10.1098/ spa.2002.1003
The wo-scale con e gence me hod applied o
gene alized Besico i ch spaces
By J u an C a sa d o-D¶i a z a n d Inmacul ad a Ga y e
Depa amen o de Ecuaciones Di e enciales y An¶alisis Num¶e ico,
Facul ad de Ma em¶a icas, C/ Ta ¯a s/n, Se illa, CP 41012, Espa~na
(jcasado@nume .us.es; gay e@nume .us.es)
Recei ed 10 Augus 2001; e ised 19 Ma ch 2002; accep ed 16 Ap il 2002;
published online 4 Oc obe 2002
The wo-scale con e gence me hod has p o ed o be a e y use ul ool o dealing
wi h pe iodic homogeniza ion p oblems. In he p esen pape we de elop his heo y
o gene alized Besico i ch spaces, which include he almos -pe iodic unc ions. The
main di¯ cul y comes om he ac ha hese spaces a e no sepa able. We also show
how o apply hese esul s o he homogeniza ion o pa ial di¬e en ial p oblems in
his amewo k.
Keywo ds: pa ial di®e en ial equa ions; homogeniza ion; wo-scale con e gence
1. In oduc ion
I is usual in homogeniza ion heo y o deal wi h composi e pe iodic ma e ials and
s uc u es wi h e y small pe iods. In o de o s udy hei physical beha iou (elec-
ical o he mal conduc i i y, elas ic beha iou , e c.), we need o sol e a pa ial
di¬e en ial equa ion ha , in a model case, can be w i en in he o m
¡di Aµx
"¶ u"= ; (1.1)
besides some bounda y condi ions. He e he ma ix Ais pe iodic and "is a small
pa ame e . F om he nume ical poin o iew, i is e y di¯ cul o calcula e u" om
his p oblem. We need o use a disc e iza ion o size smalle han ", and he e o e
sol ing a e y la ge sys em o equa ions, which equi es a lo o compu e memo y,
is ime-consuming and in ol es se e al s abili y p oblems.
Homogeniza ion heo y seeks o ob ain an app oxima ion o u" h ough he eso-
lu ion o simple pa ial di¬e en ial equa ions. The heo y o asymp o ic expansions
(see Bensoussan e al. 1978; S´anchez Palencia 1980) p o ides us wi h
u"(x)¹u0(x) + "u1µx; x
"¶+"2u2µx; x
"¶+¢¢¢ ;(1.2)
whe e he unc ions uia e ob ained as he solu ions o pa ial di¬e en ial p oblems
much easie o sol e han (1.1). A igo ous way o ob aining his expansion and
showing i s con e gence is he wo-scale con e gence me hod o Ngue seng and Allai e
(see Allai e 1992; A bogas e al. 1990; Ngue seng 1990). I has p o ed o be e y
use ul in he homogeniza ion o pe iodic p oblems.
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2925
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2926 J. Casado-D¶³ az and I. Gay e
We no e, howe e , ha al hough pe iodic ma e ials a e common in enginee ing
applica ions, hey do no usually occu in na u e. Howe e , we can see a ecu ence
in he s uc u es which sugges s ha a be e app oxima ion is o conside hese
ma e ials as almos -pe iodic. The aim o he p esen pape is o ex end he wo-scale
con e gence me hod o he case o almos -pe iodic (o mo e gene al) coe¯ cien s, in
pa icula , he sum o pe iodic unc ions wi h di¬e en pe iods, in o de o be able
o ea mo e gene al composi e ma e ials han he pe iodic ones. Simila ly o he
pe iodic case, i is necessa y o ha e a cha ac e iza ion o he limi o he exp ession
Z«
"(x)Áµx; x
"¶dx(1.3)
when "is a bounded sequence in a Lebesgue space Lp(«), p > 1, and Áis an
almos -pe iodic smoo h unc ion o , in a wide sense, in a gene alized Besico i ch
space (see Casado D´± az & Gay e 2002; Jiko e al. 1994; Zhiko & K i enko 1983)
in i s second a iable. To p o e he co esponding esul , he  s s ep is o show
he exis ence o a subsequence o ", s ill deno ed by ", such ha he e exis s he
limi o (1.3), o e e y Áas abo e. This is easy, using a diagonal a gumen , i he
space o unc ions Áis sepa able. Howe e , he gene alized Besico i ch spaces a e no
sepa able in gene al. This is he main di¯ cul y in ob aining ou esul . To sol e his
p oblem, we p opose an abs ac heo em gene alizing he well-known esul abou
he weak sequen ial compac ness o he uni ball in a e®exi e space. In a simple
si ua ion his was ca ied ou in Casado D´± az & Gay e (1996). The case whe e he
almos -pe iodic unc ions a e in a sepa able space has been conside ed by Ngue seng
(2000).
