THE TIME-PERIODIC UNFOLDING OPERATOR AND [610103]

THE TIME-PERIODIC UNFOLDING OPERATOR AND
APPLICATIONS TO PARABOLIC HOMOGENIZATION
MICOL AMAR, DANIELE ANDREUCCI, AND DARIO BELLAVEGLIA
Abstract. We apply the method of periodic unfolding to a classical homoge-
nization problem for a parabolic equation. With respect to previous lit erature,
we allow for capacity-like coefficients in the diffusion equation oscillating both
in space and time, with general independent scales. Our approach r elies upon a
generalization of the unfolding technique to the time-periodic case.
1.Introduction
In this paper we develop an approach to the homogenization of para bolic problems
with oscillating coefficients based on the method of periodic unfolding. Specifically
we introduce operators of time-periodic unfolding modeled after th e operators of
space-periodic unfolding introduced and applied in [9, 10, 11, 8]. The fi rst part of
the paper contains results of more general interest which may pos sibly be applied
in different frameworks from the one dealt with here.
Ourinterest inproblemsexhibitingoscillationsintimeoriginallyarisedfro mmath-
ematical models with boundary conditions involving alternating pores (see [20]).
Such conditions switch between a closed state and an open one, eith er periodically
or according to a random scheme. As shown in [4], the limiting behavior o f prob-
lems of this kind sharply depends on the relative scalings of the time an d space
variables; see also [6] for a MonteCarlo test of the model. In [5] oscilla tions in the
boundary conditions have been coupled to time periodic changes in th e diffusivity
coefficient as a device to reproduce the selection capability exhibited by biological
membranes.
Let us compare our approach here to previous literature. The unf olding operator
we define is essentially a suitable extension of its purely spatial count erpart in
[9, 10], andsomeofthetheoryalreadyestablished inthequotedpap erscarries over
to our case. However some significant differences appear in the limitin g behavior
of the operator, due to the degenerate character of the availab le estimates of the
time derivative of the unknown in the approximating differential prob lem. As a
technical remark, we note that in this connection the need arises f or both the space
The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilita ` a
e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), the
second and the third author are members of the Gruppo Nazionale per la Fisica Matematica
(GNFM) of the Istituto Nazionale di Alta Matematica (INdAM)..
1

2
oscillation operator introduced in [10] and the space-time oscillation o perator, see
Definition 2.4.
In [19, 21] the authors use unfolding in the space variables with a par ametric
dependence on time, to study a parabolic problem.
The papers [14, 12, 13] study problems similar to the one investigate d in this pa-
per, by means of two-scale convergence. With respect to those p apers we cover
more general cases in the following instances: First, we allow the spa ce and time
oscillation periods, respectively τandε, to vanish in the limit according to any
rate, instead of assuming that τ=εrfor somer >0. Second, we can handle the
case when the timederivative inthe diffusionequation ismultiplied bya co efficient
oscillating in time, usually arising as a capacity coefficient in applications. On the
other hand we require more regularity for the diffusion matrix. This a ssumption
however enables us to partially answer a question raised in [12], see Re mark 4.3
below, and is used only in the estimation procedure and not in the homo geniza-
tion process. We also quote [15, 18] where the two-scales techniqu e is applied to
multiscale problems.
The cases when the capacity term is constant and τ=εr,r > 0were investi-
gated also in [7, Chapter IV] by means of asymptotic expansion tech niques. There
one can find also some formal comments on the case of an oscillating c apacity
coefficient. Here we deal rigorously with some of such problems.
A case of sign changing capacity oscillating in space was also treated in [2, 3] again
by means of asymptotic expansions. We also quote [17] for thenonlin ear case when
τ=ε.
In Section 2 we introduce the basic definitions and properties of the time-periodic
unfolding periodic operator, which are of general scope. We identif y two possible
limiting behaviors depending on the relative magnitude of τandε, which we call
fast and slow oscillations in time (subsections 2.4 and 2.5 respectively) . Notice
that this classification relies on the degeneracy of the estimate of t he time deriv-
ative, whose L2norm we assume to behave in the limit as τ−m(see (2.37)). The
parametermprovides, roughly speaking, the threshold value of εasτ1−m. As a
matter of fact, m= 1/2in the rest of the paper excepting Section 6.
In Section 3 we introduce the diffusion problem and obtain the relevan t estimates
needed for the homogenization process. Here we state precisely o ur assumptions.
In Section 4 we deal with the homogenization in the case of fast oscilla tions. Ac-
tually one has to discriminate the two subcases
τ
ε2→0,τ
ε2→ℓ>0,asε,τ→0.
In Section 5 we consider the case of slow oscillations, where
ε2
τ→0,asε,τ→0.

3
Section 6 is devoted to a case where a stable estimate in the L2norm of the time
derivative is available, i.e., m= 0, and
τ
ε→ℓ>0,asε,τ→0.
As a consequence a greater generality is possible in the choice of the capacity term
multiplying the time derivative in the equation.
In Sections 4, 5, 6 we determine weak formulations of the homogeniz ed problems.
FinallyinSection7weprovideamorepreciseformulationofsuchproble ms, relying
upon factorization and cell functions.
2.The Time-Periodic Unfolding Operator
2.1.Notation. Throughout the paper ε > 0denotes the space period of the
microstructure, and likewise τ > 0denotes its time period. Though this is not
explicitly stressed by the notation for the sake of simplicity, we alway s assume
that two sequences are given: εi→0,τi→0asi→ ∞. The limiting behavior of
quantities depending on εandτis denoted by
lim
ε ,τ→0
In the other notation for the sake of simplicity we drop as a general rule the
dependence on ε, and write uτ,Tτand so on. The symbol γdenotes a generic
positive constant independent of ε,τ.
2.2.Definitions. LetΩ⊂RNbe a bounded connected open set with Lipschitz
boundary, and set
Y= (0,1)N, Σ = (0,1), Q =Y×Σ, Ω T=Ω×(0,T).
Considering the tiling of RNgiven by the sets ε(ξ+Y),ξ∈ZNwe define
Ξε=/braceleftig
ξ∈ZN, ε(ξ+Y)⊂Ω/bracerightig
,/hatwideΩε=interior

/uniondisplay
ξ∈Ξεε(ξ+Y)

;
/hatwideTτ=/braceleftbigg
t∈(0,T)/vextendsingle/vextendsingle/vextendsingleτ/parenleftbigg/bracketleftbiggt
τ/bracketrightbigg
+ 1/parenrightbigg
≤T/bracerightbigg
, Λτ=/hatwideΩε×/hatwideTτ.
Here and in the definitions below [r]denotes the integer part of r∈R.
Forx∈RNandt∈[0,+∞)we define
/bracketleftbiggx
ε/bracketrightbigg
Y=/parenleftbigg/bracketleftbiggx1
ε/bracketrightbigg
,…,/bracketleftbiggxN
ε/bracketrightbigg/parenrightbigg
,
and also denote
x=ε/parenleftbigg/bracketleftbiggx
ε/bracketrightbigg
Y+/braceleftbiggx
ε/bracerightbigg
Y/parenrightbigg
, t =τ/parenleftbigg/bracketleftbiggt
τ/bracketrightbigg
+/braceleftbiggt
τ/bracerightbigg/parenrightbigg
.

4
Then we introduce the space and the space-time cell containing (x,t)as
Yε(x) =ε/parenleftbigg/bracketleftbiggx
ε/bracketrightbigg
Y+Y/parenrightbigg
, Q τ(x,t) =ε/parenleftbigg/bracketleftbiggx
ε/bracketrightbigg
Y+Y/parenrightbigg
×τ/parenleftbigg/bracketleftbiggt
τ/bracketrightbigg
+ Σ/parenrightbigg
.
The following operator is a space-time version of the space unfolding operator
introduced in [9].
Definition 2.1 (Time-Periodic Unfolding Operator) .ForwLebesgue-measurable
onΩTthe Time-Periodic Unfolding operator Tτis defined as
Tτ(w)(x,y,t,s ) =

w/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg
,(x,y,t,s )∈Λτ×Q,
0, otherwise.
/square
Clearly for w1,w2as in Definition 2.1
Tτ(w1w2) =Tτ(w1)Tτ(w2). (2.1)
We need both an average operator in space-time and one in space on ly.
Definition 2.2 (Local Average Operators) .Letwbeintegrable in ΩT. The space-
time average operator is defined by
Mτ(w)(x,t) =

1
εNτ/integraldisplay
Qτ(x,t)w(ζ,θ)dζdθ,if(x,t)∈Λτ,
0, otherwise.(2.2)
For¯t=τ/parenleftig/bracketleftig
t
τ/bracketrightig
+s/parenrightig
we define the space average operator as
/tildewiderMτ(w)(x,t,s ) =

1
εN/integraldisplay
Yε(x)w(ζ,¯t)dζ,if(x,t,s )∈Λτ×Σ,
0, otherwise.(2.3)
/square
Remark2.3.From our definitions it follows
Mτ(w)(x,t) =/integraldisplay/integraldisplay
QTτ(w)(x,t,y,s )dyds=MQ(Tτ(w))(x,t),(2.4)
where in general MIdenotes the integral average on the set I. We also have
/tildewiderMτ(w)(x,t,s ) =/integraldisplay
YTτ(w)(x,y,t,s )dy=MY(Tτ(w))(x,t,s ).(2.5)
/square
In practice the average operators will be mostly used in connection with the oscil-
lation operators which we define presently.

