Reiterated homogenization of degenerate nonlinear elliptic equations

REITERATED HOMOGENIZATION

OF DEGENERATE NONLINEAR

ELLIPTICEQUATIONS

J.BYSTR



OM* J. ENGSTR



OM* P.WALL*

(Dedicatedtothe memoryofJacques-LouisLions)

Abstract

Theauthorsstudyhomogenizationof somenonlinear partialdierential equations of the

form div a hx;h 2

x;Du

h

=f;whereaisperiodicinthersttwoargumentsandmonotone

inthethird.Inparticularthecasewhereasatisesdegeneratedstructureconditionsisstudied.

Itisprovedthatu

h

convergesweaklyinW 1;1

0

( )totheuniquesolutionofalimitproblemas

h!1. Moreover,explicitexpressionsforthelimitproblemareobtained.

KeywordsHomogenization, Reiterated,Monotone,Degenerated

2000MR Subject Classication35B27,35J70,74Q99

Chinese Library ClassicationO175.21,O175.25 DocumentCode A

ArticleID 0252-9599(2002)03-03251-10

x1. Introduction

Thispaperisdevotedtohomogenizationofpartialdierentialoperatorsincludingseveral

periodicallyoscillatinglengthscales. Thistypeofequationsappearinmanyeldsofphysics

andengineeringscienceswherethephysicalphenomenaoccurinhighlyheterogeneousmedia.

One example is heat conduction in composite materials involving two dierent materials

whichare periodically distributed. Thelocal characteristicsarethen described byrapidly

oscillatingfunctions. Adirectnumericaltreatmentofsuchproblemsisoftenimpossibledue

totherapidly oscillatingfunctions andonehastoapplysometypeofasymptoticanalysis.

Thebranchof mathematicsdeveloped fortheanalysisofthese typesofproblems isknown

ashomogenization.Formoreinformationconcerningthehomogenizationtheory,thereader

isreferredto [1,2,7,10]and[12].

Wewillnowgiveashortoverviewofpreviousresultsconnectedtothisworkandexplain

what our contribution is. Let us consider the class of partial dierential equations of the

form

div (a

h (x;Du

h

))=f on, u

h 2W

1;p

0

(); (1.1)

wherea

h

isincreasinglyoscillatingas h!1,isanopenboundedsubsetofR n

;1<p<

1;1=p+1=q=1andf 2W 1;q

():Thehomogenizationproblemfor(1.1)consistsofthe

study of theasymptoticbehaviorof solutionsu

h

as h!1. Inmanyimportant casesu

h

convergesweaklyin W 1;p

0

() tothesolutionuofthehomogenizedproblem

div (b(Du)=f on, u2W 1;p

0 ():

In [6] and [11] the following situation was studied: a

h

is of the form a

h

(x;) = a(hx;),

whereaismonotone,continuousandsatisessuitablecoercivenessandgrowthconditionsin

thesecondargumentandisperiodicintherstargument. Acorrespondinghomogenization

ManuscriptreceivedJanuary31,2002.

DepartmentofMathematics,LuleaUniversityofTechnology,SE-97187Lulea,Sweden.

result,withthedierencethataonlysatisesdegeneratestructureconditions,wasobtained

in[9]. Inthissituationitisnaturaltoworkwithweightedspaceswhichmeansthatinstead

of(1.1)wehave

div(a(hx;Du

h

))=f on, u

h 2W

1;p

0 (;

h );

where(

h

)isasequenceofperiodicweights.

Inthe casewhen a

h

(x;)= a(hx;h 2

x;);where a isperiodicin the rsttwovariables,

onespeaks aboutreiteratedhomogenization. This conceptwas introducedbyBensoussan,

LionsandPapanicolaouin[1],whereitwasstatedaresultforlinearoperators. Concerning

reiterated homogenization ofnonlinear problems werefer to[13] and [14]. One important

application of reiterated homogenization is that it has been an indispensable tool in the

constructionofstructureswithextremeeectivematerialproperties. Concerningthistopic

we refer to the collection of classical papers in [5], where the introduction gives a good

selection of references. We remark that some of the homogenization problems abovealso

havebeenstudiedby -convergenceforthecorrespondingvariationalproblemsandbytwo-

scaleconvergence,butleaveoutthis discussionsinceitisoutofthescopeofthiswork.

