A ailable online a www.sciencedi ec .com
Ann. I. H. Poinca é – AN 30 (2013) 519–545
www.else ie .com/loca e/anihpc
Homogeniza ion and co ec o s o mono one p oblems in cylinde s
o small diame e
Juan Casado-Díaz a,∗, F ançois Mu a b, Ali Sili c
aDp o. de Ecuaciones Di e enciales y Análisis Numé ico, Fac. de Ma emá icas C. Ta ia s/n, 41012 Se illa, Spain
bLabo a oi e Jacques-Louis Lions, Uni e si é Pie e e Ma ie Cu ie, Boî e cou ie 187, 75252 Pa is Cedex 05, F ance
cDépa emen de Ma héma iques, Uni e si é du Sud Toulon-Va , BP 20132, 83957 La Ga de Cedex, F ance
Recei ed 20 July 2010; ecei ed in e ised o m 19 Oc obe 2012; accep ed 19 Oc obe 2012
A ailable online 9 No embe 2012
Abs ac
In his pape we s udy he homogeniza ion o mono one di usion equa ions posed in an N-dimensional cylinde which con e ges
o a (one-dimensional) segmen line. In o he e ms, we pass o he limi in di usion mono one equa ions posed in a cylinde whose
diame e ends o ze o, when simul aneously he coe icien s o he equa ions (which a e no necessa ily pe iodic) a e also a ying.
We ob ain a limi sys em in bo h he mac oscopic (one-dimensional) a iable and he mic oscopic a iable. This sys em is nonlocal.
F om his sys em we ob ain by elimina ion an equa ion in he mac oscopic a iable which is local, bu in con as wi h usual esul s,
he ope a o depends on he igh -hand side o he equa ions. We also ob ain a co ec o esul , i.e. an app oxima ion o he g adien s
o he solu ions in he s ong opology o he space Lpin which he mono one ope a o s a e de ined.
Résumé
Dans ce a icle nous é udions l’homogénéisa ion d’équa ions de di usion mono ones posées dans un cylind e de dimension N
qui con e ge e s un segmen (qui es donc unidimensionnel). En d’au es e mes, nous passons à la limi e dans des équa ions de
di usion mono ones posées dans un cylind e don le diamè e end e s zé o, quand en même emps les coe icien s des équa ions
(qui ne son pas nécessai emen pé iodiques) a ien eux aussi. Nous ob enons un sys ème limi e en la a iable mac oscopique
(unidimensionnelle) e en la a iable mic oscopique. Ce sys ème es non local. A pa i de ce sys ème nous ob enons pa élimina ion
une équa ion en la a iable mac oscopique qui es locale, mais dans laquelle, à la di e ence des ésul a s usuels, l’opé a eu dépend
du second memb e des équa ions. Nous ob enons aussi un ésul a de co ec eu , c’es à di e une app oxima ion des g adien s des
solu ions dans la opologie o e de l’espace Lpdans lequel son dé inis les opé a eu s mono ones.
Keywo ds: Homogeniza ion; Thin domains; Mono one p oblems
*Co esponding au ho .
E-mail add esses: [email p o ec ed] (J. Casado-Díaz), [email p o ec ed] (F. Mu a ), [email p o ec ed] (A. Sili).
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©2012
L'Associa ion Publica ions de l'Ins i u Hen i Poinca é. Published by Else ie B.V. All igh s ese ed.
©2012
L'Associa ion Publica ions de l'Ins i u Hen i Poinca é. Published by Else ie B.V. All igh s ese ed.
©2012
520 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
1. In oduc ion
We conside in his pape he homogeniza ion, when he coe icien s a y, o mono one p oblems posed in a cylinde
o RNwi h ixed leng h and small diame e . Speci ically, we conside a bounded open in e al I⊂Rand a bounded
domain ω⊂RN−1. De ining he cylinde Ωεby Ωε=I×(εω), we a e in e es ed in he solu ions o he mono one
p oblem
⎧
⎪
⎨
⎪
⎩
−di ˜
Aε(˜x,∇˜uε)=˜
ε−di ˜
εin Ωε,
˜
Aε(˜x,∇˜uε)−˜
ε˜νε=0inI×(ε∂ω),
˜uε=0in∂I ×(εω),
(1.1)
whe e ˜
Aε:Ω×RN→RNa e Ca a héodo y unc ions which a e mono one, uni o mly p-coe ci e and wi h uni o m
(p −1)-g ow h, and whe e, o Ω=I×ω, he e exis ∈Lp(Ω) and F∈Lp(Ω)N, such ha
˜
ε˜x1,˜x= ˜x1,˜x
ε,˜
ε˜x1,˜x=F˜x1,˜x
ε,a.e. ˜x1,˜x∈Ωε.(1.2)
In his p oblem, he Neumann bounda y condi ion in he la e al bounda y I×(ε∂ω) is c ucial, while changing he
Di ichle bounda y condi ion on he bases ∂I ×(εω) (as a as an H1a p io i es ima e is conse ed) does no a ec
he limi equa ion.
A p oblem simila o (1.1), bu whe e he ope a o s a e linea , Fε≡0, N=3, and Ωεis a cylinde o ixed basis
ω⊂R2and small heigh (which he e o e con e ges o he wo-dimensional se ω) has been conside ed in [4] and [10]
(see also [9] o he elas ici y p oblem). In his case, he limi p oblem, which is posed in he wo-dimensional limi
domain ω, has a s uc u e which is simila o he s uc u e o he p oblem posed in Ωε. This will also be he case
o p oblem (1.1), whose limi is posed on he one-dimensional domain I, bu , in con as wi h usual esul s, he
co esponding ope a o will depend on F.
In o de o s udy he homogeniza ion o (1.1), we pe o m he change o a iables (x1,x)=(˜x1,˜x/ε), which
ans o ms Ωεin Ωas i is usual in he s udy o he beha io o solu ions o pa ial di e en ial p oblems posed in
hin domains (see e.g. [2,4,5,9–14,17–19,22,23,25]). De ining uεby uε(x1,x)=˜uε(x1,εx), p oblem (1.1) is hen
ans o med in o a new p oblem which can be w i en in he a ia ional o m:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
uε∈W1,p(Ω), uε=0on∂I ×ω,

Ω
Aε(x, Dεuε)Dε dx=
Ω
dx+
Ω
FDε dx,
∀ ∈W1,p(Ω), =0on∂I ×ω,
(1.3)
whe e Dεis he di e en ial ope a o Dε=(∂
∂x1,1
ε∇x)and whe e Aε:Ω×RN→RNa e Ca a héodo y unc ions
which, simila ly o ˜
Aε, a e uni o mly p-coe ci e and wi h uni o m (p −1)-g ow h. These condi ions on Aεimply
ha Dεuεis bounded in Lp(Ω)N. Thus (see e.g. [17]) he e exis u0∈W1,p
0(I) and u1∈Lp(I, W1,p(ω)/R)such
ha uεcon e ges weakly o u0in W1,p(Ω) and Dεuεcon e ges weakly o D0(u0,u
1)=(du0
dx1,∇xu1)in Lp(Ω)N.
When Aε=Ais ixed, i has been p o ed in [17] (see also [18,19] o he elas ici y p oblem) ha (u0,u
1)is he
solu ion o he ollowing p oblem
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
(u0,u
1)∈W1,p
0(I) ×LpI,W1,p(ω)/R,

Ω
Ax,D0(u0,u
1)D0( 0,
1)dx =
Ω
0dx +
Ω
FD0( 0,
1)dx,
∀( 0,
1)∈W1,p
0(I) ×LpI,W1,p(ω)/R,
(1.4)
whe e D0( 0,
1)=(d 0
dx1,∇x 1); in his p oblem bo h he mac oscopic and mic oscopic a iables x1and xappea .
One can hen wonde whe he , when Aεdepends on ε, he e exis a subsequence o ε, s ill deno ed by ε, and a
Ca a héodo y unc ion A:Ω×RN→RNsa is ying he same condi ions as Aε, such ha o e e y ∈Lp(Ω) and
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 521
e e y F∈Lp(Ω)N, he limi (u0,u
1)o he solu ions uεo (1.1) is he solu ion o (1.4). We show in he p esen
pape ha his is no he case. In con as we p o e (Theo em 3.1 below) ha he limi o (1.1) is he nonlocal p oblem
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
(u0,u
1)∈W1,p
0(I) ×LpI,W1,p(ω)/R,