5
Definition 2.4. Letwbe as in Definition 2.2. The space-time oscillation operator
is defined as
Zτ(w)(x,y,t,s ) = [Tτ(w)− M τ(w)] (x,y,t,s ), (2.6)
and the space oscillation operator is defined as
/tildewideZτ(w)(x,y,t,s ) =/bracketleftig
Tτ(w)−/tildewiderMτ(w)/bracketrightig
(x,y,t,s ). (2.7)
/square
Notice that
/tildewideZτ(w) =Zτ(w)− M Y(Zτ(w)). (2.8)
2.3.Basic Properties of the Operator Tτ.In this Subsection we collect some
properties of the operators defined in Subsection 2.2. First we sta te a list of results
for the sake of further reference; their proofs can be given ess entially as in [10] and
are therefore mostly omitted. Indeed in them the time variable does not play any
special role.
In the following p∈[1,∞)unless otherwise noted. Also, functions depending only
on the microscopic variables (y,s), or only on (x,t), are often considered trivially
extended to ΩT×Q.
Proposition 2.5. The operator Tτ:Lp(ΩT)→Lp(ΩT×Q)is linear and contin-
uous.
In addition for all w∈Lp(ΩT)we have
/ba∇dblTτ(w)/ba∇dblLp(ΩT×Q)≤ /ba∇dblw/ba∇dblLp(ΩT), (2.9)
and/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
ΩTwdxdt−/integraldisplay/integraldisplay
ΩT×QTτ(w)dxdtdyds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay
Λτ|w|dxdt. (2.10)
Lemma 2.6. Letφ∈W1,1(ΩT×Q), and define
φτ(x,t) =φ/parenleftbigg
x,t,x
ε,t
τ/parenrightbigg
, (x,t)∈ΩT, (2.11)
whereφhas been extended by Q-periodicity to ΩT×RN+1. Then inΩT×Q
∂
∂sTτ(φτ) =τTτ/parenleftigg∂φ
∂t/parenrightigg
+Tτ/parenleftigg∂φ
∂s/parenrightigg
, (2.12)
and
∇yTτ(φτ) =εTτ(∇xφ) +Tτ(∇yφ). (2.13)

6
Proof.To prove (2.12) we note
∂
∂sTτ(φτ)(x,t,y,s ) =∂
∂s/bracketleftbigg
φ/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs,/bracketleftbiggx
ε/bracketrightbigg
+y,/bracketleftbiggt
τ/bracketrightbigg
+s/parenrightbigg/bracketrightbigg
χΛτ
=∂
∂s/bracketleftbigg
φ/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs,y,s/parenrightbigg/bracketrightbigg
χΛτ
=τ∂φ
∂t/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs,y,s/parenrightbigg
χΛτ
+∂φ
∂s/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs,y,s/parenrightbigg
χΛτ
=τTτ/parenleftigg∂φ
∂t/parenrightigg
+Tτ/parenleftigg∂φ
∂s/parenrightigg
.
Equation (2.13) can be proved similarly. /square
Proposition 2.7. Forφmeasurable on Q, extended by Q-periodicity to the whole
ofRN×R, define the sequence
φτ(x,t) =φ/parenleftbiggx
ε,t
τ/parenrightbigg
, (x,t)∈RN×R.
Then
Tτ(φτ)(x,y,t,s ) =/braceleftiggφ(y,s), (x,y,t,s )∈Λτ,
0, otherwise.(2.14)
Moreover, if φ∈Lp(Q)asε,τ→0
Tτ(φτ)→φ,strongly inLp(ΩT×Q). (2.15)
If there exist ∇yφ,∂φ
∂s∈Lp(Q)then
∇y(Tτ(φτ))→ ∇ yφ,strongly inLp(ΩT×Q), (2.16)
∂
∂s(Tτ(φτ))→∂
∂sφ,strongly inLp(ΩT×Q). (2.17)
Proposition 2.8 (Convergences) .Let{wτ}be a sequence of functions in Lp(ΩT).
Ifwτ→wstrongly in Lp(ΩT)asε,τ→0, then
Tτ(wτ)→w,strongly inLp(ΩT×Q). (2.18)
If we only assume that (2.18)holds true and that wτ≥C >0, then we have
Tτ(w−1
τ)→w−1,strongly in Lp(ΩT×Q). (2.19)
Ifwτis a bounded sequence of functions in Lp(ΩT),p>1, then up to a subsequence
Tτ(wτ)⇀/hatwidew,weakly inLp(ΩT×Q), (2.20)
and
wτ⇀MQ(/hatwidew),weakly inLp(ΩT). (2.21)

7
Remark 2.9.We apply (2.19) to the case wτ=φτ,φτas in (2.11). Actually
the only classes for which (2.18) is known to hold in this context, are s ums of the
following cases: φ=f1(x,t)f2(y,s),φ∈Lp(Y×Σ;C(ΩT)),φ∈Lp(ΩT;C(Y×Σ)).
In all such cases Tτ(φτ)→φstrongly in Lp(ΩT×Q)See [9, 10, 1]. /square
Thefollowingresultmaygiveafairlyprecisepictureofthecompactne ssofunfolded
sequences of functions.
Proposition 2.10. Letw∈Lp(ΩT). Assume that if h∈RN,z∈R,E⊂ΩT
with |h|+|z|+|E| ≤δthen
/integraldisplay
ΩT|w(x+h,t+z)−w(x,t)|pdxdt+/integraldisplay
E|w(x,t)|pdxdt≤ω(δ),(2.22)
whereω: [0,+∞)→[0,+∞)is an increasing function with ω(0) = 0. In(2.22)
wis extended to 0out ofΩT. Then if |h1|+|h2|+|z1|+|z2| ≤δ,
/integraldisplay
R2N+2|Tτ(w)(x+h1,t+z1,y+h2,s+z2)− T τ(uτ)(x,t,y,s )| ≤γω/parenleftig
γ(δ+ε+τ)/parenrightig
.
(2.23)
Proof.We give the details of the proof for translations in the space variable s; the
general case is similar. Let us denote here for all v:R2N→R,h∈RN,δ>0
vh(x,y) =v(x+h,y), v h(x,y) =v(x,y+h) ;
ΩT(δ) =/braceleftig
(x,t)∈ΩT|dist((x,t),∂Ω T)<δ/bracerightig
;
Λτ−h={(x,t)∈RN+1|(x+h,t)∈Λτ}.
Then we compute
/integraldisplay
R2N+2/vextendsingle/vextendsingle/vextendsingleTτ(w)h− T τ(w)/vextendsingle/vextendsingle/vextendsinglep=/integraldisplay
(Λτ−h)\Λτ/integraldisplay
Q/vextendsingle/vextendsingle/vextendsingleTτ(w)h/vextendsingle/vextendsingle/vextendsinglep+/integraldisplay
Λτ\(Λτ−h)/integraldisplay
Q|Tτ(w)|p
+/integraldisplay
(Λτ−h)∩Λτ/integraldisplay
Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglew/parenleftigg
ε/bracketleftiggx+h
ε/bracketrightigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightigg
−w/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
.
(2.24)
Notice that for a suitable γ=γ(N)
Qτ(x,t)∩[Λτ\(Λτ−h)]/\e}atio\slash=∅impliesQτ(x,t)⊂ΩT/parenleftig
γ(|h|+ε)/parenrightig
;
then by means of a standard change of variable (see [10, Propositio n 2.5]) the
second integral on the right hand side of (2.24) is bounded by
/integraldisplay
ΩT(γ(|h|+ε))|w|pdxdt≤ω(γ(|h|+ε)).
The first integral there can be majorized in the same way.

8
As to the last integral in (2.24) we recall that for any two real numb ersr1,r2we
have
[r1+r2] = [r1] + [r2] +j, j ∈ {0,1}.
Thus
ε/bracketleftiggx+h
ε/bracketrightigg
Y+εy=ε/bracketleftbiggx
ε/bracketrightbigg
Y+ε/bracketleftiggh
ε/bracketrightigg
Y+εξ+εy,
for aξ=ξ(ε,x,h )∈ {0,1}N. Then we have in any case
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglew/parenleftigg
ε/bracketleftiggx+h
ε/bracketrightigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightigg
−w/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
≤2N/summationdisplay
i=1/vextendsingle/vextendsingle/vextendsingle/vextendsinglew/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+ki+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg
−w/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
,
(2.25)
where denoting {ξi}={0,1}Nwe have set
ki=ε/bracketleftiggh
ε/bracketrightigg
Y+εξi, |ki| ≤γ(N)(|h|+ε).
On the right hand side of (2.25) wis defined as 0if its arguments is outside of
ΩT. With this convention, the integral of each summand on the right ha nd side of
(2.25) can be bounded above by
/integraldisplay
Λτ|w(x+ki,t)−w(x,t)|pdxdt≤ω(|ki|)≤ω(γ(|h|+ε)).(2.26)
Next we consider translations in the microscopic space variables; we have
/integraldisplay
R2N+2|Tτ(w)h− T τ(w)|p=/integraldisplay
Λτ×[Q\Qh]/vextendsingle/vextendsingle/vextendsingle/vextendsinglew/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
+/integraldisplay
Λτ×Qh/vextendsingle/vextendsingle/vextendsingle/vextendsinglew/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy+εh,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg
−w/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τs/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
,
(2.27)
where
Qh=Yh×Σ, Y h={y∈Y|y+h∈Y}.
Thelastintegralin(2.27)canbeboundedessentially asin(2.26). The first integral
on the right hand side in (2.27) equals after a change of variable
/integraldisplay
Λhτ|w(x,t)|pdxdt,where Λh
τ=/braceleftbigg
(x,t)∈Λτ|/braceleftbiggx
ε/bracerightbigg
Y∈Y\Yh/bracerightbigg
.

9
We conclude by observing that (the sum below is extended to all cells c ontained
inΛτ)
|Λh
τ| ≤/summationdisplay
i/vextendsingle/vextendsingle/vextendsingleQτ(xi,ti)∩Λh
τ/vextendsingle/vextendsingle/vextendsingle≤γ/summationdisplay
iεNτ|Q\Qh| ≤γ|h|.
/square
Notice that as a consequence of Definition 2.2 and of Lemma 2.6, if w∈W1,p(ΩT)
∇yZτ(w) =∇y/tildewideZτ(w) =εTτ(∇xw), (2.28)
∂
∂sZτ(w) =τTτ(wt). (2.29)
Theorem 2.11. Let{wτ}be a sequence converging strongly to win
Lp(0,T;W1,p(Ω)), asε,τ→0, then
Tτ(∇wτ)→ ∇w, strongly inLp(ΩT×Q), (2.30)
1
ε/tildewideZτ(wτ)→yc· ∇w,strongly inLp(ΩT×Σ;W1,p(Y)),(2.31)
where
yc=/parenleftbigg
y1−1
2,y2−1
2,· · ·,yN−1
2/parenrightbigg
.
Letp>1and let {wτ}be a sequence converging weakly to winLp(0,T,W1,p(Ω)).
Then, up to a subsequence, there exists /tildewidew=/tildewidew(x,y,t,s )∈Lp(ΩT×Σ;W1,p
per(Y)),
MY(/tildewidew) = 0, such that as ε,τ→0
Tτ(∇wτ)⇀∇w+∇y/tildewidew,weakly inLp(ΩT×Q), (2.32)
1
ε/tildewideZτ(wτ)⇀yc· ∇w+/tildewidew,weakly inLp(ΩT×Σ;W1,p(Y)).(2.33)
Proof.Thelimitin(2.30)followsfromthestrongconvergenceof ∇wτandProposi-
tion 2.8. To prove (2.31) we note that, applying the Poincar´ e-Wirtin ger inequality
inYto the function1
ε/tildewideZτ(wτ)−yc· ∇wand (2.30), we get
/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
ε/tildewideZτ(wτ)−yc· ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)≤γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇y/parenleftbigg1
ε/tildewideZτ(wτ)/parenrightbigg
− ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)→0.(2.34)
Now we turn to the proof of (2.32) and (2.33). Since ∇y/parenleftig
1
ε/tildewideZτ(wτ)/parenrightig
=Tτ(∇wτ),
thelimit relation(2.33)implies (2.32). Then notingthat ∇y/parenleftig
1
ε/tildewideZτ(wτ)/parenrightig
isbounded
inLp(ΩT×Q)we have
/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
ε/tildewideZτ(wτ)−yc· ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)≤γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇y/parenleftbigg1
ε/tildewideZτ(wτ)/parenrightbigg
− ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)≤K,(2.35)
whereKis a positive constant independent of εandτ. Then there exists
/tildewidew(x,y,t,s )∈Lp(ΩT×Σ;W1,p(Y))such that, up to a subsequence
1
ε/tildewideZτ(wτ)−yc· ∇w⇀/tildewidew,weakly inLp(ΩT×Σ;W1,p(Y)).(2.36)