As an example o how ou esul s can be used in he s udy o he asymp o ic
beha iou o composi e ma e ials, we s udy he nonlinea p oblem
¡di aµx
"; u"; u"¶= in « ;
u"= 0 on @« ;
whe e ais a Ca a h´eodo y unc ion which de nes a pseudomono one ope a o o o de
pand belongs o a Besico i ch space in i s second a iable. In his case we ob ain he
limi equa ion and a co ec o esul ela ed o (1.2). To comple e his in oduc ion,
we men ion ha an adap a ion o he wo-scale con e gence o s ochas ic homoge-
niza ion p oblems has been gi en in Bou gea e al. (1994) (assuming sepa abili y).
The no ion o a s ochas ic weak de i a i e gi en in his a icle is s ongly ela ed o
he mean de i a i e we use in he p esen pape . To he s udy o homogeniza ion
p oblems in a s ochas ic ame, we also e e o Dal Maso & Modica (1986) and
Abddaimi e al. (1997).
2. A compac ness heo em
I is well known ha , o a bounded sequence ngin he dual space X0o a e®exi e
space X, he e exis s a subsequence o ngwhich con e ges weakly-¤ o some 2X0,
i.e. ngpoin wise con e ges o . The pu pose o he p esen sec ion is o gene alize
his esul , by showing ha i is necessa y o assume nei he ncon inuous no X
comple e.
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Two-scale me hod applied o Besico i ch spaces 2927
Theo em 2.1. Le Xbe a subspace (no necessa ily closed) o a e°exi e space Y
and le n:X7! Rbe a sequence o linea unc ionals (no necessa ily con inuous).
Assume he e exis s a cons an C > 0which sa is¯es
lim sup
n
n(x)6Ckxk;8x2X: (2.1)
The e hen exis a subsequence nkgo ngand a unc ional 2Y0such ha
9lim
k nk(x) = h ; xi;8x2X: (2.2)
Rema k 2.2. I Xis comple e and nis con inuous, heo em 2.1 easily ollows
om he Banach{S einhauss heo em and he weak-¤sequen ial compac ness o he
uni ball in a e®exi e space. I is also clea ha heo em 2.1 holds i we eplace he
hypo hesis Xincluded in a e®exi e space by Xsepa able. The aim o heo em 2.1
is p ecisely he applica ion o spaces ha a e no sepa able.
Rema k 2.3. Theo em 2.1 has been es ablished in Casado D´± az & Gay e (1996)
when Yis a Hilbe space.
In o de o p o e heo em 2.1, we need o ecall some esul s abou smoo h no ms
(see Cio anescu 1990).
De¯ni ion 2.4. Le Ybe a Banach space. The no m in Yis called smoo h i
o e e y y2Ywi h kyk= 1 he e exis s a unique 2Y0such ha k k= 1 and
h ; yi= 1.
The ollowing heo em is due o Asplund and Lindens auss (see Cio anescu 1990;
Lindens auss 1966).
Theo em 2.5. E e y e°exi e Banach space has an equi alen smoo h no m.
P oo o heo em 2.1. By heo em 2.5, i is no es ic i e o assume ha he no m
in Yis smoo h.
Fi s s ep. Le us p o e ha he e exis a subsequence nkgo ng, a cons an
·
C>0 and a sequence zjg » Xsuch ha
kzjk= 1;(2.3)
lim sup
k
nk(x)6·
Ckxk;8x2X; (2.4)
9lim
k nk(zj)>·
C¡1
j;8j2N:(2.5)
To his end, we de ne
C1= supnlim sup
n
n(x) : x2X; kxk= 1o:
This sup emum is  ni e because o (2.1).
By de ni ion o C1 he e exis z12Xwi h kz1k= 1 and a subsequence n1(k)gk
o ngsuch ha
9lim
k n1(k)(z1)>C1¡1:
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2928 J. Casado-D¶³ az and I. Gay e
Then we de ne C2by
C2= supnlim sup
k
n1(k)(x) : x2X; kxk= 1o:
Ob iously, C26C1and he e exis z22Xwi h kz2k= 1 and a subsequence
n2(k)gko n1(k)gksuch ha
9lim
k n2(k)(z2)>C2¡1
2:
Repea ing his easoning, we deduce ha o e e y j2N he e exis Cj2R,zj2X
and nj(k)gksuch ha , deno ing n0(k) = k o e e y k2N, we ha e
kzjk= 1;(2.6)
Cj= supnlim sup
k
nj¡1(k)(x) : x2X; kxk= 1o;(2.7)
nj(k)gkis a subsequence o nj¡1(k)gk;(2.8)
9lim
k nj(k)(zj)>Cj¡1
j;(2.9)
06Cj+ 1 6Cj:(2.10)
Taking he diagonal subsequence nk(k)g, which we deno e by nkg, we ha e
lim sup
k
nk(x)6Cjkxk;8x2X; 8j2N;
9lim
k nk(zj)>Cj¡1
j;8j2N:
The e o e, o ·
C= limjCjs a emen s (2.4) and (2.5) hold.