Ω
Ax1,D
0(u0,u
1)(x1,.)
D0( 0,
1)dx =
Ω
0dx +
Ω
FD0( 0,
1)dx,
∀( 0,
1)∈W1,p
0(I) ×LpI,W1,p(ω)/R,
(1.5)
whe e Ais no mo e a Ca a héodo y unc ion A:Ω×RN→RN, bu a nonlocal Ca a héodo y ope a o A:I×
R×∇
W1,p(ω) →Lp(ω)N, which is measu able in he i s a iable and con inuous in he wo o he ones (whe e
∇W1,p(ω) deno es he space o he de i a i es ∇x o unc ions ∈W1,p(ω)), bu such ha o a.e. x1∈I, he
unc ion
Ax1,D
0(u0,u
1)(x1,.)
=Ax1,du0
dx1
(x1), ∇xu1(x1,.)
∈Lp(ω)N
a he poin x∈ωdepends no only on ∇xu1(x1,x)bu on all he poin s o ∇xu1(x1,z) o z∈ω. A simila e ec
has been ob ained in [6] o he homogeniza ion o ellip ic pe iodic equa ions o he ype −di A(x, x
ε)∇uε.
Elimina ing u1in unc ion o u0in he sys em (1.5), we ob ain a local equa ion o u0, namely
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
u0∈W1,p
0(I),

I
aFx1,du0
dx1d 0
dx1
dx1=
I
ω
dx
 0dx1+
I
ω
F1dxd 0
dx1
dx,
∀ 0∈W1,p
0(I),
(1.6)
whe e now aF:I×R→Ris a Ca a héodo y unc ion, bu which depends on he (N −1)-las en ies Fo he
igh -side Fo (1.3). Thus, p oblem (1.6) is no su icien o s udying he e ec o he igh -hand side Fon he
solu ions o (1.3), and we mus emain wi h (1.5) o his s udy.
In addi ion o sea ching o he limi p oblem o (1.1), we a e also in e es ed in he p esen pape in ob aining
a co ec o esul o (1.1). Ou main esul in his di ec ion essen ially es ablishes (see Theo em 3.8 below o he
p ecise o mula ion) he exis ence o a (sub-)sequence o nonlocal ope a o s Pε:I×R×∇
W1,p(ω) →Lp(ω)N
such ha Dεuε−Pε(x1,D
0(u0,u
1)) con e ges s ongly o ze o in Lp(Ω)N. Le us emphasize ha he e again he
co ec o Pεis nonlocal.
In Sec ion 4below we p o e ha when Aεdoes no depend on x1, he ope a o Aand he co ec o Pεa e ac ually
local (o he assump ions which also p o ide a local ope a o Acan be ound in [8,13,14]). In con as , we show in
Sec ion 5by means o wo (pe iodic in x1) examples ha e en i Aεdoes no depend on x, he ope a o Ais nonlocal.
Le us conclude his in oduc ion by poin ing now ha we conside in he p esen pape he case o cylinde s wi h
ixed leng h and small diame e wi h Neumann bounda y condi ion on he la e al bounda y. Analogous esul s can
be ob ained by he same p oo s in he case o cylinde s wi h ixed bases and small heigh wi h Neumann bounda y
condi ions on he wo bases.
2. No a ion and p elimina ies
We conside an in ege numbe N⩾2.
The ec o s xo RNwill be decomposed as x=(x1,x), wi h x1∈R,x∈RN−1.
The ec o s o RN−1will be conside ed as elemen s o RNby iden i ying x∈RN−1wi h (0,x)∈RN.
We de ine MNas he space o ma ices o o de N.
We deno e by e1∈RN he ec o (1,0).
The N-dimensional measu e o a se B⊂RNwill be deno ed by |B|, while he (N −1)-dimensional measu e o a
se D⊂RN−1will be deno ed by |D|N−1.
I Xis a no med space and Xi s dual, we deno e by x,x he duali y pai ing be ween x∈Xand x∈X.
522 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
Fo an open se Θ⊂Rmand a numbe q∈[1,+∞], we deno e by W1,q (Θ) he usual Sobole space. I Υis
a subse o he bounda y ∂Θ o Θ, we deno e by W1,p
Υ(Θ) he space o hose unc ions o W1,p(Θ) which anish
on Υ. When m=N−1, he space o he g adien s o he unc ions o W1,p(Θ) will be deno ed by ∇W1,p(Θ).
When p=2, we w i e H1(Θ) =W1,2(Θ),H1
Υ(θ) =W1,2
Υ(Θ),∇H1(Θ) =∇W1,2(Θ).
We use he index  o deno e pe iodici y, o example C∞
([0,1])is he space o he unc ions o C∞(R)which a e
pe iodic o pe iod 1.
Fo a bounded smoo h connec ed open se ω⊂RN−1and a bounded in e al I=]b,d[⊂R,wese Ω=I×ω,
Γ=({b}∪{d})×ω,Ωε=I×(εω),Γε=({b}∪{d})×(εω).
In wha ollows, we conside a sequence o Ca a héodo y unc ions Aε:Ω×RN→RN. We de ine ˆ
Eε:Ω×
RN×RN→Rand ˇ
Eε:Ω×RN→R(whe e E e e s o ene gy) by
ˆ
Eε(x,ξ,ζ)=Aε(x, ξ) −Aε(x, ζ )(ξ −ζ), ˇ
Eε(x, ξ ) =Aε(x, ξ )ξ,
o e e y ξ,ζ ∈RN,a.e.x∈Ω. We will assume ha he e exis p∈(1,+∞),α>0,β >0, σ∈(0,min{1,p−1})
and h1,h
2∈L1(Ω),h1,h
2⩾0, such ha o e e y ξ,ζ ∈RNand a.e. x∈Ω,weha e
Aε(x, 0)=0,(2.1)
α|ξ−ζ|p⩽ˆ
Eε(x,ξ,ζ), i p∈[2,+∞), (2.2)
α|ξ−ζ|p⩽ˆ
Eε(x,ξ,ζ)p
2h1+ˇ
Eε(x, ξ ) +ˇ
Eε(x, ζ )2−p
2,i p∈(1,2],(2.3)
Aε(x, ξ) −Aε(x, ζ )
p⩽βh2+ˇ
Eε(x, ξ ) +ˇ
Eε(x, ζ )p−1−σˆ
Eε(x,ξ,ζ)
σ.(2.4)
Rema k 2.1. Using he ac ha (2.1), and (2.2)o (2.3) imply ha
α|ξ|p⩽ˇ
Eε(x, ξ ), i p∈[2,+∞), and α|ξ|p⩽h1+ˇ
Eε(x, ξ ), i p∈(1,2],
and he ac ha (2.1) and (2.4) imply ha he e exis β∗>0 and h∗∈L1(Ω),h∗⩾0, such ha
ˇ
Eε(x, ξ ) ⩽β∗|ξ|p+h∗,
one can p o e he ollowing equi alences:
In he case p∈[2,+∞),i Aεsa is ies (2.1), (2.2), and (2.4), hen he e exis ¯
β>0 and ¯
h2∈L1(Ω),¯
h2⩾0, such
ha o e e y ξ,ζ ∈RNand a.e. x∈Ω
Aε(x, ξ) −Aε(x, ζ )⩽¯
β¯
h2+|ξ|p+|ζ|pp−1−σ
p−σ|ξ−ζ|σ
p−σ.(2.5)
Recip ocally, i Aεsa is ies (2.1), (2.2), and i he e exis ¯
β>0, σ∈(0,1)and ¯
h2∈L1(Ω),¯
h2⩾0, such ha o
e e y ξ,ζ ∈RNand a.e. x∈Ω
Aε(x, ξ) −Aε(x, ζ )⩽¯
β¯
h2+|ξ|p+|ζ|pp−1−σ
p|ξ−ζ|σ,(2.6)
hen he e exis β>0 and h2∈L1(Ω),h2⩾0, such ha Aεsa is ies (2.4).
In he case p∈(1,2],i Aεsa is ies (2.1), (2.3), and (2.4), hen he e exis ¯α>0, ¯
β>0 and ¯
h1,¯
h2∈L1(Ω),
¯
h1,¯
h2⩾0, such ha Aεsa is ies (2.5) and is such ha o e e y ξ,ζ ∈RNand a.e. x∈Ω
¯α|ξ−ζ|2⩽ˆ
Eε(x,ξ,ζ)
¯
h1+|ξ|p+|ζ|p2−p
p.(2.7)
Recip ocally, i Aεsa is ies (2.1), (2.6), and (2.7) o some ¯α>0, ¯
β>0, σ∈(0,p −1), and ¯
h1,¯
h2∈L1(Ω),
¯
h1,¯
h2⩾0, hen he e exis α>0, β>0 and h
1,h