10
It is easy to show that MY/parenleftig
1
ε/tildewideZτ(wτ)−yc· ∇w/parenrightig
= 0, so that MY(/tildewidew) = 0. The
Yperiodicity of /tildewidewcan be proven following the lines of the proof in Theorem 3.5
of [10]. /square
Next we deal with results connected with scalings specific to parabo lic problems.
For example the following Proposition should be compared with Propos ition 3.1
of [10], where a different scaling appears.
Proposition 2.12. Letp >1and let {wτ}be a sequence converging weakly in
Lp(0,T;W1,p(Ω))tow, and also satisfying the estimate
/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleτm∂wτ
∂t/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT)≤γ, (2.37)
with 0≤m< 1. Then
Tτ(wτ)⇀w,weakly inLp(ΩT;W1,p(Q)). (2.38)
Proof.Using (2.9), (2.12) and (2.13) owing to the stated weak convergenc e of{wτ}
we have the estimates
/ba∇dblTτ(wτ)/ba∇dblLp(ΩT×Q)≤ /ba∇dblwτ/ba∇dblLp(ΩT)≤γ, (2.39)
/ba∇dbl∇yTτ(wτ)/ba∇dblLp(ΩT×Q)≤ε/ba∇dbl∇wτ/ba∇dblLp(ΩT)≤γε, (2.40)
/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂sTτ(wτ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)≤τ1−m/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleτm∂wτ
∂t/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT)≤γτ1−m,(2.41)
so that there exist a subsequence and /hatwidew∈Lp(ΩT;W1,p(Q))such that
Tτ(wτ)⇀/hatwidew,weakly inLp(Ω;W1,p(Q)), (2.42)
∇yTτ(wτ)→0,strongly inLp(ΩT×Q), (2.43)
∂
∂sTτ(wτ)→0,strongly inLp(ΩT×Q), (2.44)
and∇y/hatwidew=∂/hatwidew
∂s= 0, so that/hatwidewdoes not depend on yands. Then from (2.21) we
have
w(x,t) =MQ(/hatwidew)(x,t) =/hatwidew(x,t).
/square
Next we prove the following
Lemma 2.13. If(2.37)is in force with p>1,0≤m≤1, then
/integraldisplay
ΩT/integraldisplay
Q/vextendsingle/vextendsingle/vextendsingleMτ(w)(x,t)−/tildewiderMτ(w)(x,t,s )/vextendsingle/vextendsingle/vextendsinglepdxdtdyds≤γτp(1−m).(2.45)

11
Proof.For(x,t)∈Λτand recalling that ¯t=τ/parenleftig/bracketleftig
t
τ/bracketrightig
+s/parenrightig
, we have on applying
twice H¨older inequality
Mτ(w)(x,t)−/tildewiderMτ(w)(x,t,s )
=1
εNτ/integraldisplay
Qτ(x,t)/bracketleftig
w(ζ,θ)−w(ζ,¯t)/bracketrightig
dζdθ
≤τ1−1
p
εNτ
/integraldisplay
Qτ(x,t)θ/integraldisplay
¯t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂w
∂λ(ζ,λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
dλdζdθ
1
p/parenleftig
εNτ/parenrightig1−1
p
≤τ1−1
p
εN
p
/integraldisplay
Qτ(x,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂w
∂λ(ζ,λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
dλdζ
1
p
.(2.46)
Then after integrating over ΩT×Qand changing variables
z=ζ−ε/bracketleftig
x
ε/bracketrightig
Y
ε, σ =λ−τ/bracketleftig
t
τ/bracketrightig
τ
we find/integraldisplay
ΩT/integraldisplay
Q/vextendsingle/vextendsingle/vextendsingleMτ(w)(x,t)−/tildewiderMτ(w)(x,t,s )/vextendsingle/vextendsingle/vextendsinglepdxdtdyds
≤/integraldisplay
ΩT/integraldisplay
Qτp/integraldisplay
Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂w
∂λ/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+zε,τ/bracketleftbiggt
τ/bracketrightbigg
+στ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
dσdzdxdtdyds
=τp/integraldisplay
ΩT/integraldisplay
QTτ/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂w
∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep/parenrightigg
(x,z,t,σ )dxdtdzdσ≤γτp(1−m),(2.47)
where we have made use of (2.9) and of (2.37). /square
2.4.Fast oscillations. We assume here that
τ1−m
ε≤C∈(0,+∞), (2.48)
where 0≤m < 1. Actually there are different subcases which are treated in the
following results.
Proposition 2.14. Let{wτ}be a sequence converging strongly to win
Lp(0,T;W1,p(Ω))and satisfying the estimate (2.37)with 0≤m< 1. If
lim
ε ,τ→0τ1−m
ε= 0, (2.49)
then1
εZτ(wτ)→yc· ∇wstrongly inLp(ΩT;W1,p(Q)). (2.50)

12
Proof.To prove (2.50) we first note that
∇y/parenleftbigg1
εZτ(wτ)/parenrightbigg
=∇y/parenleftbigg1
εTτ(wτ)/parenrightbigg
=Tτ(∇xwτ)→ ∇ xw, (2.51)
inLp(ΩT×Q)where we used property (2.13) and (2.18) applied to ∇xw. From
(2.9), (2.12) and assumptions (2.37), (2.49) we obtain
/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂s/parenleftbigg1
εZτ(wτ)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂s/parenleftbigg1
εTτ(wτ)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)=τ
ε/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleTτ/parenleftigg∂wτ
∂t/parenrightigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)
≤τ
ε/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂wτ
∂t/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT)≤γτ1−m
ε→0,(2.52)
asε,τ→0. Having disposed of the convergence of the derivatives, we turn t o
the sequence itself. We may apply the Poincar´ e-Wirtinger inequality inQto the
function Zτ(wτ)/ε−yc· ∇w, since its mean value in Qvanishes. We obtain that
/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
εZτ(wτ)−yc· ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)
≤γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇y/parenleftbigg1
εZτ(wτ)/parenrightbigg
− ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)+γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂s/parenleftbigg1
εZτ(wτ)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)(2.53)
goes to 0asε,τ→0as a consequence of (2.51) and (2.52). /square
Proposition 2.15. Let{wτ}be a sequence converging strongly to winW1,p(ΩT).
Ifτ
ε→ℓasε,τ→0, then
Tτ/parenleftigg∂wτ
∂t/parenrightigg
→∂w
∂t, strongly inLp(ΩT×Q) (2.54)
1
εZτ(wτ)→yc· ∇w+ℓ/parenleftbigg
s−1
2/parenrightbigg∂w
∂tstrongly inLp(ΩT;W1,p(Q)).(2.55)
Proof.Convergence (2.54) follows from the assumed strong convergenc e and from
Proposition 2.8. Equation (2.55) can be proven reasoning as in the pr oof of Propo-
sition 2.14. /square
Theorem 2.16. Letp>1and let {wτ}be a sequence converging weakly to win
Lp(0,T;W1,p(Ω))and satisfying the estimate (2.37)with 0≤m< 1. If(2.49)is
in force, then up to a subsequence there exists /hatwidew∈Lp(ΩT;W1,p
per(Q))such that as
ε,τ→0
Tτ(∇wτ)⇀∇w+∇y/hatwidew,weakly inLp(ΩT×Q), (2.56)
1
εZτ(wτ)⇀yc· ∇w+/hatwidew,weakly inLp(ΩT;W1,p(Q)).(2.57)
Actually MQ(/hatwidew) = 0and
∂/hatwidew
∂s= 0, (2.58)

13
so that/hatwidew=/hatwidew(x,t,y ).
Proof.In order to prove (2.57) we appeal to Poincar´ e-Wirtinger inequalit y as in
the proof of Proposition 2.14. Indeed we have
/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
εZτ(wτ)−yc· ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)≤γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇y/parenleftbigg1
εZτ(wτ)/parenrightbigg
− ∇w/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)
+γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂s/parenleftbigg1
εZτ(wτ)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q).(2.59)
Recalling (2.28) and the stated weak convergence, the first term o n the right hand
side of (2.59) is uniformly bounded on ε,τ. When we recall also (2.29), we have
/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂s/parenleftbigg1
εZτ(wτ)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleτ
εTτ/parenleftigg∂wτ
∂t/parenrightigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)≤γτ1−m
ε.(2.60)
Then the whole right hand side of (2.59) is uniformly bounded on ε,τand there
exists/hatwidew∈Lp(ΩT;W1,p(Q))such that, up to a subsequence
1
εZτ(wτ)−yc· ∇w⇀/hatwidew,weakly inLp(ΩT;W1,p(Q)),(2.61)
that is (2.57).
Since by construction MQ/parenleftig
1
εZτ(wτ)−yc· ∇w/parenrightig
= 0, then MQ(/hatwidew) = 0.
Taking intoaccount(2.28)againweseethatthelimit relation(2.57)imp lies (2.56).
On invoking (2.49), we see that (2.60) and (2.57) imply (2.58).
It remains to prove the Y-periodicity of /hatwidew, which can be done following [10]. /square
Remark2.17.Under the assumptions of Theorem 2.16, we can of course apply also
Theorem 2.11. The two functions /hatwidewand/tildewidewso determined however coincide, since
we may apply Lemma 2.13 in
1
ε/tildewideZτ(wτ)−1
εZτ(wτ) =τ1−m
ε/bracketleftig
Mτ(wτ)−/tildewiderMτ(wτ)/bracketrightig
τ1−m.
/square
Theorem 2.18. Letp>1and let {wτ}be a sequence converging weakly to win
Lp(0,T;W1,p(Ω))and satisfying the estimate (2.37)with 0<m< 1. If(2.48)is
in force, then up to a subsequence, there exists /hatwidew∈Lp(ΩT;W1,p
per(Q))such that as
ε,τ→0(2.56)and(2.57)hold true and MQ(/hatwidew) = 0. Moreover
τ
εTτ/parenleftigg∂wτ
∂t/parenrightigg
→∂/hatwidew
∂s,weakly inLp(ΩT×Q). (2.62)
Proof.TheproofstaysessentiallyunchangedfromtheoneofTheorem2.1 6. Indeed
the only difference is that the rightmost hand side in (2.60) does not t end to 0.
Actually this was used only to prove (2.58) which is not relevant here.

