Applications of Mathematics

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1 Applications of Mathematics Liselott Flodén; Marianne Olsson Homogenization of some parabolic operators with several time scales Applications of Mathematics, Vol , No. 5, Persistent URL: Terms of use: Institute of Mathematics AS CR, 2007 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 APPLICATIONS OF MATHEMATICS No. 5, HOMOGENIZATION OF SOME PARABOLIC OPERATORS WITH SEVERAL TIME SCALES Liselott Flodén, Marianne Olsson, Östersund Received June 28, 2006, in revised version October 24, 2006 Abstract. The main focus in this paper is on homogenization of the parabolic problem t u ax/,t/, t/ r u = f. Undercertainassumptionson a, thereexistsa G-limit b,whichwecharacterizebymeansofmultiscaletechniquesfor r >0, r 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made. Keywords: homogenization, G-convergence, multiscale convergence, parabolic, asymptotic expansion MSC2000:35B27 1. Introduction The main source of inspiration for the development of the modern theory of convergence for sequences of differential operators, is to find methods to compute effective properties of heterogenous materials. The most general strategies designed for this purpose are the two closely related concepts of G-convergence and H-convergence. The special case where the structure is built by small identical cubes is particularly well studied. This kind of problems is usually named periodic homogenization problems. In this paper we study the homogenization of a parabolic problem with rapid oscillations in one spatial scale and two time scales. More precisely, we homogenize 431

3 the equation t u x, t a, t, t 1 r u x, t = fx, t in Ω 0, T, u x, 0 = u 0 x in Ω, u x, t = 0 on Ω 0, T, where Ωisanopenboundedsetin Ê N and rand Tarepositiveconstants.For {} tending to zero we get a sequence of equations. The homogenization problem consists instudyingtheasymptoticbehaviorofthecorrespondingsequenceofsolutions u andfindingalimitequationwhichadmitsthelimit uof {u }asitsuniquesolution. This so-called homogenized problem is governed by a coefficient matrix without rapid oscillations, which we characterize by means of multiscale convergence techniques for differentvalueson r.wedistinguishthethreedifferentcases; 0 < r < 2where r 1, r = 2and r > 2. In Section 2 we study some features of asymptotic expansion. We discuss in which sense of convergence the expansion is valid, interpreting the significance of the firstterms. Ageneralizationoftheseobservationswillturnouttobeusefulinthe homogenization procedure in Section 4. and Notation1.Wedefine = Ω 0, T Y k,n = Y k 0, 1 n, where Y k = Y 1... Y k and Y 1 =... = Y k = Y = 0, 1 N. Let FÊ kn+n beaspaceofrealvaluedfunctionsdefinedon Ê kn+n.by F Y k,n wedenoteallfunctionsin F loc Ê kn+n whicharetheperiodicrepetitionofsome functionin FY k,n. Furthermore, L 2 0, T; Xdenotesallfunctions u: 0, T Xsuchthat T 1/2 u L 2 0,T;X = u, t 2 X dt <, 0 where XisaBanachspace. Finally,fortheevolutiontriple H 1 0Ω L 2 Ω H 1 Ω,weintroducethespace H 1 0, T; H 1 0 Ω, H 1 Ωofallfunctions uwhichbelongto L 2 0, T; H 1 0 Ωwith tu in L 2 0, T; H 1 Ω.Thisspaceisequippedwiththenorm u H1 0,T;H 1 0 Ω,H 1 Ω = u L2 0,T;H 1 0 Ω + t u L 2 0,T;H 1 Ω. 432

