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ComputersandMathematicswithApplications66(2013)1573–1580
Contents lists available at ScienceDirect
ComputersandMathematicswithApplications
journal homepage: www.elsevier.com/locate/camwa
Multiscaleanalysisofdiffusionprocessesincompositemedia
ClaudiaTimofte∗
FacultyofPhysics,UniversityofBucharest,P.O.BoxMG-11,Bucharest,Romania
a r t i c l e i n f o
Keywords:
Homogenization
Theperiodicunfoldingmethod
Dynamicalboundaryconditiona b s t r a c t
Thegoalofthispaperistopresentsomehomogenizationresultsforanonlinearproblem
arisinginthemodelingofdiffusioninaperiodicstructureformedbytwomediawith
differentproperties,separatedbyanactiveinterface.Oursettingisrelevantformodeling
heatdiffusionincompositematerialswithimperfectinterfacesorelectricalconduction
inbiologicaltissues.Theapproachwefollowisbasedontheperiodicunfoldingmethod,
whichallowsustodealwithgeneralmedia.
©2012ElsevierLtd.Allrightsreserved.
1. Introduction
Inthelastdecades,theproblemofdiffusioninhighlyheterogeneousmaterialshasbeenasubjectofhugeinterestfor
abroadcategoryofresearchers:mathematicians,physicists,biologists,etc.Themaingoalofthispaperistoanalyzethe
effectivebehaviorofthesolutionofsomenonlinearproblemsarisinginthemodelingofdiffusioninaperiodicstructure
formedbytwomediawithdifferentproperties,separatedbyanactiveinterface(see(1)).Oursettingprovestoberelevant
formodelingheatconductionincompositematerialswithimperfectinterfacesorelectricalconductioninbiologicaltissues.
We assume first that both media are connected. Using the periodic unfolding method recently introduced by
D.Cioranescu,A.Damlamian,G.Griso,P.DonatoandR.Zaki(see[1,2]),wecanprovethattheasymptoticbehaviorofthe
solutionofourproblemisgovernedbyanewnonlinearsystem(see(4)).Atamacroscopiclevel,thecompositemedium,
despiteitsdiscretestructure,canberepresentedbyacontinuousmodel,whichdescribesitasthesuperimpositionoftwo
interpenetratingcontinuousmedia,coexistingateverypointofthedomain.Hence,atthemacroscale,weobtaina bidomain
model,whichisageneralizationoftheso-called Barenblattmodel ,arisinginthecontextofdiffusioninpartiallyfissured
media(see,forinstance,[3,4]).
Asimilarmodelappearsalsointhestudyofthebioelectricalactivityoftheheartatamacroscopiclevel(see[5]).In
thiscase,atthemicroscopicscale,wedealwithamediumcomposedoftwodifferentconductivephases(theintracellular
andextracellularspaces),separatedbyadielectricinterface(modelingthecellularmembranes),whichhasacapacitiveand
anonlinearconductivebehavior.Theelectricpotentialverifiesellipticequationsinthetwoconductiveregions,coupled
byanevolutiveboundaryconditioninvolvingthepotentialjumpattheinterfacesbetweenthetwophases.Theevolution
intimeofthehomogenizedpotentialisgovernedexactlybya bidomainmodel.Suchamodel,proposedinthelate1970s
(see[6,7]),isconsideredtobeoneofthemostaccurateandrealisticmacroscopicdescriptionsoftheelectricalactivityof
cardiactissue,undernormalorpathologicalconditions.Asweshallsee,itsmathematicalformulationinvolvesasystemof
twodegenerateparabolicreaction–diffusionequations.Ithasbeenshownin[8]thatitcanalsobereformulatedintoone
parabolicsemi-linearpartialdifferentialequation,exhibitingnon-localityinspace.
Weshallalsobrieflydiscussadifferentgeometricsituation,inwhichonlyonephaseisconnected,whiletheotherone
isdisconnected.Inthiscase,weareledtoadifferentmacroscopicmodel(seeRemark2).
∗Tel.:+40721498466.
E-mailaddress:claudiatimofte@yahoo.com.