14
However, by the same token, we have to provide an argument to pr ove the Σ-
periodicity of /hatwidew. We introduce a test function ψ∈ D(ΩT×Y), and compute
/integraldisplay
ΩT/integraldisplay
Y1
ε[Zτ(x,y,t, 1)− Z τ(x,y,t, 0)]ψ(x,y,t )dxdydt
=1
ε/integraldisplay
ΩT/integraldisplay
Y/bracketleftbigg
wτ/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg
+τ/parenrightbigg
−wτ/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg/parenrightbigg/bracketrightbigg
ψ(x,y,t )dxdydt
=1
ε/integraldisplay
ΩT/integraldisplay
Ywτ/parenleftbigg
ε/bracketleftbiggx
ε/bracketrightbigg
Y+εy,τ/bracketleftbiggt
τ/bracketrightbigg/parenrightbigg
[ψ(x,y,t −τ)−ψ(x,y,t )]dxdydt
=1
ε/integraldisplay
ΩT/integraldisplay
YTτ(wτ)(x,y,t, 0)[ψ(x,y,t −τ)−ψ(x,y,t )]dxdydt
=τ
ε/integraldisplay
ΩT/integraldisplay
YTτ(wτ)(x,y,t, 0)ψ(x,y,t −τ)−ψ(x,y,t )
τdxdydt.(2.63)
Next we observe the trace inequality
/ba∇dblTτ(wτ)(·,·,·,0)/ba∇dblLp(ΩT×Y)≤γ/ba∇dblTτ(wτ)/ba∇dblLp(ΩT×Q)+γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂sTτ(wτ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)
≤γ/ba∇dblwτ/ba∇dblLp(ΩT)+γτ1−m/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleτm∂wτ
∂t/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleLp(ΩT)≤K.(2.64)
Then using H ¨older inequality we get
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
ΩT/integraldisplay
Y1
ε[Zτ(x,y,t, 1)− Z τ(x,y,t, 0)]ψ(x,y,t )dxdydt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Kτ
ε→0,(2.65)
asε,τ→0. Combining this with (2.57), considering that yc·∇wdoes not depend
ons, and that∂/hatwidew
∂s∈Lp(ΩT×Q), we get
/integraldisplay
ΩT/integraldisplay
Y[/hatwidew(x,y,t, 1)−/hatwidew(x,y,t, 0)]ψ(x,y,t )dxdydt= 0, (2.66)
implying that /hatwidewisΣ-periodic. /square
Remark 2.19.The convergence in (2.62) allows us to avoid the unloading of the
time derivative onto the test function in the homogenization proces s, see (4.17),
(4.18). In turn this avoids the appearance of non-local terms as in [12], see Re-
mark 4.3 below. /square
Inthecasem= 0Theorem2.18isreplacedwiththefollowingstrongerformulation.

15
Theorem 2.20. Letp>1and let {wτ}be a sequence converging weakly to win
W1,p(ΩT), and satisfying estimate (2.37)withm= 0. If
lim
ε ,τ→0τ
ε=ℓ∈(0,+∞), (2.67)
then there exists ˚w∈W1,p(ΩT×Q)such that
Tτ(∇wτ)⇀∇w+∇y˚w, weakly inLp(ΩT×Q),(2.68)
Tτ/parenleftigg∂wτ
∂t/parenrightigg
⇀w t+˚ws
ℓ, weakly inLp(ΩT×Q),(2.69)
1
εZτ(wτ)⇀yc· ∇w+ℓ/parenleftbigg
s−1
2/parenrightbigg
wt+ ˚w,weakly inLp(ΩT;W1,p(Q)).(2.70)
Here ˚wis periodic in Qand is such that MQ(˚w) = 0.
The proof of Theorem 2.20 can be easily given along the lines of the pro of of
Theorem 2.16. The periodicity of ˚wfollows reasoning as in Theorem 3.5 in [10],
since in this case the time derivative is controlled as the space gradien t.
Remark2.21.In fact even under the assumptions of Theorem 2.20, Theorem 2.18
is valid excepting the periodicity of /hatwidew. In fact in Theorem 2.20 ˚w=/hatwidew−ℓ(s−
1/2)wt. /square
2.5.Slow oscillations. We assume here that
lim
ε,τ→0ε
τ1−m= 0, (2.71)
for a given 0≤m< 1.
Proposition 2.22. Let{wτ}be a bounded sequence in Lp(0,T;W1,p(Ω)), satisfy-
ing(2.37)and(2.71). Then
/parenleftbiggε
τr/parenrightbigg1+α1
εZτ(wτ)→0,strongly inLp(ΩT;W1,p(Q)),(2.72)
for allα>0,0<r≤1−m. We can take α= 0ifr<1−m.
Proof.Since Zτ(wτ)has zero average in Q, we may apply Poincar´ e-Wirtinger in-
equality to it, obtaining
/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
εZτ(wτ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)
≤γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇y/parenleftbigg1
εZτ(wτ)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)+γ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂
∂s/parenleftbigg1
εZτ(wτ)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q),(2.73)

16
which yields, when invoking the properties of Tτand (2.37),
/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
εZτ(wτ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)
≤γ/ba∇dblTτ(∇wτ)/ba∇dblLp(ΩT×Q)+γτ
ε/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleTτ/parenleftbigg∂wτ
∂t/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(ΩT×Q)≤K/parenleftbigg
1 +τ1−m
ε/parenrightbigg
.(2.74)
We multiply (2.74) by (ε/τr)1+α,r,αas in the statement, and infer the sought
after limiting relation. /square
Remark 2.23.We single out for future reference (proof of Theorem 5.2) the fol-
lowing immediate consequences of the results in this Section: if w∈W1,p(ΩT)and
ε/τr→0, then for any given α>0,0<r≤1asε,τ→0
Tτ/parenleftigg∂w
∂t/parenrightigg
→∂w
∂t,strongly inLp(ΩT×Q), (2.75)
Tτ(∇w)→ ∇w,strongly inLp(ΩT×Q), (2.76)
1
ε/tildewideZτ(w)→yc· ∇w,strongly inLp(ΩT×Σ;W1,p(Y)),(2.77)
/parenleftbiggε
τr/parenrightbigg1+α1
εZτ(w)→0, strongly inLp(ΩT;W1,p(Q)). (2.78)
/square
3.A Parabolic Homogenization Problem
3.1.Assumptions. Leta:ΩT×Q→R,a1:ΩT×Y→R,a2:ΩT×Σ→R
be measurable functions. We assume that they satisfy the uniform estimates
0<C−1≤a,a 1,a2≤C <∞, (3.1)
for someC >1. Let then
Aτ(x,t) =Aτ/parenleftbigg
x,t,x
ε,t
τ/parenrightbigg
, Aτ=Aτ(x,t,y,s ),
be a sequence of N×Nmatrices such that for all τ >0
/ba∇dblAτ
ij/ba∇dbl∞≤C, i,j = 1,…,N ; Aτξ·ξ≥C−1|ξ|2, ξ ∈RN.(3.2)
We also assume that a1[respectively a2,Aτ] are Lipschitz continuos in t[respec-
tively inx,(t,s)] and that
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂a1
∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂a2
∂xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂Aτ
∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂Aτ
∂s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C, (3.3)
for alli= 1, …,Nand for all relevant arguments. We denote
aτ(x,t) =a/parenleftbigg
x,t,x
ε,t
τ/parenrightbigg
, aτ
1(x,t) =a1/parenleftbigg
x,t,x
ε/parenrightbigg
, aτ
2(x,t) =a2/parenleftbigg
x,t,t
τ/parenrightbigg
.

17
We always assume that Aτ,aτ,aτ
1,aτ
2are measurable in ΩT. This is known to be
the case for functions in the classes of Remark 2.9.
Letf∈L2(ΩT)be the source term in the diffusion equation (see (3.5)). In fact
all our results in this paper are still valid if we more generally allow fto depend
on the unknown, i.e., if we let f:ΩT×R→Rbe measurable and
|f(x,t,u )| ≤g(x,t) +C|u|, (x,t,u )∈ΩT×R, (3.4)
whereg∈L2(ΩT). Moreover we assume fto be Lipschitz continuous in its last
variable uniformly with respect to (x,t)∈ΩT. Essentially this greater generality
is possible owing to the strong convergence result of Corollary 3.4. W e have chosen
to present the proofs in the slightly simpler case above in order to ac hieve a more
compact presentation.
3.2.Estimates. Consider the parabolic problem
aτ
1aτ
2∂uτ
∂t−div(Aτ∇uτ) =f, (x,t)∈ΩT, (3.5)
uτ(x,t) = 0, (x,t)∈∂Ω×[0,T],(3.6)
uτ(x,0) =uτ
0(x), x ∈Ω. (3.7)
Here for a given initial data u0∈L2(Ω), we let {uτ
0}be a sequence in W1,2
0(Ω)
such thatuτ
0→u0strongly in L2(Ω), and /ba∇dbluτ
0/ba∇dblW1,2(Ω)≤C/√τ, whereCdepends
on/ba∇dblu0/ba∇dblL2(Ω).
In the following propositions we assume all the needed smoothness o f the solution
uτ, whose existence is classical under standard regularity assumptio ns (see [16]).
This can be done by means of an approximation procedure of the dat a and coef-
ficients in the equation; the final estimates will depend only on the co nstantCas
above.
Proposition 3.1. Letuτbe the solution to problem (3.5)–(3.7). We have the
standard energy estimate
max
0≤t≤T/integraldisplay
Ωu2
τdx+T/integraldisplay
0/integraldisplay
Ω|∇uτ|2dxdt≤γ, (3.8)
whereγis a constant independent of τ.
Proof.Chooseuτ/aτ
2as a test function in (3.5) and integrate by parts in ΩT, to
get for all ¯t∈(0,T)
1
2/integraldisplay
Ωaτ
1u2
τ(¯t)−1
2/integraldisplay
Ω¯t/integraldisplay
0(aτ
1)tu2
τ+/integraldisplay
Ω¯t/integraldisplay
0Aτ
aτ
2∇uτ· ∇uτ−/integraldisplay
Ω¯t/integraldisplay
0uτAτ
|aτ
2|2∇uτ· ∇aτ
2
=/integraldisplay
Ω¯t/integraldisplay
0fuτ
aτ
2+1
2/integraldisplay
Ωaτ
1(uτ
0)2.