4 2. Two-scale convergence and asymptotic expansions To enhance the comprehension of the two-scale convergence method and its relation to the multiple-scale expansion we first study a simple and illuminating homogenization example, namely the linear elliptic problem 2 x a u x = fx in Ω, u x = 0 on Ω. Here ayisperiodicwithperiod Y and Ω Ê N isopenandbounded. For 0 we have u x ux in H0 1 Ω, where u is the unique solution to the limit equation 3 b ux = fx in Ω, ux = 0 on Ω. The coefficient b is obtained from the local problem 4 y ay ux + y u 1 x, y = 0, where u 1 is Y-periodicinitssecondargument. Theexactsolution u togetherwiththesolution utothehomogenizedproblem arefoundinfig.1foraonedimensionalexamplewith Ω = 0, 1, fx = x 2 and 5 ay = sin2πy. To achieve the homogenization result above rigorously, one can use a method based on two-scale convergence. An adaptation of this technique to certain evolution cases isusedtoprovethehomogenizationresultinsection4. Thisconceptwasfirst introduced by Nguetseng in 1989, see[11]. Definition2. Asequence {u }in L 2 Ωissaidtotwo-scaleconvergetoalimit u 0 L 2 Ω Y if lim u xv x, x dx = u 0 x, yvx, ydy dx 0 Ω Y forall v L 2 Ω; C Y.Wewrite Ω u x u 0 x, y. 433

5 u u x Figure1. Thesolutions u and u. The definition is motivated by a compactness result which states that a sequence {u }boundedin L 2 Ωpossessesatwo-scaleconvergentsubsequence,see[11], [1] and[10]. The following theorem characterizes limits for sequences of gradients. Thisresultallowsustopasstothelimitintheweakformof2forcertainchoices of test functions and obtain the local problem4 and the homogenized problem3. Theorem3. Let {u }beasequenceboundedin H 1 Ω. Then,uptoasubsequence, it holds that u x ux in L 2 Ω and u x ux + y u 1 x, y, where u H 1 Ωand u 1 L 2 Ω; H 1 Y /Ê. Proof. See[1]or[10]. An earlier method for the determination of the local and the homogenized problem is the multiple-scale, or asymptotic expansion, method; see e.g.[4] and[5]. In this approach,oneassumesthatthesolution u canbeexpandedinapowerseriesin oftheform 6 u x = ux + u 1 x, x + 2 u 2 x, x +..., where u k is Y-periodicinthesecondargument.Plugging6intoequation2and equating equal powers of, we formally obtain both the homogenized problem3 434

6 andthelocalproblem4,solvedby uand u 1 respectively. Furthermore, uand u 1 areidenticaltothefunctionswiththesamenameswhichappearintheorem3. Weknowthat u x ux in L 2 Ω butitremainstoexploreifalso u 1 hassignificanceinthesenseofsomesuitablekind of limit. The truncated form of6canbewritten u x ux + u 1 x, x 7 u x ux u 1 x, x, whichindicatesthat {u u/}approaches u 1 insomesensesimilartothetwoscaleconvergence.accordingtotheorem3.3in[2]wecanidentifyaclassofsmooth functions v: Ω Y Êsuchthat F = 1 Ω v x, x dx isboundedin H 1 Ω. Thekeypropertiesofthesefunctionsarethattheyare Y-periodicforanyfixed x Ωandhaveintegralmeanvaluezeroover Y intheir secondargument.hence,foranyboundedsequence {α }in H 1 Ωand β = 1 Ω v x, x α xdx, {β }convergesuptoasubsequence.thismeansthat,stilluptoasubsequence,the limit u x ux lim v x, x dx 0 Ω exists for suitable test functions v. Itturnsoutthatthechoiceoftestfunctionsiscrucial.Fig.2showstheleft-hand sidetogetherwiththeright-handsideof7,where u and uarethesolutionsto2 and3, respectively, for the coefficient a given by5. Apparently, they have similar patterns of oscillations but differ in the global tendency. Fig. 3 displays the difference between these two functions, i.e. g x = u x ux u 1 x, x. 435

7 u u u x Figure2. u x ux/togetherwith u 1 x,x/ x Figure3. Thefunction g x. For vx, y = x 2 cos2πy thegraphof g xvx, x/isfoundinfig.4.theintegralof g xvx, x/over Ω iscloseto zeroforsmall. Thehighfrequencyvariationsof g arenegligible. Furthermore, the function v has mean value zero in its second variable and replacing y by x/with smalltheslowerglobaltendencyof g isfilteredawaybytheoscillations. 436