Second s ep. Le us now p o e ha he e exis s 2Y0such ha nkgand sa is y
(2.2). No e ha we can suppose ·
C > 0 because i no , by (2.4) we immedia ely ge
(2.2) wi h = 0. Since Yis e®exi e and zjgis bounded, he e exis a subsequence,
s ill deno ed by zjg, and z02Ysuch ha
zj* z0in Y: (2.11)
By (2.6), z0sa is es
kz0k61:(2.12)
Le x2Xbe a bi a y. Since nk(x)gkis bounded, he e exis s a subsequence
nk(j)gjo nkgk, depending on x, such ha
9lim
j nk(j)(x):(2.13)
Deno ing S= span( xgS zn:n2Ng), s a emen s (2.13), (2.5) and nk(j)linea
imply
9lim
j nk(j)(s);8s2S:
We de ne :S7! Rby
(s) = lim
j nk(j)(s);8s2S:
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Two-scale me hod applied o Besico i ch spaces 2929
By (2.4) and because nk(j)is a subsequence o nk o e e y s2S»X, we ha e
(s) = lim
j nk(j)(s)6lim sup
k
nk(s)6·
CkskX;
and so belongs o S0and sa is es k kS06·
C. On he o he hand, by (2.5) and
(2.6), o e e y j2Nwe ha e
(zj)>·
C¡1
j=µ·
C¡1
j¶kzjk:(2.14)
Thus k k>·
C¡1=j o e e y j2Nand so
k kS0=·
C: (2.15)
By he Hahn{Banach heo em, we can ex end o a unc ional o Y0, s ill deno ed
by , which sa is es k kY0=·
C. By (2.14), (2.15), (2.11) and (2.6), we deduce
·
C= lim
j (zj) = h ; z0i:(2.16)
Using (2.12) and (2.15), we hen ha e
h ; z0i=·
C; k k=·
C; kz0k= 1:(2.17)
Since Yis smoo h he e exis s a unique elemen 2Y0sa is ying (2.17), so in
(2.13) i is no necessa y o ake a subsequence, and he whole o he sequence nkg
poin wise con e ges o in X.¥
Following he idea o heo em 2.1, we can also p o e he ollowing heo em, which
gene alizes heo em 2.1 and con ains he case whe e Xis sepa able. The p oo can
be ound in Gay e Delgado (1998).
Theo em 2.6. Le Xbe a no med space (no necessa ily comple e) such ha
he uni sphe e o X0endowed wi h he weak-¤ opology is ¯ s coun able. Le
n:X7! Rbe a sequence o linea unc ionals (no necessa ily con inuous) sa is ying
(2.1). The e hen exis a subsequence nkgo ngand a unc ional 2X0such ha
(2.2) holds.
Rema k 2.7. I can be p o ed ha he uni sphe e o X0endowed wi h he weak-¤
opology is  s coun able i and only i o e e y 2X0,k k= 1, he e exis s a
sequence zng » Xsuch ha i g2X0sa is es kgk= 1 and hg; zni=h ; zni, o
e e y n2N. Then g= .
3. P elimina ies on gene alized Besico i ch spaces
In his sec ion we ecall some esul s on gene alized Besico i ch spaces we will need
la e . They ha e been p o ed in Casado D´± az & Gay e (2002) (see also Gay e Delgado
(1998); Jiko e al. (1994)). We ecall he de ni ion o he mean alue.
De¯ni ion 3.1. We say ha a unc ion :RN!Rhas a mean alue i he e
exis s a eal numbe M gsuch ha o e e y bounded measu able se K»RN
wi h jKj>0 we ha e
M g= lim
T!+1
1
jT KjZT K
(y) dy: (3.1)
In his case, we say ha M gis he mean alue o .
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2930 J. Casado-D¶³ az and I. Gay e
Following Jiko e al. (1994, 7.5, p. 242), we now gi e he de ni ion o an algeb a
wi h mean alue.
De¯ni ion 3.2. A linea space, X, o eal- alued unc ions de ned in RNis a
Banach algeb a wi h mean alue i he ollowing condi ions a e sa is ed.
(i) The elemen s o Xa e bounded, uni o mly con inuous and possess a mean
alue (see (3.1)).
(ii) The cons an unc ions belong o X.
(iii) Xis an algeb a.
(i ) Xendowed wi h he uni o m con e gence opology is comple e.
( ) Fo e e y 2Xand s2R, he unc ion (¢+s) belongs o X.
Rema k 3.3. As examples o X, we ha e he space o con inuous (0;1)N-pe iodic
unc ions and he space o uni o mly almos -pe iodic unc ions.
The abo e de ni ion allows us o de ne he gene alized Besico i ch spaces in he
ollowing way.
De¯ni ion 3.4. We de ne he gene alized Besico i ch space o o de p( ela i e o
X), wi h 1 6p < +1, and we deno e i by Bp, as he closu e o X o he semino m
[ ]p=µlim sup
T!+1
1
jBTjZBT
j (x)jpdx¶1=p
;(3.2)
i.e.