0898-1221/$–seefrontmatter ©2012ElsevierLtd.Allrightsreserved.
doi:10.1016/j.camwa.2012.12.003

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1574 C.Timofte/ComputersandMathematicswithApplications66(2013)1573–1580
Fig. 1.Thegeometryoftheperiodicunitcell.
Similarproblemshavebeenconsidered,usingdifferenttechniques,in[5,8,9],forstudyingelectricalconductionin
biologicaltissues.Ourapproach,asalreadymentioned,isbasedonadifferentmethod,theperiodicunfoldingmethod,
whichallowsustoavoidtheuseofextensionoperatorsand,hence,todealwithmoregeneralmedia(seeSection2fora
briefdiscussionconcerningtheadvantagesofferedbytheuseofhomogenizationtechniquesbasedonthisgeneralmethod).
Theresultspresentedinthispaperconstitutealsoageneralizationofthoseobtainedin[3,4,10,11].Asamatteroffact,some
ofthemwereannounced,withoutdetailedproofs,in[11].Here,weofferdetailedproofsandconsidermuchmoregeneral
cases(forinstance,wedealherewiththecaseofmoregeneralnonlinearities β(x,uε),g(x, vε−uε)).Correctorresultsand
resultsforthecaseofnonsymmetricmatriceswillbepresentedinafuturepaper.
Theplanofthepaperisasfollows:inSection2,wegivethesettingoftheproblemweaddresshere.Ourmainconvergence
resultisgiveninSection3.Section4isdevotedtotheproofofthisresult.Thepaperendswithafewconclusionsandsome
references.
2. Setting of the problem
LetΩbeaboundeddomainin Rn(n≥3),withaLipschitzboundary ∂Ωconsistingofafinitenumberofconnected
components.Weconsiderthecaseinwhich Ωisaperiodicstructureformedbytwocomponents, Ωε
1andΩε
2,representing
twomaterialswithdifferentfeatures,separatedbyaninterface Γε.Weassumethatboth Ωε
1andΩε
2=Ω\Ωε
1are
connected,butonlyonephase, Ωε
1,reachestheexternalfixedboundaryofthedomain Ω.Here, εrepresentsasmall
parameterrelatedtothecharacteristicsizeofourtworegions.
Moreprecisely,let Y1beaLipschitzopenconnectedsubsetoftheunitcell Y=(0,1)n.LetY2=Y\Y1(seeFig.1).We
supposethatY2hasalocallyLipschitzboundary Γandweassumethattheintersectionsoftheboundaryof Y2withthe
boundaryofYareidenticallyreproducedonoppositefacesofthecell.
Wesupposethatrepeating Ybyperiodicity,theunionofallthesets Y1isconnectedandhasalocally C2boundary.Also,
weassumethattheoriginofthecoordinatesystemissetinaballcontainedinthisunion(see[4]).
Let
Zε= {k∈Zn|εk+εY⊆Ω},
Kε= {k∈Zε|εk±εei+εY⊆Ω,∀i=1,n},
whereeiaretheelementsofthecanonicalbasisof Rn.
Wedefine(seeFig.2)
Ωε
2=int
k∈Kε(εk+εY2)
and
Ωε
1=Ω\Ωε
2
andweset
θ=Y\Y2.
Letα1, β1∈Rsuchthat0 < α1< β1.Wedenoteby M(α1, β1,Y)thesetofallthesquarematrices A∈(L∞(Y))n×n
suchthat,forany ξ∈Rn,wehave (A(y)ξ, ξ) ≥α1|ξ|2,|A(y)ξ| ≤β1|ξ|,almosteverywherein Y.Weconsiderafamilyof
matricesAε(x)=A(x/ε)definedon Ω,whereA∈M(α1, β1,Y)isasymmetricsmooth Y-periodicmatrix.Weshalldenote
thematrixAbyA1inY1andbyA2,respectively,in Y2.

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Fig. 2.Abidimensionalsectionoftheperiodiccellstructure.
