transcript
Editors: Assyr Abdulle, Jacek Banasiak, Alain Damlamian and Mamadou
Sango
GAKUTO International Series Math. Sci. Appl., Vol.** (2009)
Multiple scales problems in Biomathematics, Mechanics, Physics and
Numerics, pp. i-v
GAKKOTOSHO
Preface
This volume contains a collection of lectures presented at the 2007
CIMPA-UNESCO- South Africa School “Multiple scales problems in
Biomathematics, Mechanics, Physics and Numerics” held at the
African Institute of Mathematical Sciences in Muizenberg, South
Africa on 6th-18th of August 2007. The School primarily focused on
the presentation of the state of the art in homogenization theory,
multiscale methods for homogenization problems, asymptotic methods
as well as modeling issues in a variety of applications in physics
and biology involving multiple scales.
This School was organized under the auspices of the CIMPA (Centre
International de Mathematiques Pures et Appliquees), a non-profit
international organization based in Nice, France, whose purpose, as
a Category 2 Institute of UNESCO, is to promote international
cooperation in higher education and research in mathematics for the
benefit of developing countries (see
http://www.cimpa-icpam.org).
The School consisted of 8 courses delivered by 10 invited lecturers
coming from Cameroon, France, Scotland and South Africa and it
attracted 43 participants from countries as diverse as Burkina
Faso, Cameroon, the Democratic Republic of Congo, Morocco, Nigeria,
Scotland, Slovakia, South Africa, Tunisia and Zimbabwe. The
lectures targeted postgraduate students and young researchers and
thus contained a blend of educational material with survey of
cutting edge research.
In parallel with the School, two mini-workshops were organized: one
on functional ana- lytic methods in applied sciences and the other
on numerical methods for problems arising from multiple scales
models. The talks at the mini-workshops were presented by regular
participants of the School and invited speakers.
This volume is based on the courses given by the invited lecturers.
It also contains three invited lectures delivered during the
workshop. The first part of the book gives a thorough survey of
recent developments in homogenization theory including the periodic
unfolding method and the sigma-convergence theory. This provides an
introduction to these topics as well as an account of its recent
developments. New results in the homogenization of linear and
nonlinear elliptic eigenvalue problems in domains with fine grained
boundaries are also presented. On the computational side, new
numerical methods, the so-called heterogeneous multiscale method,
is discussed for homogenization problems. Several numerical
examples and a detailed convergence theory of various numerical
methods based on finite elements are presented. Here again, the
lecture allows for a rather complete presentation of the
developments of this new method, successful in several applications
over the past few years.
ii CIMPA School Cape Town 2007
The second part of the book discusses the asymptotic analysis of
singularly perturbed problems, applications of the asymptotic
analysis to biology as well as numerical methods for singularly
perturbed problems. Here the reader can see how modeling based on
the recognition of multiple time scales in a complex model allows
to construct a systematic way of aggregating variables leading to a
significant decrease in the dimension of the models. This, in turn,
greatly facilitates its robust analysis without compromising
accuracy of the results. The lectures of this second part cover
aggregation methods in discrete time population models and describe
a modified classical Chapman-Enskog asymptotic procedure which can
be used for aggregation of variables in continuous time population
models and kinetic models. They also provide a survey of numerical
methods designed to treat singularly perturbed problems.
A. Damlamian was the main organizer of the School from the CIMPA
side, which funded participation of African students from outside
South Africa, including two scholarships specif- ically targeted
for young women mathematicians. The local organizing committee
consisted of M. Sango and J. Banasiak. However, the School would
not have been possible without the significant help and
contributions of many other people and institutions. Special thanks
must go to Prof F. Hahne and the staff of AIMS for providing an
excellent infrastructure and support throughout the School. Thus,
the organizers were able to focus purely on academic matters - a
rare feat as far as organization of conferences is concerned. AIMS
also provided financial support for some of the participants. The
organizers are also extremely grateful for financial support
received from the National Research Foundation of South Africa,
which funded all South African participants as well as supported
two lecturers of the School. Or- ganizers received generous support
from the French Embassy in South Africa, the Hanno Rund Fund of the
School of Mathematical Sciences of the University of KwaZulu-Natal
and the Commission for Development and Exchange of the
International Mathematical Union. Last but not least, the School
would not have been successful without the enthusiasm of the
students who duly attended and actively participated in all
lectures and activities for the full two weeks.
December 2008
The web page of the school is :
http://maths.za.net/index.php?cf=5.
Multiple scales problems in Biomathematics, Mechanics, Physics and
Numerics iii
List of participants
School Lecturers
A. Abdulle (UK) P. Auger (France) J. Banasiak (SA) A. Damlamian
(France) P. Donato (France) F. Ebobisse Bille (SA) G. Nguetseng
(Cameroon) E. Perrier (France) B.D. Reddy (SA) M. Sango(SA)
Invited Workshop Lecturers
School Participants
J. Absalom (Zimbabwe) V. Aizebeokhai (Nigeria) A. Al Ahouel
(Tunisia) A. Barka (Morocco) J. Busa (Slovakia) A. Chama (SA) A.
Chirigo (Zimbabwe) A. Goswami (SA) S. Faleye (SA) W. Lamb (UK) M.
A. Luruli (SA) J. Malka Koubemba (DRC) A. Masekela (SA) B. Matadi
Maba (SA) K. Matlawa (SA)
V. Melicher (Slovakia) F. Minani (SA) J. Mtimunye (SA) B. Nana
Nbendjo (Cameroon) L. Nkague Nkamba (Cameroon) K. Okosun (Nigeria)
S. C. Oukoumie Noutchie (SA) J. Y. Semegni (SA) L. Signing
(Cameroon) J. M. Tchoukouegno Ngnotchouye (SA) A. Traore (Ivory
Coast) A. Udomene (Nigeria) J. Urombo (Zimbabwe) T.T. Yusuf
(Nigeria) J. d.D. Zabsonre Jean de Dieu (Burkina Faso)
iv CIMPA School Cape Town 2007
Multiple scales problems in Biomathematics, Mechanics, Physics and
Numerics v
CONTENTS
Part I. Homogenization, elasticity and multiscale methods
1) The Periodic Unfolding Method in Homogenization (D. Cioranescu,
Alain Damla- mian and G. Griso) page 1
2) The periodic unfolding method in perforated domains and
applications to Robin problems (D. Cioranescu, P. Donato and R.
Zaki) page 37
3) Homogenization of linear and nonlinear spectral problems for
higher-order elliptic problems in varying domains (M. Sango) page
69
4) Σ-Convergence of Parabolic Differential Operators (G. Nguetseng)
page 95
5) The Finite Element Heterogeneous Multiscale Method: a
computational strategy for multiscale PDEs (A. Abdulle) page
135
6) Mathematical Aspects of Elastoplasticity (F. Ebobisse Bille and
B. D. Reddy) page 185
Part II. Asymptotic analysis, numerical methods and
applications
1) Asymptotic analysis of singularly perturbed dynamical systems of
kinetic type (J. Banasiak) page 221
2) Aggregation methods of time discrete models: review and
application to host- parasitoid interactions (P. Auger, C. Lett and
T. Nguyen-Huu) page 257
3) Numerical schemes that preserve properties of the solutions of
the Burgers equation for small viscosity (R. Anguelov, J.K. Djoko
and J.M.-S. Lubuma) page 279
4) Implicit-Explicit (IMEX) Schemes and Relaxation Systems (M.K.
Banda) page 303
5) Numerical Methods for Multi-Parameter Singular Perturbation
Problems (K.C. Patidar) page 329
PART I
Homogenization, elasticity
Multiple scales problems in Biomathematics, Mechanics, Physics and
Numerics, pp. 1–35
The Periodic Unfolding Method in Homogenization
D. Cioranescu, Alain Damlamian & G. Griso
Abstract: We give a detailed presentation of the Periodic Unfolding
Method and how it applies to periodic homogenization problems. All
the proofs are included as well as some examples.
1. Introduction.
The notion of two-scale convergence was introduced in 1989 by G.
Nguetseng [27] and further developed by G. Allaire [1] with
applications to periodic homogenization. It was generalized to some
multi-scale problems by A. I. Ene and J. Saint Jean Paulin [17], G.
Allaire and M. Briane in [2], J. L. Lions, D. Lukkassen L. Persson
and P. Wall [25],
In 1990, T. Arbogast, J. Douglas and U. Hornung [5] defined a
“dilation” operation to study homogenization for a periodic medium
with double porosity. This technique was used again by A. Bourgeat,
S. Luckhaus and A. Mikelic [7], G. Allaire and C. Conca [3], G.
Allaire, C. Conca and M. Vanninathan [4], M. Lenczner [20-21], M.
Lenczner et all [22-24], D. Lukkassen, G. Nguetseng and P. Wall
[26], by J. Casado Daz [8] , J. Casado Daz et al. [9-11].
It turns out that the dilation technique reduces two-scale
convergence to weak conver- gence in an appropriate space.
Combining this approach with ideas from Finite Element
approximations, we give here a very general and quite simple
method, the “periodic unfold- ing method”, to study classical or
multi-scale periodic homogenization. It a fixed-domain method (the
dimension of the fixed domain depends on the number of scales) that
applies as well to problems with holes and truss-like structures or
in linearized elasticity. We pre-
2 D. Cioranescu, Alain Damlamian & G. Griso
sented this method in [12]. A preliminary version of the proofs can
be found in the survey of A. Damlamian [15]. Here the complete
proofs of the results announced in [12] are given as well as more
recent developments.