2∈L1(Ω),h
1,h

2⩾0, such ha o e e y ξ,ζ ∈RNand a.e.
x∈Ω
α|ξ−ζ|p⩽ˆ
Eε(x,ξ,ζ)p
2h
1+ˇ
Eε(x, ξ ) +ˇ
Eε(x, ζ )2−p
2,i p∈(1,2],(2.8)
Aε(x, ξ) −Aε(x, ζ )
p⩽βh
2+ˇ
Eε(x, ξ ) +ˇ
Eε(x, ζ )2(p−1)−pσ
2ˆ
Eε(x,ξ,ζ)pσ
2.(2.9)
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 523
Rema k 2.2. As a consequence o Rema k 2.1 we ge ha he class o Ca a héodo y unc ions sa is ying (2.1), (2.2)
o (2.3), and (2.4) is no emp y. Indeed he unc ion de ined by Aε(x, ξ) =aε(x)|ξ|p−2ξ, wi h p∈(1,+∞)and
aε∈L∞(Ω) such ha 0 <ˇα⩽aε(x) ⩽ˇ
β<+∞, sa is ies (2.1), (2.2)o (2.3), and (2.4) o someα>0, β>0,
h=0 and σ=1 o p⩾2, σ=p(p −1)/2 o 1<p<2.
Classes o Ca a héodo y unc ions sa is ying assump ions sligh ly mo e gene al han (2.1), (2.2)o (2.3), and (2.4)
ha e been in oduced in Sec ion 7 o [7], whe e obse a ions simila o he ones made in he abo e Rema ks 2.2 and
2.1 can also be ound.
As i was done in [7], we p e e he e o impose (2.1), (2.2)o (2.3), and (2.4) in place o he mo e classical assump-
ions (2.1), (2.2)o (2.7), and (2.6), because he assump ions w i en in he i s o m a e s able by homogeniza ion
(see Theo em 3.1 below).
We deno e by Dε:W1,p(Ω) →Lp(Ω)Nand D0:W1,p(I) ×Lp(I, W 1,p(ω)) →Lp(Ω)N, he di e en ial ope -
a o s de ined by
Dεu=∂1ue1+1
ε∇xu, ∀u∈W1,p(Ω), (2.10)
D0(u0,u
1)=du0
dx1
e1+∇
xu1,∀(u0,u
1)∈W1,p(I) ×LpI,W1,p(ω).(2.11)
We deno e by Ca gene ic posi i e cons an , which only depends on p,N,α,β,σ,h1,h2,|ω|and |I|and can
change om a line o ano he one.
Ou aim is o s udy he asymp o ic beha io o he solu ions uεo
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
uε∈W1,p
Γ(Ω),

Ω
Aε(x, Dεuε)Dε dx=
Ω
dx+
Ω
FDε dx,
∀ ∈W1,p
Γ(Ω),
(2.12)
whe e ∈Lp(Ω),F∈Lp(Ω)N(we will see la e ha he bounda y condi ion uε∈W1,p
Γ(Ω) is no e y impo an ).
As we al eady said in he In oduc ion, p oblem (2.12) is equi alen o (1.1) wi h ˜
εand ˜
εgi en by (1.2).
Taking uεas es unc ion in (2.12) and using Poinca é’s inequali y, we deduce ha he solu ions uεo (2.12) sa is y

Ω
|Dεuε|pdx ⩽C p
Lp(Ω) +Fp
Lp(Ω)N.
In wha ollows, we will use he ollowing lemma (see [17]).
Theo em 2.3. I uεis a sequence in W1,p(Ω) such ha

Ω
|Dεuε|pdx ⩽C,
hen he e exis u0∈W1,p(I),u1∈Lp(I, W 1,p(ω)) and a subsequence o uε(s ill deno ed by uε)such ha
uεu
0in W1,p(Ω), (2.13)
DεuεD
0(u0,u
1)in Lp(Ω)N.(2.14)
3. Homogeniza ion
In his sec ion we pe o m he homogeniza ion o (2.12). The main esul o he p esen pape is con ained in he
ollowing heo em which desc ibes he asymp o ic beha io o he solu ions o (2.12).

524 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
Theo em 3.1. The e exis an ope a o A:I×R×∇W1,p(ω) →Lp(ω)Nand a subsequence o ε, s ill deno ed by ε,
such ha o e e y uε∈W1,p(Ω),u0∈W1,p(I),u1∈Lp(I, W1,p(ω)), ε∈Lp(Ω),Fε∈Lp(Ω)N, ∈Lp(Ω)
and F∈Lp(Ω) which sa is y
uεu
0in W1,p(Ω), (3.1)
1
ε∇xuε∇xu1in Lp(Ω)N−1,(3.2)
ε in Lp(Ω), (3.3)
Fε→Fin Lp(Ω)N,(3.4)

Ω
Aε(x, Dεuε)Dε dx=
Ω
ε dx+
Ω
FεDε dx, ∀ ∈W1,p
Γ(Ω), (3.5)
we ha e
Aε(x, Dεuε)Ax1,D
0(u0,u
1)in Lp(Ω)N.(3.6)
Mo eo e , he unc ions u0,u1, and Fa e ela ed by

Ω
Ax1,D
0(u0,u
1)D0( 0,
1)dx =
Ω
0dx +
Ω
FD0( 0,
1)dx,
∀( 0,
1)∈W1,p
0(I) ×LpI,W1,p(ω).(3.7)
The ope a o Aalso sa is ies he ollowing p ope ies:
The applica ion x1→A(x1,s,∇xψ)∈Lp(ω)Nis measu able ∀(s, ψ) ∈R×W1,p(ω). (3.8)
A(., 0,0)=0a.e. in I. (3.9)
Deno ing by ˆ
E:I×R×∇
W1,p(ω) ×R×∇
W1,p(ω) →L1(ω)N,ˇ
E:I×R×∇
W1,p(ω) →L1(ω)N, he
ope a o s
ˆ
E(x1,s
1,∇xψ1,s
2,∇xψ2)=A(x1,s
1,∇xψ1)−A(x1,s
2,∇xψ2)(s1−s2)e1+∇
x(ψ1−ψ2),
∀(s1,ψ
1), (s2,ψ
2)∈R×W1,p(ω), a.e. x1∈I, (3.10)
ˇ
E(x1,s,∇xψ)=A(x1,s,∇xψ)(se1+∇
xψ), ∀(s, ψ) ∈R×W1,p(ω), a.e. x1∈I, (3.11)
we ha e o e e y s1,s
2∈R,ψ1,ψ
2∈W1,p(ω) and a.e. x1∈I
α
{x1}×ω|s1−s2|p+∇x(ψ1−ψ2)
pdx
⩽
{x1}×ω
ˆ
E(x1,s
1,∇xψ1,s
2,∇xψ2)dx,i p∈[2,+∞), (3.12)
α
{x1}×ω|s1−s2|p+∇x(ψ1−ψ2)
pdx
⩽
{x1}×ω
ˆ
E(x1,s
1,∇xψ1,s
2,∇xψ2)dxp
2
·
{x1}×ωh1+ˇ
E(x1,s
1,∇xψ1)+ˇ
E(x1,s
2,∇xψ2)dx2−p
2
,i p∈(1,2],(3.13)
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 525