18
Thus by means of Cauchy-Schwarz inequality and of our structura l assumptions
we infer
/integraldisplay
Ω|uτ|2(¯t) +/integraldisplay
Ω¯t/integraldisplay
0|∇uτ|2≤γ
/integraldisplay
Ω¯t/integraldisplay
0|uτ|2+/integraldisplay
Ω¯t/integraldisplay
0|f|2+/integraldisplay
Ω(uτ
0)2
.(3.9)
Next Gronwall’s inequality yields
/integraldisplay
Ω|uτ|2(¯t)≤γ, 0<¯t<T.
Finally on letting ¯t→Tin (3.9) we obtain (3.8). /square
Proposition 3.2. For allτ >0
τT/integraldisplay
0/integraldisplay
Ω/parenleftbigg∂uτ
∂t/parenrightbigg2
dxdt+τmax
0≤t≤T/integraldisplay
Ω|∇uτ|2(t)dx≤γ, (3.10)
whereγis a constant independent of τ.
Proof.We select∂uτ
∂tas a testing function in (3.5) so that on integrating by parts
in the space variables we get
T/integraldisplay
0/integraldisplay
Ωaτ
1aτ
2/parenleftbigg∂uτ
∂t/parenrightbigg2
+T/integraldisplay
0/integraldisplay
ΩAτ∇uτ· ∇∂uτ
∂t=T/integraldisplay
0/integraldisplay
Ωf∂uτ
∂t.(3.11)
Hence
T/integraldisplay
0/integraldisplay
Ωaτ
1aτ
2/parenleftbigg∂uτ
∂t/parenrightbigg2
+1
2T/integraldisplay
0/integraldisplay
Ω∂
∂t(Aτ∇uτ· ∇uτ)
−1
2T/integraldisplay
0/integraldisplay
Ω∂Aτ
∂t∇uτ· ∇uτ−1
2T/integraldisplay
0/integraldisplay
Ω∂Aτ
∂s1
τ∇uτ· ∇uτ=T/integraldisplay
0/integraldisplay
Ωf∂uτ
∂t.(3.12)
The second integral on the left hand side of (3.12) can be evaluated exactly. After
an application of Cauchy-Schwarz inequality we are led to
T/integraldisplay
0/integraldisplay
Ω/parenleftbigg∂uτ
∂t/parenrightbigg2
+/integraldisplay
Ω|∇uτ|2(T)≤γ
τT/integraldisplay
0/integraldisplay
Ω|∇uτ|2+γT/integraldisplay
0/integraldisplay
Ωf2+γ/integraldisplay
Ω|∇uτ
0|2,(3.13)
whence (3.10) follows by taking (3.8) and our assumption /ba∇dbl∇uτ
0/ba∇dblL2(Ω)≤C/√τ
into account, and making the obvious remark that Tin the proof can be replaced
with anyt∈(0,T). /square

19
Proposition 3.3 (Time Compactness) .If0<σ<T/ 2there exists γ=γ(σ)>0
such that for any 0<h<σ/ 2we have
T−σ/integraldisplay
σ/integraldisplay
Ω|uτ(x,t+h)−uτ(x,t)|2dxdt≤γ/parenleftig
1 +/ba∇dbluτ/ba∇dbl2
L2(ΩT)+/ba∇dbl∇uτ/ba∇dbl2
L2(ΩT)/parenrightig√
h.
(3.14)
Proof.Letσ∈(0,T/2),0<h<σ/ 2, and define
ϕh(x,t) =−ζ(t)t+h/integraldisplay
tuτ(x,s)ds,
whereζ∈C1
0(σ/2,T−σ/2)is a nonnegative function such that ζ= 1in(σ,T−σ)
and|ζ′| ≤γ/σ. In this proof for any v=v(x,t)we denote/tildewidev(x,t) =v(x,t+h).
Testing equation (3.5) written at times tand respectively t+hwithϕh/aτ
2and
respectively ϕh//tildewideraτ
2we get on subtracting the two integral formulations
−T/integraldisplay
0/integraldisplay
Ω/bracketleftig
/tildewideraτ
1/tildewideruτ−aτ
1uτ/bracketrightig∂ϕh
∂t−T/integraldisplay
0/integraldisplay
Ω/bracketleftig
/tildewideraτ
1t/tildewideruτ−aτ
1tuτ/bracketrightig
ϕh
+T/integraldisplay
0/integraldisplay
Ω/tildewiderAτ∇/tildewideruτ· ∇/parenleftiggϕh
/tildewideraτ
2/parenrightigg
−T/integraldisplay
0/integraldisplay
ΩAτ∇uτ· ∇/parenleftiggϕh
aτ
2/parenrightigg
=T/integraldisplay
0/integraldisplay
Ω/bracketleftigg/tildewidef
/tildewideraτ
2−f
aτ
2/bracketrightigg
ϕh.(3.15)
The first integral on the left hand side of (3.15) equals
T/integraldisplay
0/integraldisplay
Ω/bracketleftig
/tildewideraτ
1/tildewideruτ−aτ
1uτ/bracketrightig

ζ′t+h/integraldisplay
tuτ(x,s)ds+ζ[/tildewideruτ−uτ]

=T/integraldisplay
0/integraldisplay
Ω[/tildewideruτ−uτ]2ζaτ
1
+T/integraldisplay
0/integraldisplay
Ω/bracketleftig
/tildewideraτ
1/tildewideruτ−aτ
1uτ/bracketrightig
ζ′t+h/integraldisplay
tuτ(x,s)ds+T/integraldisplay
0/integraldisplay
Ω/tildewideruτ[/tildewideruτ−uτ]ζ[/tildewideraτ
1−aτ
1].(3.16)
Clearly the first term on the right hand side of (3.16) essentially is the one esti-
mated in the statement. The second integral there can be majoriz ed by means of
H¨older inequality by
γ/ba∇dblζ′/ba∇dbl∞
/integraldisplay
ΩT−σ/2/integraldisplay
σ/2|/tildewideruτ|2+|uτ|2
1
2
/integraldisplay
ΩT−σ/2/integraldisplay
σ/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+h/integraldisplay
tuτ(x,s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
1
2
≤γ/ba∇dbluτ/ba∇dbl2
L2(ΩT)√
h.(3.17)

20
The third term on the right hand side of (3.16) is estimated, invoking t he time
regularity of a1, by
γ/integraldisplay
ΩT−σ/2/integraldisplay
σ/2|/tildewideruτ|/parenleftig
|/tildewideruτ|+|uτ|/parenrightig/vextendsingle/vextendsingle/vextendsingle/tildewideraτ
1−a1/vextendsingle/vextendsingle/vextendsingle≤γ/ba∇dbluτ/ba∇dbl2
L2(ΩT)h. (3.18)
We turn to estimating the other terms in (3.15). The second integra l there can be
treated as in (3.17). The third and fourth integrals in (3.15) can be b ounded in
the same way, that is
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay
0/integraldisplay
Ω/tildewiderAτ∇/tildewideruτ· ∇/parenleftiggϕh
/tildewideraτ
2/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤γ/integraldisplay
ΩT−σ/2/integraldisplay
σ/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/tildewideruτ
/tildewideraτ
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+h/integraldisplay
t∇uτ(x,s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+γ/integraldisplay
ΩT−σ/2/integraldisplay
σ/2|∇/tildewideruτ|/vextendsingle/vextendsingle/vextendsingle∇/tildewideraτ
2/vextendsingle/vextendsingle/vextendsingle
/tildewideraτ
22/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+h/integraldisplay
tuτ(x,s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤γ/ba∇dbl∇uτ/ba∇dblL2(ΩT)
/integraldisplay
ΩT−σ/2/integraldisplay
σ/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+h/integraldisplay
t∇uτ(x,s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+h/integraldisplay
tuτ(x,s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
1
2
≤γ/ba∇dbl∇uτ/ba∇dblL2(ΩT)/parenleftig
/ba∇dbl∇uτ/ba∇dblL2(ΩT)+/ba∇dbluτ/ba∇dblL2(ΩT)/parenrightig√
h.
(3.19)
Finally the integral on the right hand side of (3.15) is dealt with by a simila r and
even simpler argument, contributing a quantity
γ/ba∇dblf/ba∇dblL2(ΩT)/ba∇dbluτ/ba∇dblL2(ΩT)√
h. (3.20)
Collecting estimates (3.15)–(3.20) we prove at once (3.14). /square
Corollary 3.4. By extracting a subsequence if needed we may assume
uτ→u,strongly in L2(ΩT)and weakly in C(0,T;L2(Ω));(3.21)
∇uτ→ ∇u,weakly inL2(ΩT). (3.22)
Proof.Both claims follow straightforwardly from Propositions 3.1 and 3.3, an d
from classical results. /square
3.3.A special case. Let us look also at the following problem
aτ(x,t)∂uτ
∂t−div/parenleftbigg
Aτ/parenleftbigg
x,x
ε/parenrightbigg
∇uτ/parenrightbigg
=f, (x,t)∈ΩT, (3.23)
uτ(x,t) = 0, (x,t)∈∂Ω×[0,T],(3.24)
uτ(x,0) =u0(x), x ∈Ω. (3.25)
Here we assume u0∈W1,2
0(Ω).

21
Proposition 3.5. Ifu0∈W1,2
0(Ω)then the solution to (3.23)–(3.25)satisfies
T/integraldisplay
0/integraldisplay
Ω/parenleftbigg∂uτ
∂t/parenrightbigg2
dxdt+ max
0≤t≤T/integraldisplay
Ω(uτ(x,t)2+|∇uτ(x,t)|2)dx≤γ, (3.26)
whereγis independent of τ.
Proof.Wemultiply (3.23)against∂uτ
∂tandintegratebypartsinthespacevariables,
reasoning as in the proof of Proposition 3.2. In the simpler case at ha nd we
immediately get
T/integraldisplay
0/integraldisplay
Ωaτ/parenleftbigg∂uτ
∂t/parenrightbigg2
+1
2/integraldisplay
ΩAτ∇uτ(T)· ∇uτ(T)≤1
2/integraldisplay
ΩAτ∇u0· ∇u0+T/integraldisplay
0/integraldisplay
Ωf∂uτ
∂t.
(3.27)
The estimates of∂uτ
∂tand∇uτfollow upon an application of Cauchy-Schwarz
inequality. Finally the estimate of uτis simply a consequence of the standard
formula
uτ(x,t) =u0(x) +t/integraldisplay
0∂uτ
∂z(x,z)dz.
/square
Remark3.6.Clearly Corollary 3.4 is still in force under the assumptions of Propo-
sition 3.5. /square
4.The limit problem in the case of fast oscillations
We look here at the case
τ≤γε2, (4.1)
and assume throughout that there exist bounded functions B:ΩT×Q→RN2,
b1:ΩT×Y→Randb2:ΩT×Σ→Rsuch that
Tτ(Aτ)→B,strongly in L1(ΩT×Q), (4.2)
Tτ(aτ
1)→b1,strongly in L1(ΩT×Y), (4.3)
Tτ(aτ
2)→b2,strongly in L1(ΩT×Σ). (4.4)
We also need assume
Tτ(aτ
1t)⇀b 1t,weakly inL2(ΩT×Y), (4.5)
Tτ(∇aτ
2)→ ∇b2,strongly in L1(ΩT×Σ), (4.6)
and the convergence at time t= 0
Tε(aτ
1(0))⇀b 1(0),weakly inL2(Ω×Y). (4.7)
About the known cases of convergences of the type above see Re mark 2.9.