If(0,T)isthetimeinterval,weshallanalyzethemacroscopicbehaviorofthesolutionsofthefollowingsystem:


−div(Aε
1∇uε)+β(x,uε)=finΩε
1×(0,T),
−div(Aε
2∇vε)=finΩε
2×(0,T),
Aε
1∇uε·ν=Aε
2∇vε·νonΓε×(0,T),
Aε
1∇uε·ν=αε∂
∂t(vε−uε)+εg(x, vε−uε)onΓε×(0,T),
uε=0 on ∂Ω×(0,T),
uε(0,x)−vε(0,x)=c0(x)onΓε.(1)
Here,νistheunitoutwardnormalto Ωε
1,f∈L2(0,T;L2(Ω)),c0∈H1
0(Ω)andα >0.
Weassumethatthefunction β=β(x, v)iscontinuous,monotonouslynon-decreasingwithrespectto vforanyx
andsuchthat β(x,0)=0.Also,thefunction g=g(x, v)isconsideredtobecontinuouslydifferentiable,monotonously
non-decreasingwithrespectto vforanyxandwithg(x,0)=0.
Wesupposethatthereexist C≥0andtwoexponents qandrsuchthat
|β(x, v)|≤C(1+|v|q) (2)
and


∂g
∂v≤C(1+|v|q),
∂g
∂xi≤C(1+|v|r)1≤i≤n,(3)
with0 ≤q<n/(n−2)andwith0 ≤r<n/(n−2)+q.
Asexamplesofsuchfunctions,wementionthecaseofLangmuirorFreundlichkinetics(see[12]).
Forthespecialcaseofelectricalconductioninbiologicaltissues,wemayconsiderthat f=0, β=0andgis,inR3,
acubicfunction,likeintheFitzHugh–Nagumomodel(see,forinstance,[5]).Thus,forthiscase,in(1)weassumethatthe
normalfluxiscontinuousacrossthecellmembranesandthetransmembranepotential vε−uεonΓεsatisfiesadynamic
condition.Infact,thepotentialhasajumpacrossthemembranes,duetoitscapacitiveandconductivebehavior.Here, αis
themembranecapacityperunitofareaandthetermcontaining gin(1)takesintoaccounttheconductivebehaviorofthe
membrane.
Forresultsconcerningthewellposednessofproblem(1),wereferto[5,9,10,13].
Unfortunately,itisimpossibletosolveacomplicatedmodelsuchas(1)thattakesintoaccountthefine-structuredetails
ofthegeometryofthecompositemedium.Therefore,thecomplicatedmicrostructuremustbeaveraged(homogenized)to
yieldamodelthatdescribesitsaveragedproperties.
UsingtheperiodicunfoldingmethodintroducedbyD.Cioranescu,A.Damlamian,G.Griso,P.DonatoandR.Zaki
(see[1,2]),wecanprovethattheasymptoticbehaviorofthesolutionofourproblemisgovernedbyanewnonlinearsystem
(see(4)).Atamacroscopicscale,thecompositemediumcanberepresentedbyacontinuousmodel,whichdescribesitasthe

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1576 C.Timofte/ComputersandMathematicswithApplications66(2013)1573–1580
superimpositionoftwointerpenetratingcontinuousmedia,coexistingateverypointofthedomain.Intheparticularcase
oftheelectricalconductioninbiologicaltissues,thetissueisconsideredtobeatwo-phasemedium(everypointinspace
iscomposedofacertainfractionofintracellularspaceandafractionofextracellularspaceandateachpointinspacethere
aretwoelectricalpotentialsandtwocurrents).
Oneadvantageofferedbythisapproachisthat,allowingustopassfromthemicroscopicscaletothemacroscopicone,
wecandescribethemacroscopicpropertiesofournonhomogeneoussystemintermsofthepropertiesofitsmicroscopic
structure.Intuitively,thenonhomogeneousrealsystem,havingaverycomplicatedmicrostructure,isreplacedbyafictitious
homogeneousone,whoseglobalcharacteristicsrepresentagoodapproximationoftheinitialsystem.Thehomogenization
methodeliminatesthedifficultiesrelatedtotheexplicitdeterminationofasolutionoftheproblematthemicroscaleand
offersalessdetaileddescription,butonewhichisapplicabletomuchmorecomplexsystems.