The periodic unfolding method is essentially based on two
ingredients. The first one is the unfolding operator Tε defined in
Section 2, where its properties are investigated. Let be a bounded
open set and Y a reference cell in IRn. By definition, the operator
Tε associates to any function v in Lp(), a function Tεv in Lp(ε×Y
), where ε is the smallest finite union of cells εY containing . An
immediate (and very interesting) property of Tε is that it enables
to transform any integral over in an integral over ε × Y . Indeed,
one has (Proposition 2.6)∫
Tε(w)(x, y) dx dy, ∀w ∈ L1(). (1.1)
If {vε} is a bounded sequence in Lp(), weakly converging to v in
Lp(), the sequence {Tε(vε)} weakly converges to v in Lp(× Y )
(Proposition 2.10). This allows to show that the two-scale
convergence in the Lp()-sense of a sequence of functions {vε}, is
equivalent to the weak convergence of the sequence of unfolded
functions {Tε(vε)} in Lp( × Y ) (Proposition 2.12).
In Section 2 are also introduce an local average operatorMε
Y and an averaging operator
Uε, the later being in some sense, the inverse of the unfolding
operator Tε. The second ingredient of the periodic unfolding method
consists of separating the char-
acteristic scales by decomposing every function belonging to W
1,p() in two parts. In Section 3 this is achieved by using the
local average. In Section 4, the original proof of this
scale-splitting, inspired by the Finite Element Method, is given.
The confrontation of the two method of Sections 3 and 4, is
interesting in itself (Theorem 3.5 and Proposition 4.7). In both
approaches, is written as = 1
ε + ε2 ε where 1
ε is a macroscopic part, designed not to capture the oscillations
of order ε (if there are any ), while the microscopic part 2
ε is designed to do so. The main result states that from any
bounded sequence {wε} in W 1,p(), weakly convergent to some w, one
can always subtract a subsequence (still denoted {wε}) such that wε
= w1
ε + εw2 ε with
(ii) Tε(wε) w weakly in Lp(;W 1,p(Y )),
(iii) Tε(w2 ε) w weakly in Lp(;W 1,p(Y )),
(iv) Tε(∇wε) ∇xw +∇yw weakly in Lp(× Y ),
(1.2)
where w belongs to Lp(;W 1,p per(Y )). Convergence (1.2)(iii) shows
that if the proper scaling
is used, oscillatory behaviour can be turned into weak (or even
strong) convergence, at the price of an increase in the dimension
of the problem.
The Periodic Unfolding Method in Homogenization 3
In Section 5 we apply the periodic unfolding method to a classical
periodic homogeniza- tion problem. We point out that in the
framework of this method, the proof of homoge- nization result is
elementary. It relies essentially on formula (1.1), on the
properties of Tε, and on convergences (1.2).
Section 6 is devoted to a corrector result which holds without any
additional regularity on the data (contrary to all previous proof,
see [6], [13] or [28]). This result follows from the use of the
averaging operator Uε. The idea of using averages to improve
corrector results first appeared in G. Dal Maso and A. Defranceschi
[14].
The periodic unfolding method is particularly well-suited for the
case of multi-scale problems. This is shown in Section 7 by a
simple reiteration argument.
2. Unfolding in Lp−spaces
2.1. The unfolding operator Tε. In IRn, let be an open set and Y a
reference cell (ex. ]0, 1[n, or more generally a set having the
paving property with respect to a basis (b1, · · · , bn) defining
the periods).
By analogy with the notation in the one-dimensional case, for z ∈
IRn, [z]Y denotes the unique integer combination
∑n j=1 kjbj of the periods such that z− [z]Y belongs to Y ,
and
set
{z}Y = z − [z]Y ∈ Y a.e. for z ∈ IRn.
Then for each x ∈ IRn, one has
x = ε ([x ε
Definition of [z]Y and {z}Y
4 D. Cioranescu, Alain Damlamian & G. Griso
We use the following notations: ε = interior
{ x ∈ ,
} .
(2.2)
The set ε is the smallest finite union of εY cells contained in ,
while Λε is the subset of containing the parts from εY cells
intersecting the boundary ∂ (See Figure below).
DEFINITION 2.1. For φ Lebesgue-measurable on , the unfolding
operator Tε is defined
as follows:
Tε(φ)(x, y) =
0 a.e. for (x, y) ∈ Λε × Y.
Observe that the function Tε(φ) is Lebesgue-measurable on × Y and
vanishes for x outside of the set ε.
The domains ε and Λε
The following property of Tε is a simple consequence of Definition
2.1 for v and w
Lebesgue-measurable, it will be used extensively :
Tε(vw) = Tε(v) Tε(w). (2.3)
REMARK 2.2. For f measurable on Y , in order to define the sequence
fε given by fε(x) = f (x ε
) , it is customary to extend the function f by Y−periodicity to
the whole of IRn. With
The Periodic Unfolding Method in Homogenization 5
notation (2.1), it seems simpler to define fε by fε(x) = f
({x
ε
} Y
extend f .
PROPOSITION 2.3. For f measurable on Y , and for fε defined in the
previous remark,
one has
0 a.e. for (x, y) ∈ Λε × Y. (2.4)
If f belongs to Lp(Y ), 1 ≤ p <∞, and if is bounded, then
Tε(fε|)→ f strongly in Lp(× Y ).
REMARK 2.4. For f in Lp(Y ), 1 < p < ∞, it is well-known that
fε| converges weakly
in Lp() to the mean value of f on Y , and not strongly unless f is
a constant. Therefore,
Proposition 2.3 shows that the strong convergence of the unfolding
of a sequence does not
imply strong convergence of the sequence itself.
Like in classical periodic homogenization, two different scales
appear in Definition 2.1: x, the “macroscopic” scale gives the
position of a point in the domain , while
x
ε , the
“microscopic” one, gives the position of a point in the cell Y .
The unfolding operator doubles the dimension of the space and put
all the oscillations in the second variable, separating in this
way, the two scales (see Figures below).
fε(x) = 1 4
Tε(fε) for the function fε(x) above and ε = 1 6
The next two results, essential in the study of the properties of
the unfolding operator, are also straightforward from Definition
2.1.
PROPOSITION 2.5. For p ∈ [1,+∞[, the operator Tε is linear and
continuous from Lp() to Lp(× Y ) . For every φ in L1() and w in
Lp()
(i) 1 |Y |
Proof. Recalling Definition 2.2 of ε, one has
1 |Y |
∫ ×Y
∫ ε×Y
The Periodic Unfolding Method in Homogenization 7
On each (εξ + εY ) × Y , by definition, Tε(φ)(x, y) = φ(εξ + εy) is
constant in x. Hence, each integral in the sum on the right hand
side successively equals∫
(εξ+εY )×Y Tε(φ)(x, y) dx dy = |εξ + εY |
∫ Y
ε
φ(x) dx which gives the result.
Property (iii) in Proposition 2.5 shows that any integral of a
function on , is “almost equivalent” to the integral of its
unfolded on × Y , the ”integration defect” arises only from the
cells intersecting the boundary ∂ and is controlled by its integral
over Λε.
The next proposition, which we call unfolding criterion for
integrals (u.c.i.), is a very useful tool when treating
homogenization problems.
PROPOSITION 2.6. (u.c.i.) If {φε} is a sequence in L1() satisfying∫
Λε
|φε| dx→ 0,
Based on this result, we introduce the following notation:
Notation. If {wε} is a sequence satisfying u.c.i., we write∫
wεdx Tε' 1 |Y |
∫ ×Y
Tε(wε) dxdy.
PROPOSITION 2.7. Let {uε} be a bounded sequence in Lp() with p
∈]1,+∞] and v ∈ Lp ′ () (1/p+ 1/p′ = 1), then∫
∫ ×Y
Tε(uε)Tε(v) dxdy. (2.5)
Suppose ∂ bounded. Let {uε} be a bounded sequence in Lp() and {vε}
a bounded
sequence in Lq() with 1/p+ 1/q < 1, then∫
uεvεdx Tε' 1 |Y |
Proof. Observe that
Consequently, by the Lebesgue’s Dominated Convergence Theorem one
gets ∫
Λε
Λε
|uεv| → 0. This proves (2.5). If ∂ is bounded,
then one immediately has 1Λε → 0, when ε→ 0 in Lr() for every r ∈
[1,∞), in particular
for 1 r = 1
p + 1 q , and this implies (2.6).
COROLLARY 2.8. Let p belong to ]1,+∞[, let {uε} be a sequence in
Lp() such that
Tε(uε) u weakly in Lp(× Y ),
and {vε} be a sequence in Lp ′ () (1/p+ 1/p′ = 1) with
Tε(vε)→ v strongly in Lp ′ (× Y ) and
∫ Λε
∫ ×Y
u v dxdy.
Proof. The result follows from the fact that the sequence {uε vε}
satisfies the u.c.i. by the Holder inequality.
We now investigate the convergence properties related to the
unfolded operator when ε→ 0. For φ uniformly continuous on , with
modulus mφ, it is easy to see that
sup x∈ε,y∈Y
|Tε(φ)(x, y)− φ(x)| ≤ mφ(ε).
So, as ε goes to zero, even though Tε(φ) is not continuous, it
converges to φ uniformly on . By density, and making use of
Proposition 2.5, further convergence properties can be expressed
using the mean value of a function defined on × Y :
DEFINITION 2.9. For Φ ∈ Lp( × Y ), the mean value M Y
(Φ) : Lp( × Y ) → Lp() is
defined as follows:
Φ(x, y) dy. a.e. for x ∈ . (2.7)
Observe that an immediate consequence of this definition is the
estimate
M Y
(Φ)Lp() ≤ |Y |− 1 p ΦLp(×Y ), for every Φ ∈ Lp(× Y ).
The Periodic Unfolding Method in Homogenization 9
PROPOSITION 2.10. Let p belong to [1,+∞[.
(i) For w ∈ Lp(),
(ii) Let {wε} be a sequence in Lp() such that
wε → w strongly in Lp().
Then
Tε(wε)→ w strongly in Lp(× Y ).
(iii) For every relatively weakly compact sequence {wε} in Lp() the
corresponding Tε(wε) is relatively weakly compact in Lp(× Y ).
