess (S)thereexistsa sequen efu n g2L 2 (R ) su hthat  ku n k L 2 (R) =1; u n !0in L 2 (R ) weakly; (S Id)u n !0inL 2 (R ) strongly:

Proposition5.1tellsus inparti ularthat

ess

(S)isnotemptyandthatanydis reteeigenve torde ays

exponentiallyatinnity. Remarkthat equation(5.4)issimilar to(2.1).

Proof. Thestudyofthespe trumofS is lassi al. Theexponentialde ayofthedis reteeigenfun tions

isobtainedthroughFloquetTheory(see, e.g.,[MR73℄, [RS78℄). The sametoolyields thelowerbound

ofthe essentialspe trum(see [AC98℄, [CPV95℄). Note that these results areobtained under themere

assumptionthat the oeÆ ients of equation (5.3)are positivemeasurable fun tions (nosmoothness is

required).

Inordertopasstothelimit!0in (5.1),wealsointrodu eanoperatorS

 a tingin L 2 (R) denedby S  :L 2 (R) ! L 2 (R ) f ! u  uniquesolutionin H 1 0 (  )of 8 > > < > > : d dx  a(x) du  dx  +(x)u  =(x)f; in  u  (x)=0on  : (5.5) The operator S 

is ompa t and its eigenvalues are the inverses of that of (5.1). Unfortunately, the

onvergen e of thesequen e S



toS is notuniform, so that the limitof the spe trum ofS



is notthe

spe trumofS. Nevertheless,thislimit anbe hara terizedexpli itlyandwere allthefollowingresult

thatmaybefoundin[AC98℄.

Proposition5.3. For allf 2L 2 (R) , S  (f) onvergesstronglytoS(f)inL 2 (R ), andwehave lim !0 (S  )=(S)[ BL :

Furthermore, therst eigenvalue



1

onvergestoa limit

1

whi h does belong tothe spe trumof S and

isthusthe smallestelement of(S). Wealso have

min BL =min ess (S)=min( 1 ; 2 ): (5.6)

That part of the limit spe trum, denoted by 

BL

, is alled the boundary layer spe trum. It an be

hara terized ompletely in terms of an equation similar to (5.2) but in the half-line (for details, see

[AC98℄). We do notdwell on this boundary layerspe trum sin e we only need to know (5.6) in the

sequel.

Lemma5.4. Let 

0

be dened asin Lemma3.9, i.e., 

1 ( 0 )= 2 ,and 1;0

the orresponding

eigen-ve tordenedby (2.1). Let(

0 )bedenedby ( 0 )=a 1 (0) d 1; 0 dy (0) a 2 (0) d 2 dy (0):

( 0 )<0; (5.7) thelimit  1 ofthe rsteigenvalue  1 of problem (1.1)satises  1 <min( 1 ; 2 ):

Remark5.5. Inparti ular,thisLemmaapplieswhen

1

=

2

,and(

0

)<0. Itimpliesthat, when

thedis ontinuity onstantisnegative,thelimitrsteigenvalue annotbepredi tedbythehomogenized

modelsobtainedunderstri tperiodi ityassumptiononea hsideoftheinterfa e.TheproofofLemma5.4

reliesonProposition2.2,thatwehavenotbeenabletoproveinthegeneral ase,butundertheadditional

assumptionthatthe oeÆ ientsareC

2

,orpie ewise onstant.

Proof ofLemma5.4. Forall 2 [

0

;+1[, where 

0

isdened in Lemma 3.9, be auseof the on avity

of

1

()and 

2

(), we anasso iatetoea haunique

0 0su hthat  1 ()= 2 ( 0 )and 2; 0 isthe rsteigenve tordened by 8 > > > > > > < > > > > > > : d dx  a 2 (x) d 2; 0 dx  + 2 (x) 2; 0 = 2 ( 0 ) 2 (x) 2; 0 in[0;1℄ x! 2; (x)e  0 x 1 periodi ; 2; 0(0) =1: (5.8)

Notethatfor=

0

,wehave

0

=0. Thepair(; ) denedby

= 1 ()= 2 ( 0 ) = 1; forx>0; = 2;0 forx<0;

isaneigen oupleforproblem(5.2)ifand onlyif

()=a 1 (0) d 1; dx (0) a 2 (0) d 2; 0 dx (0)=0: (5.9)

Thanksto Proposition2.2,wehave

lim !+1 ()= lim !+1 a 1 (0) d 1; dx (0) lim  0 ! 1 a 2 (0) d 2; 0 dx (0)=+1: Therefore,ifweassume( 0

)<0then Equation(5.9)admitsasolution,for some <

0

, and

0 >0.