Anotheradvantageofferedbyourapproach,basedontheperiodicunfoldingmethod,isthatitallowsustoworkwith
generalmedia(seeRemark5).Fordealingwithsuchtwo-componentdomains,weuseunfoldingoperators,whichmap
functionsdefinedonoscillatingdomainsintofunctionsdefinedonfixeddomains.Insuchaway,wecanavoidtheuse
ofextensionoperatorsand,therefore,wecandealwithmediawithlessregularitythanthoseusuallyconsideredinthe
literature(compositematerialsandbiologicaltissuesarehighlyheterogeneousandtheirinterfacesarenotverysmooth,in
general).
Also,fromthepointofviewofnumericalcomputation,thehomogenizedequations,definedonafixeddomain Ωand
describingtheeffectivebehaviorofoursystem,willhavesimplercoefficients(evenconstantcoefficientsinourmaincase
describedbyTheorem1)and,therefore,itwillbeeasiertobesolvednumericallythantheoriginalequations,whichhave
rapidlyoscillatingcoefficients,aredefinedonacomplicateddomainandsatisfynonlinearconditionsontheboundaries.
Thedependenceontherealmicrostructureisgiventhroughthehomogenizedcoefficients(whichnolongerinvolveany
rapidoscillations).Thehomogenizedsolutioncanbeusedasagoodapproximationofthemicroscopicsolutionforsmall
valuesof ε.Inordertoimprovetheconvergencerate,weplantoobtain,inaforthcomingpaper,correctorresults.Still,
thedegeneracyofthissystembringssomenumericaldifficulties,simulatingthecompositemediumwiththeaidofthe
bidomainmodelturningouttobequiteexpensive.Toincreasetheefficiencyofthehomogenizationmethods,onecanuse
parallelcomputationsorsomereducedalternativehomogenizationmodels.Isthebidomainmodelthe‘‘best’’one?Onehas
tofacetwooppositeviewpoints:tomodelasaccuratelyaspossibleagivenphenomenonandtokeepaslowaspossiblethe
complexityofthemodel,intermsofcomputationaleffortsandinnumberofparameters.Apossiblealternativeisoffered
bythemonodomainmodel ,whichisasimplifiedversionofthebidomainmodel,commonlyusedbecauseitleadstolower
computationalcoststhanthebidomainmodel(see,forinstance,[8]).
Ontheotherhand,itcouldbearguedthattheassumedperiodicityofthemicrostructureisnotaveryrealistichypothesis.
Forinstance,inporouscompositemediaorinbiologicaltissuesitwouldbeinterestingtotakeintoaccounttherandomness
ofthemicrostructure.However,aperiodicrepresentationofsuchmediaprovidesaverygooddescription,inagreement
withtheexperimentalfindings(see[14]).
Wecantreatinaquitesimilarmannerthemoregeneralcaseofaheterogeneousmediummodeledbyamatrix
Aε
0=A0(x,x/ε)orbyamatrixDε=D(t,x/ε),undersuitablehypothesesonthematrices A0andD.Forinstance,wecan
supposethatDisasymmetricmatrix,with D, ∂tD∈L∞(0;T;L∞per(Y))n×nandsuchthat,forany ξ∈Rn, (D(t,x)ξ, ξ) ≥
α2|ξ|2,|D(t,x)ξ| ≤β2|ξ|,almosteverywherein (0,T)×Y,for0 < α2< β2.
Also,letusmentionthatusingtheperiodicunfoldingmethod,wecanrigorouslydeal(see[15])withthecaseinwhich
weconsideranotherrelevantscaling,i.e.thecaseinwhichthedynamictermandtheterminvolving gontheinterfaces
Γεareoftheorderof1 /ε.Suchaproblem,addressedusingdifferenttechniquesin[9],isrelevantfordescribingthe
responseofbiologicaltissuestotheinjectionofelectricalcurrentsintheradiofrequencyrange.Inthiscase,oneobtains,
atthemacroscale,acompletelydifferentmodel(anellipticequationwithmemoryterms).