Bp= :RN!Rmeasu able: 8" > 0;9’2Xwi h [ ¡’]p< "g:
The gene alized Besico i ch space o o de 1( ela i e o X), B1, is de ned by
B1=n 2B1such ha [ ]1= sup
p>1
[ ]p<+1o:
The spaces Bpa e semino med spaces. The quo ien o Bpwi h he ke nel o [¢]pis
deno ed by Bpand i is a no med space.
Rema k 3.5. When Xis he space o con inuous (0;1)N-pe iodic unc ions, Bpis
he space o unc ions in Lp
loc(RN) which a e (0;1)N-pe iodic. The space o almos -
pe iodic unc ions in he sense o Besico i ch (see, o example, Besico i ch 1954;
Boh 1951) is ob ained by aking Xas he space o uni o mly almos -pe iodic unc-
ions.
The ollowing heo em shows ha he spaces Bpa e analogous o he spaces Lp o
a p obabili y measu e (see Casado D´± az & Gay e (2002) and Gay e Delgado (1998)
o he p oo ).
Theo em 3.6. The spaces Bpsa is y he ollowing p ope ies.
(i) Fo 16p6+1,Bpand hen Bpa e comple e.
(ii) Fo e e y 2Bp,16p < +1, he e exis M gand M j jpg. Besides,
[ ]p=M j jpg1=p.
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Two-scale me hod applied o Besico i ch spaces 2931
(iii) Fo 2B1and ¬= [ ]1, he unc ion T¬( )2L1(RN)sa is¯es
kT¬kL1(
R
N)= [ ]1;[T¬( )¡ ]1= 0;
whe e T¬is de¯ned by
T¬(s) = 8
>
<
>
:
¬i s > ¬ ;
si jsj6¬ ;
¡¬i s < ¡¬ :
(i ) I p < q, hen Bq»Bpand [¢]p6[¢]q. Mo eo e , i ; g 2Bqsa is y [ ¡g]p= 0,
hen [ ¡g]q= 0, and we can also hen see ha Bqis a subspace o Bp.
( ) The dual space o Bp, o 16p < +1, can be iden i¯ed wi h Bp0 h ough he
ollowing isome ic isomo phism:
F:Bp0!(Bp)0
hF( ); gi=M ~
~gg 8 ~
2 2 Bp0;8~g2g2 Bp;i 1<p<+1;
and
hF( ); gi=M T[ ]1(~
)~gg 8 ~
2 2 B1;8~g2g2 B1;i p= 1:
To  nish his sec ion, we ecall some esul s ela ed o he de i a ion heo y o
gene alized Besico i ch spaces (see Casado D´± az & Gay e 2002).
We s a by in oducing he space D1, which plays in he spaces Bp he same ole
as he spaces C10(RN) in he dis ibu ional heo y.
De¯ni ion 3.7. We de ne D1as
D1= ’2C1(RN) : D¬’2B1 L1(RN)8¬2(N[ 0g)Ng:
Reasoning by con olu ion (see Casado D´± az & Gay e 2002), we can show he
ollowing p oposi ion.
P oposi ion 3.8. The space D1is dense in Bp o 16p < +1.
Fo e e y 2B1, he e exis s a sequence ngin D1which con e ges o in
B1and is bounded in B1.
Analogously o dis ibu ional heo y, we use he spaces D1 o gi e a de ni ion o
he de i a i e in Bp.
De¯ni ion 3.9. Fo 2B1, we de ne he mean pa ial ide i a i e o , 1 6i6
N, and we deno e i by @i;m , as he linea applica ion o D1in Rgi en by
@i;m (’) = ¡M½ @’
@xi¾;8’2D1:(3.3)
We also de ne he mean g adien o 2B1, m , as m = (@1;m ; : : : ; @N;m ) and
he mean di e gence o F2(B1)N, di mF, as di mF=PN
i= 1 @i;mFi:Clea ly, hese
de ni ions can also be ex ended o B1.
The ollowing esul , which ela es he dis ibu ional de i a i e wi h he mean
de i a i e, is shown in Casado D´± az & Gay e (2002).
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2932 J. Casado-D¶³ az and I. Gay e
P oposi ion 3.10. I 2B1is such ha he e exis s i2 1; : : : ; Ngwi h
@ =@xi2B1, hen @i;m =@ =@xiin he ollowing sense:
h@i;m ; ’i=M½@
@xi
’¾;8’2D1:
The ollowing space plays a e y impo an ole in applica ions.
De¯ni ion 3.11. Fo 1 6p < +1we de ne
Wp= 2W1;p
loc (RN) : 2(Bp)N; M g= 0g
and
Wp= : 2Wpg:
Iden i ying an elemen o Wpwi h i s class in (Bp)N, Wpwill be conside ed as
a subspace o (Bp)N. Mo eo e , we iden i y w1; w22Wpi [ (w1¡w2)]p= 0, and
hen we can conside Wpas a no med space o he no m kwk= [ w]p.