Inthispaperwestudyreiteratedhomogenizationwhereaonlysatisesdegeneratestruc-

tureconditions. Morepreciselyweprovethatthesolutionsu

h of

div (a(hx;h 2

x;Du

h

))=f on, u

h 2W

1;p

0 (;

h );

convergesweaklyto uin W 1;1

0

(); whereuisthesolutionofahomogenizedproblem

div (b(Du)=f on, u2W 1;p

0 ():

Thispaperisorganizedinthefollowingway:InSection2wexsomenotationandpresent

necessarypreliminaryresults. Section3containsthehomogenizationresultdescribedabove,

whichalsoisthemainresultofthispaper. InSection4wederiveahomogenizationresultfor

anauxiliaryproblem. Akeyingredientin theproofofthemainresultisthatthesolutions

oftheauxiliaryproblemareusedtodeneaspecialtypeoftestfunction. Finally,inSection

5wegivesomepropertiesofthehomogenizedoperatorb.

x2. Preliminariesand Notation

Letbearegularbounded opensubsetofR n

and jEjdenotetheLebesguemeasureof

thesetE in R n

: Moreoverleth
;
idenotetheEuclideanscalarproductonR n

and

E the

characteristic function of the set E: Let pbe areal constant1 <p< 1and letq beits

conjugateexponent,1=p+1=q=1:WewilldenotebyC andC

i

constantsthatmaychange

fromoneplacetoanother.

Furthermore,letY =Z =(0;1) n

be theunit cubein R n

. Letf

i

Y :i=1;


;Ng

beafamilyofdisjointopensetssuchthat 



Yn N

S

i=1

i 



=0andj@

i j=0.

LetbeaweightonR n

,i.e. ismeasurableand

>0a.e., and 

1=( p 1)

arein L 1

loc ( R

n

): (2.1)

Wedenote by L p

( ;)the setof realfunctions uinL 1

loc

() such thatu

1=p

isin L p

( ),

by W 1;p

( ;) the set of the functions u in W 1;1

loc

() such that u 2 L p

(;) and Du 2

[L p

(;) ] n

: Moreover,wedenotebyW 1;p

0

(;) thecompletion ofC 1

0

( )withrespectto

thenorminW 1;p

(;) ,i.e.

kuk

W 1;p

(;)

=

 Z



juj p

+jDuj p



dx



1=p

:

ByC 1

per

(Y)wemeantheset ofall Y-periodicfunctions in C 1

(R n

)withmean valuezero.

WealsodeneW 1;p

per

(;)asthesetofrealfunctions uinW 1;1

loc (R

n

)with meanvaluezero

1;p

WenowdenetheMuckenhoupt A

p class:

Denition 2.1. Let p>1;K1andlet bea weighton R n

. Then  isin the class

A

p ( K)if



1

jQj Z

Q

dz



1

jQj Z

Q

 1

p 1

dz



p 1

K

foreverycubeQ2R n

withfacesparalleltothecoordinateplanes. WesetA

p

= S

K1 A

p (K):

Let

i

beaY-periodicweightonR n

,i.e. 

i

satises(2.1)andis Y-periodic. Wedene

theweights

h and

h as



h (x)=

N

X

i=1



i ( x)

i

(hx); 

h (x)=

N

X

i=1



i (hx)

i h

2

x

: (2.2)

Then itfollowsthat 

h

; 

h

1=(p 1)

; 

h and 

h

1=(p 1)

allare in L 1

loc (R

n

): Moreover,we

assumethat

i ,

h and

h

areinA

p

(K)forsomeK.