We have thus obtained a value of  su h that  < min(

1 ; 2 ). Finally, sin e  1  , by virtue of Proposition5.1,wehave 1 2 dis (S). Conversely, if  1 < min ( 1 ; 2

) we know from Proposition 5.6 that on both sides of the origin the

orrespondingeigenfun tion hasanexponentialde ay. ThenProposition5.1showthatitmustofthe

form =

1;

and =

2;0

for someand 

0

on ea h half line. Sin e identity (5.9)isa ne essary

then 1

isin thedis retespe trumofproblem(5.2). Theorems3.5and 3.10arethena onsequen eof

Proposition 5.6. Indeedtheeigenfun tion 



1

(x) in Theorems 3.5 and3.10 isequal to

1 p  '  1 x 
where '  1

isthersteigenfun tionin Proposition5.6. Inequality(5.10)thenbe omes

 2 d dx   1 (x) 1 p  d dx   x   L 2 () +   1 (x) 1 p   x   L 2 () Cexp    

whi hin turnimply

d dx   1 (x) 1 p  d dx   x   L 2 () +   1 (x) 1 p   x   L 2 () C 0 exp   0   forany 0 <.

Proposition5.6. Assume that problem (5.2), or equivalently operator S, admits a rst positive

nor-malized eigen ouple ( 1 ; ) su hthat  1 <min( 1 ; 2

). Then the rst positive normalized eigen ouple

(   1 ;'  1 )of (5.1), orofS  ,satises 0  1  1 Cexp     and d dx '  1 d dx L 2 (  ) +k'  1 k L 2 () Cexp     (5.10)

where C and arestri tlypositive onstant independentof .

Proof. Sin eweassumed

1 <min ess (S),wehave  1 = min '2H 1 (R) 6=0 Z R a(x)j d dx 'j 2 dx+ Z R (x) 2 dx Z R (x)' 2 dx ;

andthisminimumisattainedfor'= whi h belongstothedis retespe trumof S. Wealsohave

  1 = min '2H 1 0 (  ) 6=0 Z  a(x)j d dx 'j 2 dx+ Z  (x)' 2 dx Z  ( x)' 2 dx ;

and this implies, by in lusion of spa es that 

1

 



1

. Let be asmooth ut-o fun tion, vanishing

outside  =  l  ; L   ,equalto1on  l  +1; L  1 

,su hthat 01;and

d

dx

doesnotdependon

(seeFigure5.1). Wethenhave 2H

1 0 (  ),and   1  Z  a(x)j d dx ( )j 2 dx+ Z  (x)( ) 2 dx Z  ( x)( ) 2 dx : (5.11)

            N  N  +1 N  1 N 

Figure5.1: Cut-ofun tion .

By onstru tion

d

dx

hasitssupportin[

l  ; l  +1℄[[ L  1; L 

℄andinequality(5.11)be omes

  1  Z R a(x)j d dx j 2 dx+ Z R (x) 2 dx+R  1 (1 R  2 ) Z R (x) 2 dx ; (5.12) with R  1 =2 Z [ l  ; l  +1℄[[ L  1; L  ℄ a(x)j (x)j     d dx           d dx     +     d dx       dx and R  2 = Z l  +1 1 (x) (x) 2 + Z +1 L  1 (x) (x) 2 Z R (x) (x) 2 :

Thanksto Proposition5.1,weknowthat

sup x2( 1; l  ) j (x)jCexp   1 l   ;and sup x2( L  ;+1) j (x)jCexp   2 L   with  1 > 0 and  2

