Homogenization ..Multiscale Analysis of ….Methods … Results ..Modeling …. Tech- niques for Composite Media …Heterogeneous Media… [610111]

UNIVERSITY OF BUCHAREST
HABILITATION THESIS
HOMOGENIZATION RESULTS FOR
HETEROGENEOUS MEDIA
Homogenization ..Multiscale Analysis of ….Methods … Results ..Modeling …. Tech-
niques for Composite Media …Heterogeneous Media …..Structures….
CLAUDIA TIMOFTE
Specialization: Applied Mathematics????
Bucharest, 2016

Acknowledgments
This work could not have been accomplished without the support of ….
I would like to express my deep gratitude to Professor Horia I. Ene from the Institute
of Mathematics "Simion Stoilow" of the Romanian Academy for ….
I am very grateful to … for accepting to be members of my jury. I am extremely
honored by their interest in my work …
I am truly honored that … have accepted to participate at my thesis committee.
This thesis is based on some … the joint work that I have done over the years together
with: ……… To all of them, I want to express my gratitude for their kindness, patience
and for the love for mathematics that we shared over the years.
I am grateful to ……..
I am very indebted to my collaborators ………..
… his remarkable ability to connect dierent elds of research ….
During the last years, I have benetted a lot from inspiring discussions with the …IMAR
….
… research visits … fellowships …Santiago, Madrid, Pisa, Metz ..Santander … .I
had a wonderful time in … where I enjoyed the fruitful collaboration with …. Madrid
…hospitality I received there.
working in a stimulating environment is indispensable …for … successful …. .
Special thanks to ……
I am indebted ???Multumiri membrilor comisiei …..care au acceptat cu … sa ……Apre-
ciez efortul depus si timpul consacrat …
This thesis is just a … point dtape dun parcours jalonn de nombreuses rencontres. It
is impossible to me to mention all the people …. scientic journey … joint adventure ….
Essential contributions …
The work presented here represents a collective eort, the fruit of many exchanges I
had over the years with …. .
I appreciate a lot all the … "rencontres", I am fully conscious about their importance
… a huge …. stimulating environment ……….
I would also like to thank to …. colleagues … who accompanied me in this ??? trans-
disciplinary journey … exciting … In am convinced that neither the research activity nor
the teaching …. should be an individual … lonely ..solitary …. adventure … journey …..
Also, I would like to thank my colleagues from the Department of Theoretical Physics,
Mathematics, Optics, Plasma and Lasers of the Faculty of Physics of the University of
7

Bucharest, for their support and for the emulating atmosphere they have always created
in our department.
I had the chance to discover … fellowships … universities … post-doctoral .. and
research visits ..
o sursa inepuizabila de inspiratie …
Last, but not least, I am grateful to my family and to my friends for their unwavering
support and understanding.
There are many other people who … helped me at one stage or another in my work.
… and are not mentioned here. I express my deep gratitude to all of them!
Bucharest, March 2016 Claudia Timofte
9

10

Preface
……………… Foreword
…The aim of this manuscript, prepared to defend my Habilitation …thesis, is to ….
This habilitation thesis summarizes my research during the last … years that have
passed since I obtained my Ph.D. The main … of my research activity … work is ….ho-
mogenization theory, which have played a central role in almost all my research work.
This habilitation summarizes the works that I … The main subject of this habilitation
is the …… Many of the results presented herein are closely related to or motivated by
practical applications to real-life problems. I
This thesis was prepared to defend my Habilitation. I shall start by a brief overview of
the eld of homogenization … and then summarizes some of my research works performed
after completing my PhD …, with some additional ??? hindsights. The thesis, starting by
a summary in Romanian, is written in English.
……..This habilitation thesis is …. I decided to prepare a cumulative habilitation thesis.
Most of this work is already published or submitted for publication.
…..Introduction…..???
11

12

Contents
Preface 11
Rezumat 17
Abstract 21
I Main Scientic Achievements 13
1 Introduction 15
2 Homogenization of reactive
ows in porous media 19
2.1 Upscaling in stationary reactive
ows in periodic porous media . . . . . . . 20
2.1.1 The model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 The case of a smooth function g. The macroscopic model . . . . . . 22
2.1.3 The case of a non-smooth function g. The macroscopic model . . . . 29
2.1.4 Chemical reactions inside the grains of a porous medium . . . . . . . 34
2.2 Nonlinear adsorption of chemicals in porous media . . . . . . . . . . . . . . 37
2.2.1 The microscopic model and its weak solvability . . . . . . . . . . . . 38
2.2.2 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.3 The case of a non-smooth boundary condition . . . . . . . . . . . . . 43
2.2.4 Laplace-Beltrami model with oscillating coecients . . . . . . . . . . 44
3 Homogenization results for unilateral problems 51
3.1 Homogenization results for Signorini's type problems . . . . . . . . . . . . . 51
3.1.1 The microscopic problem . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.2 The macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Homogenization results for elliptic problems in perforated domains with
mixed-type boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Setting of the microscopic problem . . . . . . . . . . . . . . . . . . . 57
3.2.2 The limit problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
13

CONTENTS CONTENTS
4 Mathematical models in biology 69
4.1 Homogenization results for ionic transport phenomena in periodic charged
media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.2 The homogenized problem . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Multiscale Analysis of a Carcinogenesis Model . . . . . . . . . . . . . . . . . 78
4.2.1 The microscopic problem . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.2 The macroscopic model . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2.3 A nonlinear carcinogenesis model involving free receptors . . . . . . 91
4.3 Homogenization results for the calcium dynamics in living cells . . . . . . . 93
4.3.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.2 The periodic unfolding method for a two-component domain . . . . 98
4.3.3 The main convergence results . . . . . . . . . . . . . . . . . . . . . . 99
5 Multiscale modeling of heat transfer in composite materials with imper-
fect interfaces 103
5.1 Multiscale analysis in nonlinear thermal diusion problems in composite
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.1.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.1.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Diusion problems with dynamical boundary conditions . . . . . . . . . . . 108
5.3 Homogenization of a thermal problem with
ux jump . . . . . . . . . . . . 111
5.3.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3.2 The periodic unfolding method for a two-component domain . . . . 114
5.3.3 The macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . 116
II Career Evolution and Development Plans 127
6 ..???. 129
6.1 Academic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 Further Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
14

Rezumat 15
List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …???..
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …???…
15

16 Rezumat
16

Rezumat
………………
17

18 Rezumat
……….
18

Rezumat 19
……..
19

Abstract
jeowfj vmioervj
21

22 ABSTRACT
jeowfj vmioervj
22

ABSTRACT 23
jeowfj vmioervj
23

CONTENTS 25
fmc;vejrpoe
;rjvovmer;
fmc;vejrpoe
;rjvovmer;
;rjvovmer;
25

26 CONTENTS
……….
26

Abstract
Summary …
The present thesis is devoted to the study of ……… We shall present some ……………..
a brief introduction … motivation …we shall present some ideas that have encouraged the
study of …….The theory of ….. history ……..
The interest in the theory of …. crossed the border of … functional analysis and the
study of PDEs.
This habilitation thesis deals with the broad topic of …
In this habilitation thesis, I present some of my recent contributions to … in the areas
of …
This thesis reports on research carried out by the author over the last … years. The
research comprises four … major areas:…….. .
This work is based on a close collaboration with several …
…….. structure of the thesis …………
This habilitation thesis summarizes a selection of my research results obtained after
my PhD thesis, defended in … .
The main motivation behind …..is to give an overview of my contributions in the eld of
homogenization … I will try to present my main results and to … make them ….. audience
with strong, general mathematical background, but not necessarily experts in the specic
eld of homogenization ….
The goal of such an overview is to delineate the broad eld of homogenization theory,
to brie
y describe the state of the art and to discuss some open problems in this eld …
to summarize the contributions of the author of this thesis to the eld. I shall try to make
a self-contained overview and, where necessary, to give more details that are not present
in the corresponding published papers.
…………..the main ideas and techniques……
I shall include the results obtained in …. correspond to my last 15 years of scientic
work.
Author contributions …….???
This overview shows that I was involved in this eld in the last 15 years … I co-authored
papers, and, also, I wrote … single author …
The thesis is a self-contained, independent work, mostly based on my published research
11

12 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
over the last 15 years.
This manuscript is divided in two parts. The rst part, containing … splits into …
chapters. The rst part is devoted to the … investigations on ………
The second chapter presents various works on …. research. It is divided into … chap-
ters. The rst chapter presents some homogenization models for …. Finally, the … chapter
…….
We now turn our attention to ……….
The manuscript ends with a description of some questions I would like to study in the
future.
The main contribution of this thesis are: …….
Role of co-authors
I am the rst author in .. of …. the …. papers in this thesis. I acknowledge the …
equal ….contribution of all my co-authors: ……..
……… Part I, …II, …???
Preface … "This habilitation thesis is designed as a cumulative monograph and a review
which deals with the …………"
This habilitation thesis comprises a series of papers in the eld of homogenization
theory ….
….Research works overview ….
Conclusions and Perspectives
……open problems ……..short term, medium and long term ……. work in progress with
Renata, David, Iulian .. Ene ….Anca, ..Italy…..thank them ….
… promising direction ….
Finally, investigating …. could be a promising direction …….. a future direction
………interesting issues to be investigated …….
Enhancement of participation in national/international projects …
… involve future PhD students ……..
– Enhancement ??? of the collaboration with ….
In the same vein ??? of what I have presented in this manuscript, the questions I would
like to deal with in the future are related to …. collaborative works, which I consider very
important because they allow … interdisciplinary projects. ……..
……references at each part …….brief introduction with the results of other researchers
…state of the art ….???
The Research Activity
After defending my PhD thesis in 1996, dealing with ………., I investigated several
aspects concerning …………. I published … scientic papers either in …. journals or
in books or proceedings of international conferences. With few exceptions, these papers
belong to the following main directions of research: 1. ……..; 2. ……..;
Contributions to …….
…. scientic journey …. focusing on the achievements ………….

ABSTRACT 13
In Introduction … brief presentation of the articles included in the thesis … main
themes after the PhD program and results published in … articles ….
… mention fellowships … research visits, programs, collaborations ……..
… results in … papers … nantate de proiecte ….
De mentionat si alte articole si teme, necuprinse in teza.
…general conclusions at the end of the thesis …
Homogenization theory. Applications in the study of composite materials
Reaction-diusion ….
Homogenization in chemical reactive
ows through porous media
Inequalities ….
Optimization …………

14 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA

Rezumat
Lucrarea de fat  a este dedicat a prezent arii unor rezultate de omogenizare pentru … studiului
……………
The present thesis is devoted to the study of …… First, we shall present some ideas
that have encouraged the study of ………..
I hope that the … complete … accomplish … end of this thesis is just the beginning …
starting point of a new adventure.
I shall propose … some routes … for my scientic … research activity in … for the
years to come.
List of publications
Specialization: Mathematics … Applied Mathematics ……
Curriculum Vitae
At the end …Summary and Outlook ???
11

12 …???? ..HOMOGENIZATION … heterogeneous structures …media ….

Part I
Main Scientic Achievements
13

Chapter 1
Introduction
In the last decades, there has been an explosive growth of interest in studying the macro-
scopic properties of systems having a very complicated microscopic structure. In mecha-
nics, physics, chemistry, engineering, in material science or in biology, we are often led
to consider boundary-value problems in periodic media exhibiting multiple scales. It is
widely recognized that multiscale techniques represent an essential tool for understanding
the macroscopic properties of such systems having a very complicated microscopic struc-
ture. A periodic distribution is sometimes a realistic hypothesis which might be useful in
many practical applications. Typically, in periodic heterogeneous structures, the physical
parameters, such as the electrical or thermal conductivity or the elastic coecients, are
discontinuous and, moreover, highly oscillating. For example, in a composite material,
constituted by the ne mixing of two ore more components, the physical parameters are
obviously discontinuous and they are highly oscillating between dierent values character-
izing each distinct component. As a result, the microscopic structure becomes extremely
complicated. If the period of the structure is small compared to the region where we study
the given system or, in other words, if the nonhomogeneities are small compared to the
global dimension of the structure, then an asymptotic analysis becomes necessary. Two
scales are important for a suitable description of the given structure: one which is compa-
rable with the dimension of the period, called the microscopic scale and another one which
is comparable (of the same order of magnitude) with the global dimension of our system,
called the macroscopic scale. The main goal of the homogenization methods is to pass
from the microscopic scale to the macroscopic one; more precisely, using homogenization
methods we try to describe the macroscopic properties of the nonhomogeneous system
in terms of the properties of its microscopic structure. Intuitively, the nonhomogeneous
system is replaced by a ctitious homogeneous one, whose global characteristics represent
a good approximation of the initial system. Hence, the homogenization methods provide
a general framework for obtaining these macroscale properties, eliminating therefore the
diculties related to the explicit determination of a solution of the problem at the mi-
croscale and oering a less detailed description, but one which is applicable to much more
15

16
complex systems. Also, from the point of view of numerical computation, the homogenized
equations will be easier to solve. This is due to the fact that they are dened on a xed
domain and they have, in general, simpler or even constant coecients (the eective or
homogenized coecients), while the original equations have rapidly oscillating coecients,
they are dened on a complicated domain and satisfy nonlinear boundary conditions. The
dependence on the real microstructure is given through the homogenized coecients.
The analysis of the macroscopic properties of composite media was initiated by Rayleigh,
Maxwell and Einstein. Around 1970, scientists managed to formulate the physical problems
of composites in such a way that this eld became interesting from a purely mathemat-
ical point of view. This gave rise to a new mathematical discipline, the homogenization
theory. The rst rigorous developments of this theory appeared with the seminal works
of Y. Babuka [26], E. De Giorgi and S. Spagnolo [93], A. Bensoussan, J. L. Lions and G.
Papanicolaou [32], and L. Tartar [206]. De Giorgi's notion of Gamma-convergence marked
also an important step in the development of this theory. F. Murat [ ?] and L. Tartar [207]
introduced the notion of compensated compactness, which is an important tool to prove
convergence results. A rigorous method, the two-scale convergence method, was introduced
by G. Nguetseng in 1989 [ ?] and was further developed by Allaire in [1]. An extension to
multiscale problems was obtained by A.I. Ene and J. Saint Jean Paulin [ ?] and by G.
Allaire and M. Briane [ ?]. In 1990, T. Arbogast, J. Douglas and U. Hornung [11] dened
a dilation operator in order to study homogenization problems in a periodic medium with
double porosity. An alternative approach was oered by the Bloch-wave homogenization
method [ ?], which is a high frequency method that can provide dispersion relations for
wave propagation in periodic structures. Recently, D. Cioranescu, A. Damlamian and G.
Griso combined the dilation technique with ideas from nite element approximations to
give rise to a very general method for studying classical or multiscale periodic homoge-
nization problems: the periodic unfolding method [28, 29]. Let us nally mention that
probabilistic and numerical methods, such as the heterogeneous multiscale method, have
been recently developed and successfully applied to a broad category of problems of both
practical and theoretical interest (see [48]). It is important to emphasize that homogeniza-
tion theory can be applied to non-periodic media, as well. To this end, one can use G- or
H-convergence techniques. Also, it is possible to deal with general geometrical settings,
without assuming periodicity or randomness.
Homogenization methods have been successfully applied to various problems: the
convective-diusive transport in porous media, nonlinear elasticity, the study of composite
polymers, the study of nanocomposite materials, the modeling of interface phenomena in
biology and chemistry, or the problem of obtaining new composite materials with applica-
tions in modern technology. The literature on this subject is vast (see, e.g., [70], [82], [71],
[80], [83], ….). We also mention here some remarkable monographs dedicated to the math-
ematical problems of homogenization: Jikov et al. [ ?], Bakhvalov and Panasenko, Ben-
soussan, Lions and Papanicolaou, Lions, Marchenko and Khruslov, Oleinik, Shamaev and

17
Yosian, Sanchez-Palencia, Cioranescu-Donato, Cioranescu- J. Saint Jean Paulin, …Ene
and Pasa, ………
Multiscale methods oer multiple possibilities for further developments and for useful
applications in many domains of the contemporary science and technology. Their study
is one of the most active and fastest growing areas of modern applied mathematics, and
denitely one of the most interdisciplinary eld of mathematics.
Introd la ecare capitol?
Motivation…… the general framework of my research ……….
My research activity in this very … eld of homogenization theory started after de-
fending my Ph.D. thesis at "Simion Stoilow" Institute of Mathematics of the Romanian
Academy under the supervision of Professor Horia Ene.
Mainly, I was interested in the applications of the homogenization theory to a broad
category of …. To summarize, my main research interests have been related to the follow-
ing areas:
…….
……..
……..
My research activity in the eld of homogenization is interdisciplinary in its nature and
in the last years I tried to publish my results in more application-oriented high quality
journals, such as …….. These journals have a broad audience, including not only mathe-
maticians, but also physicists, engineers, and scientists from various applied elds, such as
biology or geology.
The autonomy and the visibility of my research activity performed after the completion
of my Ph.D. thesis is …. by the following arguments … facts …
I published, as main author, … papers in … journals and … papers in proceedings of
international conferences …. national … I … conferences …. invited … Invited seminars
and research visits abroad. I obtained four post-doctoral fellowships (….). All these visits
helped me to enlarge my horizon …. to collaborate … … established solid …. I was director
of … research projects and member of ….other … projects.
I am a member of the American Mathematical Society, the … SIAM and of RMS and
a reviewer for Mathematical Reviews and for more than 30 international journals, such as
…….. I was
Teaching experience …. Between 1991-2008, I was assistant professor, lecturer, and
associate professor at the Faculty of Physics of the University of Bucharest. Since 2008, I
am professor at the same faculty. I … courses and seminaries … various …at undergraduate
and also at graduate level.