8 x Figure4. Theproduct g xvx,x/, Y vx, ydy=0. Forthischoiceof vand = 0.05weobtain u x ux v x, x dx u 1 x, yvx, ydy dx , Ω Y Ω and hence a limit of two-scale type consistent with the approach of asymptotic expansionseemstobeathandforthiskindoftestfunctions.ifweomittherequirement that vhaveintegralmeanvaluezeroover Y andchoosee.g. vx, y = x cos2πy, we obtain the function illustrated in Fig. 5, whose integral over Ω does obviously not vanish. Inthiscase u x ux v x, x dx u 1 x, yvx, ydy dx Ω Ω Y for = 0.05.Thisshowsthattheclassoftestfunctionsmustbemorerestrictedthan fortheusualtwo-scaleconvergence.ourinvestigationaboverevealsthat {u u/} isapproaching u 1 onlyinacertainweaksense,namely lim 0 Ω u x ux v x, x dx = u 1 x, yvx, ydy dx, Ω Y where the delicate question is to identify the appropriate class of test functions. 437

9 x Figure5. Theproduct g xvx,x/, Y vx,ydy 0. In[9]itisproventhatforaboundedsequence {u }in H 1 Ωandwith uand u 1 definedasintheorem3itholdsthat,uptoasubsequence, lim 0 Ω u x ux v 1 xv 2 dx = u 1 x, yv 1 xv 2 ydy dx Ω Y forall v 1 DΩand v 2 C Y /Ê. Thus u 1appearsasalimitoftwo-scaletype to {u u/}whichcontributestotheinterpretationoftheasymptoticexpansion6. Ageneralizationofthisresult,seeTheorem9,willbeusedtoprovethe homogenization result in Section Multiscale convergence In[2], Allaire and Briane introduced so-called multiscale convergence. This generalization of two-scale convergence is suitable when studying problems with multiple scales of periodic oscillations. We give the following definition. Definition 4. Asequence {u }in L 2 Ωissaidto n + 1-scaleconvergeto u 0 L 2 Ω Y n if = Ω lim u xv x, x,..., x dx 0 Ω 1 n Y n u 0 x, y 1,..., y n vx, y 1,..., y n dy n... dy 1 dx

10 forall v L 2 Ω; C Y n,where k 0when 0.Wewrite u x n+1 u 0 x, y 1,..., y n. In[2]itisproventhataboundedsequence {u }in L 2 Ωhasasubsequence which n + 1-scaleconvergestosomelimit u 0 L 2 Ω Y n ifthescalesobey certain separation requirements. Adapting to our problem1 we define 2, 3-scale convergence which includes two spatial scales and three time scales. Definition5. Asequence {u }in L 2 issaidto2,3-scaleconvergeto u 0 L 2 if lim u x, t v x, t, x t t, 0 1, 1 dxdt 2 = u 0 x, t, y, s 1, s 2 vx, t, y, s 1, s 2 dy ds 2 ds 1 dxdt forall v L 2 ; C.Thisisdenotedby u x, t 2,3 u 0 x, t, y, s 1, s 2. The following compactness result holds true. Theorem 6. Let {u }beaboundedsequencein L 2 andassumethat 1 =, 1 = and 2 = r,where r > 0and r 1. Thenthereexistsafunction u 0 L 2 suchthat,uptoasubsequence, {u }2,3-scaleconvergesto u 0. Proof. ThisisanimmediateconsequenceofTheorem2.4in[2]. Remark 7. Weomitthecase r = 1toobtainseparationbetweenthescales, see[2] for details. Theorem 8 deals with 2, 3-scale convergence of gradients. The proof follows directly alongthelinesoftheproofoftheorem3.1in[8]. Theorem8. Let {u }beasequenceboundedin H 1 0, T; H 1 0 Ω, H 1 Ωand assumethat 1 =, 1 = and 2 = r,where r > 0and r 1.Thenitholdsupto a subsequence that u x, t ux, t in L 2 and u x, t 2,3 ux, t + y u 1 x, t, y, s 1, s 2, where u L 2 0, T; H0Ωand 1 u 1 L 2 0, 1 2 ; H 1 Y /Ê. 439