< 0. We an dedu e that R

 1  Cexp  
and R  2  Cexp  
with  = min (lj 1 j;Lj 2

j),andinsertingtheseinequalitiesin (5.12)weobtain

  1  1  1+Cexp     : (5.13)

Letusnowshowthat'



1

onvergesto . Inordertoobtainanapproximationof thatvanishesonthe

boundariesofthedomain



,weaddto anaÆnefun tionwhi h ompensatesitsvaluesatbothends

ofthedomain. Wedene



(x)= (x)+`



(x)where`



istheaÆnefun tionsu hthat

 l   +`   l   =0and  L   +`   L   =0: By onstru tion,  2H 1 0 (  ),and 

issolutionofthesameproblemthan'

 1 uptoaperturbationr  . 8 > > > < > > > : d dx  a(x) d  dx  +(x)  =  1 ( x)  +r  in℄ l  ; L  [  ( l  )=  ( L  )=0: (5.14)

Theperturbation is r  =( 1   1 )  +`   1 `  d dx a d`  dx 2H 1 ( 

). The oeÆ ientsbeing

bounded, weobtainthat forall2H

1 0 (  ),     Z  r       C  j   1 j+sup  j`  j  kk L 2 () +C d`  dx L 2 () d dx L 2 () ;

whereCis a onstantwhi hdoesnotdependon. Fromtheexponentialde ayof wededu ethat

sup  j`  jCexp     and d`  dx L 2 () C p exp     ; (5.15)

andwiththehelp ofestimate(5.13)weobtain

    Z  r       Cexp     kk L 2 (  ) + d dx L 2 () ! : (5.16) Thersteigenvalue  1

beingsimple,byaFredholmalternativewe ande ompose



intoa omponent

proportionalto'



1

anda omponentorthogonalto'

 1 . Wewrite  =    1 +g  ,where  isa onstant, and kg  k L 2 (  ) + dg  dx L 2 () C  kr  k H 1 (  ) where C  is the norm of S  1   1 Id
1

, a bounded operator dened on the orthogonal of the line

generated by '  1 . We haveC   C j   1   2 j

, where C is a onstantindependent of , and 

 2 is the next eigenvalueofS  . Ifweobtainthatj  1   2

j> >0,with independentof,wethendedu e,withthe

helpofinequality(5.16) kg  k L 2 (  ) + dg  dx L 2 ()  C exp     : (5.17)

Fromthede omposition

 =  '  1 +g  ,weget j  jk'  1 k L 2 (  ) kg  k L 2 (  ) k  k L 2 (  ) j  jk'  1 k L 2 (  ) +kg  k L 2 (  ) : Wehavek'  1 k L 2 (  ) =1andk k L 2 (R) =1thus   k  k L 2 () 1    =   k`  k L 2 () k k L 2 (Rn)    k`  k L 2 () +k k L 2 (Rn) C 1  exp    

thankstoestimate (5.15)and theexponentialde ayof . Asa onsequen e,jj

 j 1jC 1  exp  
and  and'  1

beingpositives,wealsohave

j  1jC 1  exp     : (5.18) Finally,ifwewrite'  1 (x) (x)=(1   )'  1 (x) g  (x)+`  (x)on 

andusingestimates(5.15),(5.17)

and(5.18)weobtain d'  1 dx d dx L 2 () +k'  1 k L 2 (  ) C 1  exp     Cexp   0  

Letusnowshowthatthespe tralgapisuniformly bounded, i.e., 0< <  2   1 <C. Weknowthat   2 onvergestoalimit 2

whi h either belongs to 

BL

[

ess

(S)orto

dis

(S). In thelatter ase,the

eigenvaluesofthedis rete spe trum areisolated so that 0< <

2 

1

<C. In theformer ase, we

know from (5.6) that 

2

 min(

1 ;

2

) whi h is stri tly larger than 

1

by assumption, so that again

0< < 2

 1

<C. ThisyieldsthedesiredresultforsuÆ ientlysmall.

6 Proofs in the ase  < 0 and  (

0

)  0.