The ollowing heo em gi es some in e es ing p ope ies o Wp(see Casado D´± az
& Gay e (2002) and Gay e Delgado (1998) o he p oo ).
Theo em 3.12. The subspace Wpis closed in (Bp)N(and hen Wpis Banach).
I 2Bp,16p < +1, is such ha m belongs o (Bp)N hen he e exis s g2Wp
such ha m = gin (Bp)N.
To ob ain u he p ope ies o Wp he algeb a mus sa is y ano he p ope y.
De¯ni ion 3.13. An algeb a Xis called e godic i o e e y 2B1such ha
[ ¡ (¢+s)]1= 0 o e e y s2RN(equi alen ly m = 0), we ha e [ ¡M g]1= 0.
P oposi ion 3.14. An algeb a is e godic i and only i
lim
R!+1·1
jBRjZBR
(x+y) dy¡M g¸p
= 0;8 2Bp;16p < +1:(3.4)
In he e godic algeb as we ha e he ollowing densi y esul .
Theo em 3.15. I he algeb a is e godic, hen D1is dense in Wp.
4. The wo-scale con e gence me hod
In his sec ion, we p esen he ex ension o he wo-scale con e gence heo y (see
Allai e 1992; Ngue seng 1990) o he gene alized Besico i ch spaces Bp ela i e o
an algeb a wi h mean alue X. We s a by gi ing he de ni ion o wo-scale con-
e gence.
De¯ni ion 4.1. Le «»RNbe open. We say ha a sequence u"g » L1
loc(«)
wo-scale con e ges o u2L1
loc(«;B1) i o e e y g2B1 L1(RN) and e e y
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Two-scale me hod applied o Besico i ch spaces 2933
E»» «bounded, measu able, we ha e
9lim
"!0Z«
u"(x)Áµx; x
"¶dx=Z«
My u(x; y)Á(x; y)gdx; (4.1)
whe e Á(x; y) = g(y)XE(x) o x2«,y2RN. We will deno e u"
2e
* u.
Rema k 4.2. The le -hand side o (4.1) makes no sense i gis only in B1since
wo ep esen a i es o gmay di¬e in e e y poin o RN. The igh -hand side does
no depend on he ep esen a i e o uchosen.
Rema k 4.3. Since B1can be iden i ed wi h he dual o B1, i is easy o deduce
ha he wo-scale limi , i ha exis s, is unique.
Rema k 4.4. Ou de ni ion o wo-scale con e gence can seem di¬e en o he
usual one o he pe iodic case, which es ablishes ha (4.1) holds o e e y Ápe iodic
in he second a iable and smoo h enough (in gene al, in he space o admissible
unc ions (see Allai e 1992)). As es ablished in p oposi ion 4.6, his is equi alen o
ou de ni ion, because i (4.1) holds o Áas in de ni ion 4.1, hen i holds o Áin all
o he spaces which appea wi h he usual de ni ion. We ha e chosen he de ni ion
gi en abo e because i makes i easie o check i a sequence wo-scale con e ges,
and when u"is bounded in Lp(«) o some p2(1;+1) (usual si ua ion), i does
no depend on p.
Al hough we ha e de ned he wo-scale con e gence me ely o a sequence u"gin
L1
loc(«), in he applica ions we will usually ha e a bounded sequence in Lp(«) o
some p2[1;+1]. In his case, we ha e he ollowing esul .
P oposi ion 4.5. Le u"gbe a bounded sequence in Lp(«) o some p2
[1;+1], which wo-scale con e ges o a unc ion u2L1
loc(«;B1). Then ubelongs
o Lp(«;Bp), he sequence u"gcon e ges weakly in Lp(«) (weakly-¤i p= +1)
o u0=My u(¢; y)gand we ha e
lim in
"!0ku"kLp(«)>kukLp(«;Bp)>ku0kLp(«):(4.2)
P oo . We deno e by S c(«;Bp0) he se o simple unc ions which ha e he sup-
po s ic ly included in «. Fo Á2S c(«;Bp0), jÁj6kÁkL1(«;B1)a.e. in «£RN
i p= 1, we ha e
¯¯¯¯Z«
u"(x)Áµx; x
"¶dx¯¯¯¯
6ku"kLp(«)°
°
°
°
Áµx; x
"¶°
°
°
°Lp0(«)
;
whe e passing o he limi when " ends o ze o, we deduce
¯¯¯¯Z«
My u(x; y)Á(x; y)gdx¯¯¯¯
6lim in
"!0ku"kLp(«)kÁkLp0(«;Bp0)8Á2S c(«;B1):
(4.3)
F om his inequali y and heo em 3.6 we easily deduce ha ubelongs o Lp(«;Bp)
and ha (4.2) holds. ¥
The ollowing esul , which is easy o p o e, ex ends de ni ion 4.1 o a wide class
o admissible unc ion Á.