11 The following theorem generalizes our observations in Section 2 to a certain evolution case. Theorem9. Let {u }beaboundedsequencein H 1 0, T; H 1 0 Ω, H 1 Ω,and let uand u 1 bedefinedasintheorem8.then,uptoasubsequence, u x, t ux, t t t lim v 1 xv 2 c 1 tc 2 c 3 0 r dxdt = u 1 x, t, y, s 1, s 2 v 1 xv 2 yc 1 tc 2 s 1 c 3 s 2 dy ds 2 ds 1 dxdt forall v 1 DΩ, v 2 L 2 Y /Ê, c 1 D0, Tand c 2, c 3 L 2 0, 1if r > 0, r 1. Proof. TheproofisperformedinexactlythesamewayastheproofofCorollary3.3in[8]. The theorems above will be used while proving the homogenization result in Theorem Homogenization In this section we study the homogenization of the parabolic equation t u x, t a, t, t r u x, t = fx, t in Ω 0, T, u x, 0 = u 0 x in Ω, u x, t = 0 on Ω 0, T, introducedinsection1,where f L 2 and u 0 L 2 Ω. Thisproblemallowsa uniquesolution u H 1 0, T; H 1 0 Ω, H 1 Ωforanyfixed > 0if aisassumedto belongto L andtosatisfythecoercivitycondition ay, s 1, s 2 ξ ξ α ξ 2, α > 0 forany y, s 1, s 2 Ê N+2 andall ξ Ê N.Undertheseassumptions, ax/, t/, t/ r satisfies the standard conditions for G-convergence for linear parabolic problems, see[6]and[14].thismeansthat,atleastforasuitablesubsequence,thereexistsa wellposedlimitproblemofthesametypeas9governedbyacoefficientmatrix b. Moreover, 10 u L 0,T;L 2 Ω C 440

12 and 11 u H 1 0,T;H 1 0 Ω,H 1 Ω C forsomepositiveconstant C. Ouraimistocharacterizethe G-limit bfurtherby means of homogenization procedures. Theorem10. Let {u }beasequenceofsolutionsin H 1 0, T; H 1 0Ω, H 1 Ω to9.then u x, t ux, t in L 2 and u x, t 2,3 ux, t + y u 1 x, t, y, s 1, s 2, where u H 1 0, T; H0 1Ω, H 1 Ωand u 1 L 2 0, 1 2 ; H 1 Y /Ê. Furthermore, u is the unique solution to the homogenized problem 12 t ux, t b ux, t = fx, t in, ux, 0 = u 0 x in Ω, ux, t = 0 on Ω 0, T where b ux, t = ay, s 1, s 2 ux, t + y u 1 x, t, y, s 1, s 2 dy ds 2 ds 1. For 0 < r < 2, r 1,thefunction u 1 isdeterminedbythelocalproblem 13 y ay, s 1, s 2 ux, t + y u 1 x, t, y, s 1, s 2 = 0, for r = 2by 14 s2 u 1 y ay, s 1, s 2 ux, t + y u 1 x, t, y, s 1, s 2 = 0, andfor r > 2bythesystemoflocalproblems ux, y ay, s 1, s 2 ds 2 t + y u 1 x, t, y, s 1 = 0, 0 s2 u 1 x, t, y, s 1, s 2 =

13 Proof. ThroughouttheproofweusetheEinsteintensorsummationconvention. Westudytheweakformof9,i.e.werequirethat 17 u x, tvx t ct + a ij, t, t r xj u x, t xi vxctdxdt = fx, tvxctdxdt forall v H0 1 Ωand c D0, T.Theaprioriestimates10and11allowusto apply Theorem 8. Hence, up to a subsequence, u x, t ux, t in L 2 and u x, t 2,3 ux, t + y u 1 x, t, y, s 1, s 2, where u L 2 0, T; H0 1Ωand u 1 L 2 0, 1 2 ; H 1 Y /Ê. Choosing v H0Ωand 1 c D0, Tindependentof in17andletting tendtozeroweget, according to Theorem 8, 18 ux, tvx t ct + a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 dy ds 2 ds 1 = fx, tvxctdxdt. xi vxctdxdt Inordertofindthelocalproblemswestudythedifferencebetween17and18, i.e. 19 u x, t ux, t vx t ct + a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 dy ds 2 ds 1 a ij, t, t r xj u x, t xi vxctdxdt = 0. We choose the test functions 442 vx = v 1 xv 2 t t, ct = c 1 tc 2 c 3 r,