{x1}×ωA(x1,s
1,∇xψ1)−A(x1,s
2,∇xψ2)
p
dx
⩽β
{x1}×ωh2+ˇ
E(x1,s
1,∇xψ1)+ˇ
E(x1,s
2,∇xψ2)dxp−1−σ
p−1
·
{x1}×ω
ˆ
E(x1,s
1,∇xψ1,s
2,∇xψ2)dxσ
p−1
.(3.14)
Rema k 3.2. The p ope ies (3.9), (3.12), (3.13) and (3.14) imply he exis ence o h0∈L1(I) and C>0, such ha
o e e y s1,s
2∈R,ψ1,ψ
2∈W1,p(ω) and a.e. x1∈I, he ope a o Asa is ies

{x1}×ωA(x1,s
1,∇xψ1)
p
dx⩽h0+C|s1|p+
{x1}×ω
|∇xψ1|pdx,(3.15)

{x1}×ωA(x1,s
1,∇xψ1)−A(x1,s
2,∇xψ2)
p
dx
⩽h0+C|s1|+|s2|p+
{x1}×ω|∇xψ1|+|∇
xψ2|pdxp(p−1−σ)
(p−1)(p−σ)
·|s1−s2|p+
{x1}×ω∇x(ψ1−ψ2)
pdxσ
(p−1)(p−σ)
.(3.16)
Rema k 3.3. Thanks o (3.8), (3.9) and (3.14), we deduce ha o e e y φ1∈Lp(I) and e e y φ∈Lp(I;W1,p(ω)),
he unc ion x1∈I→ A(x1,φ
1(x1), ∇xφ(x1, .))(x)is in Lp(I;Lp(ω))N. Thus, he e m A(x1,D
0(u0,u
1)) which
appea s in (3.6), (3.7) has a meaning as a unc ion o Lp(Ω)N.
Rema k 3.4. Obse e ha in Theo em 3.1 he sequence uεis no supposed o anish on Γ. Thus, he bounda y con-
di ion in Γis no impo an in he homogeniza ion esul , o example, i can be subs i u ed by a Neumann condi ion.
Elimina ing u1 om (3.7), Theo em 3.1 gi es in pa icula he p oblem sa is ied by he limi uo he sequence uε
o solu ions o (2.12). This is gi en by
Co olla y 3.5. We conside he subsequence o εand he ope a o A=(A1,A)gi en by Theo em 3.1. We de ine he
ope a o R:Lp(ω)N−1→(W1,p(ω)/R)by
RG,
1=
ω
G∇x 1dx,∀G∈Lp(ω)N−1,∀ 1∈W1,p(ω)/R.
We also in oduce U:I×R×(W 1,p(ω)/R)→W1,p(ω)/Rand a:I×R×(W1,p(ω)/R)→Rby

ω
Ax1,s,∇xU(x1,s,η)
∇x 1dx=η, 1,∀ 1∈W1,p(ω), a.e. x1∈I,
a(x1,s,η)=A1x1,s,∇xU(x1,s,η)
,∀(s, η) ∈R×W1,p(ω)/R,a.e. x1∈I.
Then, i uε,u0,u1, ε,Fε, and Fa e as in he s a emen o Theo em 3.1, he unc ion u0sa is ies he equa ion
−d
dx1
ax1,du0
dx1
,RF(x1,.)
=
ω x1,x−∂1F1x1,xdxin I.
526 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
Rema k 3.6. Co olla y 3.5 shows ha o η∈Lp(I, (W 1,p(ω)/R)) ixed, and F∈Lp(Ω)N−1such ha RF=η
in W1,p(ω)/R, o a.e. x1∈I, he limi p oblem o (2.12) is local in x1. In pa icula , de ining a0:I×R→Rby
a0(x1,s)=a(x1,s,0), ∀s∈R,a.e. x1∈R,
Co olla y 3.5 shows ha o e e y ,F1∈Lp(Ω), he solu ion uεo
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
uε∈W1,p
Γ(Ω),

Ω
Aε(Dεuε)Dε dx=
Ω
dx+
Ω
F1∂1 dx,
∀ ∈W1,p
Γ(Ω),
con e ges weakly in W1,p(Ω) o he unique solu ion u0o
−d
dx1
a0x1,du0
dx1=
ω x1,x−∂1F1x1,xdxin I, u0∈W1,p
0(I).
This is simila o he homogeniza ion esul gi en in [10] o he case o a pla e.
In addi ion o Theo em 3.1, we also ha e a co ec o esul o he sequence o solu ions uεo (2.12). This is gi en
by Theo em 3.8 below, i s we need o gi e he ollowing de ini ion.
De ini ion 3.7. We conside he subsequence o εand he ope a o Agi en by Theo em 3.1. Fo e e y (s, ψ) ∈
R×W1,p(ω) and a.e. x1∈I, we de ine Wε(x1,s,∇xψ) as he solu ion o
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Wε(x1,s,∇xψ)∈W1,p(Ω)/R,

Ω
Aεx1,D
εWε(x1,s,∇xψ)Dε dx=
Ω
A(x1,s,∇xψ)Dε dx,
∀ ∈W1,p(Ω)/R.
(3.17)
We hen de ine Pε:I×R×∇W1,p(ω) →Lp(ω)Nby
Pε(x1,s,∇xψ)=DεWε(x1,s,∇xψ).
Theo em 3.8. We conside he subsequence o εand he ope a o Agi en by Theo em 3.1. Then, he e exis a cons an
C>0and a unc ion h0∈L1(I), such ha o uε,u0,u1, ε,Fε, and Fas in he s a emen o Theo em 3.1 and
o e e y s ep unc ion Ψ=m
j=1(sje1+∇
xψj)χ(ij−1,ij)×ω, wi h sj∈R,ψj∈W1,p(ω),1⩽j⩽m,b<i
0<···<
im<d, we ha e
limsup
ε→0
(i0,im)×ωDεuε−Pε(x1,Ψ)

pdx
⩽im

i0h0+C
du0
dx1
+|Ψ1|p
+C
ω|∇xu1|+Ψpdxdx1q
·
(i0,im)×ωD0(u0,u
1)−Ψ
pdx1−q
,(3.18)
wi h q=(p −1−σ)/(p−σ)i p∈[2,+∞),q=(p −2σ)/(2(p −σ)) i p∈(1,2].
Mo eo e , i uε=0on Γo i (3.5)holds o e e y ∈W1,p
Γ(Ω), hen we can ake i0=b,im=d.
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 527
Rema k 3.9. The meaning o Theo em 3.8 is ha aking Ψclose enough o D0(u0,u
1),Pε(x1,Ψ)is a good app ox-
ima ion o Dεuεin he s ong opology o Lp(Ω)N(co ec o esul ). In ac , i we o mally ake Ψ=D0(u0,u
1)
in (3.18), we will deduce
Dεuε−Pεx1,D
0(u0,u
1)→0inLp(Ω)N.
Howe e , we do no know i Pεis a Ca a héodo y unc ion and hus Pε(x1,D
0(u0,u
1)) is no well de ined.
P oo o he esul s o Sec ion 3
The p oo o ou esul s is an adap a ion o L. Ta a ’s me hod (see [20,24]). We s a wi h he ollowing esul .
Lemma 3.10. We conside uε,w
ε∈W1,p(Ω), ε,g
ε∈Lp(Ω),Fε,G
ε∈Lp(Ω)N, which sa is y

Ω
Aε(x, Dεuε)Dε dx=
Ω
ε dx+
Ω
FεDε dx, ∀ ∈W1,p
Γ(Ω), (3.19)