Leta:R n

R n

R n

!R n

beafunctionsuchthat

a(y;z;)= N

X

i=1



i (y)a

i

(z;): (2.3)

Weassumethat a(
;z;) isY-periodic anda(y;
;)is Z-periodic. Wealso assumethata

satises certaincontinuityand monotonicityconditions. Tobe morespecic, assumethat

there exist constants c i

1

; c i

2

>0 and constants  and  with 0    min(1;p 1) and

max( p;2)<1suchthat

ja

i (z;

1 ) a

i (z;

2 )jc

i

1



i

(z)( 1+j

1 j+j

2 j)

p 1 

j

1



2 j



; (2.4)

ha

i (z;

1 ) a

i (z;

2 );

1



2 ic

i

2



i

(z)( 1+j

1 j+j

2 j)

p 

j

1



2 j



; (2.5)

fora.e. z2R n

,every 2R n

:Moreoverweassumethat

a

i

( z;0)=0 (2.6)

fora.e. z2R n

:

Asadirect consequenceof(2.4),(2.5),and(2.6)thefollowinginequalitieshold:

ja

i

(z;)jc i

a



i (z)



1+jj p 1



; (2.7)



i (z)jj

p

c i

b (

i

(z)+h a

i

(z;);i); (2.8)

Z

Z

j+Dv



i j

p



i

(z)dzc i

c

(1+jj p

): (2.9)

In[8]thefollowingresultisproved.

Lemma2.1. Letp>1andK1:ThenthereexisttwopositiveconstantsÆ=Æ(n;p;K)

andC=C( n;p;K)suchthat



1

jQj Z

Q

 1+Æ

dy

 1

1+Æ

C 1

jQj Z

Q

dy; (2.10)



1

jQj Z

Q



(1+Æ) =(p 1)

dy

 1

1+Æ

C 1

jQj Z

Q



1=(p 1)

dy; (2.11)

for everycube withfaces parallel tothe coordinateplanes andevery 2A

p (K):

In[9]thefollowingweightedcompensatedcompactnessresultisproved.

Lemma 2.2. Let  2A

p

; K 1; let(

h

)be a family in A

p

(K) andlet bean open

boundedset. Let(u

h

)beafamilyof functionssatisfying

(1) R

jDu

h j

p



h

dyC

1

<1for everyh2N;

(2)there existsafunctionu2W 1;p

( ;)suchthatu !uinL 1

():

Moreover,let(a

h

)beafamily ofvectorfunctionsin R n

suchthat

(3) R

ja

h j

q



1=(p 1)

h

dyC

2

<1for everyh2N;

(4)div( a

h

)=f 2L 1

( )onC 1

0

() for everyh2N;

(5)there existsa2



L q

;

1=(p 1)



n

suchthat a

h

!aweakly in



L 1

()



n

:

Then

Z

ha

h

;Du

h

idy! Z

ha;Duidy

for every2C 1

0 ():

In[15]thefollowingconvergenceresultforperiodicfunctions isproved.

Lemma 2.3. Let 1 p 1 and letu

h 2L

p

loc (R

n

) be Y-periodic for h 2 N: More-

over, suppose that u

h

! u weakly in L p

( Y) (weakly



if p = 1) as h ! 1: Let w

h be

dened by w

h

(x)=u

h

(hx): Then ash!1 it holds that w

h

! 1

jYj R

Y

u(y)dy weakly in

L p

() (weakly



if p=1):

Weendthissectionwithasimpleextensionlemma.