14 where v 1 DΩ, v 2 C Y /Ê, c 1 D0, Tand c 2, c 3 C 0, 1in19,and get 1 u x, t ux, t t t 20 v 1 xv 2 2 t c 1 tc 2 c 3 r t t t t + c 1 t s1 c 2 c 3 r + 2 r c 1 tc 2 s2 c 3 r + a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 dy ds 2 ds 1 a ij, t, t r xj u x, t xi v 1 xv 2 + v 1 x yi v 2 c 1 tc 2 t c 3 t r dxdt = 0. Nextwelet passtozero.forthecasewhen 0 < r < 2wehave a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 dy ds 2 ds 1 a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 andduetotheperiodicityof v 2 weobtain v 1 x yi v 2 yc 1 tc 2 s 1 c 3 s 2 dy ds 2 ds 1 dxdt = 0, a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 v 1 x yi v 2 yc 1 tc 2 s 1 c 3 s 2 dy ds 2 ds 1 dxdt = 0. By the variational lemma a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 yi v 2 ydy = 0 Y for all v 2 C Y /Ê, and hence, by density, for all v 2 H 1 Y /Ê, a.e. in 0, 1 2. Thisistheweakformof13. For r = 2,accordingtoTheorem9, 20 approaches u 1 x, t, y, s 1, s 2 v 1 xv 2 yc 1 tc 2 s 1 s2 c 3 s 2 dy ds 2 ds 1 dxdt + a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 dy ds 2 ds 1 v 1 x yi v 2 yc 1 tc 2 s 1 c 3 s 2 a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 v 1 x yi v 2 yc 1 tc 2 s 1 c 3 s 2 dy ds 2 ds 1 dxdt = 0, 443

15 when tendstozero.since v 2 isperiodicthemiddletermvanishesandwehave u 1 x, t, y, s 1, s 2 v 1 xv 2 yc 1 tc 2 s 1 s2 c 3 s 2 a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 v 1 x yi v 2 yc 1 tc 2 s 1 c 3 s 2 dy ds 2 ds 1 dxdt = 0. Applying the variational lemma we arrive at Y 1,1 u 1 x, t, y, s 1, s 2 v 2 y s2 c 3 s 2 a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 yi v 2 yc 3 s 2 dy ds 2 = 0, forall v 2 H 1Y /Êandall c 3 C 0, 1,a.e.in 0, 1.Wehavefoundthe weakformof14. Forthecasewhen r > 2wechoosethetestfunctions t vx = v 1 xv 2, ct = c 1 tc 2 in17,where v 1 DΩ, v 2 C Y, c 1 D0, Tand c 2 C t + c 1 t s1 c 2 x t u x, tv 1 xv 2 t c 1 tc 2 + a ij, t, t 2 xj u x, t xi v 1 xv 2 x t = fx, tv 1 xv 2 c 1 tc 2 dxdt andwhen goestozeroweget ux, tv 1 xv 2 yc 1 t s1 c 2 s 1 0, 1.Weobtain + v 1 x yi v 2 c 1 tc 2 t + a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 v 1 x yi v 2 yc 1 tc 2 s 1 dy ds 2 ds 1 dxdt = 0. Theperiodicityof c 2 andthevariationallemmaimplythat dxdt Y 1,1 a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 yi v 2 ydy ds 2 = 0 forall v 2 H 1 Y /Êa.e.in 0, 1.Thisistheweakformof