Inthis se tion we proveTheorems 3.11 and 3.12 following the strategy used in se tion 4for the ase

 0. A ording to Proposition 4.2 and Remark 4.3, the original problem (1.1) is equivalent to the

fa torizedproblem (4.9) forany valueof thedis ontinuity onstant. Introdu ing, asin Lemma 4.5,

anoperatorS



,the onvergen eof(4.9)isgovernedbythehomogenizationofproblem(4.10)withgiven

right hand side. The key element for the proof of Proposition 4.6, and in turn Theorem 3.8, is thea

prioriestimategivenbyProposition4.4. Itdoesnotholdfor<0. Nevertheless,theargumentsofthe

proofofProposition4.4yieldsasimilarresultthat westatein Proposition6.1below.

Proposition6.1. Thesolution u

 of equation (4.10) satises ku  k 2 H 1 0 () +  1  2  2 ku  k 2 L 2 (1) +   ju  (0)j 2 Ckf  k L 2 () ku  k L 2 () (6.1)

where C isa onstant independentof .

Sin eweassumed<0,(6.1)alonedoesnotfurnish suÆ ientaprioriestimatesfor on luding. Thus,

fortheproofofTheorems3.11and3.12weneedanadditionallemma.

Lemma6.2. Assume that 

1 > 

2

,  <0 and (

0

) >0. Then, the solution u

 of equation (4.10) satises du  dx L 2 () Ckf  k L 2 () ; ku  k L 2 ( 1 ) Ckf  k L 2 () ; and j u  (0)jC p kf  k L 2 () : Assumethat  1 > 2 ,<0and( 0

)=0. Then, the solutionu

 ofequation(4.10) satises du  dx L 2 ( 2 ) Ckf  k L 2 () ; ku  k L 2 (1) C p kf  k L 2 () ; and e 0 x  dv  dx L 2 ( 1 ) Ckf  k L 2 () ; where v  =u  1 (x;x=)= 1;0 (x;x=)in 1 .

Proof ofTheorem3.11. Thanks to the a priori estimate of Lemma 6.2, the ase 

1 >  2 ,  < 0 and ( 0

)>0is ompletelysimilarto the ase

1 >

2

and0,whi hisalreadysolvedinse tion4.

Proof of Theorem 3.12. Let u



be the solution of (4.10) with right hand side f



whi h is a bounded

sequen ein L

2

(). Weintrodu ethefun tion

0 (x;y)= 1 (x) 1;0 (y)+ 2 (x) 2 (y); (6.2)

v  (x)=u  (x) x; x 
 0 x; x 
: Remarkthat v  =u  in 2 , and v  (0)=u 

(0) (be ause ofthenormalization ondition(2.2)). Testing

variationallyequation(4.10)against

 0 (x; x  ) (x; x  )   (x),where  isatest fun tionin H 1 0 (), weobtain Z D  x; x   du  dx d dx 0 x; x 
x; x 
  ! dx+  1  2  2 Z 1 B  x; x   u  0 x; x 
x; x 
  ! dx (6.3) + 1  u  (0)  (0)= Z f  0 x; x 
x; x 
  ! dx: Repla ingu  byv 

in itslefthandside,identity(6.3)be omes

Z 1 a 1  x   2 1  x   d dx v  1; 0 x 
1 x 
! d dx   1; 0 x 
1 x 
! dx+ Z 2 D  x; x   dv  dx d  dx dx (6.4) +  1  2  2 Z 1  1  x   2 1;0  x   v    dx+ 1  v  (0)u  (0)= Z 1; 0 x 
1 x 
f    dx: Notethat Z 1 a 1  x   2 1  x   d dx v  1; 0 x 
1 x 
! d dx   1; 0 x 
1 x 
! dx= Z 1 a 1  x   2 1;0  x   dv  dx d  dx dx + Z 1 a 1  x   d dx  1;0  x   d dx  v    1;0  x   Z 1 a 1  x   d dx  1  x   d dx 1;0 x 

2 1 x 
v    ! ;

and,byintegrationbyparts anddenition(2.1)of

1; ,wehave Z 1 a 1  x   d dx  1;0  x   d dx  v    1;0  x   Z 1 a 1  x   d dx  1  x   d dx 2 1;0 x 
1 x 
v    ! = 1  ( ( 0 ) )v  (0)  (0)+  2  1  2 Z 1  1  x   2 1; 0  x   v    :