Lemma 2.4. Let  be a Y-periodic weight on R n

; let g : Y !R n

be a function such

that

g2[L q

(Y;

1=(p 1)

] n

; Z

Y

hg;Dwidy=0; 8w2W 1;p

per (Y;);

andleteg bethe Y-periodicextension toR n

ofg: Then wehave

e g2[L

q

loc (R

n

;

1=(p 1)

)]

n

; Z

R n

h eg;Dvidy=0; 8v2C 1

0 (R

n

):

Insituationswherenoconfusioncanoccurwewillusethesamenotationfortheextended

function asfortheoriginalone.

x3. The MainTheorem

LetusconsiderthefollowingDirichletproblems:

 R

a hx;h 2

x;Du

h

;D



dx= R

fdx; 82W 1;p

0 (;

h );

u

h 2W

1;p

0 (;

h );

(3.1)

wheref 2L 1

( ):Bystandardresultsin existencetheorythere existuniquesolutionsfor

eachh:Belowwestatethemainresultofthispaper.

Theorem3.1. Let(u

h

)bethesolutions of (3.1). Then

u

h

!uweakly inW 1;1

0 ( );

a hx;h 2

x;Du

h

!b(Du) weaklyin



L 1

( )



n

;

ash!1;where uistheunique solutionof

 R

hb(Du);Didx= R

fdx for every2W 1;p

0 ();

u2W 1;p

0 ( ):

(3.2)

The operator b:R n

!R n

isdenedas

b()= 1

jYj Z

Y b

Y

y;+Du



(y)

dy;

whereu



isthe uniquesolutionof the Y-cell problem

 R

Y

b

Y

y;+Du



( y)

;D



dy=0for every2W 1;p

per (Y);

u



2W 1;p

per (Y):

The operator b

Y

:Y R n

!R n

isdenedasb

Y

(y;)= N

P

i=1



i (y)b

i

(); where

b

i ()=

1

jZj Z

a

i



z;+Dv



i ( z)



dz;

andv



i

arethe uniquesolutions ofthe Z-cellproblems

(

R

Z D

a

i



z;+Dv



i (z)



;D( z) E

dz=0for every2W 1;p

per ( Z ;

i );

v



i 2W

1;p

per ( Z ;

i ):

Proof. Letusrstprovethat kDu

h k

L p

(;h)

C :Byperiodicityitfollowsthat

Z



h

(x)dxC ; Z

(

h (x) )

1=(p 1)

dxC : (3.3)

Thus,by(2.8),(3.1),(3.3),Poincare'sandHolder'sinequalitiesweobtain

Z

j Du

h j

p



h

(x)dxc

b

 Z



h

( x)dx+ Z

fu

h dx



C

 Z



h

( x)dx+kfk

L 1

() Z

jDu

h jdx



C



1+

 Z

jDu

h j

p



h (x)dx



1=p



:

Henceitisclearthat

kDu

h k

L p

(;

h )

C : (3.4)

Let

h

bedened as

h

=a hx;h 2

x;Du

h

:Then(2.7)and(3.4)implies

k

h k

[L q

(;

1=( p 1)

h )]

n

C : (3.5)

TakeÆ>0suchthat (2.11)holds. Nowchoose

1

suchthat

1+

1

p 1 

1

= 1+Æ

p 1 :

Then

1

>0and p 1 

1

>0:LetQbeacubeinR n

containing. Holder'sinequality

and(3.4)thengives

Z

jDu

h j

1+1

dx

 Z

jDu

h j

p



h dx

 1+

1

p

 Z



h 1+

1

p 1 

1

dx

 p 1 

1

p

C

 Z



h 1+Æ

p 1

dx

 p 1

p+Æ

C

 Z

Q



h 1+Æ

p 1

dx

 p 1

p+Æ

:

Byapplying(2.11) and(3.3)in theinequalityabove,weobtain

Z

jDu

h j

1+1

dxC

 Z

Q



h 1=(1 p)

dx



(p 1)(1+Æ)=(p+Æ)

C : (3.6)

Next,chooseÆ>0suchthat (2.10)holdsandchoose

2

suchthat

1+Æ=( 1+

2 )

q 1

q 1 

2 :

Then

2

>0andq 1 

2

>0:Byusing(2.10)andarguingsimilarlyasfor(3.6)weobtain

Z

j

h j

1+2

dxC :