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2940 J. Casado-D¶³ az and I. Gay e
By (4.18), we ha e
u"¡1
jB" jZB"
u"(¢+») d»!0 in Lp
loc(«) s ongly.
So, aking he limi in (4.20), and using (4.19) and lemma 4.12(b), we ge
Z«
My (x; y) Á(y)g’(x) dx
=¡Z«
My½µ¹(x; y)¡1
jB jZB
¹(x; y +») d»¶Á(y)¾’(x) dx;
o e e y ’2C10(«), Á2D1. This implies ha o a.e. x2«,
y;m (x; ¢) = ¹(x; ¢)¡1
jB jZB
¹(x; ¢+») d» ; (4.21)
and hen he igh -hand side o (4.21) belongs o Wp o a.e. x2«. Since i
con e ges a.e. in « o ¹(x; ¢)¡My ¹(x; ¢)gand since Wpis closed, we deduce ha
he e exis s u12Lp(«;Wp) such ha
¹¡My ¹g= yu1:
By p oposi ion 4.5, i is also clea ha My ¹g= u, which oge he wi h he abo e
equali y gi es (4.17). ¥
Analogously o p oposi ion 4.11, we ha e he ollowing esul , which implies ha
he egula i y o u1in heo em 4.13 is op imal a leas o 1 6p < +1.
P oposi ion 4.14. Le u2W1;p(«)and u12Lp(«;Wp), wi h 16p < +1.
Then he e exis s a bounded sequence u"ngin W1;p(«), such ha
u"n* u in W1;p(«)-weak;
u"n
2e
* u+ yu1:)(4.22)
P oo . By heo em 3.15 i is easy o check ha he e exis s a sequence Áng »
C1
0(«;C1(RN)) such ha yÁn2C0(«; (Bp)N),
yÁn! yu1in Lp(«; (Bp)N) (4.23)
and
lim
"!0µZ«¯¯¯¯
yÁnµx; x
"¶¡ yÁmµx; x
"¶¯¯¯¯
p
dx¶1=p
=µZ«
[ y(Án(x; y)¡Ám(x; y))]p
pdx¶1=p
<1
2m8n>m: (4.24)
Thus, o e e y n2N he e exis s a dec easing sequence "ng » (0;+1) such ha
µZ«¯¯¯¯
yÁnµx; x
"n¶¡ yÁmµx; x
"n¶¯¯¯¯
p
dx¶1=p
<1
2m8m; n 2N;16m6n;
(4.25)
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Two-scale me hod applied o Besico i ch spaces 2941
and
µZ«¯¯¯¯
"nÁnµx; x
"n¶¯¯¯¯
p
dx¶1=p
<1
2n;µZ«¯¯¯¯
"n xÁnµx; x
"n¶¯¯¯¯
p
dx¶1=p
<1
2n:
(4.26)
De ning
u"n(x) = u+"nÁnµx; x
"n¶
we deduce (4.14). ¥
5. Applica ions o homogeniza ion p oblems
Simila ly o he `classical’ wo-scale con e gence o pe iodic unc ions (see Allai e
1992; Ngue seng 1990), he main applica ion o he esul s ob ained in he p esen
pape is he homogeniza ion o pa ial di¬e en ial p oblems wi h coe¯ cien s in he
spaces Bp, gene a ed by an e godic algeb a. As an example, o 1 <p<+1, le us
conside he nonlinea p oblem
¡di (a(x="; u"; u")) = in W¡1;p0(«);
u"2W1;p
0(«);)(5.1)
whe e «»RNis a bounded open se , belongs o W¡1;p0(«) and a:RN£R£RN7!
RNsa is es
(1) Fo e e y (s; ¹ )2R£RN,a(¢; s; ¹ ) belongs o (Bp0)N. Fo a.e. x2RNand
e e y s2R,a(x; s; ¢) is con inuous.
(2) The e exis s ¬ > 0 such ha
a(x; s; ¹ )¢¹>¬j¹jp;8(s; ¹ )2R£RN;a.e. x2RN:(5.2)
(3) The e exis h2Bp0and  > 0 such ha
ja(x; s; ¹ )j6h(x) + (jsj+j¹j)p¡18(s; ¹ )2R£RN;a.e. x2RN:(5.3)
(4) We ha e
(a(x; s; ¹ 1)¡a(x; s; ¹ 2)) ¢(¹1¡¹2)>08s2R;8¹1; ¹ 22RN;a.e. x2RN:
(5.4)
(5) The e exis ® > 0, 0 < ¼ 6min p¡1;1gand k2Bp0such ha
ja(x; s1; ¹ )¡a(x; s2; ¹ )j6k(x)+®(j¹j+js1j+js2j)p¡1min js1¡s2j;1g¼;(5.5)
o e e y s1; s22R, e e y ¹2RNand a.e. x2RN.