Ω
Aε(x, Dεwε)Dε dx=
Ω
gε dx+
Ω
GεDε dx, ∀ ∈W1,p
Γ(Ω). (3.20)
We assume ha he e exis u0,w
0∈W1,p(I),u1,w
1∈Lp(I, W1,p(ω)),T,S ∈Lp(Ω)N, ,g ∈Lp(Ω),F,G ∈
Lp(Ω)N, such ha
uεu
0,w
εw
0in W1,p(Ω), (3.21)
1
ε∇xuε∇xu1,1
ε∇xwε∇xw1in Lp(Ω)N−1,(3.22)
Aε(x, Dεuε)T, A
ε(x, Dεwε)S in Lp(Ω)N,(3.23)
ε , g
εg in Lp(Ω), (3.24)
Fε→F, Gε→Gin Lp(Ω)N.(3.25)
Then, T,Ssa is y he ollowing p ope ies
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩

Ω
TD
0( 0,
1)dx =
Ω
0dx +
Ω
FD0( 0,
1)dx,

Ω
SD0( 0,
1)dx =
Ω
g 0dx +
Ω
GD0( 0,
1)dx,
∀( 0,
1)∈W1,p
0(I) ×LpI,W1,p(ω).
(3.26)
Fo a.e. x1∈I, we ha e
α
{x1}×ωD0(u0−w0,u
1−w1)
pdx⩽
{x1}×ω
(T −S)D0(u0−w0,u
1−w1)dx,i p∈[2,+∞), (3.27)
α
{x1}×ωD0(u0−w0,u
1−w1)
pdx
⩽
{x1}×ω
(T −S)D0(u0−w0,u
1−w1)dxp
2
·
{x1}×ωh1+TD
0(u0,u
1)+SD0(w0,w
1)dx2−p
2
,i p∈(1,2],(3.28)
534 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
De ini ion 4.2. We assume ha he unc ions Aεdo no depend on x1and we conside he subsequence o εand he
unc ion Agi en by Lemma 4.5. We de ine P
ε:ω×RN→RN−1by
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
P
ε(., ξ) ∈∇W1,p(ω),

ω
A
εx,ξ
1+P
εx,ξ∇xψdx
=
ω
Ax,ξ∇xψdx
,
∀ψ∈W1,p(ω),
(4.8)
o e e y ξ∈RN.
Theo em 4.3. Unde he assump ions o De ini ion 4.2, he e exis a cons an C>0and a unc ion h0∈L1(I) such
ha o uε,u0,u1, ε,Fε, and Fas in he s a emen o Theo em 3.1 and o e e y s ep unc ion Φ=m
j=1(sje1+
n
l=1ηjlχKl)χ(ij−1,ij), wi h sj∈R,ηjl ∈RN−1,b<i
0<···<i
m<d,Kl⊂¯ωcompac , |Kl1∩Kl2|N−1=0i
l1= l2,¯ω=n
l=1Kl, we ha e
lim
ε→0
(i0,im)×ω
∂1uε−du0
dx1
p
dx =0,(4.9)
limsup
ε→0
(i0,im)×ω
1
ε∇xuε−P
εx,Φ
p
dx
⩽
(i0,im)×ωhc+C
du0
dx1
+|∇
xu1|+|Φ|pdxq
·
(i0,im)×ω
du0
dx1
−Φ1
p
+∇xu1−Φ
pdx1−q
,(4.10)
wi h q=(p −1−σ)/(p−σ)i p∈[2,+∞),q=(p −2σ)/(2(p −σ)) i p∈(1,2].
P oo o he esul s o Sec ion 4
We s a wi h he ollowing lemma which can be p o ed easoning simila ly o Lemma 3.10.
Lemma 4.4. We assume ha he unc ions Aε(and hen h)do no depend on x1. We conside ψε,η
ε∈W1,p(ω),
s1,s
2∈R,F,G
∈Lp(ω)N−1, which sa is y

ω
A
εx,s
1e1+∇
xψε∇x dx=
ω
F∇x dx,∀ ∈W1,p(ω), (4.11)

ω
A
εx,s
2e1+∇
xηε∇x dx=
ω
G∇x dx,∀ ∈W1,p(ω). (4.12)
We assume he e exis ψ,η ∈W1,p(ω),T=(T1,T), S =(S1,S)∈Lp(ω)N, such ha
∇xψε∇xψ, ∇xηε∇xηin Lp(ω)N−1,(4.13)
Aεx,s
1e1+∇
xψεT, A
εx,s
2e1+∇
xηεS in Lp(ω)N.(4.14)
Then, T,Ssa is y

ω
T∇x dx=
ω
F∇x dx,
ω
S∇x dx=
ω
G∇x dx,∀ ∈W1,p(ω). (4.15)
The unc ions Tand Ssa is y he ollowing inequali ies a.e. in ω

J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 535
|T−S|p⩽βh2+T(s
1e1+∇
xψ)+S(s2e1+∇
xη)p−1−σ
p−1
·(T −S)(s1−s2)e1+∇
x(ψ −η)
σ
p−1,(4.16)
α|s1−s2|p+∇x(ψ −η)
p⩽(T −S)(s1−s2)e1+∇
x(ψ −η),(4.17)
i p∈[2,∞), and
α|s1−s2|p+∇x(ψ −η)
p⩽(T −S)(s1−s2)e1+∇
x(ψ −η)p
2
·h1+T(s
1e1+∇
xψ)+S(s2e1+∇
xη)2−p
2,(4.18)
i p∈(1,2].
Mo eo e , o e e y ϑ∈W1,∞(¯ω), we ha e
lim
ε→0
ω
ˆ
Eεx,s
1e1+∇
xψε,s
2e1+∇
xηεϑdx
=
ω
(T −S)(s1−s2)e1+∇
x(ψ −η)ϑdx. (4.19)
Using his lemma we can also p o e he ollowing esul easoning simila ly o he p oo o Theo em 3.1.
Lemma 4.5. We assume ha he unc ions Aεdo no depend on x1. Then, he e exis a subsequence o ε, s ill deno ed
by ε, and Ca a héodo y unc ion A:ω×RN→RNsa is ying (4.4), (4.7)and (4.5)o (4.6)depending i p∈[2,+∞)
o p∈(1,2], such ha o e e y (s, F )∈R×Lp(ω)N−1, he sequence ψε∈W1,p(ω)/Ro solu ions o he Neumann
p oblems

ω
A
εx,se
1+∇
xψε∇x 1dx=
ω
F∇x 1dx,∀ 1∈W1,p(ω)/R,(4.20)
con e ges weakly in W1,p(ω)/R o he solu ion ψo

ω
Ax,se
1+∇
xψ∇x 1dx=
ω
F∇x 1dx,∀ 1∈W1,p(ω)/R,(4.21)
and sa is ies
Aεx,se
1+∇
xψεA
x,se
1+∇
xψin Lp(ω)N.(4.22)
Rema k 4.6. Fo e e y s∈R, he unc ion A
s:ω×RN−1→RN−1de ined by
A
sx,η=Ax,se
1+η,∀η∈RN−1,a.e. x∈ω,
is he H-limi (see [20]) o he sequence (A
ε)s:ω×RN−1→RN−1de ined as
A
εsx,η=A
εx,se
1+η,∀η∈RN−1,a.e. x∈ω.
P oo o Theo em 4.1.We conside he subsequence o εgi en by Lemma 4.5, ex ac ing a subsequence i necessa y,
we can assume ha Theo em 3.1 holds. Fo s∈R,we akeφs(x1)=sx1. Then, o ψ∈W1,p(ω), we de ine uε∈
W1,p(Ω) as uε(x) =φs(x1)+εψε(x),a.e.inω, wi h ψε∈W1,p(ω)/R he solu ion o

ω
A
εx,se
1+∇
xψε∇x 1dx=
ω
Ax,se
1+∇
xψ∇x 1dx,∀ 1∈W1,p(ω)/R.(4.23)
By Lemma 4.5,Dεuε=se1+∇
xψεcon e ges weakly in Lp(ω)N o D0(φs(x1), ψ) =se1+∇
xψand
Aεx,se
1+∇
xψεA
x,D
0(φs,ψ)
in Lp(ω). (4.24)
Mo eo e , we ha e