22
Proposition 4.1. Let(4.1)be in force and let uτbe the solution of problem (3.5)–
(3.7). Then there exist u∈L2(0,T,W1,2(Ω))and/hatwideu∈L2(ΩT;W1,2
per(Q))such that
MQ(/hatwideu) = 0and up to a subsequence
uτ⇀u, weakly inL2(0,T;W1,2(Ω)),(4.8)
Tτ(uτ)⇀u, weakly inL2(ΩT;W1,2(Q)),(4.9)
Tτ(∇uτ)⇀∇u+∇y/hatwideu,weakly inL2(ΩT×Q), (4.10)
τ
εTτ/parenleftigg∂uτ
∂t/parenrightigg
⇀∂/hatwideu
∂s, weakly inL2(ΩT×Q). (4.11)
The convergence uτ→uis in fact strong in L2(ΩT), so that from Proposition 2.8
follows Tτ(uτ)→ustrongly in L2(ΩT×Q).
Proof.The claim readily follows from Proposition 2.12, Theorem 2.18 and Corol-
lary 3.4, by taking into account the estimates proved in Section 3. /square
Theorem 4.2. Let(4.1)–(4.7)be in force, and assume
lim
ε ,τ→0τ
ε2=ℓ∈(0,∞). (4.12)
Then the pair (u,/hatwideu)as in Proposition 4.1 is the unique solution (in the class
specified in the Proposition) of
/integraldisplay
ΩT/integraldisplay
Q/braceleftigg
−u(b1φ)t+ℓ−1b1/hatwideusΨ +B[∇xu+∇y/hatwideu]/bracketleftigg
∇x/parenleftiggφ
b2/parenrightigg
+1
b2∇yΨ/bracketrightigg/bracerightigg
dxdtdyds
=/integraldisplay
ΩT/integraldisplay
Σf
b2φdxdtds+/integraldisplay
Ω/integraldisplay
Yu0(x)φ(x,0)b1(x,0,y)dxdy,(4.13)
for allφ∈W1,2(ΩT)withφ(x,t) = 0on∂Ω×[0,T]andφ(x,T) = 0, and
Ψ∈L2(ΩT×Σ;W1,2
per(Y)).
Proof.First we prove the macroscopic part of (4.13), i.e., the equality itself with
Ψ = 0. To this end we do not need (4.12).
We useφ/aτ
2as a test function for equation (3.5), where φ∈ C∞(ΩT)withφ= 0
on∂Ω×[0,T]andφ(x,T) = 0. Integrating by parts in ΩTwe get
−/integraldisplay
ΩTuτ[aτ
1φt+aτ
1tφ]dxdt+/integraldisplay
ΩTAτ∇uτ·/bracketleftigg∇φ
aτ
2−φ∇aτ
2
(aτ
2)2/bracketrightigg
dxdt
=/integraldisplay
ΩTfφ
aτ
2dxdt+/integraldisplay
Ωuτ
0(x)φ(x,0)aτ
1(x,0)dx.(4.14)

23
Unfolding the equation above we get
−/integraldisplay
ΩT/integraldisplay
Q{Tτ(aτ
1)Tτ(uτ)Tτ(φt) +Tτ(aτ
1t)Tτ(uτ)Tτ(φ)}
+/integraldisplay
ΩT/integraldisplay
QTτ(Aτ)Tτ(∇uτ)Tτ(∇φ)Tτ/parenleftigg1
aτ
2/parenrightigg
−/integraldisplay
ΩT/integraldisplay
QTτ(Aτ)Tτ(∇uτ)Tτ(φ)Tτ(∇aτ
2)/parenleftigg
Tτ/parenleftigg1
aτ
2/parenrightigg/parenrightigg2
=/integraldisplay
ΩT/integraldisplay
QTτ(f)Tτ(φ)Tτ/parenleftigg1
aτ
2/parenrightigg
+/integraldisplay
Ω/integraldisplay
YTτ(uτ
0)Tτ(φ(0))Tτ(aτ
1(0))dxdy+Rτ,
(4.15)
whereRτ=o(1), asε,τ→0. Then taking the limit ε,τ→0and recalling
Proposition 4.1 as well as (2.18), (2.19), (4.2)–(4.7), we get
/integraldisplay
ΩT/integraldisplay
Q/braceleftigg
−b1uφt−b1tuφ+B(∇xu+∇y/hatwideu)/parenleftigg∇φ
b2−φ∇b2
b2
2/parenrightigg/bracerightigg
dxdtdyds
=/integraldisplay
ΩT/integraldisplay
Σf
b2φdxdtds+/integraldisplay
Ω/integraldisplay
Yu0φ(x,0)b1(x,0,y)dxdy,(4.16)
amounting to the differential equation in the macroscopic variables.
Next we turn to the proof of the equation in the microscopic quantit ies, where we
first appeal to (4.12). We use a test function
Φ =τ
εφ(x,t)ψ/parenleftbiggx
ε,t
τ/parenrightbigg
,
whereφis the same as above and ψ∈W1,2
per(Q)is extended periodically both in y
andsto the whole RN+1. So testing (3.5) with Φand then integrating by parts
gives
/integraldisplay
ΩT/braceleftiggτ
εaτ
1aτ
2∂uτ
∂tφψ+τ
εAτ∇uτ·(∇φ)ψ+τ
ε2Aτ∇uτ·(∇yψ)φ/bracerightigg
dxdt
=/integraldisplay
ΩTτ
εfφψdxdt.(4.17)

24
Then unfolding we get
/integraldisplay
ΩT/integraldisplay
Qτ
εTτ(aτ
1)Tτ(aτ
2)Tτ/parenleftbigg∂uτ
∂t/parenrightbigg
Tτ(φ)Tτ(ψ)dxdtdyds
+/integraldisplay
ΩT/integraldisplay
Qτ
ε2Tτ(Aτ)Tτ(∇uτ)Tτ(φ)Tτ(∇yψ)dxdtdyds
=−/integraldisplay
ΩT/integraldisplay
Qτ
εTτ(Aτ)Tτ(∇uτ)Tτ(∇φ)Tτ(ψ)dxdtdyds
+/integraldisplay
ΩTτ
εfφψdxdt+Rτ.(4.18)
In the limit as ε,τ→0the right hand side vanishes. The left hand side, by virtue
of Proposition 4.1, converges to
/integraldisplay
ΩT/integraldisplay
Q{/hatwideusb1b2φψ+ℓB(∇u+∇y/hatwideu)φ∇yψ}dxdtdyds. (4.19)
By the density of the tensor product C∞(ΩT)⊗W1,2
per(Q)inL2(ΩT×Σ;W1,2
per(Y))
these results hold for every Ψ∈L2(ΩT×Σ;W1,2
per(Y)). If we select for the sake of
formal symmetry a test function Φ/aτ
2, minor modifications to the argument above
yield (4.13).
Theuniquenessisprovedinaratherstandardfashion,exploitingth elinearityofthe
problem. If (ui,/hatwideui),i= 1,2, are two solutions of the problem, we essentially take
astestingfunctions φ=u1−u2,Ψ =/hatwideu1−/hatwideu2, afterunloadingthetimederivative on
uiagain, via a Steklov averaging procedure. Here we use also Gronwall’s theorem
and thus, if fis allowed to depend on u, its Lipschitz continuity in u. /square
Remark 4.3.The microscopic part of our homogenized equation (4.13) does not
contain any non-local term, as found instead in [12, formula (3.5)]. I ndeed [12,
Remark 3.4] stated that such a term is zero if, in our notation,∂uτ
∂tis uniformly
bounded in L2(ΩT). Therefore, here we have shown that the non-local term van-
ishes also under the weaker estimate (2.37), m= 1/2.
In our approach this bound follows from the regularity in (3.3). Howe ver the
regularity of the matrix Aτis used only to estimate∂uτ
∂tand is irrelevant in the
homogenization process.
Finally, we remark that the function u1in the notation of [12] can be written as
u1(x,t,y,s ) =/hatwideu(x,y,t,s )− M Y(/hatwideu)(x,t,s ),
for/hatwideuas in Theorem 4.2. /square
Theorem 4.4. Let(4.1)–(4.4)be in force, and assume
lim
ε ,τ→0τ
ε2= 0. (4.20)

25
Let the pair (u,/hatwideu)be as in Proposition 4.1. Then /hatwideu=/hatwideu(x,t,y ), i.e.,/hatwideus= 0, and
(u,/hatwideu)is the solution of
/integraldisplay
ΩT/integraldisplay
Q/braceleftigg
−u(b1φ)t+B[∇xu+∇y/hatwideu]/bracketleftigg
∇x/parenleftiggφ
b2/parenrightigg
+1
b2∇yΨ/bracketrightigg/bracerightigg
dxdtdyds=
/integraldisplay
ΩT/integraldisplay
Σf
b2φdxdt+/integraldisplay
Ω/integraldisplay
Yu0(x)φ(x,0)b1(x,0,y)dxdy,(4.21)
for allφ∈W1,2(ΩT)withφ(x,t) = 0on∂Ω×[0,T]andφ(x,T) = 0, and
Ψ∈L2(ΩT;W1,2
per(Y)).
Proof.The proof of the macroscopic differential equation is the same as in T heo-
rem 4.2.
Concerning the microscopic equation, we remark that when (4.20) is in force, then
/hatwideudoes not depend on s(see Theorem 2.16). Then we test the equation (3.5) with
a function
Φ =εφ(x,t)
aτ
2(x,t)ψ/parenleftbiggx
ε/parenrightbigg
,
withφ∈ C∞(ΩT),φ(x,T) = 0andψ∈W1,2
per(Y)extended periodically to the
whole RN, obtaining
/integraldisplay
ΩTAτ∇uτ(∇yψ)φ
aτ
2=ε/integraldisplay
ΩT/braceleftbigg
aτ
1uτφtψ+aτ
1tuτφψ
−Aτ∇uτ·/parenleftigg∇φ
aτ
2−φ∇aτ
2
(aτ
2)2/parenrightigg
ψ+fφψ
a2/bracerightbigg
dxdt+ε/integraldisplay
Ωuτ
0(x)aτ
1(x,0)φ(x,0)ψ/parenleftbiggx
ε/parenrightbigg
dx.
(4.22)
Clearly the right hand side of (4.22) vanishes as ε,τ→0. When we unfold the
left hand side we obtain in the limit
/integraldisplay
ΩT/integraldisplay
QTτ(Aτ)Tτ(∇uτ)Tτ(φ)Tτ(∇yψ)Tτ/parenleftigg1
aτ
2/parenrightigg
→/integraldisplay
ΩT/integraldisplay
QB
b2(∇u+∇y/hatwideu)φ∇yψ= 0.
(4.23)
Owing to the density of the tensor product C∞(ΩT)⊗W1,2
per(Y)inL2(ΩT;W1,2
per(Y)),
from (4.23) we obtain (4.21) for every Ψ∈L2(ΩT;W1,2
per(Y)), concluding the proof.
The uniqueness of solutions follows as in Theorem 4.2. /square
5.The limit problem in the case of slow oscillations
We consider here the case
lim
ε ,τ→0ε2
τ= 0. (5.1)