The exis ence o a solu ion u"o (5.1) can be ound in Lions (1969). The nex
heo em gi es he asymp o ic beha iou o u".
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2942 J. Casado-D¶³ az and I. Gay e
Theo em 5.1. Assume he abo e hypo hesis and le u"be a solu ion o (5.1)
o e e y " > 0. The e hen exis a subsequence, s ill deno ed by "g, a unc ion
u2W1;p
0(«)and a unc ion u12Lp(«;Wp)such ha
u"* u in W1;p
0(«)-weak;
u"
2e
* u+ yu1;
whe e (u; u1)2W1;p
0(«)£Lp(«;Wp)is a solu ion o he wo-scale homogenized
sys em
¡di xMy a(y; u; u+ yu1)g= in W¡1;p0(«);
¡di m;y a(y; u; u+ yu1)g= 0 a.e. x2« : )(5.6)
Rema k 5.2. In gene al, he p oblem (5.6) does no ha e a unique solu ion, so
he con e gence o u"gis only o a subsequence. Assuming a u he hypo hesis,
o example, ha a(x; s; ¹ ) does no depend on sand ha in (5.4) he inequali y
is s ic o ¹16=¹2, we ha e he uniqueness o solu ion o he limi p oblem which
assu es ha he whole o he sequence u"gcon e ges.
Rema k 5.3. De ning b:R£RN!Rby
b(s; ¹ ) = My a(y; s; ¹ + y s;¹ )g 8s; ¹ 2R£RN;
wi h s;¹ he solu ion o
¡di m(a(y; s; ¹ + y s;¹ )) = 0 in (Wp)0;
s;¹ 2Wp;¾
he unc ion uin he s a emen o heo em 5.1 sa is es
¡di b(u; u) = in W¡1;p0(«);
u2W1;p
0(«):)
Rema k 5.4. In he pa icula case o a linea equa ion, a(x="; u"; u") =
A(x=") u"and A, a ma ix whose coe¯ cien s a e almos -pe iodic unc ions, he
p oblem has been s udied in Oleinik & Zhiko (1982).
In he case when he equa ion is mono one, wi h a(x="; u") almos -pe iodic in
he  s a iable, he homogeniza ion o (5.1) has been done in B aides e al. (1992),
using app oxima ion esul s in smoo he almos -pe iodic spaces. Co ec o esul s
o hese ope a o s a e p o ed in B aides (1991), exploi ing he geome ic p ope ies
o a.
P oo . By (5.2) he sequence u"gis bounded in W1;p
0(«), and hen by (5.3), he
sequence g"gde ned by
g"=aµx
"; u"; u"¶
is bounded in Lp0(«)N. Theo ems 4.8 and 4.13 hen imply ha he e exis s a sub-
sequence, s ill deno ed by "g, a unc ion u2W1;p
0(«), a unc ion u12Lp(«;Wp)
and a unc ion g02Lp0(«;Bp0)Nsuch ha
u"* u in W1;p
0(«);(5.7)
u"
2e
* u+ yu1;(5.8)
g"
2e
* g0:(5.9)
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Two-scale me hod applied o Besico i ch spaces 2943
Le ’2C10(«), Á2C10(«), 2D1, and conside ’(x) + "Á(x) (x=") as a es
unc ion in (5.1). This gi es
Z«
g"(x)µ ’(x) + " µx
"¶ Á(x) + Á(x) y µx
"¶¶dx=¿ ; ’(x) + "Á(x) µx
"¶À;
and hen, aking he limi when " ends o ze o, we deduce
Z«
My g0(x; y)( ’(x) + Á(x) y (y))gdx=h ; ’i;
o e e y ’,Á, as abo e. Reasoning by linea i y and densi y, g0sa is es
Z«
My g0(x; y)( (x) + y 1(x; y)gdx=h ; i;
8 2W1;p
0(«);8 12Lp(«;Wp);9
=
;
(5.10)
i.e.