Furthermore, if
Tε(wε) w weakly in Lp(× Y ),
then
(iv) If Tε(wε) w weakly in Lp(× Y ), then
wLp(×Y ) ≤ lim inf ε→0
|Y | 1 p wεLp(). (2.8)
(v) Suppose p > 1 and let {wε} be a bounded sequence in Lp().
Then, the following
assertions are equivalent:
(a). Tε(wε) w weakly in Lp(× Y ) and lim sup ε→0
|Y | 1 p wεLp() ≤ wLp(×Y ),
(b). Tε(wε)→ w strongly in Lp(× Y ) and
∫ Λε
|wε|p → 0.
Proof. (i) The result is obvious for any w ∈ D(). If w ∈ Lp(), let
φ ∈ D(). Then, by using (iv) from Proposition 2.5,
Tε(w)− wLp(×Y ) = Tε(w − φ) + ( Tε(φ)− φ
) + (φ− w)Lp(×Y )
≤ 2|Y | 1 p w − φLp() + Tε(φ)− φLp(×Y ),
hence,
lim sup ε→0
Tε(w)− wLp(×Y ) ≤ 2|Y | 1 p w − φLp(),
from which statement (i) follows by density.
10 D. Cioranescu, Alain Damlamian & G. Griso
(ii) From Proposition 2.5 (iv), one has the estimate
Tε(wε)− Tε(w)Lp(×Y ) ≤ | Y | 1 p wε − wLp(), ∀w ∈ Lp(),
hence (ii). (iii) For p ∈ (1,∞), by Proposition 2.5 (iv),
boundedness is preserved by Tε. Suppose that Tε(wε) w weakly in
Lp(× Y ) and let ψ ∈ Lp′(). From Proposition 2.7∫
∫ ×Y
Tε(wε)(x, y) Tε(ψ)(x, y) dx dy.
In view of (i), one can pass to the limit in the right-hand side to
obtain
lim ε→0
w(x, y) dy } ψ(x) dx.
For p = 1, one uses the extra property satisfied by weakly
convergent sequences in L1(), in the form of the De La
Vallee-Poussin criterion (which is equivalent to relative weak
compactness): there exists a continuous convex function Φ : IR+ 7→
IR+, such that
lim t→+∞
Φ(t) t
{∫ ×Y
( Φ |Tε(wε)|
} is bounded,
which completes the proof of weak compactness of Tε(wε) in L1( × Y
) in the case of with finite measure. For the case where the
measure of is not finite, a similar argument shows that the
equi-integrability at infinity of the sequence {wε} carries over to
{Tε(wε)}.
If Tε(wε) w weakly in L1(× Y ), let ψ be in D(). For ε sufficiently
small, one has∫
wε(x)ψ(x) dx = 1 |Y |
Tε(wε)(x, y) Tε(ψ)(x, y) dx dy.
In view of (i), one can pass to the limit in the right-hand side to
obtain
lim ε→0
w(x, y) dy } ψ(x) dx.
(iv) Inequality (2.8) is a simple consequence of Proposition 2.5
(ii).
The Periodic Unfolding Method in Homogenization 11
(v) From Proposition 2.5 (i), one has for any φ in L1(),
1 |Y |
∫ ×Y
1 |Y | Tε(wε)pLp(×Y ) +
∫ Λε
This identity implies the required equivalence.
Concerning the converse of (ii) in Proposition 2.10, Remark 2.4
shows that it is not true.
REMARK 2.11. A consequence of (iii) of Proposition 2.10, together
with (iv) of Propo-
sition 2.5, is the following. Suppose the sequence {wε} converges
weakly to w in Lp(). Then Tε(wε) is relatively weakly compact in
Lp(× Y ), and each of its weak-limit points
w, satisfies M Y
(w) = w.
Now recall the following definition from G. Nguetseng [27 ] and G.
Allaire [1 ]: Two-scale convergence. Let p ∈]1,∞[. A bounded
sequence {wε} in Lp() two-scale converges to some w belonging to
Lp( × Y ), whenever, for every smooth function on × Y , the
following convergence holds:∫
w(x, y)(x, y) dxdy.
The following result reduces two-scale convergence to a mere weak
Lp(×Y )-convergence of the unfolded.
PROPOSITION 2.12. Let {wε} be a bounded sequence in Lp() with p
∈]1,∞[. The
following assertions are equivalent :
i). {Tε(wε)} converges weakly to w in Lp(× Y ), ii). {wε} two-scale
converges to w.
Proof. To prove this equivalence, it is enough to check that for
every in a set of ad- missible test-functions for two-scale
convergence (for instance, D(, Lq(Y ))), Tε[(x, x/ε)] converges
strongly to in Lq(× Y ). This follows from the definition of Tε,
indeed
Tε [ ( x, x
+ εy, y ) .
REMARK 2.13. Proposition 2.12 shows that the two-scale convergence
of a sequence in
Lp(), p ∈]1,∞[, is equivalent to the weak−Lp( × Y ) convergence of
the unfolded se-
quence. Notice that by definition, to check the two-scale
convergence one has to use
12 D. Cioranescu, Alain Damlamian & G. Griso
special test functions. To check a weak convergence in the space
Lp( × Y ), one makes
simply use of functions in the dual space Lp ′ (×Y ). Moreover, due
to density properties,
it is sufficient to check this convergence only on smooth functions
from D(× Y ).
2.2. The averaging operator Uε In this section, we consider the
adjoint Uε of Tε which we call averaging operator. To do so, let v
be in Lp(× Y ) and let u be in Lp
′ (). We have successively,
∫ ε×Y
This gives the formula for the averaging operator Uε.
DEFINITION 2.14. For p in [1,∞], the averaging operator Uε : Lp( ×
Y ) → Lp() is
defined as
) dz a.e. for x ∈ ε,
0 a.e. for x ∈ Λε.
Consequently, for ψ ∈ Lp() and Φ ∈ Lp′(× Y ), one has
∫
Φ(x, y) Tε(ψ)(x, y) dxdy. (2.9)
Note that if Φ is continuous on × Y , it is not the case for Uε(Φ)
on . As consequence of the duality (Holder’s inequality), and of
Proposition 2.5 (iv), we get
immediately
PROPOSITION 2.15. Let p belong to [1,∞]. The averaging operator is
linear and contin-
uous from Lp(× Y ) to Lp(). Moreover, for 1 p + 1
p′ = 1,
The Periodic Unfolding Method in Homogenization 13
The operator Uε maps Lp(×Y ) into the space Lp(). It allows to
replace the function x 7→ Φ
( x, {x ε
} Y
) which is meaningless in general, by a function which always makes
sense.
This implies that the largest set of test functions for two-scale
convergence is actually the set Uε(Φ) with Φ in Lp
′ (× Y ).
It is immediate from its definition, that Uε is almost a
left-inverse of Tε since
Uε ( Tε(φ)
0 a.e. for x ∈ Λε, (2.11)
for every φ in Lp(), while
Tε(Uε(Φ))(x, y) =
1 | Y |
0 a.e. for (x, y) ∈ Λε × Y, (2.12)
for every Φ in Lp(× Y ).
PROPOSITION 2.16. (Properties of Uε). Suppose that p is in [1,+∞[.
(i) Let {Φε} be a bounded sequence in Lp(×Y ) such that Φε Φ weakly
in Lp(×Y ). Then
Uε(Φε) MY (Φ) =
Φ( · , y) dy weakly in Lp().
(ii) Let {Φε} be a sequence such that Φε → Φ strongly in Lp(× Y ).
Then
Tε(Uε(Φε))→ Φ strongly in Lp(× Y ).
(iii) Suppose that {wε} is a sequence in Lp(). Then, the following
assertions are
equivalent:
(b) wε 1 ε − Uε(w)→ 0 strongly in Lp().
(iv) Suppose that {wε} is a sequence in Lp(). Then, the following
assertions are
equivalent:
∫ Λε
(d) wε − Uε(w)→ 0 strongly in Lp().
Proof.(i) This follows from Proposition 2.10(ii) by duality for p
> 1. It still holds for p = 1 in the same way as the proof of
Proposition 2.10(ii). Indeed, if the De La Vallee-Poussin
14 D. Cioranescu, Alain Damlamian & G. Griso
criterion is satisfied by the sequence {Φε}, it is also satisfied
by the sequence {Uε(Φε)}, since for F convex and continuous,
Jensen’s inequality implies
F (Uε(Φε))(x) ≤ Uε(F (Φε))(x).
(ii) The proof follows the same lines as that of (i)-(ii) of
Proposition 2.10. (iii) (a)=⇒(b) simply follows from the
application of inequality (2.10) to the function Φ .= Tε(wε)− w,
making use of (2.11).
(b)=⇒(a): by Proposition 2.10 (ii), Tε(wε − Uε(w)) → 0 strongly in
Lp( × Y ), then from the result of (ii) above, Tε(wε)→ w strongly
in Lp(× Y ). (iv) (c)=⇒(d) follows from (iii) and the second
condition of (a).
(d)=⇒(c) follows from (iii) since Uε(w) vanishes on Λε by
definition.
REMARK 2.17. In view of Proposition 2.16 (i), if Tε(wε)→ w weakly
in Lp(×Y ), then
wε 1 ε − Uε(w) converges weakly to 0 in Lp().
The converse cannot make sense. Indeed, let (wε) be such that wε 1
ε − Uε(w) con-
verges weakly to 0 in Lp(). Choose any non-zero v with M Y
(v) = 0. Since Uε(v)
converges weakly to M Y
(v) = 0 by Proposition 2.16 (i), it follows that the weak
limit
of wε 1 ε − Uε(w) is also the weak limit of wε 1
ε − Uε(w + v) making it impossible to
conclude that Tε(wε) converges weakly (would it be to w or to w +
v?)
Comparing the situations for strong and weak convergences, if v is
such that wε 1 ε −
Uε(w + v) and wε 1 ε − Uε(w) converge strongly to 0, then v = 0,
while a weak conver-
gence will only imply that M Y
(v) = 0 .