26
Proposition 5.1. Let(5.1)be in force and let uτbe the solution of problem (3.5)–
(3.7). Then there exist u∈L2(0,T,W1,2(Ω))and/tildewideu∈L2(ΩT×Σ;W1,2
per(Y))such
thatMY(/tildewideu) = 0and up to a subsequence
uτ⇀u, weakly inL2(0,T;W1,2(Ω)), (5.2)
Tτ(uτ)⇀u, weakly inL2(ΩT;W1,2(Q)), (5.3)
Tτ(∇uτ)⇀∇u+∇y/tildewideu,weakly inL2(ΩT×Q). (5.4)
The convergence uτ→uis in fact strong in L2(ΩT), so that from Proposition 2.8
follows Tτ(uτ)→ustrongly in L2(ΩT×Q).
Proof.The claim follows at once from Theorem 2.11, Proposition 2.12 and Coro l-
lary 3.4, on invoking the estimates of Section 3. /square
Theorem 5.2. Let(4.2)–(4.7)be in force, and also assume (5.1). Then the pair
(u,/tildewideu)as in Proposition 5.1 is the unique solution of the problem
/integraldisplay
ΩT/integraldisplay
Q/braceleftigg
−u(b1φ)t+B[∇xu+∇y/tildewideu]/bracketleftigg
∇x/parenleftiggφ
b2/parenrightigg
+1
b2∇yΨ/bracketrightigg/bracerightigg
dxdtdyds
=/integraldisplay
ΩT/integraldisplay
Σf
b2φdxdt+/integraldisplay
Ω/integraldisplay
Yu0(x)φ(x,0)b1(x,0,y)dxdy,(5.5)
for allφ∈W1,2(ΩT)withφ= 0on∂Ω×[0,T]andφ(x,T) = 0, and Ψ∈
L2(ΩT×Σ;W1,2
per(Y)).
Proof.The macroscopic differential equation (4.16) can be proved as in The o-
rem 4.2.
Next we introduce a test function
Φ =εφ(x,t)ψ/parenleftbiggx
ε,t
τ/parenrightbigg
aτ
2(x,t)−1,
whereφ∈ C∞(ΩT)withφ= 0on∂Ω×[0,T]andφ(x,T) = 0, andψ∈W1,2
per(Q),
withψ(y,0) = 0,ψ(y,1) = 0. We understand ψto be extended periodically both
inyandsto the whole RN+1. Then testing (3.5) with Φand integrating by parts
we get
/integraldisplay
ΩT/braceleftbigg
−ε
τaτ
1uτφψs+Aτ∇uτ·(∇yψ)φ
aτ
2/bracerightbigg
dxdt
=ε/integraldisplay
ΩT/braceleftbigg
uτ(aτ
1φ)tψ−Aτ∇uτ·/parenleftbigg
∇φ
aτ
2/parenrightbigg
ψ+fφψ
aτ
2/bracerightbigg
dxdt.(5.6)

27
The right hand side of (5.6) goes to zero as ε,τ→0. Unfolding the left hand side
we see that it equals
/integraldisplay
ΩT/integraldisplay
Q/braceleftbigg
−ε
τTτ(aτ
1)Tτ(uτ)Tτ(φ)Tτ(ψs)/bracerightbigg
+/integraldisplay
ΩT/integraldisplay
Q/braceleftbigg
Tτ(Aτ)Tτ(∇uτ)Tτ(φ)Tτ(∇yψ)Tτ/parenleftbigg1
aτ
2/parenrightbigg/bracerightbigg
+Rτ=:J1+J2+Rτ.(5.7)
As a consequence of Theorem 2.11 we have as ε,τ→0
J2→/integraldisplay
ΩT/integraldisplay
QB(∇u+∇y/tildewideu)·(∇yψ)φ
b2. (5.8)
Next we show that the term J1is vanishing in the limit. By recalling Definitions
2.2 and 2.4 we find
J1=−ε2
τ/integraldisplay
ΩT/integraldisplay
Q1
εZτ(uτ)Tτ(aτ
1)Tτ(φ)Tτ(ψs)
−ε2
τ/integraldisplay
ΩT/integraldisplay
Q1
εMτ(uτ)Tτ(aτ
1)Tτ(φ)Tτ(ψs) =:J11+J12.(5.9)
By taking into account Proposition 2.22 with m=r= 1/2andα= 1, we see that
J11→0asε,τ→0. We splitJ12again, as in
J12=−ε2
τ/integraldisplay
ΩT/integraldisplay
Q1
εMτ(uτ)Mτ(φ)Tτ(aτ
1)Tτ(ψs)
−ε2
τ/integraldisplay
ΩT/integraldisplay
Q1
εZτ(φ)Mτ(uτ)Tτ(aτ
1)Tτ(ψs) =:J121+J122.(5.10)
Again we have J122→0asε,τ→0by virtue of Remark 2.23 (with m=r= 1/2
andα= 1).
Next we invoke Lemma 2.6 and specifically (2.12) to write
J121=−ε2
τ/integraldisplay
ΩT/integraldisplay
Q1
εMτ(uτ)Mτ(φ)Tτ(aτ
1)∂
∂sTτ(ψ). (5.11)

28
Recalling that ψis zero both at s= 0and ats= 1we have after integrating by
parts and using (2.12)
J121=ε2
τ/integraldisplay
ΩT/integraldisplay
Q1
εMτ(uτ)Mτ(φ)/parenleftbigg∂
∂sTτ(aτ
1)/parenrightbigg
Tτ(ψ)
=ε/integraldisplay
ΩT/integraldisplay
QMτ(uτ)Mτ(φ)Tτ(aτ
1t)Tτ(ψ)→0.(5.12)
By a routine density argument, we see that (5.5) is in force for all te st functions
as claimed in the statement.
The uniqueness of solutions follows as in Theorem 4.2. /square
6.The limit problem in the case m= 0,τ∼ε.
Here we find a homogenized formulation for problem (3.23)–(3.25) in t he special
case where (2.67) holds true. We assume throughout that the req uirements in
Subsection 3.1 are fulfilled. We are also going to require that there ex ist bounded
functionsB:Ω×Y→RN2,b:Q→Rsuch that
Tτ(Aτ)→B,strongly in L1(Ω×Y), (6.1)
Tτ(aτ)→b,strongly in L1(ΩT×Q). (6.2)
Owing to estimate (3.26), in the notation of Section 2 we may take m= 0.
Proposition 6.1. Let(2.67)be in force and let uτbe the solution of problem
(3.23)–(3.25). Then there exist u∈L2(0,T;W1,2(Ω))and˚u∈L2(ΩT;W1,2
per(Q))
such that MQ(˚u) = 0and up to a subsequence
uτ⇀u, weakly inW1,2(ΩT), (6.3)
Tτ(uτ)⇀u, weakly inL2(ΩT;W1,2(Q)),(6.4)
Tτ(∇uτ)⇀∇u+∇y˚u,weakly inL2(ΩT×Q), (6.5)
Tτ/parenleftigg∂uτ
∂t/parenrightigg
⇀∂u
∂t+ℓ−1∂˚u
∂s,weakly inL2(ΩT×Q). (6.6)
The convergence uτ→uis in fact strong in L2(ΩT), so that from Proposition 2.8
follows Tτ(uτ)→ustrongly in L2(ΩT×Q).
Proof.The claim follows from Proposition 2.12, Theorem 2.20 and Remark 3.6, b y
taking into account the estimates proved in Subsection 3.3. /square

29
Theorem 6.2. Let(6.1)–(6.2)be in force. Then the pair (u,˚u)as in Proposi-
tion 6.1 solves
/integraldisplay
ΩT/integraldisplay
Q/braceleftig
(ut+ℓ−1˚us)bφ+B[∇xu+∇y˚u] [∇xφ+∇yΨ]/bracerightig
dxdtdyds
=/integraldisplay
ΩT/integraldisplay
Σfφdxdtds,(6.7)
for allφ∈L2((0,T)×Σ;W1,2(Ω))withφ= 0on∂Ω×[0,T]for a.e.s∈Σ, and
Ψ∈L2(ΩT×Σ;W1,2
per(Y)).
If∇xb∈L∞(ΩT×Q)such solution is unique.
Proof.To prove the macroscopic part of equation (6.7), we test (3.23) wit h the
time-oscillating function
φτ(x,t) =φ/parenleftbigg
x,t,t
τ/parenrightbigg
,
forφ∈L2(Σ;C∞(ΩT)), andφ= 0for(x,t)∈∂Ω×[0,T], obtaining
/integraldisplay
ΩT∂uτ
∂taτφτ+/integraldisplay
ΩTAτ∇uτ· ∇φτ=/integraldisplay
ΩTfφτ. (6.8)
On unfolding we are led to
/integraldisplay
ΩT/integraldisplay
QTτ/parenleftbigg∂uτ
∂t/parenrightbigg
Tτ(aτ)Tτ(φτ) +/integraldisplay
ΩT/integraldisplay
QTτ(Aτ)Tτ(∇uτ)Tτ(∇φτ)
=/integraldisplay
ΩT/integraldisplay
QTτ(f)Tτ(φτ) +Rτ.(6.9)
Then taking the limit ε,τ→0, recalling (6.1), (6.2), Proposition 6.1 and Re-
mark 2.9 we get
/integraldisplay
ΩT/integraldisplay
Q(ut+ℓ−1˚us)bφ+/integraldisplay
ΩT/integraldisplay
QB(∇xu+∇y˚u)∇xφ=/integraldisplay
ΩT/integraldisplay
Σfφ. (6.10)
Next we prove the microscopic part of equation (6.7). For this purp ose we test the
equation (3.23) with a function
εϕ(x,t)ψ/parenleftbiggx
ε,t
τ/parenrightbigg
whereϕ∈ C∞(ΩT)withϕ= 0on∂Ω×[0,T], andψ∈W1,2
per(Q)is extended
periodically both in yandsto the whole RN+1, obtaining
/integraldisplay
ΩTAτ∇uτ·(∇yψ)ϕ=−ε/integraldisplay
ΩTaτ∂uτ
∂tϕψ−ε/integraldisplay
ΩTAτ∇uτ·(∇ϕ)ψ+ε/integraldisplay
ΩTfϕψ.(6.11)