Ω
Aεx,D
εuεDε dx=
Ω
FDε dx,∀ ∈W1,p
Γ(Ω),
536 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
wi h F1=0 and F=A(se1+∇
xψ). Then, om Theo em 3.1, we also deduce
Aεx,D
εuεAx1,D
0(φs,ψ)
.
By (4.24), we ge
Ax,se
1+∇
xψ=A(x1,s,∇xψ),
o e e y (s, ψ) ∈R×W1,p(Ω),a.e.inΩ. This p o es (4.1). Since his equali y de ines he ope a o A, we deduce
ha he sequence gi en in Theo em 3.1 can be aken as he subsequence gi en in Lemma 4.5, wi hou ex ac ing any
subsequence. 2
P oo o Theo em 4.3.Le us only p o e he case p∈[2,+∞), he case p∈(1,2]is analogous.
Fo s∈Rand ψ∈W1,p(ω)/R, we de ine ψε∈W1,p(ω)/Ras he solu ion o (4.23). By De ini ion 3.7 o Pε,i is
hen easy o check ha
Pε(x1,s,ψ
ε)−se1−∇
xψε→0inLp(Ω). (4.25)
On he o he hand, o ξ∈RN,K⊂¯ωcompac and ϑ∈W1,p(ω), wi h ϑ⩾χK, asse ion (4.19) wi h s1=s,s2=ξ,
∇xηε=P
ε(., ξ),T=A(x,se
1+∇
xψ),S=A(x,ξ) and he p ope ies o Aεand Agi e he exis ence o C>0
and h0∈L1(ω) such ha
limsup
ε→0
ω∇xψε−P
εx,ξ
pϑdx

⩽
ωh0+C|s|+|ξ|+|∇
xψ|pp−1−σ
p−σ|s−ξ|p+∇xψ−ξ
p1
p−σϑdx
.
I ϑdec eases o χKwe ge
limsup
ε→0
K∇xψε−P
εx,ξ
pϑdx

⩽C
Kh0+C|s|+|ξ|+|∇
xψ|pp−1−σ
p−σ|s−ξ|p+∇xψ−ξ
p1
p−σdx.(4.26)
We now conside Φ=m
j=1(sje1+n
l=1ηjlχKl)χ(ij−1,ij)as in he s a emen o Theo em 4.3 and Ψ=m
j=1(sje1+
∇xψj)χ(pj−1,pj), wi h ψ1,...,ψ
m∈W1,p(ω).F om(4.26) and Holde ’s inequali y, we easily ge
limsup
ε→0
Ic×ωPε(x1,Ψ)−Φ1e1−P
εx,Φ
pdx
⩽
m

j=1
n

l=1
limsup
ε→0
(ij−1,ij)×KlPε(x1,s
j,ψ
j)−sje1−P
εx,η
jl
pdx
⩽
Ic×ωh0+C|E|+|Ψ|pdxp−1−σ
p−σ
Ic×ω
|E−Ψ|pdx1
p−σ
.
F om (3.18) we hen deduce
limsup
ε→0
Dεuε−Φ1e1−P
εx,Φ
Lp(Ic×ω)N
⩽limsup
ε→0
Dεuε−Pε(x1,Ψ)

Lp(Ic×ω)N+limsup
ε→0
Pε(x1,Ψ)−Φ1e1−P
εx,Φ
Lp(Ic×ω)N
⩽
Ich0+C
du0
dx1
+|Ψ1|p
+C
ω|∇xu1|+Ψpdxdx1p−1−σ
p(p−σ)
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 537
·
Ic×ωD0(u0,u
1)−Ψ
pdx1
p(p−σ)
+
Ic×ωh0+C|Φ|+|Ψ|pdxp−1−σ
p(p−σ) 
Ic×ω
|Φ−Ψ|pdx1
p(p−σ)
.
Taking in his inequali y Ψcon e ging o D0(u0,u
1)in Lp(Ic×ω) we deduce
limsup
ε→0
Dεuε−Φ1e1−P
εx,Φ
Lp(Ic×ω)N
⩽
Ic×ωh0+CD0(u0,u
1)+|Φ|pdxp−1−σ
p(p−σ) 
Ic×ωD0(u0,u
1)−E
pdx1
p(p−σ)
.
This p o es (4.10). To ob ain (4.9), i is enough o use
limsup
ε→0



∂1uε−du0
dx1


Lp(Ic×ω)
⩽limsup
ε→0
Dεuε−Φ1e1−P
εx,Φ
Lp(Ic×ω)N+limsup
ε→0



du0
dx1
−Φ1


Lp(Ic×ω)
,
and hen o use he p e ious inequali y wi h Econ e ging o D0(u0,u
1).2
5. Some examples wi h nonlocal limi
In he p e ious sec ion, we ha e shown ha i he unc ions Aεdo no depend on x1, he limi p oblem o (2.12)is
local. We show he e ha his asse ion is no ue when Aεdepends on x1e en, i hey do no depend on x. Fo his
pu pose, we conside a unc ion A∈L∞
(0,1;MN), such ha he e exis s α>0, which sa is ies
A(y1)ξξ ⩾α|ξ|2,∀ξ∈RN,a.e. y1∈R.(5.1)
Then, we conside he homogeniza ion p oblem
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
uε∈H1
Γ(Ω),

Ω
Ax1
δεDεuεDε dx=
Ω
dx+
Ω
FDε dx,
∀ ∈H1
Γ(Ω),
(5.2)
whe e belongs o L2(Ω),Fbelongs o L2(Ω)Nand δε>0 sa is ies
lim
ε→0δε=0.(5.3)
Rema k 5.1. The homogeniza ion o he nonlinea p oblem
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
uε∈W1,p
Γ(Ω),

Ω
Ax1
δε
,D
εuεDε dx=
Ω
dx+
Ω
FDε dx,
∀ ∈W1,p
Γ(Ω),
(5.4)
can be pe o med using he same a gumen s which we will use he e, bu his complica es he exposi ion and i is no
necessa y o ou pu pose.
To pe o m he homogeniza ion o (5.2), we will use he wo-scale con e gence me hod o G. Ngue seng and
G. Allai e (see [1,21]). The ollowing is he de ini ion o he wo-scale con e gence adap ed o ou p oblem.
538 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
De ini ion 5.2. Le uεbe a bounded sequence in L2(Ω), we say ha uε wo-scale con e ges o ˆu∈L2(I ×(0,1)×ω),
and we w i e
uε
2e
ˆu,
i o e e y φ∈L∞
(0,1), and e e y ϕ∈L2(Ω),weha e
lim
ε→0
Ω
uε(x)φx1
δεϕ(x)dx =
Ω
1

0
ˆux1,y
1,xφ(y1)ϕ(x) dy1dx.
Rema k 5.3. I a bounded sequence uεin L2(Ω) wo-scale con e ges o ˆu∈L2(I ×(0,1)×ω), hen uεcon e ges
weakly in L2(Ω) o he unc ion ugi en by
u(x) =
1

0
ˆux1,y
1,xdy1,a.e. x∈Ω.
Analogously o he well known wo-scale compac ness heo em o a sequence which is bounded in H1(Ω) (see
[1,21]), we can p o e in ou case he ollowing lemma.
Lemma 5.4. We conside a sequence uε∈H1
Γ(Ω) such ha o some u0∈H1
0(I)
uεu
0in H1
Γ(Ω), (5.5)

Ω
|Dεuε|2dx ⩽C(5.6)
and we assume
∃lim
ε→0
ε
δε
=λ∈[0,+∞].(5.7)
Then, o a subsequence (s ill deno ed by uε), we ha e:
i) I λ=0, he e exis ˆu0∈L2(I, H1
(0,1)/R)and ˆu1∈L2(I ×(0,1), H 1(ω)/R)such ha
Dεuε
2e
du0
dx1
+∂y1ˆu0e1+∇
xˆu1.(5.8)
ii) I λ∈(0,+∞), he e exis s ˆu1∈L2(I, H1
((0,1)×ω)/R)such ha
Dεuε
2e
du0
dx1
+λ∂y1ˆu1e1+∇
xˆu1.(5.9)
iii) I λ=+∞, he e exis ˆu0∈L2(I, H 1
((0,1), L2(ω)/R)) and u1∈L2(I, H1(ω)) such ha
Dεuε
2e
du0
dx1
+∂y1ˆu0e1+∇
xu1.(5.10)
Rema k 5.5. In iew o (5.6) and o Theo em 2.3, o a subsequence, he e exis s a unc ion u1∈L2(I, H1(ω)) such
ha
DεuεD
0(u0,u
1)in L2(Ω)N.
This unc ion u1appea s in (5.10) in he case iii), while in he cases i) and ii), he unc ions u1a e gi en in e ms o
he unc ions ˆu1by
u1(x) =
1