REMARK 2.18. The condition (iii) (a) of Proposition 2.16 is used by
some authors to
define the notion of “Strong two-scale convergence”. From the above
considerations, con-
dition (c) of Proposition 2.16 (iv) is a better candidate for this
definition.
2.3. The local average operator Mε
Y
Y : Lp() 7→ Lp(), is defined for any φ in
Lp(), p ∈ [1,+∞[, by
Y (φ) is indeed a local average, since
Mε
0 if x ∈ Λε.
REMARK 2.21. Note that Tε(Mε
Y (φ)) =Mε
Y (φ) on the set × Y for any φ in Lp().
PROPOSITION 2.22. (Properties of Mε
Y ). Suppose that p is in [1,+∞[.
(i) For any any φ in Lp(), one has
Mε
(ii) For φ ∈ Lp() and ψ ∈ Lp′(), one has
∫
Mε
Y (ψ) dx.
(iii) Let {wε} be a sequence such that wε → w strongly in Lp().
Then
Mε
Y (wε)→ w strongly in Lp().
The same result holds true with weak convergence in place of the
strong one.
Proof. The proofs of (i) and (ii) are straightforward. The proof of
(iii) is a simple conse- quence of (ii) of Proposition 2.10, and
for the weak topology, of duality.
COROLLARY 2.23. Suppose that p is in [1,+∞[ . Let w be in Lp() and
{wε} be a
sequence in Lp() satisfying
∫ Λε
wε → w strongly in Lp().
Proof. Since w does not depend on y, one has Uε(w) = Mε
Y (w) which, by Proposition
2.22 (iii), converges strongly to w. The conclusion follows from
Proposition 2.16 (iv).
3. Unfolding and gradients
Now, we will examine the properties of unfolding in the case of W
1,p() spaces. Some results require no extra hypotheses, but many
others are sensitive to the boundary con- ditions and the
regularity of the boundary itself. In the next subsection we
consider the former results, while the following subsections will
deal with the latter.
16 D. Cioranescu, Alain Damlamian & G. Griso
Observe first that for w in W 1,p() one has
∇y(Tε(w)) = εTε(∇xw), ∀w ∈W 1,p() a.e. for (x, y) ∈ × Y.
(3.1)
Proposition 2.5 (iv) implies that Tε maps W 1,p() into Lp(;W 1,p(Y
)) .
PROPOSITION 3.1 (gradients in the direction of a period). Let k in
[1, . . . , n] and
{wε} be a bounded sequence in Lp() with p ∈]1,+∞], satisfying
ε ∂wε ∂xk
≤ C. (3.2)
Then, there exist a subsequence (still denoted ε) and w ∈ Lp(× Y )
with ∂w
∂yk ∈ Lp(× Y ), such that
Tε(wε) w weakly in Lp(× Y ),
εTε (∂wε ∂xk
(3.3)
(weakly- ∗ for p =∞). Moreover, the limit function w is 1-periodic
with respect to the yk coordinate.
Proof. Convergences (3.3) are a simple consequence of (3.1) and
(3.2). It remains to prove the periodicity of w. For simplicity, we
assume k = n and write y = (y′, yn), with y′
in Y ′ .= (0, 1)n−1 and yn ∈ (0, 1).
Let ψ ∈ D( × Y ′). By (3.3) Tε(wε) is bounded in Lp( × Y ′;W 1,p(0,
1)) so that Tε(wε)|{yn=s} is bounded in Lp( × Y ′) for every s ∈
[0, 1]. The result follows from the following computation with an
obvious change of variable:∫
×Y ′
] ψ(x, y′) dx dy ′
= ∫
] dx dy′,
] dx dy′,
which goes to zero.
Applying the pevious result for all k = 1, ·, n at once, we
get
COROLLARY 3.2. Let {wε} in W 1,p() with p ∈]1,+∞[, and assume that
{wε} is a
bounded sequence in Lp() satisfying
ε∇wεLp() ≤ C.
The Periodic Unfolding Method in Homogenization 17
Then, there exist a subsequence (still denoted ε) and w ∈ Lp(;W
1,p(Y )), such that
{ Tε(wε) w weakly in Lp(;W 1,p(Y )),
εTε(∇xwε) ∇yw weakly in Lp(× Y ).
Moreover, the limit function w is Y−periodic, i.e. belongs to Lp(;W
1,p per(Y )), where
W 1,p per(Y ) denotes the Banach space of Y−periodic functions inW
1,p
loc (IRn) with theW 1,p(Y ) norm.
COROLLARY 3.3. Let p be in ]1,+∞[ and {wε} be a sequence converging
weakly in
W 1,p() to w. Then,
Tε(wε) w weakly in Lp(;W 1,p(Y )).
Furthermore, if wε converges strongly to w in Lp() (e.g. W 1,p() is
compact in Lp()), the above convergence is strong.
Proof. By hypothesis, using (3.1) gives estimates
Tε(wε)Lp(×Y ) ≤ C,
∇y(Tε(wε))Lp(×Y ) ≤ εC,
so that there exist a subsequence (still denoted ε) and w in Lp(;W
1,p(Y )) such that
Tε(wε) w weakly in Lp(;W 1,p(Y )), (3.4)
with ∇yw = 0. Consequently, w does not depend on y, and Proposition
2.10 (iii) im- mediately gives that w = M
Y (w) = w. Moreover, convergence (3.4) holds for the entire
sequence ε. If wε converges strongly to w in Lp(), so does Tε(wε)
by Proposition 2.10 (ii).
PROPOSITION 3.4. Suppose that p is in [1,+∞[. Let (wε) be a
sequence which converges
strongly to some w in W 1,p(). Then,
(i) Tε(∇wε)→ ∇w strongly in Lp(× Y ),
(ii) 1 ε
where
Proof. Set
) ,
which has mean value zero in Y . Since ∇yZε = 1 ε ∇y ( Tε ( wε
))
= Tε ( ∇wε
∇yZε → ∇w strongly in Lp(× Y ),
which is the first assertion of the proposition. To prove (ii),
recall the Poincare-Wirtinger inequality in Y
∀ψ ∈W 1,p(Y ), ψ −M
Y (ψ) Lp(Y )
Applying it to the function Zε − yc · ∇w, gives
Zε − yc · ∇wLp(×Y ) ≤ C∇yZε −∇wLp(×Y ), (3.6)
which concludes the proof.
THEOREM 3.5. Suppose that p is in ]1,+∞[. Let {wε} be a sequence
which converges
weakly to some w in W 1,p() and strongly in Lp(). Up to a
subsequence (still denoted
ε), there exists some w in Lp(;W 1,p per(Y )) such that
(i) Tε(∇wε) ∇w +∇yw weakly in Lp(× Y ),
(ii) 1 ε
(3.7)
(w) = 0.
Proof. Following the same lines as in the previous proof,
introduce
Zε = 1 ε
) ,
which has mean value zero in Y . Since ∇yZε = Tε ( ∇wε
) , (ii) implies (i).
To prove (ii), note that the sequence {∇yZε} is bounded in Lp(×Y ).
Hence, by (3.5),Zε− yc · ∇wLp(×Y ) is bounded, and there exists w
in Lp(;W 1,p(Y )) such that, up to
a subsequence,
Since, by construction, M Y
(yc) vanishes, so does M Y
(w).
The Periodic Unfolding Method in Homogenization 19
It remains to prove the Y−periodicity of w. This is obtained in the
same way as in the proof of Proposition 3.1 using a test function ψ
∈ D(× Y ′). One has successively,∫
×Y ′
] ψ(x, y′) dx dy ′
= ∫
] dx dy′,
1 ε
] dx dy′.
− ∫
Similarly, noticing that (yc · ∇w)(y ′, 1)− (yc · ∇w)(y ′, 0) =
∂w
∂xn , we obtain∫
= ∫
∫ ×Y ′
w(x) ∂ψ
∂xn (x, y′) dx dy′.
This, with (3.8) and using convergence (3.7) (ii), shows that∫ ×Y
′
[ w(x, (y ′, 1))− w(x, (y ′, 0)
] ψ(x, y′) dx dy ′ = 0,
so that w is yn−periodic. The same holds in the directions of all
the other periods.
Theorem 3.5 can be generalized to the case of W k,p()−spaces with k
≥ 1 and p ∈ ]1,+∞[ . To do so, introduce the notation Dr, r = (r1,
. . . , rn) ∈ INn with |r| = r1 + . . .+ rn ≤ k:
Dr x =
Actually, the following result holds:
THEOREM 3.6. Let {wε} be a sequence converging weakly in W k,p() to
w, k ≥ 1 and p ∈]1,+∞[. Then, there exist a subsequence (still
denoted ε) and w in the space
Lp(;W k,p per (Y )) such that the following convergence
holds:{
Tε(Dl xwε) Dl
Tε(Dl xwε) Dl
(3.9)
20 D. Cioranescu, Alain Damlamian & G. Griso
Furthermore, if wε converges strongly to w in W k−1,p() (e.g. W
1,p() is compact in
Lp()), the above convergences for |l| ≤ k − 1 are strong in Lp(;W
k−l,p(Y )).
Proof. We briefly prove the result for k = 2. The same argument
generalizes for k > 2. If |l| = 1, the first convergence in
(3.9) follows directly from Proposition 2.11. Set
Wε = 1 ε2
Y
( ∇wε
)] The sequence {wε} is bounded in W 2,p(), hence proceding as in
the proof of Proposition 2.22(iii), one obtains Wε
Lp(×Y )
) with |l| = 2.
This implies that the sequence {Wε} is bounded in Lp(;W 2,p(Y )).