Structure of the thesis
Perspectives ……..??? Here?
18

Chapter 2
Homogenization of reactive
ows
in porous media
The problem of obtaining suitable global descriptions for some complex reactive
ows in
porous media was addressed in the literature by using various upscaling methods: heuristic
and empirically based methods, variational methods, stochastic methods, methods based
on homogenization, mixture theories, or volume averaging techniques. Also, the use of
numerical models for studying single-phase or multi-phase
ows in heterogeneous porous
media has received considerable attention in the last decades. However, even with the
increases in the power of computers, the complex multiscale structure of the analyzed
media constitutes a critical problem in the numerical treatment of such models and there
is a considerable interest in the development of upscaled or homogenized models in which
the eective properties of the medium vary on a coarse scale which proves to be suitable
for ecient computation, but enough accurately to capture the in
uence of the ne-scale
structure on the coarse-scale properties of the solution.
Porous media play an important role in many areas, such as hydrology (groundwater

ow, salt water intrusion into coastal aquifers), geology (petroleum reservoir engineering,
geothermal energy), chemical engineering (packed bed rectors, drying of granular materi-
als), mechanical engineering (heat exchangers, porous gas burners), the … . of industrial
materials (glass ber materials, brick manufacturing). There is an extensive literature on
the determination of the eective properties of heterogeneous porous media (see, e.g., ……..
[21] …… and [29] …..20] ??? and the references therein).
Transport processes in porous media have been extensively studied in last decades by
engineers, geologists, hydrologists, mathematicians, physicists. In particular, mathemati-
cal modeling of chemical reactive
ows through porous media is a topic of huge practical
importance in many engineering, physical, chemical, and biological applications. Obtain-
ing suitable macroscopic laws for the processes in geometrically complex porous media
(such as soil, concrete, rock, or pellets) involving
ow, diusion, convection, and chemical
reactions is a dicult task. The homogenization theory proves to be a very ecient tool by
19

20 HOMOGENIZATION … heterogeneous structures …media ….
providing suitable techniques allowing us to pass from the microscopic scale to the macro-
scopic one and to obtain suitable macroscale models. Since the seminal work of G.I. Taylor
[208], dispersion phenomena in porous media have attracted a lot of attention. There are
many formal or rigorous methods in the literature. We refer to [142] and [141] as one of
the rst works containing rigorous homogenization results for reactive
ows in porous me-
dia. By using the two-scale convergence method, coupled with monotonicity methods and
compensated compactness, the convergence of the homogenization procedure was proven
for problems with nonlinear reactive terms and nonlinear transmission conditions. Since
then, many works have been devoted to the homogenization of reactive transport in porous
media (see [162], [163], [164], [165], [161], [179], [228], [229], [196], [182] and the references
therein). For instance, rigorous homogenization results for reactive
ows with adsorption
and desorption at the boundaries of the perforations, for dominant P eclet numbers and
Damkohler numbers, are obtained in [4], [5], and [168]. For reactive
ows combined with
the mechanics of cells, we refer to [146]. Rigorous homogenization techniques for obtaining
the eective model for dissolution and precipitation in a complex porous medium were
successfully applied in [196]. Solute transport in porous media is also a topic of interest for
chemists, geologists and environmental scientists (see, e.g., [7] and [159]). Related prob-
lems, such that single or two-phase
ow or miscible displacement problems were addressed
in various papers (see, for instance, [22], [11], [12], [13], [167]). For an interesting survey on
homogenization techniques applied to problems involving
ow, diusion, convection, and
reactions in porous media, we refer to [140].
In this chapter, some applications of the homogenization method to the study of reac-
tive
ows in periodic porous media will be presented. The chapter represents a summary
of the results I obtained in this area and is based on the following papers:
1…..
2…………..
3………….
The chapter is organized as follows: in the rst section, …………… .
2.1 Upscaling in stationary reactive
ows in periodic porous
media
We shall discuss now, following [79] and [210], some homogenization results for chemi-
cal reactive
ows through porous media. For a nice presentation of the chemical aspects
involved in this kind of problems and also for some mathematical and historical back-
grounds, we refer to S. N. Antontsev et al. [10], J. Bear [164], J. I. D az [94], [95], [96], and

HOMOGENIZATION OF REACTIVE FLOWS 21
U. Hornung [140] and the references therein. We shall start with a problem modeling the
stationary reactive
ow of a
uid conned in the exterior of some periodically distributed
obstacles, reacting on the boundaries of the obstacles. More precisely, the challenge in our
rst paper dedicated to this subject, namely [79], consists in dealing with Lipschitz or even
non-Lipschitz continuous reaction rates such as Langmuir or Freundlich kinetics, which, at
that time, were open cases in the literature. Our results represent a generalization of some
of the results in [140]. Using rigorous multiscale techniques, we derive a macroscopic model
system for such elliptic problems modeling chemical reactions on the grains of a porous
medium. The eective model preserves all the relevant features .. information … from the
microscopic level. The case in which chemical reactions arise inside the grains of a porous
medium will be also discussed. Also, we shall present some results obtained in [210], where
we have analyzed the eective behavior of the solution of a nonlinear problem arising in
the modeling of enzyme catalyzed reactions through the exterior of a domain containing
periodically distributed reactive solid obstacles.
2.1.1 The model problem
Let
be an open smooth connected bounded set in Rn(n3) and let us insert in it a set
of periodically distributed reactive obstacles. As a result, we obtain an open set
"which
will be referred to as being the exterior domain ;"represents a small parameter related
to the characteristic size of the reactive obstacles. More precisely, let Y= (0;1)nbe the
representative cell in Rn. Denote by Fan open subset of Ywith smooth boundary @F
such thatFY. We shall refer to Fas being the elementary obstacle . We setY=YnF.
If"is a real parameter taking values in a sequence of positive numbers converging to zero,
for each"and for any integer vector k2Zn, setF"
kthe translated image of "Fby the
vectork,F"
k="(k+F). The setF"
krepresents the obstacles in Rn. Also, let us denote by
F"the set of all the obstacles contained in
, i.e.
F"=[
F"
kjF"
k
; k2Zn
:
Set
"=
nF". Hence,
"is a periodic domain with periodically distributed obstacles
of size of the same order as the period. We remark that the obstacles do not intersect the
boundary@
. LetS"=[f@F"
kjF"
k
; k2Zng. So,@
"=@
[S".
We denote byj!jthe Lebesgue measure of any measurable subset !Rnand, for an
arbitrary function 2L2(
"), we denote by e its extension by zero to the whole of
, i.e.
inside the obstacles. Also, throughout this thesis, by Cwe denote a generic xed strictly
positive constant, whose value can change from line to line.
The rst problem we present in this section concerns the stationary reactive
ow of
a
uid conned in
", of concentration u", reacting on the boundary of the obstacles. A

22 HOMOGENIZATION … heterogeneous structures …media ….
simplied version of this kind of problem can be written as follows:
8
>>><
>>>:Df
u"=fin
";
Df@u"
@="g(u") onS";
u"= 0 on@
:(2.1)
Here,is the exterior unit normal to
",f2L2(
) andS"is the boundary of our exterior
medium
n
". Moreover, for simplicity, the
uid is assumed to be homogeneous and
isotropic, with a constant diusion coecient Df>0. We can treat in a similar manner
the more general case in which, instead of considering constant diusion coecients, we
work with an heterogeneous medium represented by periodic symmetric bounded matrices
which are assumed to be uniformly coercive.
The semilinear boundary condition on S"in problem (2.1) describes the chemical re-
actions which take place locally at the interface between the reactive
uid and the grains.
From strictly chemical point of view, this situation represents, equivalently, the eective
reaction on the walls of the chemical reactor between the
uid lling
"and a chemical
reactant located in the rigid solid grains.
The function gis assumed to be given. Two representative situations will be considered;
the case in which gis a monotone smooth function satisfying the condition g(0) = 0 and
the case of a maximal monotone graph with g(0) = 0, i.e. the case in which gis the
subdierential of a convex lower semicontinuous function G. These two general situations
are well illustrated by the following important practical examples:
a)g(v) =v
1 +v; ; > 0 (Langmuir kinetics) (2.2)
and
b)g(v) =jvjp1v;0<p< 1 (Freundlich kinetics) : (2.3)
The exponent pis called the order of the reaction . We point out that if we assume
f0, one can prove (see, e.g. [96]) that u"0 in
n
"andu">0 in
", although
u"is not uniformly positive except in the case in which gis a monotone smooth function
satisfying the condition g(0) = 0, as, for instance, in example a). Besides, since uis, in
practical applications, a concentration, it could be natural to suppose that f1, and,
then, we can prove that, in this case, u1.
As usual in homogenization, our goal is to obtain a suitable description of the asymp-
totic behavior, as the small parameter "tends to zero, of the solution u"of problem (2.1)
in such domains.
2.1.2 The case of a smooth function g. The macroscopic model
Let us deal rst with the case of a smooth function g. We consider that gis a continuously
dierentiable function, monotonously non-decreasing and such that g(v) = 0 if and only if

HOMOGENIZATION OF REACTIVE FLOWS 23
v= 0. Moreover, we suppose that there exist a positive constant Cand an exponent q,
with 0q<n= (n2), such that
dg
dvC(1 +jvjq): (2.4)
We introduce the functional space V"=
v2H1(
")jv= 0 on@

, endowed with the
normkvkV"=krvkL2(
"). The weak formulation of problem (2.1) is:
8
><
>:Findu"2V"such that
DfZ

"ru"
r'dx+"Z
S"g(u")'d=Z

"f'dx8'2V":(2.5)
By classical existence results (see [43]), there exists a unique weak solution u"2V"\H2(
")
of problem (2.1). This solution being dened only on
", we need to extend it to the whole
of
to be able to state the convergence result. To this end, let us recall the following
well-known extension result (see [73]):
Lemma 2.1 There exists a linear continuous extension operator
P"2L(L2(
");L2(
))\L(V";H1
0(
))
and a positive constant C, independent of ", such that, for any v2V",
kP"vkL2(
)CkvkL2(
")
and
krP"vkL2(
)CkrvkL2(
"):
Therefore, we have the following Poincar e's inequality in V":
Lemma 2.2 There exists a positive constant C, independent of ", such that
kvkL2(
")CkrvkL2(
")for anyv2V":
The main convergence result for this case is stated in the following theorem.
Theorem 2.3 There exists an extension P"u"of the solution u"of the variational problem
(2.5) such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of
8
><
>:nX
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =f in
;
u= 0 on@
:(2.6)

24 HOMOGENIZATION … heterogeneous structures …media ….
Here,Q= ((qij))is the standard homogenized matrix, whose entries are dened as follows:
qij=Df
ij+1
jYjZ
Y@j
@yidy
(2.7)
in terms of the functions i; i= 1;:::;n; solutions of the cell problems
8
>>>><
>>>>:
i= 0 inY;
@(i+yi)
@= 0 on@F;
iYperiodic.(2.8)
The constant matrix Qis symmetric and positive-denite.
Proof. We divide the proof of this theorem into four steps.
First step. Letu"2V"be the solution of the variational problem (2.5) and let P"u"
be the extension of u"inside the obstacles given by Lemma 2.1. Taking '=u"as a test
function in (2.5), using Schwartz and Poincar e's inequalities, we easily get
kP"u"kH1
0(
)C:
Then, by passing to a subsequence, still denoted by P"u", we can suppose that there exists
u2H1
0(
) such that
P"u"*u weakly inH1
0(
): (2.9)
It remains to determine the limit equation satised by u.
Second step . In order to get the limit equation satised by u, we have to pass to the
limit in (2.5). The most delicate part, and, in fact, the main novelty brought by our paper,
is the passage to the limit, in the variational formulation (2.5) of problem (2.1), in the
nonlinear term on the boundary of the grains, i.e. in the second term in the left-hand side
of (2.5). To this end, we introduce, for any h2Ls0(@F), 1s01 , the linear form "
h
onW1;s
0(
) dened by
h"
h;'i="Z
S"hx
"
'd8'2W1;s
0(
);
with 1=s+ 1=s0= 1. Then (see [70]),
"
h!hstrongly in ( W1;s
0(
))0; (2.10)
where
hh;'i=hZ

'dx;withh=1
jYjZ
@Fh(y) d:
Ifh2L1(@F) or ifhis constant, we have "
h!hstrongly in W1;1(
) and we denote
by"the above introduced measure in the particular case in which h= 1. Notice that in

HOMOGENIZATION OF REACTIVE FLOWS 25
this casehbecomes1=j@Fj=jYj. Let us prove now that for any '2D(
) and for any
v"*v weakly inH1
0(
), we get
'g(v")*'g (v) weakly in W1;q
0(
); (2.11)
where
q=2n
q(n2) +n:
To prove (2.11), let us rst note that
supkrg(v")kLq(
)<1: (2.12)
Indeed, from the growth condition (2.4) imposed to g, we get
Z

@g
@xi(v")q
dxCZ


1 +jv"jqq@v"
@xiq
dx
C(1 + (Z

jv"jqq
dx)1=
)(Z

jrv"jqdx)1=;
where we took
andsuch thatq= 2, 1=
+ 1== 1 andqq
= 2n=(n2):Notice that
from here we get q=2n
q(n2) +n. Also, since 0q < n= (n2), we have q >1. Now,
since
supkv"k
L2n
n2(
)<1;
we get immediately (2.12). Hence, to get (2.11), it remains only to prove that
g(v")!g(v) strongly in Lq(
): (2.13)
But this is just a consequence of the following well-known result (see [87], [152] and [155]):
Theorem 2.4 LetG:
R!Rbe a Carath eodory function, i.e.
a) for every vthe function G(
;v)is measurable with respect to x2
:
b) for every (a.e.) x2
, the function G(x;
)is continuous with respect to v.
Moreover, if we assume that there exists a positive constant Csuch that
jG(x;v)jC
1 +jvjr=t
;
withr1andt<1, then the map v2Lr(
)7!G(x;v(x))2Lt(
)is continuous in the
strong topologies.
Indeed, since
jg(v)jC(1 +jvjq+1);
applying the above theorem for G(x;v) =g(v),t=qandr= (2n=(n2))r0, withr0>0
such thatq+1<r=t and using the compact injection H1(
),!Lr(
) we easily get (2.13).
Finally, from (2.10), written for h= 1, and (2.11) written for v"=P"u", we conclude
h";'g(P"u")i!j@Fj
jYjZ

'g(u) dx8'2D(
) (2.14)

26 HOMOGENIZATION … heterogeneous structures …media ….
and this ends this step of the proof.
Third step . Let"be the gradient of u"in
"and let us denote by e"its extension
with zero to the whole of
. Obviously, e"is bounded in ( L2(
))nand hence there exists
2(L2(
))nsuch that
e"* weakly in (L2(
))n: (2.15)
Let us see now which is the equation satised by . Take'2D(
). From (2.5) we get
Z

e"
r'dx+"Z
S"g(u")'d=Z


"f'dx: (2.16)
Now, we can pass to the limit, with "!0, in all the terms of (2.16). For the rst one, we
have
lim
"!0Z

e"
r'dx=Z


r'dx: (2.17)
For the second term, using (2.14), we get
lim
"!0"Z
S"g(u")'d=j@Fj
jYjZ

g(u)'dx: (2.18)
It is not dicult to pass to the limit in the right-hand side of (2.16). Since

"f *jYj
jYjfweakly inL2(
);
we obtain
lim
"!0Z


"f'dx=jYj
jYjZ

f'dx: (2.19)
Putting together (2.17)-(2.19), we have
Z


r'dx+j@Fj
jYjZ

g(u)'dx=jYj
jYjZ

f'dx8'2D(
):
Henceveries
div+j@Fj
jYjg(u) =jYj
jYjfin
: (2.20)
It remains now to identify .
Fourth step. In order to identify , we shall make use of the solutions of the cell-problems
(2.8). For any xed i= 1;:::;n; let us dene
i"(x) ="
ix
"
+yi
8×2
"; (2.21)
wherey=x=". By periodicity,
P"i"*xiweakly inH1(
): (2.22)

HOMOGENIZATION OF REACTIVE FLOWS 27
Let"
ibe the gradient of  i"in
":Denote bye"
ithe extension by zero of "
iinside the
holes. From (2.21), for the j-component of e"
iwe get

e"
i
j= g@i"
@xj!
=
^@i
@yj(y)!
+ijY
and hence

e"
i
j*1
jYjZ
Y@i
@yjdy+jYjij
=jYj
jYjqijweakly inL2(
): (2.23)
On the other hand, it is not dicult to see that "
isatises
(
div"
i= 0 in
";
"
i
= 0 onS":(2.24)
Now, let'2D(
). Multiplying the rst equation in (2.24) by 'u"and integrating by
parts over
"we get
Z

""
i
r'u"dx+Z

""
i
ru"'dx= 0:
So Z

e"
i
r'P"u"dx+Z

""
i
ru"'dx= 0: (2.25)
On the other hand, taking 'i"as a test function in (2.5) we obtain
Z

"(ru"
r')i"dx+Z

"(ru"
ri")'dx+"Z
S"g(u")'i"d=Z

"f'i"dx;
which, using the denitions of e"ande"
i, gives
Z

e"
r'P"i"dx+Z

"ru"
"
i'dx+"Z
S"g(u")'i"d=Z

f
"'P"i"dx:
Now, using (2.25), we get
Z

e"
r'P"i"dxZ

e"
i
r'P"i"dx+"Z
S"g(u")'i"d=Z

f
"'P"i"dx:(2.26)
Let us pass to the limit in (2.26). Firstly, using (2.15) and (2.22), we have
lim
"!0Z

e"
r'P"i"dx=Z


r'xidx: (2.27)
On the other hand, (2.9) and (2.23) imply that
lim
"!0Z

e"
i
r'P"u"dx=jYj
jYjZ

qi
r'udx; (2.28)
whereqiis the vector having the j-component equal to qij.