¡di xMy g0(x; y)g= in W¡1;p0(«);
¡di m;y g0(x; y)g= 0;a.e. x2« : )(5.11)
Le us use he Min y ule o cha ac e ize g0. Fo ª ; © 2S c(«; (Bp)N), such ha
ª(x; ¢), ©(x; ¢) belong o D1 o a.e. x2«, and 2(0;1), we de ne
·"= u(x) + ªµx; x
"¶+ © µx; x
"¶;
which wo-scale con e ges o ·0de ned by
·0(x; y) = u(x) + ª(x; y) + © (x; y):(5.12)
By (5.4) we ha e
Z«µg"¡aµx
"; u"; · "¶¶¢( u"¡·") dx>0:(5.13)
Le us pass o he limi in he di¬e en e ms o his inequali y. Taking u"as a es
unc ion in (5.1) and uas a es unc ion in he  s equa ion o (5.11), we ge
lim
"!0Z«
g" u"dx= lim
"!0h ; u"i=h ; ui=Z«
My g0 ugdx: (5.14)
By (5.9), we ha e
lim
"!0Z«
g"·"dx=Z«
My g0·0gdx: (5.15)
The hypo hesis (5.5) o aimplies
¯¯¯¯
aµx
"; u"; · "¶¡aµx
"; u; · "¶¯¯¯¯
6kµx
"¶+®(j·"j+ju"j+juj)p¡1min ju"¡uj;1g¼:
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2944 J. Casado-D¶³ az and I. Gay e
Taking he powe p0and in eg a ing in «, we deduce
Z«¯¯¯¯
aµx
"; u"; · "¶¡aµx
"; u; · "¶¯¯¯¯
p0
dx
621=(p¡1) Z«
kµx
"¶p0
+®p03p¡1(j·"jp+ju"jp+jujp) min ju"¡uj;1g¼ p0dx:
(5.16)
Since he powe p0o kand he powe po j·"jand ju"jha e a mean alue,
he sequence k(x=")p0+®p03p¡1(j·"jp+ju"jp+jujp)gcon e ges weakly in L1(«).
Mo eo e , he sequence min ju"¡uj;1g¼ p0gis bounded in L1(«) and con e ges
a.e. in « o ze o. Thus (use Ego o ’s heo em), we conclude ha he igh -hand side
o (5.16) con e ges o ze o, and hen
aµx
"; u"; · "¶¡aµx
"; u; · "¶!0 in Lp0(«):(5.17)
So, by (5.8) we deduce
lim
"!0Z«
aµx
"; u"; · "¶( u"¡·") dx= lim
"!0Z«
aµx
"; u; · "¶( u"¡·") dx
=Z«
My a(y; u; · 0)(( u+ yu1)¡·0)gdx:
(5.18)
Thus, passing o he limi when " ends o ze o in (5.13) we conclude
Z«
My g0g udx¡Z«
My g0·0gdx¡Z«
My a(y; u; · 0)(( u+ yu1)¡·0)gdx>0:
Replacing ·0by (5.12) and using he second equa ion o (5.11) we ha e
Z«
My [g0¡a(y; u; · 0)][ª+ © ¡ yu1]gdx60:(5.19)
Choosing ªcon e ging o yu1in Lp(«;Bp)Nand ©con e ging o a unc ion W2
Lp(«; (Bp)N), we deduce
Z«
My [g0¡a(y; u; u+ yu1+ W )W]gdx608W2Lp(«; (Bp)N):
Di iding by and hen aking he limi when ends o ze o, we ge
Z«
My (a(y; u; u+ yu1)¡g0)Wgdx>08W2Lp(«; (Bp)N);
i.e. g0=a(y; u; u+ yu1), which by (5.11)  nishes he p oo o heo em 5.1. ¥
In o de o ob ain a co ec o esul , le us now assume ha ais uni o mly mono-
one, i.e. he e exis s ¯ > 0 such ha o e e y s2R,¹1; ¹ 22RNand a.e. x2«we
ha e
(a(x; s; ¹ 1)¡a(x; s; ¹ 2)) ¢(¹1¡¹2)>8
>
<
>
:
¯j¹1¡¹2jpi p>2;
¯j¹1¡¹2j2
(j¹1j+j¹2j)2¡pi 1 <p<2:(5.20)
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Two-scale me hod applied o Besico i ch spaces 2945
Theo em 5.5. Unde he hypo hesis o heo em 5.1 and (5.20), i u1is
smoo h enough ( o example, u12C1(·
«£RN)wi h xu12L1(«£RN), yu12
C(·
«;B1) L1(«£RN), hen
u"(¢)¡u(¢)¡"u1µ¢;¢
"¶!0in W1;p(«):(5.21)
P oo . Le Z"= u(x)+ yu1(x; x="). Reasoning simila ly o he p oo o (5.17),
we can p o e
aµx
"; u"; Z"¶¡aµx
"; u; Z"¶!0 in Lp0(«):
So,
lim
"!0Z«µaµx
"; u"; u"¶¡aµx
"; u"; Z"¶¶¢( u"¡Z") dx
= lim
"!0Z«µaµx
"; u"; u"¶¡aµx
"; u; Z"¶¶¢( u"¡Z") dx: (5.22)
In o de o pass o he limi on he igh -hand side o his equali y, i is enough o use
he ac ha a(x="; u; u") wo-scale con e ges o a(y; u; u+ yu1) (see he p oo
o heo em 5.1), he ac ha u" wo-scale con e ges o u+ yu1and (5.14),
which imply
lim
"!0Z«µaµx
"; u"; u"¶¡aµx
"; u; Z"¶¶¢( u"¡Z") = 0:(5.23)
F om (5.22), (5.23) and (5.20) we conclude ha u"¡Z"con e ges s ongly o ze o
in Lp(«) and hen (5.21) holds. ¥
This wo k has been pa ly suppo ed by P ojec PB98-1162 o he DGESIC o Spain.
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