Therefore, there exist a subsequence (still denoted ε) and w ∈
Lp(;W 2,p(Y )) such that
Wε w weakly in Lp(;W 2,p(Y )), ∂Wε
∂yi =
(3.10)
xwε) Dl yw weakly in Lp(× Y ), |l| = 2. (3.11)
Now we apply Theorem 3.5 to each of the derivatives ∂wε ∂xi
, i ∈ {1, . . . , n}. There exist a
subsequence (still denoted ε) and wi ∈ Lp(;W 1,p per(Y )) such that
M
Y (wi) ≡ 0 and
From (3.10) follows:
Set w = w − 1 2
n∑ i,j=1
∂xi∂xj . By construction, the function w belongs
to Lp(;W 2,p(Y )). Furthermore
Y (∇yw) = 0.
The last equality implies that w belongs to Lp(;W 2,p per(Y )).
Finally from (3.12) one gets
Dl yw = Dl
which together with (3.11) proves the last convergence of
(3.9).
COROLLARY 3.7. Let {wε} be a sequence converging weakly in W 2,p()
to w, and
p ∈]1,+∞[. Then, there exist a subsequence (still denoted ε) and w
in the space
Lp(;W 2,p per(Y )) such that
1 ε2
[ Tε(wε)−Mε
∂xi∂xj + w
weakly in Lp(;W 2,p(Y )), where w is such that M
Y (w) = 0.
4. Macro–micro decomposition: the scale-splitting operators Qε and
Rε
In this section, we give a different method to prove Theorem 3.5.
It was the original proof in [12], [15], and the contruction itself
is useful later for corrector results. Since for these corrector
results, a smooth boundary of the domain is necessary, we will
assume such a regularity in this section (in the general situation,
the contruction of this section can still be carried out
locally).
The procedure is based on a splitting of functions φ in W 1,p()
as
φ = Qε(φ) +Rε(φ),
where Qε(φ) is an approximation of φ having the same behavior as φ,
while Rε(φ) is a remainder of order ε.
When considering the sequence {∇wε} where {wε} converges to w in W
1,p() we show that, while {∇wε} , {∇(Qε(wε))} and {Tε(∇Qε(wε))}
have the same weak limit ∇w in Lp(), respectively in Lp(×Y ), the
sequence {Tε(∇wε)} converges (up to a subsequence) in Lp(×Y ) to
the limit ∇w+ r where r = ∇yw and is the weak limit of Tε
( ∇(Rε(wε))
) .
From now on, we suppose that is a bounded domain such that there
exists a continuous extension operator P : W 1,p() 7→W 1,p(IRn)
satisfying
P(φ)W 1,p(IRn) ≤ C φW 1,p(), ∀φ ∈W 1,p(),
22 D. Cioranescu, Alain Damlamian & G. Griso
where C is a constant depending only upon p and ∂.
The construction of Qε is based on the Q1−interpolate of some
discrete approximation, as is customary in the Finite Element
Method (FEM). The idea of using these type of interpolate was
already present in G. Griso [19-20], for the study of truss-like
structures. For the purpose of this paper, it is enough to take the
average on εξ+ εY to construct the discrete approximations, but any
other well-behaved average will do.
DEFINITION 4.1. For any φ in Lp(IRn), p ∈ [1,+∞[, the operator Qε :
Lp(IRn) 7→ W 1,∞(IRn), is defined as follows
Qε(φ)(εξ) = Mε
Y (φ)(εξ) for ξ ∈ ε ZZ n,
and for any x ∈ IRn, we set
Qε(φ)(x) is the Q1 interpolate of the values of Qε(φ) at the
vertices
of the cell ε [x ε
] Y
+ εY. (4.1)
For any φ in W 1,p(), the operator Qε : W 1,p() 7→W 1,∞() is
defined by
Qε(φ) = Qε(P(φ))|, where Qε(P(φ)) is given by (4.1).
A straighforward computation gives the following estimates:
PROPOSITION 4.2 (properties of Qε on IRn ). For φ in Lp(IRn), 1 ≤ p
≤ ∞, there
exists a constant C depending only upon n and Y such that:
Qε(φ)Lp(IRn) ≤ CφLp(IRn), ∇Qε(φ)Lp(IRn) ≤
C
C
ε1+n/p φLp(IRn).
For φ in Lp(IRn), 1 ≤ p <∞ we have the following convergences:{
Qε(φ) −→ φ strongly in Lp(IRn),
ε∇Qε(φ) −→ 0 strongly in (Lp(IRn))n.
Furthermore, for any ψ in Lp(Y )
Qε(φ)ψ ({ ·
ε
} Y
if ψ is in W 1,p per(Y ), then
Qε(φ)ψ ({ ·
ε
} Y
DEFINITION 4.3. The remainder Rε(φ) is given by
Rε(φ) = φ−Qε(φ) for any φ ∈W 1,p().
The following proposition is well-known from the Finite Elements
Method:
PROPOSITION 4.4 (properties of Qε and Rε on W 1,p()). For any φ ∈W
1,p(), one
has
(ii). Rε(φ)Lp() ≤ εCφW 1,p(),
(iii). ∇Rε(φ)Lp() ≤ C∇φLp().
Moreover,
Lp()
≤ C
ε ∇φLp() for i, j ∈ [1, . . . , n], i 6= j. (4.4)
Up to the factor P, the constant C is the Poincare-Wirtinger
constant for Y and depends
upon neither nor ε.
Proof. We start with φ in W 1,p(IRn). From Proposition 2.5 (i) and
inequality (3.5 ), we get
φ−Mε
Y (φ)Lp(IRn×Y ) ≤ εC∇φLp(IRn). (4.5)
On the other hand, for any ψ ∈W 1,p(Y ∪ (Y + ei)), i ∈ {1, . . . ,
n}, we have
| M Y+ei
(ψ)−M Y
(ψ) |=| M Y
( ψ(·+ ei)− ψ(·)
≤ Cψ(·+ ei)− ψ(·)Lp(Y ) ≤ C∇ψLp(Y ∪(Y+ei)).
By a scaling argument and using Definition 4.1, this gives
|Qε(φ)(εξ)−Qε(φ)(εξ + εei)| ≤ εC∇φLp(ε(ξ+Y )∪ε(ξ+ei+Y )).
(4.6)
for all ξ ∈ εZZn. Let x ∈ ε
( ξ + Y
) and set for every i = (i1, . . . , in) ∈ {0, 1}n,
x (ik) k =
24 D. Cioranescu, Alain Damlamian & G. Griso
If ξ ∈ εZZn, for every i ∈ {0, 1}n by definition we have
Qε ( φ ) (x) =
∑ i∈{0,1}n
Qε(φ) ( εξ + εi
) −Qε(φ)
n ,
and a same expression for the other derivatives. This last formula
and (4.5)-(4.7) imply estimate (i) written in IRn.
Now, from (4.7), we get
φ(x)−Qε ( φ ) (x) =
n ,
and (ii) (again in IRn), follows by using estimate (4.5). Estimate
(iii) is straightforward from the previous ones. In the spirit of
Definition 4.3, if φ is in W 1,p(), estimates (i)-(iii) are simply
obtained by taking the restrictions to of Qε(P(φ)) and
Rε(P(φ)).
To finish the proof, it remains to show (4.4). To do so, it
suffices to take the derivative
with respect to any xk with k 6= 1 in the formula of ∂Qε(φ)
∂x1
above and use estimate (4.6).
REMARK 4.5. Observe that by construction (see explicite formula
(4.7)) , the function
Qε(φ) is separately piece-wise linear on each cell. Morover, the
expression of ∂Qε(φ) ∂xk
shows that this function is independent of xk in each cell ε (
ξ+Y
) , for any k ∈ {1, . . . , n}.
PROPOSITION 4.6. Let {wε} be a sequence converging weakly in W
1,p() to w. Then,
the following convergences hold:
(ii). Qε(wε) w weakly in W 1,p(),
(iii). Tε(∇Qε(wε)) ∇w weakly in Lp(× Y ).
Proof. (i) and (ii). Statement (i) is a direct consequence of
estimate (ii) in Proposition 4.4. It implies, together with
estimate (i) of Proposition 4.4, convergence (ii).
(iii). Obviously,
The Periodic Unfolding Method in Homogenization 25
From (4.4), ∂
ε for i, j ∈ [1, . . . , n], i 6= j.
Then, by Proposition 3.1, there exist a subsequence (still denoted
ε) and wj ∈ Lp(× Y )
with ∂wj ∂yi ∈ Lp(× Y ), such that
Tε (∂Qε(wε)
εTε (∂2Qε(wε) ∂xi∂xj
weakly in Lp(× Y ),
where wj is yi−periodic with i 6= j. Moreover, from Remark 4.5, the
function wj does not depend on yj , hence it is Y−periodic. But,
see again Remark 4.5, wj is also piecewise linear with respect to
any variable yi. Consequently, wj is independent of y. On the other
hand, from (ii) above we have
∂Qε(wε) ∂xj
∂xj which shows that convergence (iii) holds for the
whole sequence ε.
PROPOSITION 4.7 (Theorem 3.5 revisited). Let {wε} be a sequence
converging weakly
in W 1,p() to w. Then, up to a subsequence , and w ′
in the space Lp(;W 1,p per(Y )) such
that the following convergence holds: 1 ε Tε ( Rε(wε)
) w
′ weakly in Lp(× Y ).
Tε(∇wε) ∇w +∇yw ′
weakly in Lp(× Y ).
Actually, the connection with the w of Theorem 3.5 is given
by:
w = w ′ −M
Y (w ′ ).
Proof. Due to the estimates of Proposition 4.4, up to a
subsequence, there exists w ′
in Lp(;W 1,p
) w
Combining with convergence (iii) of Proposition 4.6, shows
that
Tε ( ∇wε
) ∇w +∇yw
Y (w ′ ).
We end this section with a new characterization of the limit
function w ′
in terms of ∇w and w given in Theorem 3.5 above.
REMARK 4.8. In the previous proposition, one can actually compute
the average of w ′ .