28 HOMOGENIZATION … heterogeneous structures …media ….
Because the boundary of Fis smooth, of class C2,P"i"2W1;1(
) andP"i"!xi
strongly inL1(
):Then, since g(P"u")P"i"!g(u)xistrongly inLq(
) andg(P"u")P"i"
is bounded in W1;q(
), we have g(P"u")P"i"*g(u)xiweakly inW1;q(
). So,
lim
"!0"Z
S"g(u")'i"d=j@Fj
jYjZ

g(u)'xidx: (2.29)
Finally, for the limit of the right-hand side of (2.26), since 
"f *jYj
jYjfweakly in
L2(
), using again (2.22) we have
lim
"!0Z

f
"'P"i"dx=jYj
jYjZ

f'xidx: (2.30)
Hence, we get
Z


r'xidxjYj
jYjZ

qi
r'udx+j@Fj
jYjZ

g(u)'xidx=jYj
jYjZ

f'xidx: (2.31)
Using Green's formula and equation (2.20), we have
Z


rxi'dx+jYj
jYjZ

qi
ru'dx= 0 in
:
The above equality holds true for any '2D(
). This implies that

rxi+jYj
jYjqi
ru= 0 in
: (2.32)
Writing (2.30) by components, derivating with respect to xi;summing after iand using
(2.19), we conclude that
jYj
jYjnX
i;j=1qij@2u
@xi@xj= div=jYj
jYjf+j@Tj
jYjg(u);
which means that usatises
nX
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =fin
:
Sinceu2H1
0(
) (i.e.u= 0 on@
) anduis uniquely determined, the whole sequence
P"u"converges to uand Theorem 2.3 is proven.
Remark 2.5 The right scaling "in front of the function gmodeling the contribution of
the nonlinear reactions on the boundary of the grains leads in the limit to the presence of
a new term distributed all over the domain
.

HOMOGENIZATION OF REACTIVE FLOWS 29
Remark 2.6 The results in [79] are obtained for the case n3. All of them are still
valid, under our assumptions, in the case n= 2. Of course, for this case, n=(n2)has
to be replaced by +1and, hence, (2.4) holds true for 0q <1. The results of this
section could be obtained, under our assumptions, without imposing any growth condition
forg(see [211]).
Remark 2.7 In [79], the proof of Theorem 2.3 was done by using the so-called energy
method of L. Tartar (see [206]). We point out that one can use also the recently developed
periodic unfolding method, introduced by Cioranescu, Damlamian, Donato, Griso and Zaki
(see, e.g., [58], [64], [65], and [74]), which, apart from a signicant simplication in the
proof, allows us to deal with more general media, since we are not forced to use extension
operators.
2.1.3 The case of a non-smooth function g. The macroscopic model
The case in which the function gappearing in (2.1) is a single-valued maximal monotone
graph in RR, satisfying the condition g(0) = 0, is also treated in [79]. If we denote by
D(g) the domain of g, i.e.D(g) =f2Rjg()6=?g, then we suppose that D(g) =R.
Moreover, we assume that gis continuous and there exist C0 and an exponent q, with
0q<n= (n2), such that
jg(v)jC(1 +jvjq): (2.33)
Notice that the second important practical example b) mentioned above is a particular
example of such a single-valued maximal monotone graph.
We know that in this case there exists a lower semicontinuous convex function Gfrom
Rto ]1;+1],Gproper, i.e. G6+1, such that gis the subdierential of G,g=@G.
Let
G(v) =Zv
0g(s) ds:
Dene the convex set
K"=
v2V"jG(v)jS"2L1(S")
:
For a given function f2L2(
), the weak solution of the problem (2.1) is also the unique
solution of the following variational inequality:
8
>>><
>>>:Findu"2K"such that
DfZ

"Du"D(v"u") dxZ

"f(v"u") dx+h";G(v")G(u")i0
8v"2K":(2.34)
First, let us notice that there exists a unique weak solution u"2V"\H2(
") of the above
variational inequality (see [43]). Also, notice that it is well-known that the solution u"of

30 HOMOGENIZATION … heterogeneous structures …media ….
the variational inequality (2.34) is also the unique solution of the minimization problem
8
<
:u"2K";
J"(u") = inf
v2K"J"(v);
where
J"(v) =1
2DfZ

"jDvj2dx+h";G(v)iZ

"fvdx:
Let us introduce the following functional dened on H1
0(
):
J0(v) =1
2Z

QDvDv dx+j@Fj
jYjZ

G(v) dxZ

fvdx:
The main convergence result for problem (2.34) can be formulated as follows:
Theorem 2.8 One can construct an extension P"u"of the solution u"of the variational
inequality (2.34) such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of the minimization problem
8
<
:Findu2H1
0(
)such that
J0(u) = inf
v2H1
0(
)J0(v):(2.35)
Moreover,G(u)2L1(
). Here,Q= ((qij))is the classical homogenized matrix, whose
entries were dened in (2.7)-(2.8).
Remark 2.9 We notice that ualso satises
8
>><
>>:nX
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =f in
;
u= 0 on@
:
Proof. Letu"be the solution of the variational inequality (2.34). We shall use the same
extensionP"u"as in the previous case. It is not dicult to see that P"u"is bounded in
H1
0(
). So, by extracting a subsequence, one has
P"u"*u weakly inH1
0(
): (2.36)
Let'2D(
). By classical regularity results i2L1. Using the boundedness of iand
', there exists M0 such that

@'
@xi

L1

i

L1<M:

HOMOGENIZATION OF REACTIVE FLOWS 31
Let
v"='+X
i"@'
@xi(x)ix
"
: (2.37)
Then,v"2K"which will allow us to take it as a test function in (2.34). Moreover, v"!'
strongly in L2(
). Further,
Dv"=X
i@'
@xi(x)
ei+Dix
"
+"X
iD@'
@xi(x)ix
"
;
where ei, 1in, are the elements of the canonical basis in Rn. Usingv"as a test
function in (2.34), we get
Z

"Du"Dv"dxZ

"f(v"u") dx+Z

"Du"Du"dxh";G(v")G(u")i:
In fact, we have
Z

DP"u"^(Dv") dxZ

"f(v"u") dx+Z

"Du"Du"dxh";G(v")G(u")i:(2.38)
Let us denote
Qej=1
jYjZ
Y(Dj+ej) dy; (2.39)
where=jYj=jYj. It is not dicult to pass to the limit in the left-hand side and in the
rst term of the right-hand side of (2.38). We have
Z

DP"u"gDv"dx!Z

QDuD' dx (2.40)
andZ

"f(v"u") dx=Z

f
"(v"P"u") dx!Z

f('u) dx: (2.41)
For the third term of the right-hand side of (2.38), assuming the growth condition (2.33)
for the single-valued maximal monotone graph gand reasoning exactly like in the previous
subsection, we get
G(P"u")*G(u) weakly in W1;q
0(
)
and then
h";G(P"u")i!j@Fj
jYjZ

G(u) dx:
In a similar manner, we obtain
h";G(v")i!j@Fj
jYjZ

G(') dx
and hence we get
h";G(v")G(P"u")i!j@Fj
jYjZ

(G(')G(u)) dx: (2.42)

32 HOMOGENIZATION … heterogeneous structures …media ….
So, it remains to pass to the limit only in the second term of the right-hand side of (2.38).
For doing this, we can write down the subdierential inequality
Z

"Du"Du"dxZ

"Dw"Dw"dx+ 2Z

"Dw"(Du"Dw") dx; (2.43)
for anyw"2H1
0(
). Reasoning as before and choosing
w"='+X
i"@'
@xi(x)ix
"
;
where'enjoys similar properties as the corresponding ', the right-hand side of the in-
equality (2.43) passes to the limit and one has
lim inf
"!0Z

"Du"Du"dxZ

QD'D'dx+ 2Z

QD'(DuD') dx;
for any'2D(
). But since u2H1
0(
), taking '!ustrongly in H1
0(
);we conclude
lim inf
"!0Z

"Du"Du"dxZ

QDuDu dx: (2.44)
Putting together (2.40)-(2.42) and (2.44), we get
Z

QDuD' dxZ

f('u) dx+Z

QDuDu dxj@Fj
jYjZ

(G(')G(u)) dx;
for any'2D(
) and hence by density for any v2H1
0(
). So, nally, we obtain
Z

QDuD (vu) dxZ

f(vu) dxj@Fj
jYjZ

(G(')G(u)) dx;
which gives exactly the limit problem (2.34). This completes the proof of Theorem 2.8.
Remark 2.10 The choice of the test function (2.37) gives, in fact, a rst-corrector term
for the weak convergence of P"u"tou.
Remark 2.11 All the results of this section can be obtained for a general diusion matrix
A"(x) =A(x="), whereA=A(y)is a smooth matrix-valued function on Rnwhich is
Y-periodic. We shall assume that
8
<
:A2L1(
)nn;
Ais a symmetric matrix,
For some 0<
<;
jj2A(y)
jj28; y2Rn:
Problems similar to the one presented here may arise in various other contexts. For
instance, in [210], we analyzed the eective behavior of the solution of a nonlinear problem
arising in the modeling of enzyme catalyzed reactions through the exterior of a domain
containing periodically distributed reactive solid obstacles, with period ". Enzymes are

HOMOGENIZATION OF REACTIVE FLOWS 33
proteins that speed up the rate of a chemical reaction without being used up. They are
specic to particular substrates. The substrates in the reaction bind to active sites on
the surface of the enzyme. The enzyme-substrate complex then undergoes a reaction to
form a product along with the original enzyme. The rate of chemical reactions increases
with the substrate concentration. However, enzymes become saturated when the substrate
concentration is high. Additionally, the reaction rate depends on the properties of the
enzyme and the enzyme concentration. We can describe the reaction rate with a simple
equation to understand how enzymes aect chemical reactions. Michaelis-Menten equation
remains the most generally applicable equation for describing enzymatic reactions. In this
case, we consider the following elliptic problem:
8
>>><
>>>:Df
u"+(u") =fin
";
Df@u"
@="g(u") onS";
u"= 0 on@
:(2.45)
Here, the function is continuously dierentiable, monotonously non-decreasing and such
that(0) = 0. For example, we can take to be a linear function, i.e. (v) =v, or we can
consider the nonlinear case in which is given by (2.2) (Langmuir kinetics). For the given
functiong, we deal here with the case of a single-valued maximal monotone graph with
g(0) = 0, i.e. the case in which gis the subdierential of a convex lower semicontinuous
functionG. More precisely, we shall consider an important practical example, arising in
the diusion of enzymes (the Michaelis-Menten model):
g(v) =8
><
>:v
v+
; v0;
0; v< 0;
for;
> 0.
The existence and uniqueness of a weak solution of (2.45) is ensured by the classical
theory of monotone problems (see [43] and [105]). Therefore, we know that there exists a
unique weak solution u"2V"TH2(
"). Moreover, u"is also the unique solution of the
following variational problem:
8
>>>>><
>>>>>:Findu"2K"such that
DfZ

"Du"D(v"u") dx+Z

"(u")(v"u") dx
Z

"f(v"u") dx+h";G(v")G(u")i0;8v"2K";(2.46)
where"is the linear form on W1;1
0(
) dened by
h";'i="Z
S"'d;8'2W1;1
0(
):
The main convergence result in this case, proven in [210], is stated in the following theorem.

34 HOMOGENIZATION … heterogeneous structures …media ….
Theorem 2.12 The solution u", properly extended to the whole of
, converges to the
unique solution of the following variational inequality:
8
>>>>>>><
>>>>>>>:u2H1
0(
);
Z

QDuD (vu) dx+Z

(u)(vu) dxZ

f(vu) dx
j@Fj
jYjZ

(G(v)G(u)) dx;8v2H1
0(
):(2.47)
Here,Q= ((qij))is the homogenized matrix, dened in (2.7).
Remark 2.13 Notice that ualso satises
8
>><
>>:nX
i;j=1qij@2u
@xi@xj+(u) +j@Fj
jYjg(u) =f in
;
u= 0 on@
:
Thus, the asymptotic behavior of the solution of the microscopic problem (2.45) is governed
by a new elliptic boundary-value problem, with an extra zero-order term that captures the
eect of the enzymatic reactions. The eect of the enzymatic reactions initially situated
on the boundaries of the grains spread out in the limit all over the domain, giving the
extra zero-order term which captures this boundary eect. In fact, one could obtain a
similar result by considering interior enzymatic nonlinear chemical reactions given by the
same well-known nonlinear function g. The only dierence in the limit equation will be
the coecient appearing in front of this extra zero-order term. So, one can control the
eective behavior of such reactive
ows by choosing dierent locations for the involved
chemical reactions. Moreover, as we shall see in the next section, we can obtain similar
eects by considering transmission problems, with an unknown
ux on the boundary of
each grain, i.e. we can consider the case in which we have chemical reactions in
", but
also inside the grains, instead on their boundaries. The dierence in the limit equation will
be the coecient appearing in front of this extra zero-order term. Hence, we can control
the eective behavior of such reactive
ows by choosing dierent locations for the involved
chemical reactions.
2.1.4 Chemical reactions inside the grains of a porous medium
We shall brie
y present now some results obtained in [79] and …. for the case in which
we assume that we have a granular material lling the obstacles and we consider some
chemical reactive
ows through the grains. In fact, we consider a perfect transmission
problem (with an unknown
ux on the boundary of each grain) between the solutions of
two separated equations (for the case of imperfect transmission problems, see Chapter 5).

HOMOGENIZATION OF REACTIVE FLOWS 35
A simplied version of this kind of models can be formulated as follows:
8
>>>>>>>>><
>>>>>>>>>:Df
u"=fin
";
Dp
v"+g(v") = 0 in "
Df@u"
@=Dp@v"
@onS";
u"=v"onS";
u"= 0 on@
:(2.48)
Here, "=
n
",is the exterior unit normal to
",u"andv"are the concentrations
in
"and, respectively, inside the grains ",Df>0,Dp>0,f2L2(
) andgis a
continuous function, monotonously non-decreasing and such that g(v) = 0 if and only if
v= 0. Moreover, we suppose that there exist a positive constant Cand an exponent q,
with 0q < n= (n2), such thatjg(v)jC(1 +jvjq+1). Notice that above mentioned
examplesa) andb) are both covered by this class of functions.
Let us introduce the space
H"=
w"= (u";v")u"2V";v"2H1("); u"=v"onS"
;
with the norm
kw"k2
H"=kru"k2
L2(
")+krv"k2
L2("):
The variational formulation of problem (2.48) is the following one:
8
>>>><
>>>>:Findw"2H"such that
DfZ

"ru"
r'dx+DpZ
"rv"
r dx+Z
"g(v") dx=Z

"f'dx
8('; )2K":(2.49)
Under the above structural hypotheses and the conditions fullled by H", it is well-known
(see [43] and [155]) that (2.49) is a well-posed problem.
If we introduce the matrix
A=(
DfId inYnF
DpId inF;
then the main result in this situation is stated in the following theorem (for a detailed
proof, see [79]):
Theorem 2.14 One can construct an extension P"u"of the solution u"of the variational
problem (2.49) such that
P"u"*u weakly inH1
0(
);

36 HOMOGENIZATION … heterogeneous structures …media ….
whereuis the unique solution of
8
><
>:nX
i;j=1a0
ij@2u
@xi@xj+jFj
jYjg(u) =f in
;
u= 0 on@
:(2.50)
Here,A0= ((a0
ij))is the homogenized matrix, whose entries are dened as follows:
a0
ij=1
jYjZ
Y
aij+aik@j
@yk
dy; (2.51)
in terms of the functions i; i= 1;:::;n; solutions of the so-called cell problems
8
<
:div(AD(yj+j)) = 0 inY;
jYperiodic.(2.52)
The constant matrix A0is symmetric and positive-denite.
Corollary 2.15 Ifu"andv"are the solutions of the problem (2.48), then, passing to a
subsequence, still denoted by ", there exist u2H1
0(
)andv2L2(
)such that
P"u"*u weakly inH1
0(
);
ev"*v weakly inL2(
)
and
v=jFj
jYju:
Corollary 2.16 Let"be dened by
"(x) =u"(x)x2
";
v"(x)x2":
Then, there exists 2H1
0(
) such that"*  weakly inH1
0(
), whereis the unique
solution of8
><
>:nX
i;j=1a0
ij@2
@xi@xj+ajFj
jYjg() =f in
;
= 0 on@
;
andA0is given by (2.51)-(2.52), i.e. =u, due to the well-posedness of problem (2.49).
Remark 2.17 As already mentioned, the approach used in [79] and [210] is the so-called
energy method or the oscillating test function method introduced by L. Tartar [206], [207]
for studying homogenization problems. It consists of constructing suitable test functions
that are used in our variational problems. We point out that another possible way to get
the limit results could be to use the two-scale convergence technique, coupled with periodic
modulation, as in [34]. Also, one can use the periodic unfolding method (see, e.g., [65],
and [74]).