Ifwedealwithadifferentgeometry,i.e.weconsiderthatonlyonephaseisconnected,whiletheotheroneisdisconnected,
weareledtoadifferentmacroscopicmodel(seeRemark2).
3. The main result
Usingtheperiodicunfoldingmethod,wecanpasstothelimitinthevariationalformulationofproblem(1)andweobtain
theeffectivebehaviorofthesolutionofourmicroscopicmodel.
Theorem 1.Thesolution (uε, vε)ofsystem(1)converges,as ε→0,totheuniquesolution (u, v),withu , v∈L2(0,T;H1
0(Ω)),
∂
∂t(u−v)∈L2(0,T;L2(Ω))andu, v∈C0([0,T];H1
0(Ω)),ofthefollowingmacroscopicproblem:


α|Γ|∂
∂t(u−v)−div(A1∇u)+θβ(x,u)− |Γ|g(x, v−u)=θf inΩ×(0,T),
α|Γ|∂
∂t(v−u)−div(A2∇v)+ |Γ|g(x, v−u)=(1−θ)f inΩ×(0,T),
u(0,x)−v(0,x)=c0(x)onΩ.(4)

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In(4),A1andA2arethehomogenizedmatrices,definedby:
A1
ij=
Y1
aij+aik∂χ1j
∂yk
dy,
A2
ij=
Y2
aij+aik∂χ2j
∂yk
dy
andχ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k=1, . . . ,n,aretheweaksolutionsofthecellproblems
−∇y·(A1(y)∇yχ1k)= ∇yA1(y)ek,y∈Y1,
(A1(y)∇yχ1k)·ν= −A1(y)ek·ν,y∈Γ,
−∇y·(A2(y)∇yχ2k)= ∇yA2(y)ek,y∈Y2,
(A2(y)∇yχ2k)·ν= −A2(y)ek·ν,y∈Γ./square
So,atamacroscopicscale,weobtainanewsystem,whichissimilartothe bidomainmodel,appearinginthecontextof
diffusioninpartiallyfissuredmediaorinthecontextofelectricalactivityoftheheart(forthiscase, f=0, β=0).
Letusnoticethatourmacroscopicmodel(4)isadegenerateparabolicsystem,asthetimederivativesinvolvethe
unknown v−u./square
Remark 2.Ifweconsiderthecaseofadifferentgeometry,i.e.ifweassumethat Ωε
1isstillconnected,but Ωε
2isdisconnected,
thenthehomogenizedmatrix A2=0andsystem(4)consistsinthecouplingofapartialdifferentialequationandanordinary
differentialone,which,inparticularcases,canbesolved.Insuchaway,weremainwithonlyonepartialdifferentialequation
withmemory. /square
Remark 3.Wecanrelaxtheconditionsimposedonnonlinearities gandβ.Forinstance,wecanconsiderthat βandgare
maximalmonotonegraphs,satisfyingsuitablegrowthconditions(see[15]). /square
Remark 4.Wecandeal,usingtheperiodicunfoldingmethod,withotherinterestingcases,obtainedforvariousscalings
ofthediffusiontermsinthetwophasesand,also,foradifferentscalingforthedynamicandthenonlineartermsonthe
imperfectinterfaces,asin[16]. /square
4. Proof of the main result
Weconsiderthevariationalformulationofproblem(1):T
0
Ωε
1Aε
1∇uε· ∇ϕdxdt+T
0
Ωε
2Aε
2∇vε· ∇ϕdxdt
+T
0
Ωε
1β(x,uε)ϕdxdt+αεT
0
Γε(uε−vε)∂
∂t[ϕ]dσdt
+αε
Γε(uε−vε)(0)[ϕ](0)dσ+εT
0
Γεg(x, vε−uε)[ϕ]dσdt=T
0
Ωfϕdxdt, (5)
forany ϕ∈L2(Ω×(0,T))suchthat
ϕ|Ωε
1∈L2(0,T;H1(Ωε
1)), ϕ |Ωε
2∈L2(0,T;H1(Ωε
2)),
[ϕ] ∈H1(0,T;L2(Γε)),
ϕvanisheson ∂Ω×(0,T)andϕvanishesatt=T.Here,wehavedenotedby [ϕ]thedifferenceofthetracesof ϕ|Ωε
1and
ϕ|Ωε
2onΓε.