0
ˆu1x1,y
1,xdy1,a.e. x∈Ω(5.11)
(see Rema k 5.3).
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 539
F om Lemma 5.4 we can now deduce he ollowing esul .
Theo em 5.6. We assume (5.7)and we conside he solu ion uεo (5.2). Then, we ha e:
i) I λ=0,wege (5.8), wi h u0,ˆu0,ˆu1 he solu ions o
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(u0,ˆu0,ˆu1)∈H1
0(I) ×L2I,H1
(0,1)/R×L2I×(0,1), H 1(ω)/R,

Ω
1

0
A(y1)du0
dx1
+∂y1ˆu0e1+∇
xˆu1d 0
dx1
+∂y1ˆ 0e1+∇
xˆ 1dy1dx
=
Ω
0dx +
Ω
1

0
Fd 0
dx1
e1+∇
xˆ 1dy1dx,
∀( 0,ˆ 0,ˆ 1)∈H1
0(I) ×L2I,H1
(0,1)/R×L2I×(0,1), H 1(ω)/R.
(5.12)
ii) I λ∈(0,+∞),wege (5.9), wi h u0,ˆu1 he solu ions o
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(u0,ˆu1)∈H1
0(I) ×L2I,H1
(0,1)×ω/R,

Ω
1

0
A(y1)du0
dx1
+λ∂y1ˆu1e1+∇
xˆu1d 0
dx1
+λ∂y1ˆ 1e1+∇
xˆ 1dy1dx
=
Ω
0dx +
Ω
1

0
Fd 0
dx1
e1+∇
xˆ 1dy1dx,
∀( 0,ˆ 1)∈H1
0(I) ×L2I,H1
(0,1)×ω/R.
(5.13)
iii) I λ=+∞,wege (5.10), wi h u0,ˆu0,u1 he solu ions o
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(u0,ˆu0,u
1)∈H1
0(I) ×L2I,H1
0,1,L
2(ω)/R×L2I,H1(ω)/R,

Ω
1

0
A(y1)du0
dx1
+∂y1ˆu0e1+∇
xu1d 0
dx1
+∂y1ˆ 0e1+∇
x 1dy1dx
=
Ω
0dx +
Ω
Fd 0
dx1
e1+∇
x 1dx,
∀( 0,ˆ 0,
1)∈H1
0(I) ×L2I,H1
(0,1), L2(ω)/R×L2I,H1(ω)/R.
(5.14)
P oo . We only p o e he case λ=0, he cases λ∈(0,+∞)and λ=+∞a e simila .
We conside a subsequence o ε,u0∈H1
0(I),ˆu0∈L2(I, H 1
(0,1)/R),ˆu1∈L2(I ×(0,1), H 1(ω)/R)such ha
(5.8) holds. Fo ϕ0,ˆϕ0∈C∞
0(I),ˆ
U0∈C∞
([0,1]),ˆϕ1∈C∞
c(I, C∞(¯ω)),ˆ
U1∈C∞
([0,1]), we ake as es unc ion
in (5.2) he sequence ε∈H1
Γ(Ω) de ined by
ε(x) =ϕ0(x1)+δεˆϕ0(x1)ˆ
U0x1
δε+εˆϕ1(x) ˆ
U1x1
δε,a.e. x∈Ω.
Using
ε(x) =ϕ0(x1)+ ε,
Dε ε(x) =dϕ0
dx1
(x1)+ˆϕ0(x1)dˆ
U0
dy1x1
δεe1+∇
xˆϕ1(x) ˆ
U1x1
δε+Rε(x),
whe e εand Rεcon e ge s ongly o ze o in L2(Ω) and L2(Ω)N espec i ely, we ge

540 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545

Ω
Ax1
δεDεuεdϕ0
dx1
(x1)+ˆϕ0(x1)dˆ
U0
dy1x1
δεe1+∇
xˆϕ1(x) ˆ
U1x1
δεdx
=
Ω
ϕ
0dx +
Ω
Fdϕ0
dx1
(x1)+ˆϕ0(x1)dˆ
U0
dy1x1
δεe1+∇
xˆϕ1(x) ˆ
U1x1
δεdx +Oε,
whe e Oε ends o ze o. Using (5.8) o pass o he limi in his equali y we deduce

Ω
1

0
A(y1)du0
dx1
+∂y1ˆu0e1+∇
xˆu1dϕ0
dx1
+ˆϕ0
dˆ
U0
dy1e1+∇
xˆϕ1ˆ
U1dxdy1
=
Ω
ϕ
0dx +
Ω
1

0
Fdϕ0
dx1
+ˆϕ0
dˆ
U0
dy1e1+∇
xˆϕ1ˆ
U1dxdy1,
o e e y ϕ0,ˆϕ0,ˆ
U0,ˆϕ1and ˆ
U1, as abo e. By linea i y and densi y, his implies ha u0,ˆu0and ˆu1a e he solu ions
o (5.12), and hen, by uniqueness, ha i is no necessa y o ex ac any subsequence. 2
Rema k 5.7. When λ∈(0,+∞), Theo em 5.6 can be deduced om he esul s ob ained in [22] (in [22] F=0, bu
o assume F= 0 does no make he p oblem mo e di icul ). O he homogeniza ion esul s o hin s uc u es wi h
pe iodic coe icien s can be ound in [2,3,5,11].
Rema k 5.8. Fo λ=0, he abo e heo em means ha he asymp o ic beha io o uεis as i we conside δε=δ
ixed, and we ake he limi i s in εand hen in δ, i.e. as we make i s he educ ion o dimension and hen he
homogeniza ion. I is possible o ob ain a gene al esul in his di ec ion assuming ha he equency o he oscilla ions
in x1is smalle han 1
ε. Speci ically, he ollowing esul holds: Assume Aεsa is ying he assump ions in Sec ion 2
and such ha
lim
ε→0sup
0⩽h⩽1
Aε(. +εhe1,ξ)−Aε(., ξ )
Lp(Ic×ω) =0,∀IcI.
Fo ∈Lp(Ω) and F∈Lp(Ω)N, we de ine u0,ε,u1,ε as he solu ions o
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(u0,ε,u
1,ε)∈W1,p(I) ×LpI,W1,p(ω)/R,

Ω
Aεdu0,ε
dx1
e1+∇
xu1,εd 0
dx1
e1+∇
x 1dx
=
Ω
0+Fd 0
dx1
e1+∇
x 1dx,
∀( 0,
1)∈W1,p(I) ×LpI,W1,p(ω)/R.
(5.15)
Then, we ha e
Dεuε−du0,ε
dx1
e1+∇
xu1,ε→0inLp(Ω),
whe e uεis he solu ion o (2.12). This educes he homogeniza ion o (2.12) o he homogeniza ion o (5.15). We
will no p o e his esul because we will no use i .
When λ∈(0,+∞), Theo em 5.6 means ha he educ ion o dimension and he homogeniza ion hold simul ane-
ously.
When λ=∞, Theo em 5.6 means ha we can pe o m i s he homogeniza ion and hen he educ ion o di-
mension. Clea ly in his case he p oblem o u0and u1is local. We will see ha he o he wo cases gi e nonlocal
p oblems in gene al. Namely we gi e wo examples, wi h λ=0 and λ=1 in which he limi p oblem o (2.12)is
nonlocal.
In he wo examples we assume N=2, ω=(0,1), and we se x=x2.
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 541
Example 1
We de ine A∈L∞
((0,1), M2)by
A(y1)=11
1γ(y
1),a.e. y1∈(0,1), (5.16)
wi h γ∈L∞
(0,1), such ha he e exis s ν>0, wi h γ>1+νa.e. in (0,1). We conside δεsuch ha
lim
ε→0
ε
δε
=0.
Theo em 5.9. Fo he abo e choice o Aand δε, he limi p oblem o (5.2)is (3.7)whe e A:R×∇H1(ω) →L2(ω)2
is he nonlocal ope a o gi en by
As, dψ
dx2=s+dψ
dx2e1+s+γ∗dψ
dx2
+ˆγ−γ∗
1