HOMOGENIZATION OF REACTIVE FLOWS 37
Remark 2.18 The two reactive
ows studied above, namely (2.1) and (2.48), lead to
completely dierent eective behaviors. The macroscopic problem (2.1) arises from the
homogenization of a boundary-value problem in the exterior of some periodically distributed
obstacles and the zero-order term occurring in (2.6) re
ects the in
uence of the chemical
reactions taking place on the boundaries of the reactive obstacles. On the other hand,
the second model is a boundary-value problem in the whole domain
, with discontinuous
coecients. Its macroscopic behavior also involves a zero-order term, but of a completely
dierent nature, emerging from the chemical reactions occurring inside the grains.
Remark 2.19 In (2.48) we considered that the ratio of the diusion coecients in the
two media is of order one in order to compare the case in which the chemical reactions take
place on the boundary of the grains with the case in which the chemical reactions occur
inside them. However, a more interesting problem arises if we consider dierent orders
for the diusion in the obstacles and in the pores. More precisely, if one takes the ratio
of the diusion coecients to be of order "2, then the limit model will be the so-called
double-porosity model . This scaling preserves the physics of the
ow inside the grains, as
"!0. The less permeable part of our medium (the grains) contributes in the limit as a
nonlinear memory term. In fact, the eective limit model includes two equations, one in F
and another one in
, the last one containing an extra-term which re
ects the remaining
in
uence of the grains (see, for instance, [11], [33], [34], [80], [142]).
We can treat in a similar manner the case of multi-valued maximal monotone graphs,
which includes various semilinear boundary-value problems, such as Dirichlet, Neumann
or Robin problems, Signorini's problems, problems arising in chemistry (see [70], [80], [83],
and [?]). Also, for the case of a dierent geometry of the perforated domain and dierent
transmission conditions, see Chapter 5.
2.2 Nonlinear adsorption of chemicals in porous media
In this section, we shall present some homogenization results, obtained in [80] (see, also,
..Timofte…), concerning the eective behavior of some chemical reactive
ows involving
diusion, dierent types of adsorption rates and chemical reactions which take place at
the boundary of the grains of a porous material. Such problems arise in many domains,
such as chemical engineering or soil sciences (see, for instance, [140], where the asymptotic
behavior of such chemical processes was analyzed and rigorous convergence results were
given for the case of linear adsorption rates and linear chemical reactions). The case
of nonlinear adsorption rates, left as open in [140], was treated in [80]. Two well-known
examples of such nonlinear models, namely the so-called Freundlich and Langmuir kinetics,
were studied. We brie
y describe here the results in [80]. In a rst step, we consider that
the surface of the grains is physically and chemically homogeneous. Then, we assume that
the surface of the solid part is physically and chemically heterogeneous and we allow also

38 HOMOGENIZATION … heterogeneous structures …media ….
a surface diusion modeled by a Laplace-Beltrami operator to take place on this surface.
In this last case, we show that the eective behavior of our system is governed by a new
boundary-value problem, with an additional microvariable and a zero-order extra term
proving that memory eects are present in this limit model.
2.2.1 The microscopic model and its weak solvability
Our main goal in [80] was to obtain the asymptotic behavior, as "!0, of the microscopic
models (2.53)-(2.55) below. The geometry of this problem is the same as the one in Section
2.1. More precisely, the domain consists of two parts: a
uid phase
"and a solid skeleton
(grains or pores),
n
". We assume that chemical substances are dissolved in the
uid
part. They are transported by diusion and also, by adsorption, they can change from
being dissolved in the
uid to residing on the surface of the pores. Here, on the boundary,
chemical reactions (which can be in
uenced by catalysts) take place. Hence, the model
consists of a diusion system in the
uid phase, a reaction system on the pore surface and
a boundary condition coupling them (see (2.54)). A simplied modeling of this situation
is as follows:
(V")8
>>><
>>>:@v"
@t(t;x)Df
v"(t;x) =h(t;x); x2
"; t> 0;
v"(t;x) = 0; x2@
; t> 0;
v"(t;x) =v1(x); x2
"; t= 0;(2.53)
Df@v"
@(t;x) ="f"(t;x); x2S"; t> 0; (2.54)
and
(W")8
<
:@w"
@t(t;x) +aw"(t;x) =f"(t;x); x2S"; t> 0;
w"(t;x) =w1(x); x2S"; t= 0;(2.55)
where
f"(t;x) =
(g(v"(t;x))w"(t;x)): (2.56)
Here,v"represents the concentration of the solute in the
uid region, w"is the con-
centration of the solute on the surface of the skeleton
n
",v12H1
0(
) is the initial
concentration of the solute and w12H1
0(
) is the initial concentration of the reactants on
the surface S"of the skeleton. Also, the
uid is assumed to be homogeneous and isotropic,
with a constant diusion coecient Df>0,a;
> 0 are the reaction and, respectively,
the adsorption factor and his an external source of energy.
The semilinear boundary condition on S"gives the exchanges of chemical
ows across
the boundary of the grains, governed by a non-linear balance law involving the adsorption
factor
(which, in a rst step, is considered to be constant) and the adsorption rate
represented by the nonlinear function g. Two model situations are considered: the case
of a monotone smooth function gwithg(0) = 0 and, respectively, the case of a maximal

HOMOGENIZATION OF REACTIVE FLOWS 39
monotone graph with g(0) = 0, i.e. the case in which gis the subdierential of a convex
lower semicontinuous function G. These two general situations are well illustrated by the
two important practical examples a) andb) (see (2.2) and (2.3)) mentioned in Section
2.1.1, namely the Langmuir and, respectively, the Freundlich kinetics.
It is worth remarking that if v"0 in
"andv">0 in
", then the function gin
examplea) is indeed a particular example of our rst model situation ( gis a monotone
smooth function satisfying the condition g(0) = 0). Also, let us note that, instead of (2.56),
we could consider a more general boundary condition, given in terms of
f"(t;x) =
1g(v"(t;x))
2w"(t;x);
where
1>0 is called adsorption factor and
2>0 is called desorption factor (see [142]).
The existence and uniqueness of a weak solution of the system (2.53)-(2.56) can be
settled by using the classical theory of semilinear monotone problems (see, for instance,
[43] and [155]). As a result, we know that there exists a unique weak solution u"= (v";w").
In order to write down the variational formulation of problem (2.53)-(2.56), let us dene
some suitable function spaces. Let H=L2(
), with the classical scalar product
(u;v)
=Z

u(x)v(x) dx;
and letH=L2(0;T;H), with the scalar product
(u;v)
;T=ZT
0(u(t);v(t))
dt;whereu(t) =u(t;
);v(t) =v(t;
):
Also, letV=H1(
), with (u;v)V= (u;v)
+ (ru;rv)
andV=L2(0;T;V), with
(u;v)V=ZT
0(u(t);v(t))Vdt:
We set
W=
v2Vdv
dt2V0
whereV0is the dual space of V;
V0=
v2Vv= 0 on@
a.e. on (0 ;T)
;W0=V0\
W:
Similarly, we dene the spaces V(
"),V(
"),V(S") andV(S"). For the latter we write
hu;viS"=Z
S"g"uvd;
whereg"is the metric tensor on S"; the rule of partial integration on S"applies and, if we
denote the gradient on S"byr"and the Laplace-Beltrami operator on S"by
", we have
(
"u;v)S"=hr"u;r"viS":
Also, for the space of test functions we use the notation D=C1
0((0;T)
)).

40 HOMOGENIZATION … heterogeneous structures …media ….
We shall start our analysis with the case in which gis a continuously dierentiable
function, monotonously non-decreasing and such that g(v) = 0 if and only if v= 0.
Also, we shall suppose that there exist a positive constant Cand an exponent q, with
0q<n= (n2), such thatdg
dvC(1 +jvjq): (2.57)
The weak formulation of problem (2.53)-(2.56) is:
8
>>>><
>>>>:Findv"2W 0(
");v"(0) =v1j
"such that

v";d'
dt

";T+"(f";')
";T=Df(rv";r')
";T+ (h;')
";T;
8'2W 0(
")(2.58)
and8
>><
>>:Findw"2W(S");w"(0) =w1jS"such that

w";d'
dt
S";T+a(w";')S";T= (f";')S";T;8'2W(S"):(2.59)
Proposition 2.20 There exists a unique weak solution u"= (v"; w")of the system (2.58)-
(2.59).
Remark 2.21 Let us notice that the solution of (2.59) can be written as
w"(t;x) =w1(x)e(a+
)t+
tZ
0e(a+
)(ts)g(v"(s;x)) ds
or, if we denote by ?the convolution with respect to time, as
w"(
;x) =w1(x)e(a+
)t+
r(
)?g(v"(
;x));
where
r() =e(a+
):
The solution v"of problem ( V") being dened only on
", we need to extend it to the
whole of
to be able to state the convergence result. In order to do that, we use Lemma
2.1. We also recall the following well-known result (see [81]):
Lemma 2.22 There exists a positive constant C, independent of ", such that
kvk2
L2(S")C("1kvk2
L2(
")+"krvk2
L2(
"));
for anyv2V":

HOMOGENIZATION OF REACTIVE FLOWS 41
2.2.2 The main result
Theorem 2.23 (Theorem 2.5 in [80]) One can construct an extension P"v"of the solution
v"of the problem (V")such that
P"v"*v weakly inV;
wherevis the unique solution of the following limit problem:
8
>>>>><
>>>>>:@v
@t(t;x) +F0(t;x)nX
i;j=1qij@2v
@xi@xj(t;x) =h(t;x); t> 0; x2
;
v(t;x) = 0; t> 0; x2@
;
v(t;x) =v1(x); t = 0; x2
;(2.60)
with
F0(t;x) =j@Fj
jY?j
h
g(v(t;x))w1(x)e(a+
)t
r(
)?g(v(
;x))(t)i
:
Here,Q= ((qij))is the classical homogenized matrix, whose entries are dened in (2.7).
Moreover, the limit problem for the surface concentration is:
8
><
>:@w
@t(t;x) + (a+
)w(t;x) =
g(v(t;x)); t> 0; x2
;
w(t;x) =w1(x); t = 0; x2
(2.61)
andwcan be written as
w(t;x) =w1(x)e(a+
)t+
r(t)?g(v(t;x)):
Remark 2.24 The weak formulation of problem (2.60) is:
8
>>>><
>>>>:Findv2W 0(
);v(0) =v1such that

v;d'
dt

;T+ (F0;')
;T=(Qrv;r')
;T+ (h;')
;T
8'2W 0(
):(2.62)
Proof. The proof of Theorem 2.23 (see [80]) consists of several steps. The rst step
is to prove the uniqueness of the limit problem (2.62). This is stated in the following
proposition, proven in [80]:
Proposition 2.25 There exists at most one solution of the weak problem (2.62).
The second step of the proof of Theorem 2.23 consists of describing the macroscopic
behavior of the solution u"= (v";w"), as"!0. To this end, some a priori estimates on
this solution are required (for a detailed proof, see [80]).

42 HOMOGENIZATION … heterogeneous structures …media ….
Proposition 2.26 Letv"andw"be the solutions of the problem (2.53)-(2.56). There
exists a positive constant C, independent of ", such that
kw"(t)k2
S"(kw"(0)k2
S"+

kg(v")k2
S";t)e
t;8t0;8>0;

@w"
@t

2
S";tC(kw"(0)k2
S"+kg(v")k2
S";t);8t0;
kv"(t)k2

"C;krv"(t)k2

";tC
and

@v"
@t(t)

2

"C:
The last step is the limit passage and the identication of the homogenized problem. Let
v"2W 0(
") be the solution of the variational problem (2.58) and let P"v"be the extension
ofv"inside the holes given by Lemma 2.1. Using the above a priori estimates, it follows
that there exists a constant Cdepending on Tand the data, but independent of "such
that
kP"v"(t)k
+krP"v"k
;t+k@t(P"v")(t)k
C;
for alltT. Consequently, by passing to a subsequence, still denoted by P"v", we can
assume that there exists v2Vsuch that the following convergence properties hold:
P"v"*v weakly inV;
@t(P"v")*@tvweakly inH;
P"v"!vstrongly inH:
It remains to identify the limit equation satised by v. To this end, we have to pass to the
limit, with "!0, in all the terms of (2.58). The most dicult part consists in passing
to the limit in the term containing the nonlinear function g. For this one, using the same
techniques as those used in Section 2.1, we can prove that
h";'g(P"v"(t))i!j@Fj
jYjZ

'g(v(t))dx8'2D:
We are now in a position to use Lebesgue's convergence theorem. Using the above pointwise
convergence, the a priori estimates stated in Proposition 2.26 and the growth condition
(2.57), we get
lim
"!0"
(g(v");')S";T=j@Fj
jYj
(g(v);')
;T:
For the rest of the terms, the proof is standard and we obtain immediately (2.60). Since
v2W 0(
) (i.e.v= 0 on@
) andvis uniquely determined, the whole sequence P"v"
converges to vand Theorem 2.23 is proven.

HOMOGENIZATION OF REACTIVE FLOWS 43
2.2.3 The case of a non-smooth boundary condition
In this subsection, we cover the case in which the function gin (2.56) is given by
g(v) =jvjp1v;0<p< 1 (Freundlich kinetics) :
For this case, which was left as an open one in [140], gis a single-valued maximal monotone
graph in RR, satisfying the condition g(0) = 0 and with D(g) =R. Moreover, gis
continuous and satises jg(v)jC(1 +jvj). As in Section 2.1.3, let Gsuch thatg=@G.
In this case, we also obtain the results stated in Theorem 2.23. The idea of the proof
is to use an approximation technique, namely Yosida regularization technique.
Let>0 be given. We consider the approximating problems:
8
>>>><
>>>>:Findv"
2W 0(
");v"
(0) =v1j
"such that

v"
;d'
dt)
";T+"(f"
;'

";T=Df(rv"
;r')
";T+ (h;')
";T;
8'2W 0(
")(2.63)
and8
><
>:Findw"
2W(S");w"
(0) =w1jS"such that

w"
;d'
dt
S";T+a(w"
;')S";T= (f"
;')S";T;8'2W(S"):(2.64)
where
f"
=
(g(v"
)w"
)
and
g=IJ
;
with
J= (I+@G)1:
Note thatgis a non-decreasing Lipschitz function, which satises the condition g(0) = 0.
Problem (2.63)-(2.64) has a unique solution ( v"
;w"
), for every  > 0 (see [43] and
[155]). As we saw in Section 2.2.1, we can express w"
in terms of v"
; therefore, it is enough
to get a problem only for v"
and in what follows we shall focus our attention only on
getting the limit problem for v"
.
Mollifying gto make it a smooth function (see [31]) and using the results of the
previous subsection, for any >0, we get
P"v"
!vstrongly inH(
):
Then, it is not dicult to see that, proving suitable a priori estimates (classical energy
estimates) on the solutions v, we can ensure, via compactness arguments (see [29]), the
strong convergence of v, as!0, tov, the unique solution of problem (2.60). Hence
v!vstrongly inH(
):

44 HOMOGENIZATION … heterogeneous structures …media ….
Finally, since
kP"v"vk
;TkP"v"P"v"
k
;T+kP"v"
vk
;T+kvvk
;T;
we get the strong convergence of P"v"tovinH(
).
Remark 2.27 The conclusion of the above theorem remains true for more general situa-
tions. It is the case of the so-called zeroth-order reactions, in which, formally, gis given
by the discontinuous function g(v) = 0 , ifv0andg(v) = 1 ifv >0. The correct math-
ematical treatment needs the problem to be reformulated by using the maximal monotone
graph of R2associated to the Heaviside function (v) =f0gifv < 0,(0) = [0;1]and
(v) =f1gifv>0. The existence and uniqueness of a solution can be found, for instance,
in [43]. The solution is obtained by passing to the limit in a sequence of problems associated
to a monotone sequence of Lipschitz functions approximating and so the results of this
section remain true.
2.2.4 Laplace-Beltrami model with oscillating coecients
In problem (2.53)-(2.56), the rate aof chemical reactions on S"and the adsorption co-
ecient
were assumed to be constant. A more realistic model implies to assume that
the surface @Fis chemically and physically heterogeneous, which means that aand
are
rapidly oscillating functions, i.e.
a"(x) =ax
"
;
"(x) =
x
"
;
withaand
positive functions in W1;1(
) which are Y-periodic (for linear adsorption
rates, see [142]). In this case, v"andw"satisfy the following system of equations:
(V")8
><
>:@v"
@t(t;x)Df
v"(t;x) =h(t;x); x2
"; t> 0;
v"(t;x) = 0; x2@
; t> 0;
v"(t;x) =v1(x); x2
"; t= 0;(2.65)
Df@v"
@(t;x) ="f"(t;x); x2S"; t> 0; (2.66)
and
(W")(@w"
@t(t;x) +a"(x)w"(t;x) =f"(t;x); x2S"; t> 0;
w"(t;x) =w1(x); x2S"; t= 0;(2.67)
where
f"(t;x) =
"(x)(g(v"(t;x))w"(t;x)): (2.68)
If we denote y=x=", then the main result in this case is the following one:

HOMOGENIZATION OF REACTIVE FLOWS 45
Theorem 2.28 (Theorem 4.1 in [80]) The eective behavior of vandwis governed by
the following system:
8
>>><
>>>:@v
@t(t;x) +G0(t;x)nX
i;j=1qij@2v
@xi@xj=h(t;x); t> 0; x2
;
v(t;x) = 0t>0; x2@
;
v(t;x) =v1(x)t= 0; x2
;(2.69)
and
(@w
@t(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(v(t;x)); t> 0; x2
; y2@F
w(t;x;y ) =w1(x)t= 0; x2
; y2@F;(2.70)
where
G0(t;x) =1
jY?jZ
@Ff0(t;x;y ) d
and
f0=
(y)(g(v(t;x))w(t;x;y )):
Here,Q= ((qij))is the classical homogenized matrix, dened by (2.7).
Obviously, the solution of (2.70) can be found using the method of variation of parameters.
Hence, we get
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)Zt
0e(a(y)+
(y))(ts)g(v(s;x)) ds;
or, using the convolution notation
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)r(
;y)?g(v(
;x))(t);
with
r(;y) =e(a(y)+
(y)):
Moreover, let us notice that (2.69)-(2.70) imply that v(t;x) satises the following equation
@v
@t(t;x)nX
i;j=1qij@2v
@xi@xj(t;x) +F0(t;x) =h(t;x); t> 0; x2
; (2.71)
with
F0(t;x) =1
jY?jZ
@Fn

(y)[g(v(t;x))w1(x)e(a(y)+
(y))t
(y)r(
;y)?g(v(
;x))(t)]o
d:

46 HOMOGENIZATION … heterogeneous structures …media ….
Remark 2.29 The above adsorption model can be slightly generalized by allowing surface
diusion on S". In fact, the chemical substances can creep on the surface and this eect
is similar to a surface-like diusion. From a mathematical point of view, we can model
this phenomenon by introducing a diusion term in the law governing the evolution of the
surface concentration w". This new term is the properly rescaled Laplace-Beltrami operator.
This implies that the rst equation in (2.67) has to be replaced by
@w"
@t(t;x)"2E
"w"(t;x) +a"(x)w"(t;x) =f"(t;x)x2S"; t> 0;
whereE > 0is the diusion constant on the surface S"and
"is the Laplace-Beltrami
operator on S".
In this case, the homogenized limit is the following one:
@w
@t(t;x;y )E
@F
yw(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(v(t;x));
fort>0; x2
; y2@F. Here,
@Fdenotes the Laplace-Beltrami operator on @Fand the
subscriptyindicates the fact that the derivatives are taken with respect to the local variable
y. The limit problem involves the solution of a reaction-diusion system with respect to an
additional microvariable. Also, notice that the local behavior is no longer governed by an
ordinary dierential equation, but by a partial dierential one.
Remark 2.30 We notice that the bulk behavior of system (V")-(W")involves an additional
microvariable y. This local phenomena yields a more complicated microstructure of the
eective medium; in (2.69)-(2.70) xplays the role of a macroscopic variable, while yis a
microscopic one. Also, we observe that the zero-order term in (2.71), namely F0involves
the convolution
r?g , which shows that we clearly have a memory term in the principal
part of our diusion-reaction equation (2.71).
The above results can be extended (see ….Timofte???) to include the case in which we
add a space-dependent nonlinear reaction rate =(x;v) in the interior of the domain and
we consider a space-dependent nonlinear adsorption rate g=g(x;v) and a non-constant
diusion matrix D"(x). More precisely, we analyze the asymptotic behavior, as "!0, of
the following coupled system of equations:
8
>>><
>>>:@v"
@t(t;x)div(D"(x)rv"(t;x)) +(x;v") =h(t;x); x2
"; t> 0;
v"(t;x) = 0; x2@
; t> 0;
v"(t;x) =v1(x); x2
"; t= 0;(2.72)
D"(x)rv"(t;x)
="f"(t;x); x2S"; t> 0; (2.73)
and 8
<
:@w"
@t(t;x) +a"(x)w"(t;x) =f"(t;x); x2S"; t> 0;
w"(t;x) =w1(x); x2S"; t= 0;(2.74)

HOMOGENIZATION OF REACTIVE FLOWS 47
where
f"(t;x) =
"(x)(g(x;v"(t;x))w"(t;x)): (2.75)
Assumptions.
1) The diusion matrix is dened as being D"(x) =D(x="), whereD=D(y) is a
smooth matrix-valued function on Rnwhich isY-periodic. We shall assume that
8
<
:D2L1(
)nn;
Dis a symmetric matrix,
For some 0 <
<;
jj2D(y)
jj28; y2Rn:
2) The function =(x;v) is continuous, monotonously non-decreasing with respect
tovfor anyxand such that (x;0) = 0.
3) The function g=g(x;v) is continuously dierentiable, monotonously non-decreasing
with respect to vfor anyxand withg(x;0) = 0. We suppose that there exist C0 and
two exponents qandrsuch that
j(x;v)jC(1 +jvjq) (2.76)
and 8
>><
>>:@g
@vC(1 +jvjq);
@g
@xiC(1 +jvjr) 1in;(2.77)
with 0q<n= (n2) and with 0r<n= (n2) +q.
Using the theory of semilinear monotone problems (see [43] and [155]), we know that
there exists a unique weak solution u"= (v";w") of system (2.72)-(2.75). Following the
same techniques as before, we obtain the following result (see ….):
Theorem 2.31 One can construct an extension P"v"of the solution v"of the problem
(2.72)-(2.75) such that
P"v"*v weakly inL2(0;T;H1(
));
wherevis the unique solution of the following limit problem:
8
><
>:@v
@t(t;x)div(D0rv) +(x;v) +F0(t;x) =h(t;x); t> 0; x2
;
v(t;x) = 0; t> 0; x2@
;
v(t;x) =v1(x); t = 0; x2
;(2.78)
with
F0(t;x) =
1
jY?jZ
@Fn

(y)[g(x;v(t;x))w1(x)e(a(y)+
(y))t
(y)r(
;y)?g(x;v(
;x))(t)]o
d:

48 HOMOGENIZATION … heterogeneous structures …media ….
The limit problem for the surface concentration is:
(@w
@t(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(x;v(t;x)); t> 0; x2
; y2@F
w(t;x;y ) =w1(x)t= 0; x2
; y2@F:(2.79)
Here,D0= ((d0
ij))is the homogenized matrix, dened by:
d0
ij=1
jYjZ
Y
dij(y) +dik(y)@j
@yk
dy;
in terms of the functions j; j= 1;:::;n; solutions of the cell problems
8
>>>>><
>>>>>:divyD(y)(ryj+ej) = 0 inY;
D(y)(rj+ej)
= 0 on@F;
j2H1
#Y(Y?);Z
Y?j= 0;
where ei,1in, are the elements of the canonical basis in I Rn. The constant matrix
D0is symmetric and positive-denite.
The solution of (2.79) can be found using the method of variation of parameters. We
have
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)tZ
0e(a(y)+
(y))(ts)g(x;v(s;x))ds;
or, using the convolution notation
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)r(
;y)?g(x;v(
;x))(t);
withr(;y) =e(a(y)+
(y)).
Remark 2.32 If we consider the case in which we have diusion of the chemical species
on the surface S", i.e.
@w"
@t(t;x)"2E
"w"(t;x) +a"(x)w"(t;x) =f"(t;x)x2S"; t> 0;
whereE > 0is the diusion constant on the surface S"and
"is the Laplace-Beltrami
operator on S", then instead of (2.79) we get the following local partial dierential equation:
@w
@t(t;x;y )E
@F
yw(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(x;v(t;x));(2.80)
fort>0; x2
; y2@F.

Related problems were addressed in the literature by many authors. Let us mention
only the papers of U. Hornung and W. J ager [142], U. Hornung [140], D. Cioranescu, P.
Donato and R. Zaki [72], C. Conca, J.I. D az and C. Timofte [80], C. Timofte … , G.
Allaire and H. Hutridurga [7]. The results presented here constitute a generalization of
some of the results obtained in [80], [140] and [ ?], by considering heterogeneous
uids,
space-dependent nonlinear reaction rates in the interior of the domain and non-smooth
reactions rates on the boundaries of the pores.
An interesting perspective is to investigate, in multi-component porous media with
imperfect interfaces, the case of systems of reaction-diusion equations involving nonlinear
reaction-terms and nonlinear boundary conditions. Also, it would be of interest to deal
with the case of various geometries … more general nonlinearities …
49

50 …..HOMOGENIZATION … heterogeneous structures …media ….

Chapter 3
Homogenization results for
unilateral problems
Our goal in this chapter is to discuss some homogenization results for a class of unilateral
problems in periodically perforated media. In a series of papers (see [83], [84], [209], [49],
and [51]), the limit behavior of the solutions of …….
For homogenization results for variational inequalities, we refer, e.g., to ……..Boccardo,
Murat, Yosian, Duvaut, Lions …………
3.1 Homogenization results for Signorini's type problems
In this section, we shall present some homogenization results, obtained in [83] by using
Tartar's oscillating test function method, for the solutions of some Signorini's type-like
problems in periodically perforated domains with period ". The main feature of these kind
of problems is the existence of a critical size of the perforations that separates dierent
emerging phenomena as "!0. In the critical case, it is shown in [83] that the solution of
such a problem converges to a Dirichlet one, associated to a new operator which is the sum
of a standard homogenized operator and an extra zero order term, known as strange term ,
coming from the geometry; its appearance is due to the special size of the holes. The limit
problem captures the two sources of oscillations involved in this kind of free boundary-value
problems, namely, those arising from the size of the holes and those due to the periodic
inhomogeneity of the medium. The main ingredient of the method used in the proof is an
explicit construction of suitable test functions which provide a good understanding of the
interactions between the above mentioned sources of oscillations.
The results in [83] constitute a generalization of those obtained in the the well-known
pioneering work of D. Cioranescu and F. Murat [63]. In their article, the authors deal
with the asymptotic behavior of solutions of Dirichlet problems in perforated domains,
showing the appearance of a strange extra-term as the period of the perforations tends
to zero and the holes are of critical size. They mainly considered the constraint u"0
51

52 …..HOMOGENIZATION … heterogeneous structures …media ….
on the holes (which includes the classical Dirichlet condition u"= 0 onS"). In [83], we
generalized their method and framework to a class of Signorini's problem, involving just a
positivity condition imposed on the boundary of the holes. Our results show that one is
led to analogous limit problems despite the fact that the constraint is only acting on S".
The classical weak formulations of such unilateral problems involve a standard varia-
tional inequality (in the sense of [157]), corresponding to a nonlinear free boundary-value
problem. Such a model was introduced in the earliest '30 by A. Signorini [204] (see also
G. Fichera [115]) for studying problems arising in elasticity, and more precisely problems
involving an elastic body under unilateral contact shear forces acting on its boundary. For
a nice presentation of the mechanical aspects behind Signorini's problem (and also for some
mathematical and historical backgrounds) we refer to [43] and [105].
3.1.1 The microscopic problem
Let
be a smooth bounded connected open subset of Rnand letFbe another open
bounded subset of Rn, with a smooth boundary @F(of classC2). We assume that 0
belongs toFand thatFis star-shaped with respect to 0. Since Fis bounded, to simplify
matters, without loss of generality, we shall assume that FY, whereY= (1
2;1
2)nis
the representative cell in Rn. We setY=YnF. Letr:R+!R+be a continuous map,
related to the size of the holes. We assume that
lim
"!0r(")
"= 0 andr(")<"=2 (3.1)
or
r(")": (3.2)
The rst situation corresponds to the case of small holes , while the last one covers the case
ofbig holes .
For each"and for any integer vector k2Zn, letF"
k="k+r(")F. Also, let us denote
byF"the set of all the holes contained in
, i.e.
F"=[
F"
kjF"
k
; k2Zn
:
Set
"=
nF"andS"=[f@F"
kjF"
k
;k2Zng. So,@
"=@
[S".
Let us consider a family of inhomogeneous media occupying the region
, parameter-
ized by"and represented by nnmatricesA"(x) of real-valued coecients dened on

. The positive parameter "will thus also dene a length scale measuring how densely
the inhomogeneities are distributed in
. Indeed, one of our main physical goals in this
paper will be a reasonable understanding of the interactions between these two sources
of oscillations represented by the parameter ", namely, those coming from the geometry
(more exactly, the size of the holes) and those due to the inhomogeneity of the medium
(the matrix A"(x) involves rapidly oscillating coecients).

HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 53
With
"we associate the following nonempty closed convex subset of H1(
"):
K"=
v2H1(
")jv= 0 on@
; v0 onS"
; (3.3)
whereS"is the boundary of the holes and @
is the external boundary of
. Our main
motivation is to study the asymptotic behavior of the solution of the following variational
problem in
":
8
<
:Findu"2K"such thatZ

"A"Du"D(v"u") dxZ

"f(v"u") dx8v"2K";(3.4)
wherefis a given function in L2(
).
The solution u"of (3.4) is also known to be characterized as being the solution of
the following non-linear free boundary-value problem: Find a smooth function u"and two
subsetsS"
0andS"
+such thatS"
0[S"
+=S"; S"
0\S"
+=;, and
8
<
:div(A"Du") =fin
";
u"= 0 onS"
0; A"Du"
0 onS"
0;
u">0 onS"
+; A"Du"
= 0 onS"
+;(3.5)
whereis the exterior unit normal to the surface S". This means that we can distinguish
onS"twoa priori unknown subsets S"
0andS"
+whereu"satises complementary boundary
conditions coming from the following global constraints:
u"; A"Du"
0 andu"A"Du"
= 0 onS": (3.6)
We shall consider periodic structures dened by
A"(x) =Ax
"
;
whereA=A(y) is a matrix-valued function on Rnwhich isY-periodic and satises the
following conditions:
8
<
:A2L1(
)nn;
Ais a symmetric matrix,
for some 0<<; jj2A(y)
jj28; y2I Rn:
For simplicity, we further assume that Ais continuous with respect to y. Under the above
structural hypotheses and the conditions fullled by K", it is well-known by a classical
existence and uniqueness result of J. L. Lions and G. Stampacchia [157] that (3.4) is a
well-posed problem.
3.1.2 The macroscopic models
Several situations occur depending on the asymptotic behavior of the size of the holes.
There exists a critical size that separates dierent behaviors of the solution u"as"!0:

54 …..HOMOGENIZATION … heterogeneous structures …media ….
This size is of order "n=(n2)ifn3 and of order exp( 1="2) ifn= 2. For simplicity,
in what follows we shall discuss only the case n3 (the case n= 2 can be treated in an
analogous manner).
In the critical case, it was proven in [83] that the solution u"of problem (3.4), properly
extended to the whole of
, converges to the unique solution of a Dirichlet problem in
associated with a new operator which is the sum of the standard homogenized one and an
extra term strange term that comes in from the special geometry (the size of the holes).
More precisely, we have the following result:
Theorem 3.1 (Theorem 4.1 in [83]) There exists an extension P"u"of the solution u"of
the variational inequality (3.4), positive inside the holes, such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of
8
<
:u2H1
0(
);Z

A0DuDv dx
0u;v
H1(
);H1
0(
)=Z

fvdx8v2H1
0(
):(3.7)
Here,A0is the classical homogenized matrix, whose entries are dened as follows:
a0
ij=1
jYjZ
Y
aij(y) +aik(y)@j
@yk
dy;
in terms of the functions j; j= 1;:::;n; solutions of the so-called cell problems
8
<
:divy(A(y)Dy(yj+j)) = 0 inY;
jYperiodic
and0is given by
0= inf
w2H1(Rn)Z
RnA(0)DwDw dxjw1q.e. onF
: (3.8)
The constant matrix A0is symmetric and positive-denite.
Remark 3.2 The limit problem takes into account all the ingredients involved in (3.4).
In (3.7) are involved two sources of oscillations and both of them are captured at the limit.
Those coming from the periodic heterogeneous structure of the medium are re
ected by the
presence of the homogenized matrix A0, and those due to the critical size of the holes are
re
ected by the appearance of a strange term 0:The other ingredient contained in (3.7)
is the spreading eect of the unilateral condition u"0onS"which can be seen by the
fact that the strange term only charges the negative part of u;it is just the negative part
uthat is penalized at the limit.

HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 55
The proof of Theorem 3.1, given in [83], is based on the use of a technical result of
E. De Giorgi [92] for matching boundary conditions for minimizing sequences. This result
allowed us to modify sequences of functions near the holes Br(")("i) and to separate the
contribution of the gradient of our solution close and far from the holes. By doing this,
we are able to capture, simultaneously, the oscillations in the functional coming from the
periodic oscillations of the medium and those arising from the oscillations of the boundaries
of the holes.
Besides this critical case , there are three other cases that are considered in [83]. The
rst one is the case in which the holes are much smaller than the critical ones. In this
case, they are too small to produce any visible contribution at the limit and the solution u"
converges to the solution of a classical homogenized Dirichlet problem in
associated to the
matrixA. The second case is that when the size of the holes is bigger than the critical one,
but still smaller than the period ". The holes being big enough, the positivity constraint
of the solution u"imposed only on S"will become a positivity condition, u0, all over
the domain. The limit problem is an obstacle problem associated to the corresponding
homogenized medium. In this case, the holes only spread the positivity condition all over
the domain. The last case that we state explicitly below is characterized by the fact that
the size of the holes is exactly of order ". The solution u", properly extended to the whole
of
, converges in this case to an obstacle problem, associated to the homogenization of a
periodic heterogeneous and perforated medium. The in
uence of the holes comes twofold:
on one hand, they spread the positivity condition on S"to the whole of
and on the other
one, their size do aect the homogenized medium. More precisely, for the case of holes of
the same size as the period, we have the following result (see [83]):
Theorem 3.3 (Theorem 4.6 in [83]) There exists an extension P"u"of the solution u"of
the variational inequality (3.4), positive inside the holes, such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of
8
>>>><
>>>>:u2H1
0(
); u0in
;
Z

A0
pDuDu dx2Z

fudxZ

A0
pDvDv dx2Z

fvdx;
8v2H1
0(
); v0in
:(3.9)
Here,A0
p= (a0
ij)is the classical homogenized matrix, whose entries are dened as follows:
a0
ij=1
jYjZ
Y
aij(y) +aik(y)@j
@yk
dy;

56 …..HOMOGENIZATION … heterogeneous structures …media ….
in terms of the functions j; j= 1;:::;n; solutions of the cell problems
8
>>><
>>>:divyA(y)(Dyj+ej) = 0 inY;
A(y)(Dj+ej)
= 0 on@F;
j2H1
#Y(Y?);R
Y?j= 0;
where ei,1in, are the elements of the canonical basis in I Rn. The constant matrix
A0is symmetric and positive-denite.
Let us notice that the variational inequality in (3.9) can be written as
Z

A0
pDuD (vu)dxZ

f(vu)dx:
As mentioned before, the method we followed in [83] is the energy method of L. Tar-
tar [206]. However, it is worth mentioning that the -convergence of integral functionals
involving oscillating obstacles is an alternative which already proved to be a successful one.
Extensive references on this topic can be found in the monographs of G. Dal Maso [87] and
of A. Braides and A. Defranceschi [42]. Also, as we shall see in the next section, another
way to obtain convergence results for such problems is to use the recently developed peri-
odic unfolding method. This method was introduced, for xed domains, by D. Cioranescu,
A. Damlamian, G. Griso in [67], [68] and by A. Damlamian in [90]. Their results were ex-
tended to perforated domains by D. Cioranescu, P. Donato, R. Zaki [74], [58] and, further,
by D. Cioranescu, A. Damlamian, G. Griso, D. Onofrei in [69], by D. Onofrei in [174] and
by A. Damlamian, N. Meunier in [91] for the case of small holes.
The periodic unfolding method brings signicant simplications in the proofs of many
convergence results and allows us to deal with media with less regularity, since we don't
need to use extension operators.
3.2 Homogenization results for elliptic problems in perfo-
rated domains with mixed-type boundary conditions
In this section, via the periodic unfolding method, we present some results obtained in [49]
and generalizing some of the results in [83]. More precisely, we shall analyze the asymptotic
behavior of a class of elliptic equations with highly oscillating coecients, in an "-periodic
perforated structure, with two holes of dierent sizes in each period. Depending on the
boundary interaction that take place at their surfaces, two distinct conditions, one of
Signorini's type and another one of Neumann type, are imposed on the corresponding
boundaries of the holes. On the exterior xed boundary of the perforated domain, an
homogeneous Dirichlet condition is prescribed. As in [49], the main feature of this kind
of problems is the existence of a critical size of the perforations that separates dierent
emerging phenomena as the small parameter "tends to zero. In this critical case, we prove

HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 57
that the homogenized problem, obtained by the periodic unfolding method, contains two
additional terms coming from the particular geometry. These new terms, a right-hand side
term and a strange one, capture the two sources of oscillations involved in this problem, i.e.
those arising from the special size of the holes and those due to the periodic heterogeneity
of the medium.
Similar problems were addressed in the literature. As mentioned in Section 3.1, the
homogenization of the Poisson equation with a Dirichlet condition for perforated domains
was treated by D. Cioranescu, F. Murat [76], putting into evidence, in the case of critical
holes, the appearance of a strange term. Their results were extended, using dierent tech-
niques, to heterogeneous media by N. Ansini, A. Braides [23], G. Dal Maso, F. Murat [89]
and D. Cioranescu, A. Damlamian, G. Griso, D. Onofrei in [69]. Recently, A. Damlamian,
N. Meunier [91] studied the periodic homogenization for multivalued Leray-Lions operators
in perforated domains. The case of non homogeneous Neumann boundary conditions was
considered, among others, by C. Conca, P. Donato [82] and D. Onofrei [174]. For problems
with Robin or nonlinear boundary conditions we refer, for instance, to D. Cioranescu, P.
Donato [70], D. Cioranescu, P. Donato, H. Ene [71], D. Cioranescu, P. Donato, R. Zaki
[74] and A. Capatina, H. Ene [50]. Also, for Signorini's type problems we mention Yu. A.
Kazmerchuk, T. A. Mel'nyk [150]. The homogenization of problems involving perforated
domains with two kinds of holes of various sizes, was recently considered by D. Cioranescu,
Hammouda [75].
The non-standard feature of the problem we present here is given by the presence, in
each period, of two holes of dierent sizes and with dierent conditions (3.10) 2;3imposed on
their boundaries. More precisely, we consider the case of Signorini and, respectively, critical
Neumann holes. The Signorini condition (3.10) 2(see [175]) implies that the variational
formulation (3.11) of our problem is expressed as an inequality, which creates further
diculties. Problems involving such boundary conditions arise in groundwater hydrology,
chemical
ows in media with semipermeable membranes, etc. For more details concerning
the physical interpretation of the above mentioned boundary conditions, the interested
reader is referred to G. Duvaut, J.L. Lions [105] and U. Hornung [140].
3.2.1 Setting of the microscopic problem
Let us brie
y describe now the new geometry of the problem (see Figure …).
….gure ….
Let
Rn,n3, be a bounded open set such that j@
j= 0 and let Y=
1
2;1
2n
be the reference cell. We consider an "Y-periodic perforated structure with two kind of
holes: some of size "1and the other ones of size "2, with1and2depending on "and
going to zero as "goes to zero. More precisely, we consider two open sets BandFwith

58 …..HOMOGENIZATION … heterogeneous structures …media ….
.BYδ1δ2δ1B.Ωε,δ1δ2δ2TT
Figure 3.1: The perforated domain
smooth boundaries such that BY,FYandB\F=  and we denote the above
mentioned holes by
B"1=[
2Zn"(+1B);
F"2=[
2Zn"(+2F):
LetY12=Yn(1B[2F) be the part occupied by the material in the cell and suppose
that it is connected. The perforated domain
";12with holes of size of order "1and of
size of order "2at the same time, is dened by

";12=
n(B"1[F"2) =n
x2
jnx
"o
Y2Y12o
:
LetA2L1(
)nnbe aY-periodic symmetric matrix. We suppose that there exist two
positive constants and, with 0<< , such that
jj2A(y)
jj282Rn;8y2Y :
Moreover, we assume that Ais continuous at the point 0.
Given aY-periodic function g2L2(@F) and a function f2L2(
), we consider the
following microscopic problem:
8
>>>>><
>>>>>:div (A"ru";12) =fin
";12;
u";120; A"ru";12
B0; u";12A"ru";12
B= 0 on@B"1;
A"ru";12
F=g"2on@F"2;
u";12= 0 on@ext
";12;(3.10)
where
A"(x) =Ax
"
and
g"2(x) =g1
2nx
"o
Y
a.e.x2@F"2:

HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 59
In (3.10),BandFare the unit exterior normals to the sets B"1and, respectively, F"2.
In order to obtain a variational formulation of problem (3.10), we introduce the space
V"
12=fv2H1(
";12)jv= 0 on@ext
";12g
and the convex set
K"
12=fv2V"
12jv0 on@B"1g:
Then, the variational formulation of (3.10) is the following variational inequality:
(P";12)8
>>>>>>><
>>>>>>>:Findu";122K"
12such that
Z

";12A"ru";12
(rvru";12) dxZ

";12f"(vu";12) dx
+Z
@F"2g"2(vu";12) ds8v2K"
12:(3.11)
Classical results for variational inequalities (see, for example, [176], [137], [44]) ensure the
existence and the uniqueness of a weak solution of the problem (3 :11).
We are interested in obtaining the asymptotic behavior of the solution of problem (3.11)
when"; 1; 2!0. Following [49], we consider the case in which
8
>>>><
>>>>:k1= lim
"!0n
21
1
";0<k1<1;
k2= lim
"!0n1
2
";0<k2<1;(3.12)
which means that we are dealing with the case of critical size, both for the Signorini holes
and, respectively, for the Neumann ones. Due to (3.12), we shall write that "!0 instead
of writing ( ";1;2)!(0;0;0).
3.2.2 The limit problem
In order to state the main convergence result for this problem, we introduce the following
functional space
KB=fv2L2(Rn) ;rv2L2(Rn); v= constant on Bg
where 2is the Sobolev exponent2n
n2associated to 2. Also, for i= 1;:::;n , let us
considerithe solution of the cell problem
8
><
>:i2H1
per(Y);
Z
YAr(iyi)
rdy= 082H1
per(Y):(3.13)

60 …..HOMOGENIZATION … heterogeneous structures …media ….
andthe solution of the problem
8
>><
>>:2KB; (B) = 1;
Z
RnnBA(0)r
rvdz= 08v2KBwithv(B) = 0:(3.14)
For the special geometry of this problem, we need to introduce, following [64] and
[69], two unfolding operators T"andT", the rst one corresponding to the case of xed
domains and the second one to the case of domains with small inclusions. For dening
the rst operator, we need to introduce some notation. For x2Rn, we denote by [ x]Yits
integer part k2Zn, such that x[x]Y2Yand we setfxgY=x[x]Yfor a.e.x2Rn.
So, for almost every x2Rn, we havex="hx
"i
+nx
"o
. We consider the following sets:
bZ"=n
k2Znj"Yk
o
;b
"= int[
k2bZ"
"Yk
;"=
nb
":
For any function '2Lp(
), withp2[1;1), we dene the periodic unfolding operator
T":Lp(
)!Lp(
Y) by the formula
T"(')(x;y) =8
<
:'
"hx
"i
+"y
for a.e. (x;y)2b
"Y
0 for a.e. ( x;y)2"Y:
The operatorT"is linear and continuous from Lp(
) toLp(
Y). We recall here some
useful properties of this operator (see, for instance, [64]):
(i) if'and are two Lebesgue measurable functions on
, one has
T"(' ) =T"(')T"( );
(ii) for every '2L1(
), one has
1
jYjZ

YT"(')(x;y) dxdy=Z
b
"'(x) dx=Z

'(x) dxZ
"'(x) dx;
(iii) iff'"gL2(
) is a sequence such that '"!'strongly in L2(
), then
T"('")!'strongly in Lp(
Y);
(iv) if'2L2(Y) isY-periodic and '"(x) ='(x="), then
T"('")!'strongly in L2(
Y);
(v) if'"*' weakly inH1(
), then there exists a subsequence and b'2L2(
;H1
per(Y))
such that
T"(r'")*rx'+ryb'weakly inL2(
Y):

HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 61
For domains with small holes, we need to introduce an unfolding operator depending
on both parameters "and. We recall now its denition (for details, see [69]). To this
end, let us consider domains with "Y-periodically distributed holes of size ", with >0
going to zero with "tending to zero. More precisely, for a given open set BY, we
denoteY
=YnB. The perforated domain

"is dened by


"=n
x2
nx
"o
2Y
o
:
If we consider functions which vanish on the whole boundary of the perforated domain,
i.e. functions belonging to H1
0(

"), then we can extend them by zero to the whole of
.
In this case, we shall not distinguish between functions in H1
0(

") and their extensions in
H1
0(
).
Denition 3.4 For any'2Lp(
), withp2[1;1), we dene the periodic unfolding
operatorT"by the formula
T"(')(x;z) =8
<
:T"(')(z;z)if(x;z)2b
"1
Y
0 otherwise:
By using the change of variable z= (1=)y, one can obtain similar properties for the
operatorT"to those stated for T"(see [69]).
The main convergence result obtained in [49] is stated in the following theorem.
Theorem 3.5 (Theorem …. in [ ?]) Letu";12be the solution of the variational inequality
(3.11). Under the above hypotheses, there exists u2H1
0(
)such that
T"(u";12)*u weakly inL2(
;H1(Y)); (3.15)
whereu2H1
0(
)is the unique solution of the homogenized problem
8
>>>>><
>>>>>:u2H1
0(
)
Z

Ahomru
r'dxk2
1Z

u'dx =Z

f'dx
+k2j@FjM@F(g)Z

'dx8'2H1
0(
):(3.16)
In (3.16),Ahomis the classical homogenized matrix dened, in terms of isolution of
(3.13), as
Ahom
ij=Z
Y
aij(y)nX
k=1aik(y)@j
@yk(y)!
dy
andis the capacity of the set B, given by
=Z
RnnBA(0)rz
rzdz;
whereveries (3.14).

62 …..HOMOGENIZATION … heterogeneous structures …media ….
Remark 3.6 In the limit problem (3.16), we can see the presence of two extra terms,
generated by the suitable sizes of our holes. Also, let us remark in (3.16) the spreading
eect of the unilateral condition imposed on the boundary of the Signorini holes: the strange
term, depending on the matrix A, charges only the negative part uof the solution.
Remark 3.7 In the case k1= 0, the extra term generated by the Signorini holes vanishes
in the limit, while for k2= 0the contribution of the Neumann holes disappears.
Proof of Theorem 3.4. It is well known that the variational inequality (3.11) is
equivalent to the following minimization problem
(Findu";122K"
12such that
J"
12(u";12)J"
12(v)8v2K"
12;(3.17)
where
J"
12(v) =1
2Z

";12A"rv
rvdxZ

";12fvdxZ
@F"2g"2vds: (3.18)
Let us prove that
lim sup
"!0J"
12(u";12)J0(')8'2D(
); (3.19)
where
J0(') =1
2Z

Ahomr'
r'dx+1
2k2
1Z

(')2dx
Z

f'dx+k2j@FjM@F(g)Z

'dx:(3.20)
For'2D(
), we put
h"(x) ='(x)"nX
i=1@'
@xi(x)ix
"
;
whereiis the solution of the problem (3.13).
If we takev"1=h+
"w"1h
", where
w"1(x) = 11
1nx
"o
Y
8x2Rn;
withgiven by (3.14), we obtain
J"
12(v"1) =I1
"I2
";
where
I1
"=1
2Z

";12A"(rh+
"w"1rh
"h
"rw"1)
(rh+
"w"1rh
"h
"rw"1) dx;
I2
"=Z

";12f"(h+
"w"1h
")dx+Z
@F"2g"(h+
"w"1h
") ds:

HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 63
Using the periodic unfolding operators T"andT"1introduced in [67] and [69], we get
T"(A")(x;y) =A(y) in
Y;
T"1(w"1)(x;z) =T"(w"1)(x;1z) = 1(z) in
Rn;
T"1(rw"1)(x;z) =1
"1rz(z) in
Rn:
We also have (
w"1*1 weakly in H1(
);
h"!'strongly in H1(
):(3.21)
Taking into account the properties of the unfolding operator T"andT"1, we get the
following convergences
8
>><
>>:T"(h")!'strongly in L2(
Y);
T"(rh")!rx'+ry'1strongly in L2(
Y);
T"1(rh")!rx'+ry'1strongly in L2(
Rn);(3.22)
where
'1=nX
i=1@'
@xii:
By unfolding and using the fact that frh
"g"is bounded in ( L2(
))n, we can pass to the
limit in the unfolded form of integral I1
"and we obtain
lim
"!0I1
"=1
2Z

YA(r'+ry'1)
(r'+ry'1) dxdy+
1
2k2
1Z

(RnnB)A(0)(')2rz
rzdxdz;
which, combined with (3.13), yields
lim
"!0I1
"=1
2Z

YAhomr'
r'dxdy +1
2k2
1Z

(RnnB)A(0)(')2rz
rzdxdz:(3.23)
Exactly like in [75], i.e. using the boundary unfolding operator Tb
"2, we can pass to the
limit inI2
"and we obtain
lim
"!0I2
"=Z

f'dx+k2j@FjM@F(g)Z

'dx: (3.24)
Therefore, putting together (3.23) and (3.24), we are led to
lim
"!0J"
12(v"1) =J0(')8'2D(
): (3.25)
Thus, we get (3.19).