16 Next we study the difference19 for the test functions vx = r 1 v 1 xv 2 t t, ct = c 1 tc 2 c 3 r, where v 1 DΩ, v 2 C Y /Ê, c 1 D0, Tand c 2, c 3 C 1 c 3 t r 0, 1,andweobtain u x, t ux, t t v 1 xv 2 r t c 1 tc 2 t t t t + r 1 c 1 t s1 c 2 c 3 r + c 1 tc 2 s2 c 3 r + a ij y, s 1, s 2 xj ux, t + yj u 1 x, t, y, s 1, s 2 dy ds 2 ds 1 a ij, t, t r xj u x, t r 1 xi v 1 xv 2 When passestozerowegetaccordingtotheorem9 t t + r 2 v 1 x yi v 2 c 1 tc 2 c 3 r dxdt = 0. u 1 x, t, y, s 1, s 2 v 1 xv 2 yc 1 tc 2 s 1 s2 c 3 s 2 dy ds 2 ds 1 dxdt = 0 and by the variational lemma 1 0 u 1 x, t, y, s 1, s 2 s2 c 3 s 2 ds 2 = 0 forall c 3 C 0, 1,a.e.in Y 0, 1. Thisistheweakformof16and meansthat u 1 doesnotdependon s 2. Remark 11. Homogenizationoflinearparabolicequationswithoscillationsin both space and time was first investigated by means of asymptotic expansions in[4]. This is also studied in[13], where a different proof is provided. The homogenization of monotone nonlinear problems with two scales in space and time, respectively, is provenbymeansof G-convergencein[15].Similarresultscanalsobefoundin[12]. For corrector results for linear parabolic equations with oscillation only in space wereferto[3]. Suchresultsforproblemswithoscillationsalsointimehavebeen proven by means of two-scale convergence methods in[8], and with H-convergence techniques in[7]. The corresponding results for nonlinear problems are obtained in[16]. In[9] reiterated homogenization results for linear parabolic operators are proven by means of multiscale convergence methods. 445

17 References [1] G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal , zbl [2] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenization. Proc. R. Soc. Edinburgh, Sect. A , zbl [3] S. Brahim-Otsmane, G. A. Francfort, and F. Murat: Correctors for the homogenization of the heat and wave equations. J. Math. Pures Appl , zbl [4] A. Bensoussan, J.-L. Lions, and G. Papanicolaou: Asymptotic Analysis for Periodic Structures. Stud. Math. Appl. North-Holland, Amsterdam-New York-Oxford, zbl [5] D. Cioranescu, P. Donato: An Introduction to Homogenization. Oxford Lecture Ser. Math. Appl. Oxford University Press, Oxford, zbl [6] F. Colombini, S. Spagnolo: Sur la convergence de solutions d équations paraboliques. J. Math. Pures Appl , In French. zbl [7] A. Dall Aglio, F. Murat: A corrector result for H-converging parabolic problems with time-dependent coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci., IV. Ser , zbl [8] A. Holmbom: Homogenization of parabolic equations. An alternative approach and some corrector-type results. Appl. Math , zbl [9] A. Holmbom, N. Svanstedt, N. Wellander: Multiscale convergence and reiterated homogenization for parabolic problems. Appl. Math , zbl [10] D. Lukkassen, G. Nguetseng, P. Wall: Two-scale convergence. Int. J. Pure Appl. Math , zbl [11] G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal , zbl [12] A. Pankov: G-convergence and Homogenization of Nonlinear Partial Differential Operators. Mathematics and its Applications. Kluwer, Dordrecht, [13] A. Profeti, B. Terreni: Uniformità per una convergenza di operatori parabolichi nel caso dell omogenizzazione. Boll. Unione Math. Ital. Ser. B , In Italian. zbl [14] S. Spagnolo: Convergence of parabolic equations. Boll. Unione Math. Ital. Ser. B , zbl [15] N. Svanstedt: G-convergence of parabolic operators. Nonlinear Anal. Theory Methods Appl , zbl [16] N. Svanstedt: Correctors for the homogenization of monotone parabolic operators. J. Nonlinear Math. Phys , zbl Authors address: L. Flodén, M. Olsson, Department of Engineering, Physics and Mathematics, Mid Sweden University, SE Östersund, Sweden, s: marianne.olsson@miun.se. 446