Asa onsequen e,identity(6.4)be omes

Z 1 a 1  x   2 1;0  x   dv  dx d  dx dx+ Z 2 D  x; x   dv  dx d  dx dx (6.5) + 1  ( 0 )v  (0)  (0)= Z 1;0 x 
1 x 
f    dx

0 Z 1 a 1  x   2 1;0  x   dv  dx d  dx dx+ Z 2 D  x; x   dv  dx d  dx dx= Z 1;0 x 
1 x 
f    dx

Notethatforanyboundedsequen e

 inW 1;1 (),     Z 1 a 1  x   2 1;0  x   dv  dx d  dx dx     C e  0 x  dv  dx L 2 ( 1 ) e  0 x  L 2 ( 1 ) !0 sin eke  0 x  dv  dx k L 2 ( 1 )

isbounded, thanksto Lemma6.2. Of ourse

R 1 1; 0 ( x  ) 1( x  ) f    goesto0

exponen-tiallyfast. Forsu hbounded

 ,(6.3)thereforewrites Z 2 D  x; x   du  dx d  dx dx= Z 2 f    dx+o(1): (6.6)

Sin ethetestfun tions inthetwo-s ale onvergen emethodareofthetype

 (x)= 0 (x)+ 1 (x;x=)

withsmoothfun tions 

0 ;

1

,theyareuniformlyboundedinW

1;1

()andone anuse(6.6)topassto

thelimit. Classi alargumentsofhomogenizationallowto on lude.

Proof ofLemma6.2. With the hoi e

 =v  in(6.5)weobtain Z 1 a 1  x   2 1;0  x    dv  dx  2 dx+ Z 2 D  x; x    dv  dx  2 dx+ 1  ( 0 )(v  ) 2 (0)= Z f  u  dx: (6.7) If( 0

)>0,thisimpliesthat

jv  (0)j 2 Ckf  k L 2 () ku  k L 2 () : (6.8) Be auseu  (0)=v 

(0),plugging(6.8)in (6.1)yieldsthedesiredresults.

If(

0

)=0,identity(6.7)onlyimpliesthat

e 0 x  dv  dx 2 L 2 (1) Ckf  k L 2 () ku  k L 2 () ; and du  dx 2 L 2 (2) Ckf  k L 2 () ku  k L 2 () : (6.9)

Wewill nextshowthat

ku  k 2 L 2 () C  kf  k 2 L 2 () +ku  k 2 L 2 ( 2 )  (6.10)

and,togetherwith(6.9)andPoin areinequalityin

2

,thisyieldsthedesiredresults.

Notethat u  (0) 2 j 2 j 2 Z 2  du  dx  2 Ckf  k L 2 () ku  k L 2 () :

Usingthisinequalityin (6.1)gives

ku  k 2 L 2 (1) Ckf  k L 2 () ku  k L 2 () C  kf  k 2 L 2 () +ku  k 2 L 2 () 

7.1 The ase of C 2

oeÆ ients.

TherststepissimilartotheproofofLemma2.1,namelywetransform(2.1)into(2.4). Ifweassumethat

the oeÆ ientsa i ; i and i areC 2

periodi fun tionson[0;1℄,thenthersteigenfun tion

i;0

isa tually

twotimesdierentiable,andthusthe oeÆ ientsb andsof(2.4)arealsoof lassC

2

. Proposition2.2is

thena onsequen eof Lemma7.1.

Lemma7.1. Let b and s be periodi positive fun tions on [0;1℄ su h that their se ond derivative b

00

ands

0 0

existand are pie ewise ontinuous. Denote by M >m>0twopositive onstant whi h arethe

upper andlower bounds of b and s. For ea h  2 R the rst eigenve tor u



of problem (2.4) with the

normalizationu  (0)=1satises C 1 C 2 p () +C 3 p ()  jj u 0  (0)C 3 p ()+C 1 + C 2 p () ; (7.1)

where the positive onstantsC

1 ;C

2

andC

3

dependonly onb ands.