Let
H1
∂Ω(Ωε
1)= {v∈H1(Ωε
1)|v=0on∂Ω∩∂Ωε
1},
endowedwiththenorm
∥v∥H1
∂Ω(Ωε
1)= ∥∇v∥L2(Ωε
1).
Letusintroducethespace Hε=L2(0,T;H1
∂Ω(Ωε
1))×L2(0,T;H1(Ωε
2)),endowedwiththescalarproduct
(u, ϕ)Hε=T
0
Ωε
1∇u1· ∇ϕ1dxdt+T
0
Ωε
2∇u2· ∇ϕ2dxdt
+εT
0
Γε(u1−u2)(ϕ1−ϕ2)dσdt,∀u=(u1,u2), ϕ=(ϕ1, ϕ2). (6)

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Exactlylikein[4,17],wecanprovethatthenormassociatedtothescalarproduct(6)isequivalenttothestandardnorm
inL2(0,T;H1
∂Ω(Ωε
1))×L2(0,T;H1(Ωε
2)),withconstantsindependentof ε.
Thereexistsauniqueweaksolution (uε, vε)of(5),withuε∈L2(0,T;H1
∂Ω(Ωε
1)), vε∈L2(0,T;H1(Ωε
2))(see,forinstance,
[4,5,9,10,17]).
Undertheabovehypothesesonthedata,usingCauchy–Schwartz,Poincaré’s,Young’sandGronwall’sinequalities,we
canobtainsuitableenergyestimates,independentof ε,foroursolution(see[5,9,10,12]).Moreprecisely,ifwemultiplythe
firstequationin(1)by uε,thesecondoneby vεandweintegrateformallybyparts,weobtain,for0 <t<T,
t
0
Ωε
1Aε
1∇uε· ∇uεdxdτ+t
0
Ωε
2Aε
2∇vε· ∇vεdxdτ+t
0
Ωε
1β(x,uε)uεdxdτ
+εα
2
Γε(vε−uε)2dσ+εt
0
Γεg(x, vε−uε)(vε−uε)dσdτ
=εα
2
Γε(c0)2dσ+t
0
Ωε
1fuεdxdτ+t
0
Ωε
2fvεdxdτ.
Usingtheconditionsimposedfor βandgandourhypothesesonthedata,wehave:
t
0
Ωε
1Aε
1∇uε· ∇uεdxdτ+t
0
Ωε
2Aε
2∇vε· ∇vεdxdτ+ε
Γε(vε−uε)2dσ≤C, (7)
whereCisindependentof ε.Thesecomputationscanbemaderigorousbyusingthetechniquesdevelopedin[5].
Therefore,itfollowsthat (uε, vε)isboundedinHεandthereexistsaconstant C>0,independentof ε,suchthat
∥uε∥L2(0,T;H1
∂Ω(Ωε
1))≤C,
∥vε∥L2(0,T;H1(Ωε
2))≤C
and
∥vε−uε∥L2(0,T;L2(Γε))≤Cε−1
2.
Forprovingourmainresult,weusetwounfoldingoperators, Tε
1andTε
2,whichmapfunctionsdefinedonoscillating
domainsintofunctionsdefinedonfixeddomains.Insuchaway,wecanavoidtheuseofextensionoperators(see[1,17,18]).
Also,weshallmakeuseoftheboundaryunfoldingoperator, Tε
b,introducedin[2].Therefore,usingour aprioriestimates
andthepropertiesoftheabovementionedunfoldingoperators,wecanprovethatthereexist u, v∈L2(0,T;H1
0(Ω)),u∈
L2((0,T)×Ω;H1
per(Y1)),v∈L2((0,T)×Ω;H1
per(Y2))suchthat,uptoasubsequence,for ε→0,wehave:
Tε
1(uε) ⇀uweaklyinL2((0,T)×Ω,H1(Y1)),
Tε
1(∇uε) ⇀∇u+ ∇yuweaklyinL2((0,T)×Ω×Y1),
Tε
2(vε) ⇀ vweaklyinL2((0,T)×Ω,H1(Y2)),
Tε
2(∇vε) ⇀∇v+ ∇yvweaklyinL2((0,T)×Ω×Y2).