64 …..HOMOGENIZATION … heterogeneous structures …media ….
Let us prove now that
lim inf
"!0J";12(u";12)J0(u): (3.26)
To this end, we decompose our solution into its positive and, respectively, its negative part,
i.e.
u";12=u+
";12u
";12:
From the problem ( P";12), it follows that there exists a constant Csuch that
ku";12kH1(
";12)C: (3.27)
Sinceu";122V"
12, we can assume that, up to a subsequence, there exists u2H1
0(
)
such that
8
<
:T"(u";12)*u weakly inL2(
;H1(Y));
ku
";12ukL2(
";12)!0;
T"(u
";12)!ustrongly in L2(
Y):(3.28)
Obviously, we have
lim inf
"!0Z

";12A"ru+
";12
ru+
";12dxZ

Ahomru+
ru+dx: (3.29)
In order to get (3.26), taking into account that the linear terms pass immediately to
the limit, it remains only to prove that
lim inf
"!0Z

";12A"ru
";12
ru
";12dxZ

Ahomru
rudx+k2
1Z

(u)2dx:(3.30)
Since
Z

";12A"r(u
";12h"w"1)
r(u
";12h"w"1) dx0;

HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 65
we haveZ

";12A"ru
";12
ru
";12dxZ

";12A"(h")2rw"1
rw"1dx
Z

";12A"(w"1)2rh"
rh"dx+ 2Z

";12A"h"ru
";12
rw"1dx
+2Z

";12A"w"1ru
";12
rh"dx2Z

";12A"h"w"1rw"1
rh"dx

n
21
1
"!2Z

RnT"1(A")(T"1(h"))2rz
rzdxdz
Z

YT"(A")T"(rh")
T"(rh")(T"(w"1))2dxdy
2n
21
1
"Z

RnT"1(A")T"1(h")h
n=2
1T"1(ru
";12)i

rz(z) dxdz
+2Z

YT"(A")T"(w"1)T"(ru
";12)
T"(rh") dx
+2n
21
1
"n=2
1Z

RnT"1(A")T"1(h")(1(z))rz
T"1(rh") dxdz:
From (3.27) and (3.28) and the properties of the unfolding operators T"andT"1(see [69]),
it follows that there exist u12L2(
;H1
per(Y)) andU12L2(
;L2
loc(Rn)) such that
8
<
:T"(ru
";12)*ru+ryu1 weakly inL2(
Y);
n
2
1T"1(ru
";12)*rzU1 weakly inL2(
Rn):(3.31)
Therefore, we obtain
lim inf
"!0Z

";12A"ru
";12
ru
";12dxk2
1Z

'2dx
Z

YA(r'+ry'1)
(r'+ry'1) dxdy2k1Z

(RnnB)A0'rzU1
rzdxdz
+2Z

YA(ru+ryu1)
(r'+ry'1)dxdy8'2H1
0(
):
(3.32)
SinceT"1(u
";12) = 0 on
B, we haveU1= 0 on
B. Therefore, W1=U1k1u2
L2(
;KB). On the other hand, from the cell problem (3.14) we get
divz(A(0)rz) = 0 inD0(
(RnnB))
which, by Stokes formula, leads to
Z
RnnBA(0)rz
rz dz= (B)Z
@BA(0)rz
Bds8 2KB: (3.33)

66 …..HOMOGENIZATION … heterogeneous structures …media ….
For almost every x2
,W1(x;
)2KB, so (3.33) gives
Z
RnnBA(0)rz
rzW1dz=W1(x;B)Z
@BA(0)rz
Bds:
Now, sincerzW1=rzU1andU1(x;B) = 0, we obtain
Z
RnnBA(0)rz
rzU1dz=k1uZ
@BA(0)rz
Bds=k1u
which gives
2k1Z

(RnnB)A(0)'rzU1
rzdx= 2k2
1Z

u'dx:
Taking'=uin (3.32) and using the fact thatnX
i=1@u
@xii=u1, we obtain (3.30).
Finally, from (3.19), (3.26) and by the density D(
),!H1
0(
) , we deduce
lim
"!0J"
12(u";12) =J0(u)J0(')8'2H1
0(
): (3.34)
Asis non-negative, by Lax-Milgram theorem, it follows that the minimum point for
the functional J0is unique. This means that the whole sequence T"(u";12) converges to
u.
Using a classical trick (see, for example, [189], [76]), one can prove that the functional
P(v) =1
2Z

(v)2dx8v2H1
0(
)
is Fr echet (and thus G^ ateaux) dierentiable and its gradient is given by
P0(u)
v=Z

uvdx8u;v2H1
0(
):
Therefore, the functional J0is G^ ateaux dierentiable on H1
0(
) and this ensures the equiv-
alence of the minimization problem
J0(u) = min
'2H1
0(
)J0(') (3.35)
with the problem (3.16) and this ends the proof of Theorem 3.5.
Remark 3.8 From (3.27), it follows that there exists an extension bu";12of our solution
to the whole of
, positive on the Signorini holes (see [83]), such that
bu";12*u weakly inH1
0(
): (3.36)
For instance, in a rst step we extend our solution inside the Signorini holes in such a way
that 
bu";12= 0 inB"1;
bu";12=u";12on@B"1;

and, then, we extend it further in a standard way (see, for instance, [82]) inside the Neu-
mann holes. As a matter of fact, the use of unfolding operators allows us to work without
extending our solution to the whole of
.
Remark 3.9 We can treat in a similar manner the problem ( ??) for a general matrix A
satisfying the usual conditions of boundedness and coercivity but we have to suppose, like
in [69] or [91], that there exist two matrix elds AandA0, such that
T"(A")(x;y)!A(x;y)a.e. in
Y
and
T"1(A")(x;z)!A0(x;z)a.e. in
(RnnB):
The only dierence is the fact that in this case the corresponding homogenized matrix, the
cell problems and the strange term depend also on x.
We end this section by pointing out that in Section …. we shall brie
y discuss a
related model, obtained via the periodic unfolding method in [51]. More precisely, we shall
be concerned with the derivation of macroscopic models for some elasticity problems in
periodically perforated domains with rigid inclusions of the same size as the period.
67

68 …..HOMOGENIZATION … heterogeneous structures …media ….

Chapter 4
Mathematical models in biology
In the last decades, there has been an explosive growth of interest in studying the macro-
scopic properties of biological systems having a very complicated microscopic structure.
When studying such systems, we are often led to consider boundary-value problems in me-
dia exhibiting multiple scales. It is widely recognized that multiscale techniques represent
an essential tool for understanding the macroscopic properties of such systems having a
very complicated microscopic structure.
In this chapter, we shall present some homogenization results for a series of problems
arising in the mathematical modeling of various reaction-diusion processes in biological
tissues. In the last years, … attention for .. mathematical models in biology … …toy
models … still, despite the all these eorts, many rigorous mathematical models can be
…..toy models , being far from capturing the complexity of . ….
Reaction-diusion problems with applications in biology
This chapter is based on the … papers ……..
1. C. Timofte, Homogenization results for a carcinogenesis model, in press???, Mathe-
matics and Computers in Simulation, 2016.
These results were announced in …. C. Timofte, Multiscale Analysis of a Carcinogenesis
Model, Biomath Communications, 2 (1), 2015.
2. C. Timofte, Homogenization results for the calcium dynamics in living cells, Math-
ematics and Computers in Simulation, in press, 2016, doi:10.1016/j.matcom.2015.06.01
2015.
…. only to mention … C. Timofte, Homogenization results for enzyme catalyzed
reactions through porous media, Acta Mathematica Scientia 29B (1), 74-82, 2009.
3. C. Timofte, Homogenization results for dynamical heat transfer problems in het-
erogeneous biological tissues, Bulletin of the Transilvania University of Brasov, 2 (51),
143-148, 2009.
4. C. Timofte, Upscaling in dynamical heat transfer problems in biological tissues,
Acta Physica Polonica B 39 (11), 2811-2822, 2008.
69

70 …..HOMOGENIZATION … heterogeneous structures …media ….
4.1 Homogenization results for ionic transport phenomena
in periodic charged media
We start this chapter by presenting some homogenization results for ionic transport phe-
nomena in periodic charged porous media. These results were obtained, via the periodic
unfolding method, in [ ?]. More precisely, we shall describe the eective behavior of the
solution of a system of coupled partial dierential equations arising in the modeling of
ionic transfer phenomena, coupled with electrocapillary eects, in periodic charged porous
media. The so-called Nernst-Planck-Poisson system was originally proposed by W. Nernst
and M. Planck (see [200]) for describing the potential dierence in galvanic cells and has
nowadays broad applicability in electrochemistry, in biology, in plasma physics or in the
semiconductor device modeling, where this system is also known as van Roosbroeck system .
For more details about the physical aspects behind these models and for a review of the
recent relevant literature, the interested reader is referred to [122], [198], [200], [201], and
[202].
We deal, at the microscale, with a periodic structure modeling a saturated charged
porous medium. In such a periodic microstructure, we shall consider the Poisson-Nernst-
Planck system, with suitable boundary and initial conditions. The diusion in the
uid
phase is governed by the Nernst-Planck equations, while the electric potential which in-

uence the ionic transfer is described by the Poisson equation. Moreover, we include in
our analysis electrocapillary eects. Arguing in a similar manner as in [122] or [198], we
get the well posedness of this problem in suitable spaces and we can obtain proper energy
estimates. The increased complexity of the geometry and of the governing equations im-
plies that an asymptotic procedure must be applied for describing the solution of such a
problem. The complicated microstructure will be homogenized in order to obtain a model
that captures its averaged properties.
Via the periodic unfolding method, we can show that the eective behavior of the
solution of our problem is described by a new coupled system of equations (see (3.11)-
(3.14)). In particular, the evolution of the macroscopic electrostatic potential is governed
by a new law, similar to Grahame's law (see [122] and [132]). Apart from a signicant
simplication in the proofs, an advantage of using such an approach based on unfolding
operators, which transform functions dened on oscillating domains into functions acting
on xed domains, is that we can avoid using extension operators and, thus, we can deal in a
rigorous manner with media possessing less regularity than those appearing usually in the
literature (it is well-known that composite materials, biological tissues or semiconductor
devices are highly heterogeneous and their interfaces are not, generally, enough smooth).
Moreover, the homogenized equations being dened on a xed domain
and having simpler
coecients will be easier to be handled numerically than the original equations. The
dependence on the initial microstructure can be seen at the limit in the homogenized
coecients.

MATHEMATICAL MODELS IN BIOLOGY 71
Related problems have been addressed, using dierent techniques, in [122], [198] or
[201]. As already mentioned, our approach in [ ?] relies on a new method, namely the
periodic unfolding method, which enables us to work with general media. Another novelty
brought by our paper resides in dealing with a general nonlinear boundary condition for
the electrostatic potential and with nonlinear reaction terms.
4.1.1 Setting of the problem
Let us describe the geometry of the problem. As customary in the literature, we assume
that the porous medium possesses a periodic microstructure. Basically, the geometry is
the one introduced in Section 1.1, but we shall use a specic terminology for this case.
Thus, we consider a bounded connected smooth open set
in Rn, withj@
j= 0 and
withn2. We shall deal here only with the physically relevant cases n= 2 orn= 3.
The reference cell Y= (0;1)nis decomposed in two smooth parts, the
uid phase Yand,
respectively, the solid phase F. We suppose that the solid part has a Lipschitz continuous
boundary and does not reach the boundary of Y. Therefore, the
uid region is connected.
We denote by
"the
uid part, by F"the solid part and by S"the inner boundary of the
porous medium, i.e. the interface between the
uid and the solid phases. Since the solid
part is not allowed to reach the outer boundary @
, it follows that S"\@
=;.
In such a periodic microstructure, we shall consider the Poisson-Nernst-Planck system,
with suitable boundary and initial conditions. The diusion in the
uid phase is governed
by the Nernst-Planck equations, while the electric potential which in
uence the ionic trans-
fer is described by the Poisson equation. Also, we include electrocapillary eects in our
analysis. More precisely, if we denote by [0 ;T], withT > 0, the time interval we are in-
terested in, we shall analyze the eective behavior, as the small parameter "!0, of the
solution of the following system:
8
>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>:
"=c+
"c
"+D in (0;T)
";
r"
=""G(x;") on (0;T)S";
r"
= 0 in (0 ;T)@
;
@c
"
@tr
(rc
"c
"r") =F(c+
";c
") in (0;T)
";
(rc
"c
"r")
= 0 on (0 ;T)S";
(rc
"c
"r")
= 0 on (0 ;T)@
;
c
"=c
0inft= 0g
":(4.1)
In (4.1),is the unit outward normal to
", "is the electrostatic potential, c
"are the
concentrations of the ions (or the density of electrons and holes in the particular case of
van Roosbroeck model), D2L1(
) is the given doping prole, Gis a nonlinear function
which captures the eect of the electrical double layer phenomenon arising at the interface
S"andFis a reaction term.

72 …..HOMOGENIZATION … heterogeneous structures …media ….
Let us notice that the scaling in the right hand side of the boundary condition on
S"for the electric potential ensures that we keep the in
uence of the double layer at the
macroscale. This scaling is, in fact, physically justied by experiments. For the case in
which one considers dierent scalings in (4.1), see [198] and [ ?].
We suppose that
"(x) =x
"
;
where(y) is aY-periodic, bounded, smooth real function such that (y) >0. Also,
we assume that the electrocapillary adsorption phenomenon at the substrate interface is
modeled by a given nonlinear function G. We address the case in which G=G(x;s) is
continuously dierentiable, monotonously increasing with respect to sfor anyxand with
G(x;0) = 0. Also, we assume that, for n3, there exist C0 and two exponents pand
msuch that 8
>>>><
>>>>:@G
@sC(1 +jsjp);
@G
@xiC(1 +jsjm) 1in;(4.2)
with 0pn=(n2) and 0m<n= (n2) +p. Moreover, by using a regularization
procedure, for example Yosida approximation, as in [80], the hypothesis concerning the
smoothness of the nonlinearity Gcan be relaxed. For instance, we can treat the case
of single or multivalued maximal monotone graphs, as in [80]. Also, our results can be
obtained, under our assumptions, without imposing any growth condition (see [211]).
In practical applications, based on the Gouy-Chapman theory, one can use the Grahame
equation (see [122] and [132]) in which
G(s) =K1sinh(K2s); K 1;K2>0:
For the case of lower potentials, sinh( x) can be expanded in a power series of the form
sinh(x) =x+x3
3!+:::
and one can use the approximations sinh xxor sinhxx+x3=3!.
Concerning the reaction terms, the case in which
F(c+
";c
") =(c+
"c
")
was treated in [216], but, as mentioned there, the case in which
F(c+
";c
") =(a"c+
"b"c
");
with
a"(x) =ax
"
; b"(x) =bx
"
;

MATHEMATICAL MODELS IN BIOLOGY 73
wherea(y) andb(y) areY-periodic, real, smooth, bounded functions with a(y)a0>0,
b(y)b0>0, can be easily treated in a similar manner.
In [?], we deal with the more general case in which
F(c+
";c
") =g(c+
"c
");
withgan increasing locally Lipschitz continuous function on R, withg(0) = 0. In partic-
ular, this setting includes the case treated in [113], i.e. the one in which
F(c+
";c
") =(f1(c+
")f2(c
"));
withfi, fori21;2, increasing Lipschitz continuous functions satisfying conditions which
guarantee the positivity and the necessary uniform upper bounds for the concentration
elds.
Using the techniques from [80], [225] or [ ?], we can study other relevant types of reaction
rates, such as those appearing in the so-called Auger generation-recombination model or
in the Shockley-Read-Hall model (see [139]). More precisely, we can deal with the case in
whichF=F(u;v) is a continuously dierentiable function on R2, which is sublinear
and globally Lipschitz continuous in both variables and such that F(u;v) = 0 foru<0
orv <0. For other nonlinear reaction rates Fand more general functions G, see [140],
[141] and ???[ ?].
We suppose that the initial data are non-negative and bounded independently with
respect to"andZ

"(c+
0c
0+D) dx="Z
S""G(x;") ds:
Moreover, we suppose that the mean value in
"of the potential  "is zero.
From the Nernst-Planck equation, it is not dicult to see that the the total mass
M=Z

"(c+
"+c
") dx
is conserved and suitable physical equilibrium conditions are veried, both at the microscale
and at the macroscale (see, for details, [122] and [ ?]). Let us mention that, for simplifying
the notation, we have eliminated in system (4.1) some constant physical relevant parame-
ters.
We consider here only two oppositely charged species, i.e. positively and negatively
charged particles, with concentrations c
", but all the results can be easily generalized for
the case ofNspecies. Also, let us notice that we deal here only with the case of an isotropic
diusivity in the
uid phase, but we can extend our analysis to the case of heterogeneous
media given by matrices D
"or to the case of the Stokes-Poisson-Nernst-Planck system,
with Neumann, Dirichlet or even Robin boundary condition. Also, let us remark that we
can address the case in which the electrostatic potential is dened all over the domain
,
with suitable transmission conditions at the interface S", as in [122] or [202].

