Homogenization and correctors for monotone problems in cylinders of small diameter

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). 0294-1449/$ – see on ma e h p://dx.doi.o g/10.1016/j.anihpc.2012.10.004 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 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)