Moreover,asin[10],
∂
∂t(u−v)∈L2(0,T;L2(Ω))
and
u, v∈C0([0,T];H1
0(Ω)).
Inordertoobtainthelimitproblem(4),wetake,inafirststep, Φ1,Φ2∈C∞
0(Ω)=D(Ω)andΨ∈C∞
0((0,T))=
D(0,T).Wehave:
T
0
Ωε
1Aε
1∇uε· ∇Φ1Ψdxdt+T
0
Ωε
2Aε
2∇vε· ∇Φ2Ψdxdt
+T
0
Ωε
1β(x,uε)Φ1Ψdxdt+αεT
0
Γε(uε−vε)(Φ2−Φ1)dΨ
dtdσdt
+εT
0
Γεg(x, vε−uε)(Φ2−Φ1)Ψdσdt
=T
0
Ωε
1fΦ1Ψdxdt+T
0
Ωε
2fΦ2Ψdxdt. (8)

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C.Timofte/ComputersandMathematicswithApplications66(2013)1573–1580 1579
Applyingthecorrespondingunfoldingoperatorsin(8),weobtain:
T
0
Ω×Y1Tε
1(Aε
1)Tε
1(∇uε)·Tε
1(∇Φ1)Ψdxdydt +T
0
Ω×Y2Tε
2(Aε
2)Tε
2(∇vε)·Tε
2(∇Φ2)Ψdxdydt
+T
0
Ω×Y1Tε
1(β(x,uε))Tε
1(Φ1)Ψdxdydt +αT
0
Ω×ΓTε
b(uε−vε)Tε
b(Φ2−Φ1)dΨ
dtdxdσdt
+T
0
Ω×ΓTε
b(g(x, vε−uε))Tε
b(Φ2−Φ1)Ψdxdσdt
=T
0
Ω×Y1Tε
1(f)Tε
1(Φ1)Ψdxdydt +T
0
Ω×Y2Tε
2(f)Tε
2(Φ2)Ψdxdydt . (9)
Inviewoftheaboveconvergenceresults,itisobvioushowtopasstothelimitinthelineartermsof(9)definedon Ω×Y1
and,respectively, Ω×Y2(see,fordetails,[1,10–12]).
Forthetermsinvolvingthenonlinearfunctions βandg,letusnoticethat,exactlylikein[12],onecanprovethat,
assuming(2),forany zε⇀zweaklyinH1
0(Ω),weget β(x,zε)→β(x,z)stronglyinLq(Ω),whereq=2n
q(n−2)+n.Notice
thatq≥1.Also,onecanprove(see[12])thatif Hisacontinuouslydifferentiablefunction,monotonouslynon-decreasing,
withH(x, v)=0ifandonlyif v=0andsatisfyingtheassumptions(3),then,forany zε⇀zweaklyinH1
0(Ω),weget
H(x,zε) ⇀H(x,z),weaklyinW1,q
0(Ω).Therefore,usingthepropertiesoftheunfoldingoperators Tε
1and,respectively, Tε
b
(see,forinstance,[2])andLebesgue’sconvergencetheorem,wehave
T
0
Ω×Y1Tε
1(β(x,uε))Tε
1(Φ1)Ψdxdydt →T
0
Ω×Y1β(x,u)Φ1Ψdxdydt ,
αT
0
Ω×ΓTε
b(uε−vε)Tε
b(Φ2−Φ1)dΨ
dtdxdσdt→αT
0
Ω×Γ(u−v)(Φ2−Φ1)dΨ
dtdxdσdt
and
T
0
Ω×ΓTε
b(g(x, vε−uε))Tε
b(Φ2−Φ1)Ψdxdσdt→T
0
Ω×Γg(x, v−u)(Φ2−Φ1)Ψdxdσdt.