Proof. Theassumedsmoothnessofb andsenablesustoperformaLiouvilletransformationofproblem

(2.4). Introdu ing t= 1 Z x 0  s(z) b(z)  1 2 dz = Z 1 0  s(z) b(z) 1 2 dz; and f  (t)=(s(x)b(x)) 1 4 u  (x); (7.2)

thetransformedequationis,see[Eas73℄,

8 < : d 2 f  dt 2 (t)+( 2 ()+Q(t))f  =0in[0;1℄; t!f  (t)e t 1 periodi ; (7.3) with Q(t)= 2 b 1 4 (x)s 3 4 (x) d dx  b(x) d dx (b(x)s(x)) 1 4  :

We anassumewithoutlossofgeneralitythat =1. Theboundary onditionsarepreservedsin ethis

hangeof variablepreservesperiodi ity. Weshalluse thefa t that Qisabounded 1-periodi fun tion.

ItissuÆ ientto prove(7.1)for>0,sin ein theother asethefun tiong



(t)=f



( t) issolutionof

(7.3),with >0,ifQis repla edbyQ( t),whi his alsoabounded1-periodi fun tion. Byaddinga

onstant to Q (andsubtra ting it from ()), we an alwaysassume that M <Q(t)< 1. On the

otherhand,thankstoLemma2.3,forsuÆ ientlylargewe analsoassumethat()translatedbythe

above onstantisnegative.

Next,weintrodu eg

1

andg

2

asthetwofundamentalsolutionsoftheCau hyproblem fortheordinary

dierentialequation d 2 g dt 2 +(()+Q(t))g=0,satisfying g 1 (0)=1; g 0 1 (0)=0;andg 2 (0)=0; g 0 2 (0)=1:

Itis a lassi alresultofFloquettheorythat X 1

=e andX

2

=e are therootsof the hara teristi

equation X 2 (g 1 (1)+g 0 2 (1))X+1=0:

By linearity, we an write f

 (t) = f  (0)g 1 (t)+f 0  (0)g 2 (t). Sin e  > 0, e  =
+
2 1
1 2 where 2
=(g 1 (1)+g 0 2 (1)) . Consequently,
>1and g 1 (1)+g 2 0(1)=2
>e  ande  >2
1
>2
2e  =g 1 (1)+g 2 0(1) 2e  : (7.4)

Fromtherelationf

 (1)=e  f  (0)wededu ethat f 0  (0) f  (0) g 2 (1)=e  g 1

(1). Usingrelation(7.4)wehave

obtainedthat g 0 2 (1)> f 0  (0) f  (0) g 2 (1)>g 0 2 (1) 2e  : (7.5)

FollowingPi ard'siterationmethod(seee.g. [MW79℄),wedenere ursivelyasequen e(v

n (t)) n2N by v 0 (t)= 1 ! sinh(!t)and v n (t)= 1 ! Z t 0 sinh( !(t ))Q()v n 1 ()d foralln1: For!= p (),wend that g 2 (x)= P +1 n=0 v n

(x). Sin e sinh(!(t ))Q()>0forall0< <t,

andv

0

(t)>0for allt >0, byindu tion,we an on ludethat W

n (x)v n (x)w n (x), foralln 0 andx0,whereW n andw n

aretwoothersequen esdened byW

0 =v 0 =w 0 , and W n = M ! Z t 0 sinh(!(t ))W n 1 ()d; w n = 1 ! Z t 0 sinh(!(t ))w n 1 ()dforn1: NotethatW(t)= P +1 0 W n (t)(resp. w(t)= P +1 0 w n (t))isasolutionof d 2 W dt 2 +( M)W =0  resp. d 2 w dt 2 +( 1)w=0  ;

andthereforeisgivenbyW(t)=sinh t

p M 
(resp. w(t)=sinh t p 1 
)and onsequently sinh  p M   =W(1)g 2 (1)w(1)=sinh  p 1   : (7.6) SimilarlyW 0 n (t)v 0 n (t)w 0 n (t),and p M  osh  p M   =W 0 (1)g 0 2 (1)w 0 (1)= p 1  osh  p 1   : (7.7)