30
The right hand side of equation (6.11) goes to zero as ε,τ→0. Unfolding the left
hand side we see that it equals
/integraldisplay
ΩT/integraldisplay
QTτ(Aτ)Tτ(∇uτ)Tτ(∇yψ)Tτ(ϕ) +Rτ. (6.12)
Recalling Proposition 6.1, as ε,τ→0we get from (6.11)–(6.12)
/integraldisplay
ΩT/integraldisplay
QB(∇xu+∇y˚u) (∇yψ)ϕ= 0. (6.13)
By a routine density argument we see that (6.7) is in force for all tes t functions as
claimed in the statement.
In order to prove uniqueness of solutions, we preliminarily remark th at by virtue
of (6.7) we may write for any solution (u,˚u)
˚u(x,t,y,s ) =u∗(x,t,y ) +u(x,t,s ), (6.14)
whereu∗is the unique solution of
divy(B(x,y)[∇xu+∇yu∗]) = 0
such that MY(u∗) = 0andu∗isY-periodic. Then MΣ(u) = 0and
∂˚u
∂s=∂u
∂s(6.15)
does not depend on y. Next we invoke again the integral equation (6.7). Take
there
φ(x,t,s ) =ϕ(x,t)
MY(b)(x,t,s ), (6.16)
forϕsatisfying the requirements in the statement, and obtain
/integraldisplay
ΩT/integraldisplay
Q/braceleftigg
(ut+ℓ−1˚us)ϕ+B[∇xu+∇y˚u]/bracketleftigg
∇x/parenleftbiggϕ
MY(b)/parenrightbigg
+∇yΨ/bracketrightigg/bracerightigg
dxdtdyds
=/integraldisplay
ΩT/integraldisplay
Σfϕ
MY(b)dxdtds.(6.17)
The contribution of the term ˚usϕvanishes by periodicity. Then we are back to a
formulation similar e.g., to (4.21), and can proceed accordingly. That is to say,
given two solutions (u1,˚u1),(u2,˚u2)we may conclude u1=u2. Thusu∗
1=u∗
2by
the definition of u∗
i. We make use a last time of (6.7), and of (6.15), to infer
/integraldisplay
ΩT/integraldisplay
Σℓ−1(u1s−u2s)MY(b)(x,t,s )φ(x,t,s )dxdtds= 0,
whence
u1s−u2s= 0.
It followsu1=u2since MΣ(u1) =MΣ(u2) = 0. /square

31
Remark6.3.One can see that as a difference fromthe other macroscopic equat ions
obtained in the homogenization limit in this paper, the macroscopic par t of (6.7)
contains a residual microscopic time derivative ˚us. In contrast, the term /hatwideusin
(4.13) belongs to the microscopic equation. See also Remark 7.6. /square
Remark 6.4.The case investigated in this Section is actually covered by Theo-
rem 5.2, if we take aτ=aτ
1aτ
2, so thatb=b1b2. Here we show how to reconcile
equations (5.5) and (6.7). In the latter take
φ(x,t,s ) =ϕ(x,t)
b2(x,t,s ),
whereϕis admissible as in Theorem 5.2. Observe that in the resulting equation,
owing to the periodicity of ˚uins, we have
/integraldisplay
ΩT/integraldisplay
Qℓ−1˚us(x,t,y,s )b1(x,t,y )ϕ(x,t)dxdtdyds= 0.
After integrating by parts the term utb1ϕwe recover (5.5), since Ψ/b2is admissible
whenever Ψis. /square
7.Reduction to the macroscopic scale
In order to reduce the homogenized problems for the three differe nt scalings of the
parameters ε,τto macroscopic formulations, we first introduce the cell functions
χi. Let us denote the elements of the limit matrix in (4.2) by
B(x,t,y,s ) = (bi,j(x,t,y,s ))1≤i,j≤N.
Definition 7.1. If (4.12) is in force, for 1≤i≤Nthe functions χi(x,t,y,s )
satisfy MQ(χi) = 0and are the Q-periodic solutions of the problem
ℓ−1b1(x,y,t )χi s−N/summationdisplay
j,k=1∂
∂yj/parenleftiggbj,k(x,t,y,s )
b2(x,t,s )∂(χi−yi)
∂yk/parenrightigg
= 0inΩT×Q. (7.1)
If (4.20) is in force, for 1≤i≤Nthe functions χi(x,t,y )satisfy MY(χi) = 0and
are theY-periodic solutions of the problem
N/summationdisplay
j,k=1∂
∂yj/parenleftigg
MΣ/parenleftiggbj,k
b2/parenrightigg
(x,t,y )∂(χi−yi)
∂yk/parenrightigg
= 0inΩT×Y. (7.2)
If (5.1) is in force, for 1≤i≤Nthe functions χi(x,t,y,s )satisfy MY(χi) = 0
and are the Y-periodic solutions of the problem
N/summationdisplay
j,k=1∂
∂yj/parenleftiggbj,k(x,t,y,s )
b2(x,t,s )∂(χi−yi)
∂yk/parenrightigg
= 0inΩT×Q. (7.3)
/square
Next we prove

32
Theorem 7.2. Letudenote the limit of the sequence {uτ}of solutions to problems
(3.5)–(3.7), obtained in Theorems 4.2, 4.4 and 5.2.
Thenuis the solution of the following homogenized problem
ahomut−div/parenleftig
Ahom∇u/parenrightig
−Ehom· ∇u=Fhomf, (x,y)∈ΩT,(7.4)
u(x,t) = 0, (x,t)∈∂Ω×(0,T),(7.5)
u(x,0) =u0(x), x ∈Ω,(7.6)
where
ahom(x,t) =MY(b1), (7.7)
Ahom(x,t) =MQ
B
b2/parenleftig
I−[∇yχ1| · · · |∇ yχN]/parenrightig
, (7.8)
Ehom(x,t) =MQ
B/parenleftig
I−[∇yχ1| · · · |∇ yχN]/parenrightig∇xb2
|b2|2
, (7.9)
Fhom(x,t) =MΣ/parenleftbigg1
b2/parenrightbigg
, (7.10)
and theχihave been introduced in Definition 7.1.
Proof.If (4.12) is in force, we factorize as
/hatwideu(x,t,y,s ) =−∇ xu(x,t)·N/summationdisplay
i=1χi(x,t,y,s )ei,(x,t,y,s )∈ΩT×Q,(7.11)
whereχiis defined by (7.1). By using (7.11) in (4.13) with φ= 0we obtain the
problem (7.1) in the microscopic space-time cell, which is satisfied than ks to our
definition of χi. Then considering (4.13) with Ψ = 0and using again (7.11), we
get equation (7.4).
The formulations in the other cases are obtained in a similar way. Name ly if (4.20)
is in force, we use the factorization
/hatwideu(x,t,y ) =−∇ xu(x,t)·N/summationdisplay
i=1χi(x,t,y )ei,(x,t,y )∈ΩT×Y , (7.12)
whereχiis defined by (7.2). If instead (5.1) is in force we write
/tildewideu(x,t,y,s ) =−∇ xu(x,t)·N/summationdisplay
i=1χi(x,t,y,s )ei,(x,y,t,s )∈ΩT×Q,(7.13)
whereχiis defined by (7.3). /square

33
7.1.The casem= 0,τ∼ε.In this case the elements bi,jof the limit matrix in
(6.1) depend only on (x,y).
Definition 7.3. If (2.67) is in force, for 1≤i≤Nthe functions χi(x,y)satisfy
MY(χi) = 0and are the Y-periodic solutions of the problem
N/summationdisplay
j,k∂
∂yj/parenleftigg
bj,k(x,y)∂(χi−yi)
∂yk/parenrightigg
= 0, (x,y)∈Ω×Y . (7.14)
/square
Theorem 7.4. Letudenote the limit of the sequence {uτ}of solutions to problems
(3.23)–(3.25), obtained in Theorem 6.2.
Thenuis the solution of the homogenized problem (7.4)–(7.6), where
ahom(x,t) = 1, (7.15)
Ahom(x,t) =MQ
B
MY(b)/parenleftig
I−[∇yχ1| · · · |∇ yχN]/parenrightig
, (7.16)
Ehom(x,t) =MQ
B/parenleftig
I−[∇yχ1| · · · |∇ yχN]/parenrightig
∇x/parenleftigg1
MY(b)/parenrightigg
,(7.17)
Fhom(x,t) =MΣ/parenleftigg1
MY(b)/parenrightigg
, (7.18)
andχiare as in Definition 7.3.
Proof.In (6.7) we split ˚u(x,t,y,s )as in (6.14), and factorize as
˚u(x,t,y,s ) =−∇ xu(x,t)·N/summationdisplay
i=1χi(x,y)ei+u(x,t,s ),(x,t,y,s )∈ΩT×Q,(7.19)
whereχiis defined by (7.14). By using (7.19) in (6.7) with φ= 0we obtain
the problem (7.14) in the microscopic space cell, which is satisfied than ks to our
definitionof χi. Thenconsidering(6.7)with Ψ = 0andrecallingthat φ=φ(x,t,s ),
we get in the distribution sense
/integraldisplay
Yut(x,t)b(x,t,y,s )dy+/integraldisplay
Yℓ−1˚us(x,t,y,s )b(x,t,y,s )dy−
/integraldisplay
Ydivx(B(∇xu(x,t) +∇y˚u)) =f(x,t).(7.20)
Then using the factorization (7.19) in (7.20) we obtain
MY(b)ut+MY(b)ℓ−1us−divx(MY(B)∇xu)+divx(MY(B∇yχi)∇xu) =f.
(7.21)

34
Next, on dividing by MY(b)and integrating in Σwe get
ut+/integraldisplay
Σℓ−1usds+MQ/parenleftigg
B/parenleftig
I−[∇yχ1| · · · | ∇ yχN]/parenrightig
∇x/parenleftigg1
MY(b)/parenrightigg/parenrightigg
∇xu−
divx/bracketleftigg
MQ/parenleftiggB
MY(b)/parenleftig
I−[∇yχ1| · · · |∇ yχN]/parenrightig/parenrightigg
∇xu/bracketrightigg
=MΣ/parenleftiggf
MY(b)/parenrightigg
.(7.22)
The thesis follows once we note that the second term on the left han d side of (7.22)
vanishes since uisΣ-periodic. /square
Remark7.5.It is worthwhile remarking the following characterization of u. From
equation (7.21) we get the differential equation in the variable s
us=ℓ/bracketleftigg
−ut+1
MY(b)divx/parenleftig
MY/parenleftig
B/parenleftig
I−[∇yχ1| · · · |∇ yχN]/parenrightig/parenrightig
∇xu/parenrightig
+f
MY(b)/bracketrightigg
.
Of course this equation should be understood in the suitable weak se nse of (6.7),
and complemented with the information that uisΣ-periodic and MΣ(u) = 0./square
Remark 7.6.Though ˚usdisappears from the single scale formulation of (6.7),
actually its presence forces one to divide by MY(b)as in (7.22), therefore implying
the structure in (7.15)–(7.18). /square
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Department of Basic and Applied Sciences for Engineering, S apienza University
of Rome, via A.Scarpa 16, 00161 Roma Italy