It depends strongly on the choice of the cell Y and of the
definition of Qε. In the case of
Y = (0, 1)n and the Definition 4.1, one can check the following
:
M Y
(w ′ ) = −1
5. Periodic unfolding and the standard homogenization problem
Let α, β ∈ IR, such that 0 < α < β. Denote by M(α, β,O) the
set of the n× n matrices A = (aij)1≤i,j≤n ∈ (L∞ (O))n×n such that
for any λ ∈ IRn and a.e. on O,{
i. (A(x)λ, λ) ≥ α|λ|2,
ii. |A(x)λ| ≤ β|λ|.
be a sequence of non constant matrices such that
Aε ∈M(α, β,). (5.2)
For f given in H−1(), consider the Dirichlet problem{ −div (Aε∇uε)
= f in
uε = 0 on ∂. (5.3)
By the Lax-Milgram theorem, there exists a unique uε ∈ H1 0 ()
satisfying∫
Aε∇uε∇v dx = f, vH−1(),H1 0 (), ∀v ∈ H1
0 (), (5.4)
The Periodic Unfolding Method in Homogenization 27
which is the variational formulation of (5.3). Moreover, one has
the apriori estimate
uεH1 0 () ≤
1 α fH−1(). (5.5)
Consequently, there exist u0 in H1 0 () and a subsequence, still
denoted ε, such that
uε u0 weakly in H1 0 (), (5.6)
We are now interested in obtaining a limit problem, the so-called
“homogenized” problem satisfied by u0. This is called standard
homogenization and the answer, for some classes of Aε, can be found
in many works , starting with the classical book A. Bensoussan,
J.L. Lions and G. Papanicolaou [6] (see, for instance D. Cioranescu
and P. Donato [13] and the references herein). We now recall
it.
THEOREM 5.1 (standard periodic homogenization). Let A =
(aij)1≤i,j≤n belong to
M(α, β, Y ), where aij = aij(y) are Y−periodic. Set
Aε(x) = ( aij
(x ε
a.e. on , (5.7)
Let uε be the solution of the corresponding problem (5.3) with f in
H−1(). Then the
whole sequence {uε} converges to a limit u0 which is the unique
solution of the homogenized
problem −div (A0∇u0) =
u0 = 0 on ∂,
(5.8)
where the constant matrix A0 = (a0 ij)1≤i,j≤n is elliptic and given
by
a0 ij =MY
∂yk
) . (5.9)
In (5.9), the functions χj (j = 1, . . . , n), often referred to as
correctors, are the solutions
of the cell systems −
(5.10)
As will be seen below, using the periodic unfolding, the proof of
this theorem is ele- mentary! Actually, with the same proof, a more
general result can be obtained, with a sequence of matrices
Aε.
28 D. Cioranescu, Alain Damlamian & G. Griso
THEOREM 5.2 (periodic unfolded homogenization). Let uε be the
solution of prob-
lem (5.3) with f in H−1() and Aε satisfying (5.1)-(5.2). Suppose
that there exists a
matrix B such that
Then there exists u0 ∈ H1 0 () and u ∈ L2(;H1
per(Y )) such that
0 (),
Tε(∇uε) ∇u0 +∇yu weakly in L2(× Y ),
(5.12)
and the pair (u0, u) is the unique solution of the problem
∀Ψ ∈ H1
1 |Y |
∫ ×Y
][ ∇Ψ(x) +∇yΦ(x, y)
(5.13)
Remark 5.3. Problem (5.13) is of standard variational form in the
space
H = H1 0 ()× L2(; H1
per(Y )/IR).
Remark 5.4. Hypothesis (5.11) implies that B ∈M(α, β,× Y ).
Remark 5.5. If Aε is of the form (5.7), then B(x, y) = A(y). In the
case where Aε(x) = A1(x)A2
(x ε
) , one has (5.11) with B(x, y) = A1(x)A2(y).
Remark 5.6. Let us point out that every matrix B ∈M(α, β,×Y ) can
be approached by the sequence of matrices Aε in M(α, β,) with Aε
defined as follows:
Aε =
Proof of Theorem 5.2. Convergences (5.12) follow from estimate
(5.5), Proposition 2.10 and Theorem 3.5, respectively.
Let us choose v = Ψ, with Ψ ∈ D() as test function in (5.4). The
integration formula (2.5) from Proposition 2.7, gives
1 |Y |
∫ ×Y
Tε' f,ΨH−1(),H1 0 (). (5.14)
We are allowed to pass to the limit in (5.14), due to (5.11) and
(5.12), to get
1 |Y |
∫ ×Y
0 () (5.15)
The Periodic Unfolding Method in Homogenization 29
which, by density, still holds for every Ψ ∈ H1 0 ().
Now, taking in (5.4), as test function vε(x) = εΨ(x)ψ (x ε
) , Ψ ∈ D(), ψ ∈ H1
1 |Y |
∫ ×Y
.
Since vε 0 in H1 0 (), we get at the limit
1 |Y |
∫ ×Y
] Ψ(x)∇yψ(y) dxdy = 0
which, due to the density of the tensor product D() × H1 per(Y ),
is valid for all Φ in
L2(;H1 per(Y )).
Remark 5.7. As in the two-scale method, (5.13) gives u in terms of
∇u0 and yields the standard form of the homogenized equation, i.e.,
(5.8). In the simple case where A(x, y) = A(y) = (aij(y))1≤i,j≤n,
it is easily seen that the limit matrix B is precisely A0
which was defined in Theorem 5.1 by (5.9)-(5.10). One also
has
u = n∑ i=1
∂xi χi. (5.16)
PROPOSITION 5.8 (convergence of the energy). Under the hypotheses
of Theorem
5.2, one has
lim ε→0
∫ ×Y
1 |Y |
∫ ×Y
f, uεH−1(),H1 0 ()
= f, u0H−1(),H1 0 () =
1 |Y |
∫ ×Y
which gives (5.17) as well as the convergence
lim sup ε→0
whence (5.18).
Remark 5.9. From the above proof, we also have the following
convergence:
lim ε→0
Tε(∇uε)→ ∇u0 +∇yu strongly in L2(× Y ). (5.19)
Proof. We have successively
] .
Each term in the right hand side converges due to (5.12), Remark
5.9 and hypothesis (5.11) so that the limit is zero. Then
convergence (5.19) follows from the ellipticity of Bε.
6. Some corrector results
Under additional regularity assumptions on the homogenized solution
u0 and the corrector functions χj , the strong convergence for the
gradient of u0 with a corrector is known (cf. [13], [14]). More
precisely, suppose that ∇yχj ∈ (Lr(Y ))n, j = 1, . . . , n and ∇u0
∈ Ls() with 1 ≤ r, s <∞ and such that 1/r + 1/s = 1/2.
Then
∇uε −∇u0 −∇yu (·, · ε
)→ 0 strongly in L2().
Proposition 2.16 however gives a corrector result without any
additional regularity as-
sumption on χj . In the fact, the proof of this corrector result
(as given below), reduces to a few lines. We also include a new
type of corrector.
THEOREM 6.1. Under the hypotheses of Theorem 5.2, one has
∇uε −∇u0 − Uε ( ∇yu
The Periodic Unfolding Method in Homogenization 31
In the case B = A0 the function u0 + ε n∑ i=1
Qε (∂u0
has
Qε (∂u0
) −→ 0 strongly in H1(). (6.2)
Proof. From (5.18), (5.19) and Proposition 2.16 (iii), one
immediately has
∇uε − Uε ( ∇u0
Uε ( ∇u0
) → ∇u0 strongly in L2(),
whence (6.1). From (4.3) in Proposition 4.2 the function u0 + ε n∑
i=1
Qε (∂u0
∇u0 + Uε ( ∇yu
∂xi
)] χ ({ ·
ε
} Y
) and one immediately has the strong convergence in L2() of the
right hand side in the above equality. Thanks to (6.1) and the
convergences in Proposition 4.2 one has (6.2).
7. Periodic unfolding and multiscales
In this section, we want to consider a “partition” of Y in two
non-empty disjoint open subsets Y1 and Y2, i.e. such that Y1∩Y2 =
/0 and Y = Y 1∪Y 2. We also introduce another unit periodicity cell
Z and consider a matrix field Aεδ is defined by
Aεδ(x) =
A1
{x ε
} Y ∈ Y2,
where the two matrix fields A1 and A2 are defined on × Y and × Y ×
Z respectively. In this problem, there are two small scales, namely
ε and εδ, associated respectively to
the cells Y and Z. Consider the solution uεδ ∈ H1 0 () of
∫
f w dx ∀w ∈ H1 0 ().
32 D. Cioranescu, Alain Damlamian & G. Griso
Suppose that A1 is in L∞(× Y ) and A2 in L∞(× Y ×Z). With standard
ellipticity hypotheses it is easy to obtain some u0 such that, up
to a subsequence,
uεδ u0 weakly in H1 0 ()).
Using the unfolding method for scale ε, as before we have
Qε ( uεδ ) u0 weakly in H1
0 (),
Tε(uεδ) u0 weakly in L2(; H1(Y )), 1 ε Tε ( Rε(uεδ)
) u weakly in L2(; H1(Y )),
Tε ( ∇uεδ
) ∇u0 +∇yu in L2(× Y ).
These convergences do not see the oscillations at the scale εδ. In
order to capture them, one considers the restrictions to the set ×
2 defined by
vεδ(x, y) .= 1 ε Tε ( Rε(uεδ)
) |2 .
Obviously,
vεδ u|2 weakly in L2(;H1(2)).
Now, we apply to vεδ, a similar unfolding operation for the
variable y, thus adding a new variable z ∈ Z, denoted T yδ .
T yδ (vεδ)(x, y, z) = vεδ ( x, δ [y δ
] Z
+ δz )
for x ∈ , y ∈ 2 and z ∈ Z.
At this point, it is essential to remark that all the estimates and
weak convergence
properties which were shown for the original unfolding Tε still
hold for T yδ with
x being a mere parameter. For example, Proposition 4.6 and Theorem
3.5 adapted to this case imply :
T yδ ( ∇yvεδ
T yδ ( Tε ( ∇uεδ
)) ∇u0 +∇yu+∇zu weakly in L2(× 2 × Z).