Usinginequalities(7.6)and(7.7)in(7.5)yields,

Ce M 1 2 p  p M  f 0  (0) f  (0)  p 1 e M 1 2 p  2e  :

Usingthe hange of variables (7.2),and using the resultof Lemma 2.3 to bound e



in termsof ()

Asin theprevioussubse tion,itissuÆ ient to onsider system(2.4),whi h isequivalentto(2.1), and

tostudythe ase goingto+1. Asin Lemma2.3,werewrite(2.4)asarst-ordersystem

 Y 0 (x)=A(x)Y(x) Y(1)=e  Y(0) ;A=  0 b 1 ()s 0  ; andY =  Y 1 =u  Y 2 =bu 0   :

Here weassume that the oeÆ ients b;sare pie ewise onstant fun tions. Morepre isely,there exists

a number N, a family of points (x

i ) 0iN satisfying x 0 = 0 < x i 1 < x i < x i+1 < x N = 1 for

2iN 2,andpositivevalues(b

i ) 1iN and( s i ) 1iN su hthat b(x)=b i ands(x)=s i forx2(x i 1 ;x i ); 1iN:

Thegoalis to provethat Y

2

(0) growslinearlyas goes to +1,whi hin turn provesProposition2.2,

sin e d i; dx (0)=b 1 (0)Y 2 (0)+ d i;0 dx (0):

By Lemma 2.3 we already know that () < 0 for  6= 0 and has quadrati growth at innity. A

straightforward omputationyieldsforanyx2(x

i 1 ;x i ) Y(x)=M i (;x)Y(x i 1 ); M i (;x)= 2 6 4 osh' i (x) 1 p ()b i s i sinh' i (x) p ()b i s i sinh' i (x) osh' i (x) 3 7 5 ; (7.8) with' i (x)= q ()si b i (x x i 1 ). Thus Y(1)=M()Y(0)=e  Y(0);with M()= N Y i=1 M i (;x i ): Ea hmatrixM i

(;x)hasitsdeterminantequalto1,aswellasM(). ThusthetwoeigenvaluesofM()

aree



ande



. Letus omparetheseexa teigenvalueswiththoseoftheleadingordertermofM()as

goesto+1. Introdu ing D()=diag

 p () ;1  ,wehave M i (;x i )=e 'i(xi) D() 1 M 0 i D() 1+O e 

;with M 0 i = 1 2 2 4 1 1 p bisi p b i s i 1 3 5 ; and=min 1iN  2(x i x i 1 ) q s i m b i M 

>0. Therefore,noti ing that

P N i=1 ' i (x i )=C p ()where

C>0doesnotdependon,weobtain

M()=e C p () D() 1 M 0 D() 1+O e 

;withM 0 = N Y i=1 M 0 i :

Uptoasmallremainder,theeigenvaluesofM()arethusequaltothoseofM timesthemultipli ative

fa torexp(C

p

() ). Sin eM

0

doesnotdepend on , this provesthat ()= 

2

+o(1) forsome

positive onstant >0. Ontheotherhand,theeigenve torsofM()areequaltoD()

1

timesthoseof

M 0

(uptoasmallremainder). ChoosingthenormalizationY

1

(0)=1,thisyieldsthatY

2

(0)=

0

+o(1)

forsome onstant

0

,whi h ispositiveasalreadyremarkedintheproofofLemma2.3.

NoteAddedin Proof. After submission of this paper for publi ation, wefound an alternative proof of

Lemma5.4,whi hdonotrelyonProposition2.2. ThisenablesustoproveTheorem3.5andTheorem3.10

assumingonlythattheperiodi oeÆ ientsarepositive,bounded,measurablefun tions. Thisproofwill

bepresentedinafutureworkin ollaborationwithA. Piatnitski.

Referen es

[All92℄ G.AllaireHomogenizationandtwo-s ale onvergen e. SIAMJ.Math.Anal.23(6):1482{1518,

(1992).

[AB99℄ G. Allaire, G.Bal Homogenization of the riti ality spe tral equation in neutrontransport.