Thismeansthat(u

h

)and(

h

)areboundedinW 1;1+

1

0

()and



L 1+

2

()



n

respectively:

Sincethesespacesarere exive,wehavethatthereexistsubsequences,stilldenotedby(u

h )

and(

h

),suchthat

u

h

!u



weaklyin W 1;1+

1

0

(); (3.7)



h

!



weaklyin



L 1+

2

()



n

: (3.8)

From(3.7)and(3.8)itfollowsthat

u

h

!u



weaklyin W 1;1

0

(); (3.9)

 ! weaklyin



L 1

()



n

: (3.10)

Fromouroriginalproblem(3.1)wehave

Z

h

h

;Didx= Z

fdxforevery2W 1;p

0 (;

h

): (3.11)

Byusing thefact C 1

0

()W 1;p

0 (;

h

)and(3.10)wecanpasstothelimitin(3.11),thus

Z

h



;Didx= Z

fdxforevery2C 1

0 ():

Densityandthefact that

 2[L

q

()]

n

(see(3.17))thengives

Z

h



;Didx= Z

fdxforevery2W 1;p

0

( ): (3.12)

Letusnowobservethat by(3.9),(3.10) and(3.12)thetheorem isprovedifweshowthat

u

 2W

1;p

0

( ); (3.13)





=b(Du



) a.e. on; (3.14)

since the uniqueness of the solution of the homogenized problem (3.2) then implies that

u



=ua.e. on:

Westartwiththeproofof(3.13). Weobservethatsinceisaregularboundedopenset

itissuÆcienttoshowthatDu

 2[L

p

()]

n

: Let2C

0

():ThenHolder'sinequalityand

(3.4)gives

kDu

h

k

[L 1

() ] n



 Z

jDu

h j

p



h (x)dx

1

p

 Z

(

h (x) )

1

p 1

jj q

dx

1

q

C

 Z

( 

h (x) )

1

p 1

jj q

dx

 1

q

: (3.15)

Applyingliminfonbothsidesof(3.15)andusingtheweaklowersemicontinuityofthenorm

ontheleft handsideandperiodicityontherighthandsideweobtain

Z

jDu



jjjdxCkk

L q

()

forevery2C

0 ():

BydensityandLandau'stheoremwethenhavethat

Du

 2[L

p

()]

n

: (3.16)

Byusing(3.5)andargumentssimilartothoseemployedintheproofof(3.16)itcanalsobe

deducedthat



 2[L

q

() ] n

: (3.17)

Itremainstoprove(3.14). Forthispurposeletusdenethetestfunction

w



h

(x)=( ;x)+ 1

h u



h (hx);

where u



h

is dened as in the auxiliaryproblem (see Section 4). To be able to apply the

compensatedcompactnessresult(Lemma2.2)wehavetoprovecertainfactsaboutw



h and

a



hx;h 2

x;Dw



h



. Therefore,byperiodicity,(4.5)andthefactthat

h

(x)=

h

(hx)weget

Z





Dw



h 



 p



h

(x)dxC : (3.18)

Moreover,byusing(2.7)and(3.18) weobtain

Z

ja(hx;h 2

x;Dw



h )j

q

(

h ( x))

1=(p 1)

dxC :

ByperiodicityandLemma 2.3wehavethat

w



h

(
)!( ;
) in L 1

();

a



hx;h 2

x;Dw



h



! 1

jYj Z

Y b

Y

y;+Du



dy=b() weaklyin



L 1

()



n

:

Finally,dueto(4.6),wecanapplyLemma2.4on(4.1)andobtain

div



a



hx;h 2

x;Dw



h



=0onC 1

0 ();

andwearenowreadytoapplythecompensatedcompactnessresult. Indeed,bythemono-

tonicityofawehaveforaxed  that

Z

D



h a



hx;h 2

x;Dw



h (x)



;Du

h

(x) Dw



h (x)