Under these conditions, the limit functions u0, u and u are
characterized in the following theorem:
Theorem 7.1. The functions
per(Y )/IR), u ∈ L2(× 2, H 1 per(Z)/IR)
The Periodic Unfolding Method in Homogenization 33
are the uniquesolutions of the following variational problem:
}{ ∇Ψ +∇yΦ +∇zΘ
per(Y )/IR),∀Θ ∈ L2(× 2, H 1 per(Z)/IR)
The proof uses test functions of the form
Ψ(x) + εΨ1(x)Φ1
(x ε
) ,
where Ψ,Ψ1,Ψ2 are in D(), Φ1 in H1 per(Y ), Φ2 ∈ D(2) and Θ2 ∈
H1
per(Z). A more general approach to multiscale periodic
homogenization in A. Damlamian and P. Donato [16] (where reiterated
H0-convergence dealing with holes, is considered).
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The Periodic Unfolding Method in Homogenization 35
[27] G. Nguetseng, A general convergence result for a functional
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Doina Cioranescu & George Griso LaboratoireJacques-Louis Lions
Universite Pierre et Marie Curie (Paris VI) Boite courrier 187 4
Place Jussieu 75252 Paris Cedex 05 France Email:
cioran@ann.jussieu.fr, ggriso@wanadoo.fr
Alain Damlamian Laboratoire d’Analyse et de Mathematiques
Appliquees Universite Paris Est 94010 Creteil Cedex France Email:
damla@univ-paris12.fr
36 D. Cioranescu, Alain Damlamian & G. Griso
GAKUTO International Series
Math. Sci. Appl., Vol.** (2009) Multiple scales problems in
Biomathematics,
Mechanics, Physics and Numerics, pp. 37–66
GAKKOTOSHO
and applications to Robin problems
D. Cioranescu, P. Donato and R. Zaki
Abstract: The periodic unfolding method was introduced in [C.R.
Acad. Sci. Paris, Ser. I 335 (2002), 99-104] by D. Cioranescu, A.
Damlamian and G. Griso for the study of classical periodic
homogenization. The main tools are the unfolding operator and a
macro-micro decomposition of functions which allows to separate the
macroscopic and microscopic scales. In this paper, we extend this
method to the homogenization in domains with holes, intro- ducing
the unfolding operator for functions defined on periodically
perforated domains as well as a boundary unfolding operator. As an
application, we study the homogenization of some elliptic problems
with a Robin condition on the boundary of the holes, proving
convergence and corrector results.
1 Introduction.
The homogenization theory is a branch of the mathematical analysis
which treats the asymp- totic behavior of differential operators
with rapidly oscillating coefficients. We have now different
methods related to this theory:
• The multiple-scale method introduced by A. Bensoussan, J.-L.
Lions and G. Papani- colaou in [2].
• The oscillating test functions method due to L. Tartar in
[16].
• The two-scale convergence method introduced by G. Nguetseng in
[15], and further developed by G. Allaire in [1].
38 D. Cioranescu, P. Donato and R. Zaki
Recently, the periodic unfolding method was introduced in [4] by D.
Cioranescu, A. Damla- mian and G. Griso for the study of classical
periodic homogenization in the case of fixed domains. This method
is based on two ingredients: the unfolding operator and a macro-
micro decomposition of functions which allows to separate the
macroscopic and microscopic scales. The interest of the method
comes from the fact that it only deals with functions and classical
notions of convergence in Lp spaces. This renders the proof of
homogenization results quite elementary. It also provides error
estimates and corrector results (see [13] for the case of fixed
domains). Here, we present the adaptation of the method to the
homogenization in domains with holes introduced in [5] and [6]. We
refer also to [7] for some complementary result and an appli-
cation to a problem with nonlinear boundary conditions. We define
in the upcoming section the unfolding operator for functions
defined on periodically perforated domains. We also define in
Section 5 a boundary unfolding operator, in order to treat problems
with nonho- mogeneous boundary conditions on the holes (Neumann or
Robin type). The main feature is that, when treating such problems,
we do not need any extension operator. Consequently, we can
consider a larger class of geometrical situations than in [2], [5],
and [9] for instance. In particular, for the homogenous Neumann
problem, we can admit some fractal holes like the two dimensional
snowflake (see [19]). For a general nonhomogeneous Robin condition,
we only assume a Lipschitz boundary, in order to give a sense to
traces in Sobolev spaces.
We also show in Section 4 a compactness result (Theorem 4.7) which
states that any sequence {vε}, with vεH1(ε) ≤ C, defined on a space
depending on ε, is mapped by the unfolding operator into a compact
set in L2
loc( × Y ). This result is crucial for proving corrector results,
as showed in Section 6.
The paper is organized as follows: In Section 2, we define the
unfolding operator and prove some linked properties. In Section 3,
we give the macro-micro decomposition of functions defined in
perforated domains and in Section 4, we introduce the averaging
operator and state a corrector result. The boundary unfolding
operator, essential in this work, is introduced in Section 5,
together with its main properties. Finally, Section 6 contains an
application to the homogenization of an elliptic problem with Robin
boundary condition.
2 The periodic unfolding operator in a perforated do-
main.
In this section, we introduce the periodic unfolding operator in
the case of perforated do- mains. In the following we denote:
• an open bounded set in RN ,
• Y = N∏ i=1
[0, li[ the reference cell, with li > 0 for all 1 ≤ i ≤ N , or
more generally a set
having the paving property with respect to a basis (b1, · · · , bN)
defining the periods,
• T an open set included in Y such that ∂T does not contain the
summits of Y . We can be, sometimes, transported to this situation
by a simple change of period,
The periodic unfolding method in perforated domains 39
• Y = Y \ T a connected open set.
We define T ε =
Figure 1: The domain ε and the reference cell Y
We assume in the following that ε is a connected set. Unlike
preceding papers treat- ing perforated domains (see for example
[5],[8],[9]) we can allow that the holes meet the boundary ∂. In
the rest of this paper, we only take the regularity
hypothesis
|∂| = 0. (1)
Remark 1 The hypothesis aforementioned is equivalent to the fact
that the number of cells intersecting the boundary of is of order
ε−N (we refer to [11, Lemma 21]).
Remark 2 An interesting example on the hypotheses aforementioned
would be the lattice- type structures for which it is not possible,
in some cases, to define extension operators. This situation
happens if the holes intersect the exterior boundary ∂ (see
[9],[10]).
In the sequel, we will use the following notation:
• for the extension by 0 outside ε (resp. ) for any function in
Lp(ε) (resp. Lp()),
• χε for the characteristic function of ε,
• θ for the proportion of the material in the elementary cell, i.e.
θ = |Y |
|Y | ,
• ρ(Y ) for the diameter of the cell Y ,
40 D. Cioranescu, P. Donato and R. Zaki
• T εint for the set of holes that do not intersect the boundary
∂.
By analogy to the 1D notation, for z ∈ RN , [z]Y denotes the unique
integer combination j=N∑ j=1
kjbj , such that z− [z]Y belongs to Y . Set {z}Y = z− [z]Y (see
Fig. 2). Then, for almost
every x ∈ RN , there exists a unique element in RN , denoted by [x
ε
] Y , such that
Figure 2: The decomposition z = [z]Y + {z}Y
Definition 1 (Unfolding operator) Let ∈ Lp(ε), p ∈ [1,+∞]. We
define the function Tε() ∈ Lp(RN × Y ) by setting
Tε()(x, y) = ( ε [x ε
] Y
Tε : ∈ Lp(ε) → Tε() ∈ Lp(RN × Y )
is called the unfolding operator.
Remark 3 Notice that the oscillations due to perforations are
shifted into the second variable y which belongs to the fixed
domain Y , while the first variable x belongs to RN . One see
immediately the interest of the unfolding operator. Indeed, when
trying to pass to the limit in a sequence defined on ε, one needs
first, while using standard methods, to extend it to a fixed
domain. With Tε, such extensions are no more necessary.
The periodic unfolding method in perforated domains 41
The main properties given in [4] for fixed domains can easily be
adapted for the perforated ones without any major difficulty in the
proofs. These properties are listed in the proposition below.
To do so, let us first define the following domain: ε = int(
ξ∈Λε
} .
The set ε is the smallest finite union of εY cells containing
.
Figure 3: The domain ε
Proposition 4 The unfolding operator Tε has the following
properties:
1. Tε is a linear operator.
2. Tε() ( x, {x ε
} Y
) = (x), ∀ ∈ Lp(ε) and x ∈ RN .
3. Tε(ψ) = Tε()Tε(ψ), ∀, ψ ∈ Lp(ε).
4. Let in Lp(Y ) or Lp(Y ) be a Y - periodic function. Set ε(x) =
(x ε
) . Then,
5. One has the integration formula
∫
42 D. Cioranescu, P. Donato and R. Zaki
6. For every ∈ L2(ε), Tε() belongs to L2(RN × Y ). It also belongs
to L2(ε × Y ).
7. For every ∈ L2(ε), one has
Tε()L2(RN×Y ) = √
|Y |L2(ε).
8. ∇yTε()(x, y) = εTε(∇x)(x, y) for every (x, y) ∈ RN × Y .
9. If ∈ H1(ε), then Tε() is in L2(RN ;H1(Y )).
10. One has the estimate
∇yTε()(L2(RN×Y ))N = ε √ |Y |∇x(L2(ε))N .
∫
+ εy ) dx dy,
since is null in the holes. The desired result is then
straightforward.
N.B. In the rest of this paper, when a function ψ is defined on a
domain containing ε, and for simplicity, we may use the notation
Tε(ψ) instead of Tε(ψ|ε).
Proposition 5 Let ∈ L2(). Then,
1. Tε() → strongly in L2(RN × Y ),
2. χε θ weakly in L2(),
3. Let (ε) be in L2() such that
ε → strongly in L2().