M2AN 33,pp.721-746,(1999).

[AC00℄ G. Allaire, Y. Capdebos q Homogenization of a spe tral problem in neutroni multigroup

diusion.Comput.MethodsAppl.Me h. Engrg.187:91-117,(2000).

[AC98℄ G. Allaire and C. Con a Blo h wavehomogenization and spe tral asymptoti analysis. J.

Math.PuresetAppli. 77:153-208,(1998).

[AM97℄ G.Allaire,andF.MaligeAnalyseasymptotiquespe traled'unproblemedediusion

neutron-ique.C.R. A ad.S i. ParisSerie I,t.324:939{944,(1997).

[AP02℄ G.Allaire,A.PiatnitskiUniformspe tralasymptoti sforsingularlyperturbedlo allyperiodi

operators,Commun.inPartial Dierential Equations,27,pp.705-725(2002).

[Ans71℄ P.AnseloneColle tivelyCompa tOperatorApproximationTheoryandAppli ationstoIntegral

Equations. Prenti e-Hall,EnglewoodClis,N.J.,(1971).

[Cap98℄ Y.Capdebos qHomogenizationofadiusionequationwithdrift.C.R.A ad.S i.ParisSerie

I,t.327:807-812,(1998).

[Cap99℄ Y.Capdebos q Homogeneisationdesmodelesdediusionen neutronique. ThesedeDo torat

del'UniversiteParisVI,(1999).

[Cap02℄ Y.Capdebos q Homogenizationofaneutroni riti aldiusionproblemwithdrift. Roy.So .

Edin.Pro .A, 132,pp.1{28(2002).

[CZ℄ C.Castro, E. Zuazua Lowfrequen y asymptoti analysis of astringwith rapidly os illating

density. SIAMJ.Appl. Math.60(4):1205-1233,(2000).

[Cha83℄ F. Chatelin Spe tral Approximation of Linear Operators. Computer S ien e and Applied

Sons,(1995).

[Eas73℄ M.S.P. Eastham The Spe tral Theory of Periodi Dierential Equations. S ottish A ademi

Press,(1973).

[GT83℄ D.Gilbarg andN. Trudinger Ellipti Partial Dierential Equations of Se ondOrder,2nd ed.

ComprehensiveStudies inMathemati s.Springer-Verlag,(1983).

[JKO95℄ V.Jikov,S.Kozlov,O.OleinikHomogenizationof dierentialoperatorsandintegral

fun tion-als,Springer,Berlin,(1995).

[KP93℄ S. Kozlov, A. Piatnitski Ee tive diusion for aparaboli operator with periodi potential,

SIAMJ.Appl.Math.53,401{418,(1993).

[Mal96℄ F. Malige Etude mathematique et numerique de l'homogeneisation des assemblages

om-bustiblesd'un urde rea teurnu leaire. ThesedeDo toratde

l'

e olePolyte hnique,(1996).

[MW79℄ W. Magnus et S. Winkler Hill's Equation. Corre ted reprint of the 1966 edition, Dover

Publi ations,(1979).

[MR73℄ J. Mawhin and N. Rou he Equations dierentielles ordinaires, Tome 1 : theorie generale.

Masson,Paris,(1973).

[MT78℄ F.Murat andL.Tartar H- onvergen e. Seminaire d'Analyse Fon tionnelle etNumerique de

l'Universited'Alger, mimeographednotes (1978). Englishtranslationin Topi s in the

mathe-mati al modellingof omposite materials, CherkaevA., KohnR.V.,Editors,Progressin

Non-linearDierentialEquationsandtheirAppli ations,31,Birkhauser,Boston,(1997).

[Ngu89℄ G.Nguetseng A general onvergen e resultfor afun tionalrelated to thetheoryof

homoge-nization. SIAMJ.Math. Anal.,20:608{623,(1989).

[Pia98℄ A.Piatnitski Asymptoti behaviourofthegroundstateofsingularlyperturbedellipti

equa-tions. Commun.Math. Phys.197:527-551,(1998).

[RS78℄ M.ReedandB.Simon Methodsofmodern mathemati alphysi s. A ademi Press,NewYork,