Thus,passingtothelimitin(9),with ε→0,weget:
T
0
Ω×Y1A1(∇u+ ∇yu)· ∇Φ1Ψdxdydt +T
0
Ω×Y2A2(∇v+ ∇yv)· ∇Φ2Ψdxdydt
+T
0
Ω×Y1β(x,u)Φ1Ψdxdydt +αT
0
Ω×Γ(u−v)(Φ2−Φ1)dΨ
dtdxdσdt
+T
0
Ω×Γg(x, v−u)(Φ2−Φ1)Ψdxdσdt
=T
0
Ω×Y1fΦ1Ψdxdydt +T
0
Ω×Y2fΦ2Ψdxdydt . (10)
Inasecondstep,wetakethetestfunctions wε
i=εΦi(x)ϕi(x
ε)Ψ(t),withi=1,2,where Φ∈D(Ω), ϕi∈H1
per(Yi),
Ψ∈D((0,T)).Observingthat Tε
i(wε
i)→0stronglyinL2((0,T)×Ω×Yi)andTε
i(∇wε
i)→Φi∇yϕi,stronglyinL2((0,T)×
Ω×Yi),wecanpasstothelimitandweget:
T
0
Ω×Y1A1(∇u+ ∇yu)· ∇yϕ1Φ1Ψdxdydt +T
0
Ω×Y2A2(∇v+ ∇yv)· ∇yϕ2Φ2Ψdxdydt =0. (11)
Adding(10)and(11)andusingstandarddensityarguments,weobtain:
T
0
Ω×Y1A1(∇u+ ∇yu)·(∇xΦ1+ ∇yϕ1)dxdydt
+T
0
Ω×Y2A2(∇v+ ∇yv)·(∇xΦ2+ ∇yϕ2)dxdydt
+T
0
Ω×Y1β(x,u)Φ1dxdydt +αT
0
Ω×Γ∂
∂t(v−u)(Φ2−Φ1)dxdσdt

Author's personal copy
1580 C.Timofte/ComputersandMathematicswithApplications66(2013)1573–1580
+T
0
Ω×Γg(x, v−u)(Φ2−Φ1)dxdσdt
=T
0
Ω×Y1fΦ1dxdydt +T
0
Ω×Y2fΦ2dxdydt , (12)
forΦ1,Φ2∈L2(0,T;H1
0(Ω)),ϕ1∈L2((0,T)×Ω;H1
per(Y1))andϕ2∈L2((0,T)×Ω;H1
per(Y2)).
Thisisexactlythevariationalformulationofthelimitproblem(4).Indeed,ifwerememberthat θ= |Y1|andwetake
Φ1=0and,respectively, Φ2=0,usingthecellproblems,wegetexactly(4).
Letusremarkthatwecaneasilypasstothelimit,with ε→0,intheinitialconditionandweobtain u(0,x)−v(0,x)=
c0(x),∀x∈Ω.
Asuandvareuniquelydetermined(see[5,10]),thewholesequencesofmicroscopicsolutionsconvergetoasolutionof
theunfoldedlimitproblemandthiscompletestheproofofTheorem1.
Remark 5.Theaboveresultscanbeextendedtothecaseinwhich Aεisasequenceofmatricesin M(α1, β1,Ω)suchthat
Tε
i(Aε)→Aa.e.in Ω×Y,
withi=1,2andA=A(x,y)∈M(α1, β1,Ω×Y).Theonlydifferenceisthatinthiscasethehomogenizedmatricesareno
longerconstantanddependon x./square
5. Conclusions
Usingtheperiodicunfoldingmethod,theeffectivebehaviorofthesolutionofsomeproblemsarisinginthemodelingof
diffusionprocessesinaperiodicstructureformedbytwomediawithdifferentproperties,separatedbyanactiveinterface,
wasanalyzed.Twointerestinggeometricsituationswerediscussed,leadingtodifferentmacroscopicmodels(seeTheorem1
andRemark2).
Oneadvantageofferedbyourapproachisthatitallowsustoavoidtheuseofextensionoperatorsand,asaresult,todeal
withmuchmoregeneralmedia.
Oursettingisrelevantforstudyingtheheatconductionincompositematerialswithimperfectinterfacesortheelectrical
conductioninbiologicaltissues.
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