Then,
The periodic unfolding method in perforated domains 43
Proof. 1. The first assertion is obvious for every ∈ D(). If ∈
L2(), let k ∈ D() such that k → in L2(). Then
Tε() − L2(RN×Y ) ≤ Tε() − Tε(k)L2(RN×Y ) + Tε(k) − kL2(RN×Y )
+k − L2(RN×Y ),
from which the result is straightforward.
∫
) .
On one hand, by using 1 and 7 of Proposition 2.5, we get as ε→ 0
∫
RN×Y
lim ε→0
(Tε() − )2 dx dy = 0.
Therefore, assertion 3 holds true.
Proposition 6 Let ε be in L2(ε) for every ε, such that
Tε( ε) weakly in L2(RN × Y ).
Then,
(·, y)dy weakly in L2(RN).
Proof. Let ψ ∈ D(). Using 3 and 5 of Proposition 2.5, one has
successively ∫
RN
This gives, using 1 of Proposition 2.6
∫
|Y |
Proposition 7 Let ε be in L2(ε) for every ε, with
ε L2(fε) ≤ C,
ε∇x ε(L2(fε))N ≤ C.
Then, there exists in L2(RN ;H1(Y )) such that, up to
subsequences
1. Tε( ε) weakly in L2(RN ;H1(Y )),
2. εTε(∇x ε) ∇y weakly in L2(RN × Y ),
where y 7→ (., y) ∈ L2(RN ;H1
per(Y )).
∫
RN×Y
=
∫
=
∫
[ψ (x− εli −→ei , y) − ψ (x, y)] dx dy.
Passing to the limit, we obtain the result since ψ(x− εli −→ei ,
y)−ψ(x, y) → 0 when ε→ 0.
3 Macro-Micro decomposition.
Following [4], we decompose any function in the form
= Qε() + Rε(),
where Rε is designed in order to capture the oscillations. As in
the case of fixed domains, we start by defining Qε() on the nodes
εξk of the εY -lattice.
The periodic unfolding method in perforated domains 45
Here, it is no longer possible to take the average on the entire
cell Y as in [4], but it will be taken on a small ball Bε centered
on εξk and not touching the holes. This is possible using the fact
that ∂T does not contain the summits of Y . However, Bε must be
entirely contained in ε. To guarantee that, we are let to define
Qε() on a subdomain of ε only. To do so, for every δ > 0, let us
set
ε δ = {x ∈ ; d(x, ∂) > δ} and ε
δ = int(
δ
δ
Qε()(εξk) = 1
(εξk + εz)dz.
Observe that by definition, any ball Bε centered in a node of ε
2ερ(Y ) is entirely con-
tained in ε, since actually they all belong to ε ερ(Y ).
• We define Qε() on the whole ε 2ερ(Y ), by taking a
Q1-interpolate, as in the finite ele-
ment method (FEM), of the discrete function Qε()(εξk).
• On ε 2ερ(Y ), Rε will be defined as the remainder: Rε() =
−Qε().
46 D. Cioranescu, P. Donato and R. Zaki
Proposition 8 For belonging to H1(ε), one has the following
properties:
1. Qε() H1(bε
2ερ(Y ) ) ≤ C
))N .
Proof. These results are straightforward from the definition of Qε.
The proof, based on some FEM properties, is very similar to the
corresponding one in the case of fixed domains (see [4]), with the
simple replacement of Y by Y .
We can now state the main result of this section.
Theorem 9 Let ε be in H1(ε) for every ε, with εH1(ε) bounded. There
exists in H1() and in L2(;H1
per(Y )) such that, up to subsequences
1. Qε( ε) weakly in H1
loc(),
loc(;H1(Y )),
4. Tε(∇x( ε)) ∇x+ ∇y weakly in L2
loc(;L2(Y )).
Remark 10 When comparing with the case of fixed domains, the main
difference is that, since the decomposition was done on ε
2ερ(Y ), we have here local convergences only.
Proof of Theorem. Assertions 2, 3 and 4 can be proved by using the
same arguments as in the corresponding proofs for the case of fixed
domains. We consider here just the first assertion.
Let K be a compact set contained in . As d(K, ∂) > 0, there
exists εK > 0 depend- ing on K, such that
∀ε ≤ εK , K ⊂ ε 2ερ(Y ).
Hence,
Qε( ε) weakly in H1
loc().
The periodic unfolding method in perforated domains 47
What remains to be proved is that ∈ H1(). To do so, we make use of
the Dominated Convergence theorem.
Let us consider the sequence (ε 1 N
)N . Observe that it is increasing. Indeed,
x ∈ ε 1 N
1 N+1
.
Moreover, for every N , there exists εN depending on ε 1 N
such that
⊂ ε 2ερ(Y ).
Let us define the sequence of functions (N)N for every N ∈ N as
follows:
N = ||2 χε 1 N
.
Let us show that
One has successively
dx =
||2 dx,
∫
ε)2 L2(bε
2ερ(Y ) ) ≤ C,
whence (4). The next step is to prove that
the sequence (N)N simply converges towards ||2 . (5)
,
and x ∈ ε 1
x ∈ ε 1 N
χε 1 N
48 D. Cioranescu, P. Donato and R. Zaki
and this ends the proof of (5). Thanks to (3),(4) and (5), we can
apply the Dominated Convergence theorem to deduce that
||2 ∈ L1() and lim N→∞
∫
4 The averaging operator Uε.
Definition 2 For ∈ L2(RN × Y ), we set
Uε()(x) = 1
) dz, for every x ∈ RN .
Remark 11 For V ∈ L1(RN × Y ), the function x 7→ V ( x, {x ε
} Y
) is generally not mea-
surable (for example, we refer to [5]-Chapter 9). Hence, it cannot
be used as a test function. We replace it by the function Uε(V
).
The next result extends the corresponding one given in [4].
Proposition 12 One has the following properties:
1. The operator Uε is linear and continuous from L2(RN ×Y ) into
L2(RN), and one has for every ∈ L2(RN × Y )
Uε()L2(RN ) ≤ L2(RN×Y ),
2. Uε is the left inverse of Tε on ε, which means that Uε Tε = Id
on ε,
3. Tε (χεUε()) (x, y) = 1
|Y |
4. Uε is the formal adjoint of Tε.
Proof. 1. It is straightforward from Definition 4.1.
The periodic unfolding method in perforated domains 49
2. For every ∈ L2(ε), one has
Uε (Tε ()) (x) = 1
|Y |
] Y
3. Let ∈ L2 ( RN ) , one has
Tε (χεUε ()) (x, y) = Uε () ( ε [x ε
] Y
+ εy )
= 1
|Y |
and ψ ∈ L2 ( RN × Y
) , we have
|Y |
= 1
|Y |
= 1
|Y |
Uε() → strongly in L2(RN).
2. Let ∈ L2(RN × Y ). Then,
Tε (χεUε()) → strongly in L2(RN × Y ),
and
50 D. Cioranescu, P. Donato and R. Zaki
Proof. 1. If ∈ L2(RN), one has by definition
Uε()(x, y) = 1
But ( ε [ x ε
] Y
+ εz ) → (x) when ε → 0, and this explains the result.
2. It is a simple consequence of 1 in Proposition 2.6, and
Proposition 2.7.
As in the case of fixed domains, one has
Theorem 14 Let ε be in L2(ε) for every ε, and let ∈ L2(RN × Y ).
Then,
1. Tε( ε) → strongly in L2(RN × Y ) ⇐⇒ ε − Uε() → 0 strongly in
L2(RN).
2. Tε( ε) → strongly in L2
loc(R N ;L2(Y )) ⇐⇒ ε−Uε() → 0 strongly in L2
loc(R N).
εUε ε)L2(RN×Y )
ε)L2(RN×Y )
ψ ≥ 0 and ψ = 1 on w.
Then, by using 1 of Proposition 2.6, one has
ε − UεL2(w) ≤ ψ ( ε − Uε
) L2(RN )
≤ C Tε (ψ) (Tε (ε) − Tε (χεUε ε)) L2(suppψ×Y )
≤ C ( Tε (ψ) (Tε (ε) − ) L2(suppψ×Y ) + Tε (ψ) (− Tε (χεUε
ε)) L2(suppψ×Y )
The converse implications are immediate.
This result is essential for proving corrector results when
studying homogenization prob- lems, as we show in Section 6. To
apply it, the compactness result given by Theorem 4.7 below is
crucial.
Let us first state the following proposition:
Proposition 15 For every ∈ H1(ε) one has
Rε()L2(ε) = −Qε()L2(ε) ≤ Cε∇(L2(ε))N .
The periodic unfolding method in perforated domains 51
Theorem 16 Let vε be in H1(ε) for every ε and v ∈ H1() such
that
• vεH1(ε) is bounded,
Then,
loc(, L 2(Y )).
∫
(∫
∫
) .
loc().
|Tε (Qεv ε) − v|2 dx dy = 0.
∫
ω×Y
|Tε (vε −Qε (vε))|2 dx dy = Cvε −Qε (vε) 2 L2(ω∩ε)
≤ Cε2∇vε2 (L2(ε))N ,
2(Y )).
Remark 17 We can stress here one of the major properties of the
unfolding operator. In- deed, it transforms any function defined on
the perforated domain ε into a function Tε() defined on the fixed
domain RN × Y . Theorem 4.7 actually states that any sequence {vε},
with vεH1(ε) ≤ C, is mapped into a compact set in L2
loc( × Y ).
5 The boundary unfolding operator.
We define here the unfolding operator on the boundary of the holes
∂T ε, which is specific to the case of perforated domains. To do
that, we need to suppose that T has a Lipschitz boundary.
Definition 3 (Unfolding boundary operator) Suppose that T has a
Lipschitz boundary, and let ∈ Lp(∂T ε), p ∈ [1,+∞]. We define the
function T b
ε () ∈ Lp(RN × ∂T ) by setting
T b ε ()(x, y) =
( ε [x ε
ε () ∈ Lp(RN × ∂T )
is called