E

( x)dx0

forevery2C 1

0

( ),0:Bythecompensated compactnesslemma (Lemma 2.2)with

theweight =1,wegetinthelimit

Z

h



b();Du



(x) i( x)dx0

forevery2C 1

0

();0:Henceforourxed2R n

wehavethat

h



b();Du



(x) i0fora.e. x2:

Bydensityandthecontinuityofb (seeLemma5.2),itfollowsthat

h



b();Du



( x) i0fora.e. x2andevery2R n

:

Since b is monotone (5.3)and continuous (5.4),wehave that b is maximalmonotone and

hence (3.14) follows. Finally, let us observe that we have provedthe theorem only up to

asubsequence;but sincethehomogenizedoperator isuniquelydened andthesolutionof

thehomogenized problemis uniquewecanconcludethat thetheorem holdsfor thewhole

sequence.

x4. An AuxiliaryProblem

Inthis section weprovea homogenizationresult forthe auxiliaryproblem. This result

was used in the denition of the special type of test functions dened in the proof of the

mainresult(Theorem3.1)ofthispaper.

Fix andconsiderthefollowingDirichletproblems:

 R

Y

ha(y;hy;+Du



h

);Didy=0,82W 1;p

per Y;

h

;

u



h 2W

1;p

per Y;

h

:

(4.1)

Bystandard resultsin existencetheory there exist uniquesolutions for each h: Belowwe

statetheauxiliaryresultofthispaper.

Theorem4.1. Let



u



h



bethe solutionsof (4.1). Wethenhave that

u



h

!u



weakly inW 1;1

per (Y);

a



y;hy;+Du



h



!b

Y

y;+Du



weakly in



L 1

(Y)



n

;

ash!1;where u



isthe uniquesolution of

 R

Y

b

Y

y;+Du



;D



dy=0for every2W 1;p

per (Y);

u



2W 1;p

per (Y):

(4.2)

The operator b

Y

:Y R n

!R n

isdenedasb

Y

(y;)= N

P

i=1



i (y)b

i

( ); where

b

i ()=

1

jZj Z

a

i

(z; +Dv



i (z))dz

andv



i

arethe uniquesolutions ofthe Z-cellproblems

 R

Z h a

i

(z; +Dv



i

(z));D( z)idz=0,82W 1;p

per ( Z ;

i );

v



i 2W

1;p

per (Z ;

i ):

(4.3)

Proof. By(2.8),(4.1)and(2.7)wehavethat

Z

Y

j+Du



h j

p



h

(y)dyC

b

 Z

Y



h

( y)dy+C

a Z

Y



1+j+Du



h j

p 1





h

(y)jjdy



:

Moreover,ifweuseYoung's inequality onthe righthand side and rearrangethe resulting

inequalityweobtain



1 C

a C

b 2

q 1



q

 Z

Y

j+Du



h j

p



h (y)dy

C

b



1+ C

a 2

q 1



q +

C

a jj

p

 (p 1)

p

 Z

Y



h ( y)dy;

where isapositiverealnumber. Bychoosing smallenoughwegetthat

Z

Y

j+Du



h j

p



h

(y)dyC : (4.4)

Inparticularthisimplies

Z

Y jDu



h j

p



h

( y)dyC : (4.5)

Letusdene i

h

=a

i

(hy;+Du



h

): Byusing(2.7)and(4.4),itfollowsthat

Z

i 



 i

h 

 q

(

i (hy))

1=(p 1)

dyC : (4.6)

Moreover,by using (2.11), (4.5),(2.10), (4.6)andargumentssimilar tothose employed in

theproofof(3.6),itcanbededucedthat

Z

Y 



Du



h 



 1+1

dyC ; Z

i 



 i

h 

 1+

2

dyC :

Thuswehavethat



u



h



and  i

h

areboundedinW 1;1+1

per

(Y)and



L 1+2

(

i )



n

respectively:

Sincethese spaces arere exive,there exist subsequences,still denotedby



u



h



and  i

h

;

suchthat

u



h

!u





weaklyinW 1;1+1

per

(Y);  i

h

! i



weaklyin



L 1+2

(

i )



n

:

Hencewecan concludethat

u



h

!u





weaklyinW 1;1

per

(Y);  i

h

! i



weaklyin



L 1

(

i )



n

: (4.7)

Usingsimilarideasasintheproofof(3.16),itcanbeshownthat

 i

 2[L

q

(Y)]

n

: (4.8)

From(4.1)and(4.7)itfollowsthat

N

X

i=1 Z

i

 i



;D



dy=0forevery2C 1

per ( Y):

Densityargumentsin conjunctionwith(4.8)thenresultsin

N

X Z

i

 i



;D



dy=0forevery2W 1;p

per ( Y):

Thusthetheoremisprovedifweshowthat

u



 2W

1;p

per

( Y); (4.9)

 i



=b

i

+Du





a.e. on

i

; (4.10)

since the uniqueness of the solution of the homogenized problem (4.2) then implies that

u





=u



a.e. onY:

Let us start with (4.9). We observe that since Y is a regular bounded open set it is

suÆcient to showthat Du



 2 [L

p

(Y)]

n

which is obtained using the sameideas asin the

proofof(3.16).

Itremainstoprove(4.10). Thereforeletusdenethetestfunction w

;i

h by

w

;i

h

(y)=(;y)+ 1

h v



i

(hy); (4.11)

where v



i 2 W

1;p

per (Z ;

i

) isdened as in (4.3). To be ableto applythe compensated com-

pactnessresult(Lemma2.2) wehaveto provecertainfacts aboutw

;i

h anda

i



hy;Dw

;i

h



.

Indeed,

Z

i jDw

;i

h j

p



i

(hy)dyC (4.12)

followsfrom(4.11). Moreover,wealsohavethat

Z

i ja

i

(hy;Dw

;i

h )j

q

(

i (hy))

1=(p 1)

dyC

by(2.7)and(4.12). Byperiodicityweobtain

w

;i

h

(
)!( ;
) stronglyinL 1

(

i );

a

i

(hy;Dw

;i

h )!

1

jZj Z

Z a

i

(z;+Dv



i

(z))dz=b

i

() weaklyin



L 1

(

i )



n

:

Applicationof Lemma 2.4on(4.3)givesdiv (a

i

(hy;Dw

;i

h

))=0onC 1

0 (

i

): Bythe mono-

tonicityofa

i

wehaveforaxed  that

Z

i h

i

 a

i

(hy;Dw

;i

h

(y));+Du



h

(y) Dw

;i

h

(y)i( y)dy0

forevery2C 1

0 (

i

),0:Bythecompensatedcompactnesslemma(Lemma 2.2)with

=1;wethengetinthelimit

Z

i

 i

 b

i

();+Du



 (y) 



(y)dy0

forevery2C 1

0 (

i

); 0:Henceforourxed 2R n

wehavethat

 i

 b

i

();+Du



 ( y) 



0fora.e. y2

i :

Bydensityandthecontinuityofb

i

(5.2),itfollowsthat

a

 b

i

( );+Du



 ( y) 



0fora.e. y2

i

andevery 2R n

.

Sinceb

i

ismonotone(5.1)andcontinuous(5.2),wehavethatb

i

ismaximalmonotoneand

hence(4.10)follows.

x5. PropertiesoftheHomogenized Operators b

i

andb

Inthissectionwelistsomepropertiesofthehomogenizedoperatorsb

i

andb. Inparticular

these properties imply the existence and uniqueness of the solution of the homogenized

problem(intheauxiliaryand mainproblemrespectively).

Lemma5.1. Letb bethe homogenizedoperator denedin Theorem 4:1. Then

(a)b

i

(
)isstrictly monotone. Inparticular, we havethat

hb

i (

1 ) b

i (

2 );

1



2 iec

2 (1+j

1 j+j

2 j)

p 

j

1



2 j



(5.1)

for every

1

;

2 2R

n

:

(b)b

i

(
)iscontinuous. Inparticular, wehave for = 

  that

jb

i ( 

1 ) b

i (

2 )jec

1 (1+j

1 j+j

2 j)

p 1

j

1



2 j

(5.2)

for every

1

;

2 2R

n

:

(c) b

i

(0)=0:

Proof. Thesepropertiesfollowbyusingthesameideasasin[3]and[4].