0
dψ
dx2
( ) d ,a.e. in ω, (5.17)
wi h
γ∗=1

0
dy1
γ(y
1)−1
,ˆγ=1

0
dy1
γ(y
1)−1−1
+1.(5.18)
P oo . Fo ∈L2(Ω) and F∈L2(Ω)2, we de ine (u0,ˆu0,ˆu1)as he solu ion o (5.12) and u1by (5.11). We know
ha i uεis he solu ion o (2.12) hen (2.13), (2.14) hold.
Taking in (5.12), 0=0, ˆ 0=0, we deduce
du0
dx1
+∂y1ˆu0+γ(y
1)∂x2ˆu1=F2a.e. in I×(0,1)×ω. (5.19)
Using now 0=0 and ˆ 1=0in(5.12) we deduce ha he e exis s a unc ion ∈L2(I) such ha
du0
dx1
+∂y1ˆu0+
1

0
∂x2ˆu1dx2= (x1)a.e. in I×(0,1).
Taking in his exp ession he alue o ∂x2ˆu1gi en by (5.19), we ge
du0
dx1
+∂y1ˆu0+1
γ¯
F2−du0
dx1
−∂y1ˆu0= (x1)a.e. in I×(0,1),
wi h
¯
F2(x1)=
1

0
F2(x1,x
2)dx
2,a.e. x1∈I.
Thus, we ha e
∂y1ˆu0=γ
γ−1 −du0
dx1
−1
γ−1¯
F2a.e. in I×(0,1). (5.20)
In eg a ing his equali y wi h espec o y1and using ha ˆu0is pe iodic wi h espec o y1, we easily deduce
(x1)=ˆγ−1
ˆγ
du0
dx1
+1
ˆγ
¯
F2,
which subs i u ed in (5.20) p o es
∂y1ˆu0=γ
ˆγ−11
γ−1¯
F2−du0
dx1.(5.21)
542 J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545
Using he exp ession (5.21)o ∂y1ˆu0in (5.19)weha e
∂x2ˆu1=1
γF2−du0
dx1−1
ˆγ−1
γ1
γ−1¯
F2−du0
dx1,
a.e. in I×(0,1)×ω, which in eg a ed in (0,1)wi h espec o y1gi es
∂x2u1=1
γ∗F2−du0
dx1+1
ˆγ−1
γ∗¯
F2−du0
dx1,a.e. in Ω. (5.22)
In eg a ing now in (0,1)wi h espec o x2, we ge
¯
F2−du0
dx1
=ˆγ
1

0
∂x2u1(x1, )d , a.e. in I,
which subs i u ed in (5.22) implies
du0
dx1
+γ∗∂x2u1+ˆγ−γ∗
1

0
∂x2u1(x1, )d =F2,a.e. in Ω. (5.23)
On he o he hand, aking in (5.12)ˆ 0=0, ˆ 1=0weha e

Ωdu0
dx1
+∂x2u1d 0
dx1
dx1=
Ω
0+F1
d 0
dx1dx1,(5.24)
o e e y 0∈H1
0(I).F om(5.23) and (5.24) we conclude ha u0,u1sa is y (3.7) wi h Agi en by (5.17). 2
Example 2
We ake δε=ε,ω=(0,1), and we de ine A∈L∞
((0,1), M2)by
A(y1)=10
0γ(y
1),a.e. in (0,1), (5.25)
wi h γ∈L∞
(0,1), such ha o some α>0, we ha e γ>αa.e. in R.
Theo em 5.10. Fo he abo e choice o Aε, he limi p oblem o (2.12)is (3.7)whe e A:R×∇H1(ω) ×L2(0,1)2
is a nonlocal ope a o gi en by
As, dψ
dx2(x) =se1+A2dψ
dx2(x2), a.e. x∈Ω, (5.26)
wi h A2:L2(ω) →L2(ω) de ined by
A2(H)(x2)=2
π2
∞

k=11
0H( )sin(kπ ) d
k21
0ψk(y1)dy
1
sin(kπx2), a.e. x2∈(0,1). (5.27)
He e ψk∈H1
(0,1)is he solu ion o
1

0
dψk
dy1
d
dy1
dy1+k2π2
1

0
γψ
k dy
1=
1

0
dy
1,∀ ∈H1
(0,1). (5.28)
P oo . Fo ∈L2(Ω) and F∈L2(Ω)2, we de ine u0,ˆu1as he solu ions o (5.13) wi h λ=1 and u1by (5.11). We
know ha i uεis he solu ion o (2.12) hen (2.13), (2.14) hold.
J. Casado-Díaz e al. / Ann. I. H. Poinca é – AN 30 (2013) 519–545 543
Taking in (5.13) 0=0, we deduce ha o a.e. x1∈I, he unc ion ˆu1(x1,.,.)∈H1((0,1)2), pe iodic wi h espec
o y1, sa is ies
1

0
1

0
(∂y1ˆu1∂y1ˆ 1+γ∂
x2ˆu1∂x2ˆ 1)dy
1dx2=
1

0
1

0
F2∂x2ˆ 1dy1dx2,
∀ˆ 1∈H1(0,1)2,pe iodic wi h espec o y1.
Using ha he unc ions cos(kπx2), wi h k∈N, a e a basis o H1(0,1)( hey a e he eigen unc ions co esponding o
he ope a o (d2
dx2
2
)−1wi h Neumann bounda y condi ion), we look o a Fou ie expansion o ˆu1,
ˆu1=
∞

k=1
ηk(y1)cos(kπx2).
We ge
∂x2ˆu1(x1,y
1,x
2)=2π2
∞

k=1
k2
1

0
F2(x1, )sin(kπ ) d ψk(y1)sin(kπx2),
a.e. in I×(0,1)2. In eg a ing wi h espec o y1, we conclude ha u1sa is ies
A2∂x2u1(x1,.)
(x2)=F2(x), a.e. x∈Ω, (5.29)
wi h A2de ined by (5.27). On he o he hand, aking ˆ 0=0in(5.13) we deduce

Ω
du0
dx1
d 0
dx1
dx =
Ω
0+F2
d 0
dx1dx. (5.30)
F om (5.29) and (5.30) we conclude ha u0,u1sa is y (3.7) wi h Agi en by (5.27). 2
Rema k 5.11. We obse e ha i he ope a o A2is local, i.e. i he e exis s c:Ω→Rsuch ha A2(H ) =c(x1,x
2)H ,
o e e y H∈L2(0,1), hen, by (5.27)cis a posi i e cons an and
1

0
ψk(y1)dy
1=1
ck2π2,∀k⩾1.(5.31)
The ollowing esul p o es ha his only holds i γis cons an .
P oposi ion 5.12. The solu ion ψko (5.28)sa is ies (5.31)i and only i γ=ca.e. in (0,1).
P oo . I is clea ha i γis cons an hen (5.31) holds.
Fo he ecip oca e, we assume ha (5.31) hold. Taking ψkas es unc ion in (5.28) and using (5.31) we deduce
k2
1

0
dψk
dy1
2
dy1+k4π2
1

0
γ|ψk|2dy1=1
cπ2.(5.32)
The e o e, up o a subsequence, he e exis s ψ∈L2(0,1)such ha k2ψkcon e ges weakly o ψin L2(0,1). Then,
aking ϕ∈H1
(I) as es unc ion in (5.28) we deduce
1

0
dψk
dy1
dϕ
dy1
dy1+π2
1

0
γk2ψkϕdy
1=
1

0
ϕdy
1.(5.33)