Lemma5.2. Letb bethe homogenizedoperator denedin Theorem 3:1. Then

(a)b(
)isstrictly monotone. Inparticular, we havethat

h b(

1 ) b(

2 );

1



2 i

e

C

2 (1+j

1 j+j

2 j)

p 

j

1



2 j



(5.3)

for every

1

;

2 2R

n

:

(b)b(
)iscontinuous. Inparticular, wehavefor Æ=



= 

( )  that

jb(

1 ) b(

2 )j

e

C

1 (1+j

1 j+j 

2 j)

p 1 Æ

j

1



2 j

Æ

(5.4)

for every

1

;

2 2R

n

:

(c) b(0)=0:

Proof. These properties follow by using (5.1) and (5.2) in the corresponding theorem

givenin forexample[3]and[4].

References

[1] Bensoussan,A.,Lions,J.L.&Papanicolaou,G.,Asymptoticanalysisforperiodicstructures[M],North

Holland,Amsterdam,1978.

[2] Braides,A.&Defranceschi,A.,Homogenizationofmultipleintegrals[M],OxfordUniversityPress,New

York,1998.

[3] Bystrom,J., Correctorsfor somenonlinear monotone operators [R],Research report, No. 11, ISSN:

1400{4003,DepartmentofMathematics,LuleaUniversityofTechnology,1999.

[4] Bystrom,J.,Correctorsforsomenonlinearmonotoneoperators[J],J.NonlinearMath.Phys.,8:1(2001),

8{30.

[5] Cherkaev,A.&Kohn,R.,Topicsinthemathematicalmodellingofcompositematerials[M],Birkhauser,

Boston,1997.

[6] Chiado Piat, V. & Defranceschi, A., Homogenization of monotone operators [J], Nonlinear Anal.,

14:9(1990),717{732.

[7] Cioranescu,D.&Donato,P.,Anintroductiontohomogenization[M],OxfordUniversityPress,Oxford,

1999.

[8] Coifman,R.R.&Feerman,C.,Weightednorminequalitiesformaximalfunctionandsingularintegrals

[J],StudiaMath.,51(1974),241{250.

[9] De Arcangelis, R. & Serra Cassano, F., Onthe homogenization of degenerate elliptic equations in

divergenceform[J],J.Math.PuresAppl.,71(1992),119{138.

[10] Defranceschi,A.,AnintroductiontohomogenizationandG-convergence[M],Lecturenotes,Schoolon

homogenization,ICTP,Trieste,1993.

[11] Fusco,N.&Moscariello,G.,Onthe homogenizationofquasilineardivergencestructureoperators[J],

AnnaliMat.Pura Appl.,146(1987),1{13.

[12] Jikov,V.,Kozlov, S.&Oleinik,O.,Homogenizationof dierentialoperators andintegralfunctionals

[M],Springer-Verlag,Berlin-Heidelberg-NewYork,1994.

[13] Lions,J.-L.,Lukkassen,D.,Persson,L.-E.&Wall,P.,Reiteratedhomogenizationofmonotoneoper-

ators[J],C.R.Acad.Sci.ParisSeriesI,330:8(2000),675{680.

[14] Lions,J.-L.,Lukkassen,D.,Persson,L.-E.&Wall,P.,Reiteratedhomogenizationofnonlinearmono-

toneoperators[J],Chin.Ann.Math.,22B:1(2001),1{12.

[15] Lukkassen,D.&Wall,P.,Onweakconvergenceoflocallyperiodicfunctions[J],J.Nonlinear. Math.

Phys.,9:1(2002),42{572002.