74 …..HOMOGENIZATION … heterogeneous structures …media ….
The weak formulation of problem (4.1) is as follows: nd ( "; c+
"; c
"), with
8
>>>><
>>>>:"2L1(0;T;H1(
"));
c
"2L1(0;T;L2(
"))\L2(0;T;H1(
"));
@c
"
@t2L2(0;T; (H1(
"))0)(4.3)
such that, for any t>0 and for any '1; '22H1(
"), the triple ( "; c+
"; c
") satises:
Z

"r"
r'1dxZ
S"r"
'1d=Z

"(c+
"c
"+D)'1dx; (4.4)
D@c
"
@t; '2E
(H1)0;H1+Z

"(rc
"c
"r")
r'2dx=Z

"F(c+
";c
")'2dx (4.5)
and
c
"(0;x) =c
0(x) in
": (4.6)
The variational problem (3.3)-(3.6) has a unique weak solution ( "; c+
"; c
") (see [122],
[198] or [ ?]). Moreover, exactly like in [198], we can prove that the concentrations are
non-negative. In fact, they are bounded from below and above, uniformly in ".
Under the above hypotheses, using a standard procedure, it follows that there exists a
constantC2R+, independent of ", such that the following a priori estimates hold true:
k"kL2((0;T)
")+kr"kL2((0;T)
")C
max
0tTkc
"kL2(
")+ max
0tTkc+
"kL2(
")+krc
"kL2((0;T)
")+krc+
"kL2((0;T)
")+

@c
"
@t

L2(0;T;(H1(
"))0)+

@c
"
@t

L2(0;T;(H1(
"))0)C:
Due to the complexity of the microscopic problem, we are interested in obtaining the
eective behavior, as "!0, of the solution ( "; c+
"; c
") of problem (3.3)-(3.6). Our
approach relies on the periodic unfolding method (see [67] and [74]). By using such a
technique, we do not need to use extension operators like in [122] or [198].
Let us brie
y recall here the denition of the unfolding operator T"introduced in [67]
and [74] for a perforated domain with holes of the same size as the period. For more details,
including complete proofs of the properties of this operator, we refer to [65], [64], [74], [58],
and [72].
For any Lebesgue measurable function 'on
", the periodic unfolding operator T"is
the linear operator dened by
T"(')(x;y) =8
<
:'
"hx
"i
+"y
for a.e. (x;y)2b
"Y
0 for a.e. ( x;y)2"Y:

MATHEMATICAL MODELS IN BIOLOGY 75
The periodic unfolding operator T"has similar properties as the corresponding operator
T"dened for xed domains in Section …
Using the properties of the unfolding operator T"and the above a priori estimates,
we can easily prove that there exist  2L2(0;T;H1(
)),b2L2((0;T)
;H1
per(Y)),
c2L2(0;T;H1(
)),bc2L2((0;T)
;H1
per(Y)), such that, up to a subsequence,
T"(")* weakly in L2((0;T)
;H1(Y)); (4.7)
T"(r")*r +ryb weakly in L2((0;T)
Y); (4.8)
T"(c
")!cstrongly in L2((0;T)
;H1(Y)); (4.9)
T"(rc
")*rc+rybcweakly inL2((0;T)
Y): (4.10)
4.1.2 The homogenized problem
In order to obtain the needed asymptotic behavior of the solution of our microscopic
model, using the periodic unfolding method, we shall pass to the limit, with "!0, in the
variational formulation of problem (3.1). We get the following convergence result:
Theorem 4.1 Under the above hypotheses, the solution ("; c+
";c
")of system (3.1) con-
verges, in the sense of (3.7)-(3.10), as "!0, to the unique solution (; c+;c)of the
following macroscopic problem in (0;T)
:
8
>><
>>:div(D0r) +1
jYj0G=c+c+D;
@c
@tdiv(D0rcD0cr) =F
0;(4.11)
with the boundary conditions on (0;T)@
:
(
D0r
= 0;
(D0rcD0cr)
= 0(4.12)
and the initial conditions
c(0;x) =c
0(x);8×2
: (4.13)
Here,
0=Z
@F(y) ds;
F
0=F(c+;c) =g(c+c)
andD0= (d0
ij)is the homogenized matrix, dened as follows:
d0
ij=1
jYjZ
Y
ij+@j
@yi(y)
dy;

76 …..HOMOGENIZATION … heterogeneous structures …media ….
withj; j= 1;:::;n; solutions of the cell problems
8
>>>><
>>>>:j2H1
per(Y);Z
Yj= 0;

j= 0 inY;
(rj+ej)
= 0 on@F(4.14)
andei,1inthe vectors in the canonical basis of Rn.
… de scris unitar … ca la EJDE Section 1.
Proof. In order to prove Theorem 3.1, let us take, in a rst step, in the Poisson
equation (3.4), the test function
'1(t;x) = 0(t;x) +" 1
t;x;x
"
;
with 02D((0;T);C1(
)) and 12D((0;T)
;H1
per(Y)). Using the unfolding ope-
ratorT"(see Appendix …), we get
ZT
0Z

YT"(r")T"(r( 0+" 1)) dxdydt+
ZT
0Z

@FT"()T"(G("))T"( 0+" 1)) dxdsdt=
ZT
0Z

YT"(c+
"c
"+D)T"( 0+" 1)) dxdydt: (4.15)
Taking into account the above convergence results, it is not dicult to compute the limit
of the linear terms in (3.15) dened on
Y(see, for instance, [67], [80] and ???? [ ?]).
For the term containing the nonlinear function G, let us remark that, exactly like in [80],
one can prove that if Ris a continuously dierentiable function, monotonously increasing,
withR(x;v) = 0 if and only if v= 0 and fullling the assumption (3.2), then, for any
w"*w weakly inH1
0(
), we get
R(x;w")*R(x;w);
weakly inW1;p
0(
), where
p=2n
q(n2) +n:
Let us observe that p1. Therefore, using the properties of the unfolding operator T"
and Lebesgue's convergence theorem, we have
ZT
0Z

@FT"()T"(G(x;"))T"( 0+" 1)) dxdsdt!0ZT
0Z

G(x;) 0dxdt:

MATHEMATICAL MODELS IN BIOLOGY 77
Thus, for"!0, we obtain:
ZT
0Z

Y(r(t;x) +ryb(t;x;y )) (r 0(t;x) +ry 1(t;x;y )) dxdydt+
0ZT
0Z

G(x;(t;x)) 0(t;x) dxdt=
ZT
0Z

Y(c+(t;x)c(t;x) +D(x)) 0(t;x) dxdydt: (4.16)
By density, (3.16) holds for any 02L2(0;T;H1(
)) and 12L2((0;T)
;H1
per(Y)).
Taking 0(t;x) = 0, we obtain
8
><
>:
yb(t;x;y ) = 0 in (0 ;T)
Y;
ryb
=rx(t;x)
on (0;T)
@F;
b(t;x;y ) periodic in y:
By linearity, we get
b(t; x; y ) =nX
j=1j(y)@
@xj(t; x); (4.17)
wherej; j=1; n, are the solutions of the local problems (3.14).
Taking 1(t;x;y ) = 0, integrating with respect to the variable xand using (3.17), we
easily get the homogenized problem for the electrostatic potential .
In a second step, taking in the Nernst-Planck equation the test function
'2(t;x) = 0(t;x) +" 1
t;x;x
"
;
with 02D((0;T);C1(
)) and 12D((0;T)
;H1
per(Y)), we have:
ZT
0Z

YT"(c
")T"@
@t( 0+" 1)
dxdydt+
ZT
0Z

YT"(rc
"c
"r")T"( 0+" 1)) dxdydt=
ZT
0Z

YT"(F(c+
";c
"))T"( 0+" 1) dxdydt:
Passing to the limit with "!0, we obtain
ZT
0Z

Yc(t;x)@
@t 0(t;x) dxdydt+
ZT
0Z

Y(rc(t;x) +rybc(t;x;y ))(r 0(t;x) +ry 1(t;x;y )) dxdydt=

78 …..HOMOGENIZATION … heterogeneous structures …media ….
ZT
0Z

YF
0(c+;c) 0(t;x) dxdydt: (4.18)
Using again standard density arguments, (3.18) can be written for any 02L2(0;T;H1(
))
and 12L2((0;T)
;H1
per(Y)).
Now, taking 0(t;x) = 0 and, then, 1(t;x;y ) = 0, we get exactly the homogenized
problem for the concentrations c.
Due to the uniqueness of the solutions  and cof problem (3.3)-(3.6) (see [198] and [ ?]),
the whole sequences of microscopic solutions converge to the solution of the homogenized
problem and this completes the proof of Theorem 3.1.
4.2 Multiscale Analysis of a Carcinogenesis Model
In this section, we shall focus on the results obtained in [218], where our goal was to analyze,
using homogenization techniques, the eective behavior of a coupled system of reaction-
diusion equations, arising in the modeling of some biochemical processes contributing to
carcinogenesis in living cells. We are concerned with the carcinogenic eects produced in
the human cells by Benzo-[a]-pyrene molecules.
Carcinogenesis is a complex multi-step process during which normal cells change their
behavior and their metabolism. As a result, they might proliferate in an uncontrolled way,
escaping to the surveillance of the immune system and producing, ultimately, metastases.
Various biochemical mechanisms, such as chemical reactions and diusion processes taking
place at the surface of the endoplasmic reticulum or binding and cleaning processes, mod-
eled by suitable nonlinear functions, must be taken into account in a realistic mathematical
model. A well-known cause for carcinogenesis is a reactive toxic molecule called Benzo-
[a]-pyrene (BP), found in coal tar, cigarette smoke, charbroiled food, etc. To understand
the complex behavior of these molecules, mathematical models including reaction-diusion
processes and binding and cleaning mechanisms have been developed. In order to reduce
the complexity of the structure of such a model and to make it numerically treatable and
not so computationally expensive, homogenization techniques have been used.
Following [128], we consider here a simplied setting in which BP molecules invade the
cytosol inside of a human cell. There, they diuse freely, but they cannot enter in the
nucleus. Also, they bind to the surface of the endoplasmic reticulum (ER), where chem-
ical reactions, produced by the enzyme system called MFO (microsomal mixed-function
oxidases), take place, BP being chemically activated to a diol epoxide molecule, Benzo-[a]-
pyrene-7,8-diol-9,10-epoxide (DE). The DE molecules can unbind from the surface of the
endoplasmic reticulum and they can diuse again in the cytosol, where they may enter
in the nucleus. These new molecules can bind to DNA, DNA damage being considered a
primary cause of cancer. Natural cleaning mechanisms occurring in the cytosol that make
the carcinogenic molecules harmless are taken into account, too. The slow diusion process
taking place at the surface of the endoplasmic reticulum is modeled with the aid of the

MATHEMATICAL MODELS IN BIOLOGY 79
Laplace-Beltrami operator. For describing the binding-unbinding process at the surface of
the endoplasmic reticulum, we consider various functions, leading to dierent homogenized
models. A dierent carcinogenesis model, introduced in [129], will be brie
y discussed in
Section 3.2.3. In this model, BP molecules can bind to the surface of the endoplasmic
reticulum by linking to receptors, the binding process being modeled, based on the law of
mass action, by the product of the concentration of molecules and that of receptors. Also,
for a receptor-based model obtained using homogenization techniques, see [85]. For more
details about the mechanisms governing carcinogenesis in human cells we refer to [124] and
[232].
Problems closed to the one we treat here were addressed in [7], [85], [128], [129] and
[131]. For papers devoted to the upscaling of reactive transport in porous media, we refer
to [5], [225], [140], [227], [228], [231], [223] and the references therein. For reaction-diusion
problems involving adsorption and desorption, we refer to [7], [79], [80], [104], [140], [196].
For proving our main convergence results, we use the periodic unfolding method devel-
oped in [64], [67], [74] and [99] and extended in [129] and [130] for dealing with gradients of
functions dened on smooth periodic manifolds. Our analysis extends some of the results
obtained in [128] and [129] and were announced in [233]. More precisely, we deal here with
the case in which the surface of the endoplasmic reticulum is supposed to be heterogeneous
and, also, with the case in which the adsorption is modeled with the aid of a nonlinear
isotherm of Langmuir type.
The non-linearity of the model requires strong compactness results for the sequence of
solutions in order to be able to pass to the limit. Also, for passing to the limit in the terms
containing gradients of functions dened on the surface of the endoplasmic reticulum, we
follow the ideas in [129].
4.2.1 The microscopic problem
Let us describe now brie
y the geometry of the problem, which is similar to the one
considered in Section 1.1. More precisely, we consider a bounded connected open set
inRn, with a Lipschitz boundary @
and with n2. The domain
, which, as in
[129], is assumed to be representable by a nite union of axis-parallel cuboids with corner
coordinates belonging to Qn, represents a human cell with the domain occupied by the
nucleus removed. We denote by Cthe cell membrane and by Nthe boundary of the
nucleus. Hence, @
= C[N. LetY= (0;1)nbe the reference cell and let FYbe an
open set with a Lipschitz continuous boundary @Fthat does not touch the boundary of
Y.@Frepresents the surface of the endoplasmic reticulum. The volume occupied by the
cytosol isY=YnF. Repeating Yby periodicity, the union of all Yis a connected set
inRn, denoted by Rn
1.
Let"2(0;1) be a small parameter related to the periodicity length, taking values
in a positive real sequence tending to zero and such that
is a nite union of cuboids
which are homothetic to the unit cell with the same ratio "(see Figure ….). We set

80 …..HOMOGENIZATION … heterogeneous structures …media ….

"=S
k2Zn"(k+Y)\
andS"=S
k2Zn"(k+@F)\
and we assume that S"\@
=;.
We notice that
is a nite union of cuboids which are homothetic to the unit cell and the
inclusions do not intersect the boundary @
.
In such a periodic microstructure, we shall consider a non-linear system of coupled
partial dierential equations describing some processes contributing to carcinogenesis in
human cells, with suitable boundary and initial conditions. More precisely, if we denote
by [0;T], with 0< T <1, the time interval we are interested in, we shall analyze the
eective behavior, as the small parameter "!0, of the solution of the following coupled
system of equations:
8
>>>>>>>>>><
>>>>>>>>>>:@u"
@tDu
u"=f(u) in (0;T)
";
u"=ubon (0;T)C;
Duru"
= 0 on (0;T)N;
Duru"
="G1(u";s") on (0;T)S";
u"(0;x) =u0(x) in
";(4.19)
8
>>>>>>>>><
>>>>>>>>>:@v"
@tDv
v"=g(v") in (0;T)
";
v"= 0 on (0;T)N;
Dvrv"
= 0 on (0;T)C;
Duru"
="G2(v";w") on (0;T)S";
v"(0;x) =v0(x) in
";(4.20)
8
<
:@s"
@t"2Ds
"s"=h(s") +G1(u";s") on (0;T)S";
s"(0;x) =s0(x) onS":(4.21)
8
<
:@w"
@t"2Dw
"w"=h(s") +G2(v";w") on (0;T)S";
w"(0;x) =w0(x) onS":(4.22)
In (3.19)-(3.22), is the unit outward normal to
",
"is the Laplace-Beltrami operator
onS",u": [0;T]
"!Randv": [0;T]
"!Rrepresent the concentrations of
BP molecules and, respectively, of DE molecules in the cytosol and s": [0;T]S"!R
andw": [0;T]S"!Rare the concentrations of BP molecules and, respectively, of DE
molecules bound to the surface of the ER.
Remark 4.2 We point out that the diusion on the surface of the endoplasmic reticulum
is scaled with "2, this scaling being the one that keeps the in
uence of the slow surface
diusion term at the macroscale. Also, let us notice that the scaling in the right hand side

MATHEMATICAL MODELS IN BIOLOGY 81
of the boundary conditions (3.19) and (3.20) on S"ensures that we keep the in
uence of
the binding processes at the macroscale. We can treat in a similar manner the case in
which the binding-unbinding term on S"corresponding to the BP molecules is scaled with
"
and the binding-unbinding term for DE molecules is multiplied by "m, with
;m2[0;1)
(for the linear case, see [128]). .
We make the following assumptions on the data:
1.The diusion coecients Du;Dv;Ds;Dw>0 are supposed to be, for simplicity,
constant (see, also, Remark 3.10).
2.f,gandhare nonlinear functions modeling the cleaning mechanisms in
"and,
respectively, the transformation of the BP molecules to DE molecules bound to the surface
of the endoplasmic reticulum. Following [129], we assume that the cleaning mechanism
is described by the following nonlinear, nonnegative, increasing, bounded and Lipschitz
continuous function:
f(x) =8
<
:ax
x+b; x0;
0; x< 0;
fora;b > 0. The functions gandhare supposed to be of the same form as f, but with
dierent parameters. We consider here Michaelis-Menten functions, but we can deal also
with the case of other bounded Lipschitz continuous monotone functions if we impose
structural conditions in order to ensure the positivity of the solution ( u";v";s";w") and
uniform upper bounds for it (see [140], [223] and [ ?]).
3.The binding-unbinding phenomenon at the surface of the endoplasmic reticulum is
modeled with the aid of two given functions G1andG2. Dierent types of such functions
can be considered, provided that, additionally, we impose suitable structural conditions for
ensuring the positivity and L1-estimates of the solution ( u";v";s";w"). A standard choice
is given in [128]. More precisely, the authors consider the linear case in which
G1(u";s") =ls(u"s"); G 2(v";w") =lw(v"w");
wherels;lw>0 represent the binding and unbinding rates to the endoplasmic reticulum,
supposed to be constant. We shall deal in this paper with two cases, namely the linear
Henry isotherm with highly oscillating coecients and the case of a Langmuir isotherm.
More precisely, we consider, in a rst situation, that
G1(u";s") =l"
uu"l"
ss"; G 2(v";w") =l"
vv"l"
ww"; (4.23)
with
l"
u(x) =lux
"
; l"
s(x) =lsx
"
; l"
v(x) =lvx
"
; l"
w(x) =lwx
"
;
wherelu(y),ls(y),lv(y) andlw(y) areY-periodic, real, smooth, bounded functions with
lu(y)l0
u>0,ls(y)l0
s>0,lv(y)l0
v>0,lw(y)l0
w>0. The fact that we consider

82 …..HOMOGENIZATION … heterogeneous structures …media ….
that the model coecients are not constant but vary with respect to the surface variable is
physically justied (for examples where the processes on the membrane are inhomogeneous,
see [86]). As a consequence, in the homogenized limit, additional integral terms are present,
capturing the eect of the cell heterogeneity on the macroscopic behavior of the solution of
system (3.19)-(3.22). However, for simplicity, we assume that all the parameters involved
in our model are time independent, but the case in which they depend on time can be also
treated.
In the second situation we shall analyze here, we consider the case in which G1andG2
are nonlinear functions dened in terms of isotherms of Langmuir type:
G1(u";s") =l"
s1u"
1 +1u"s"
; G 2(v";w") =l"
w2v"
1 +2v"w"
; (4.24)
withi;i>0, fori= 1;2. We denote
g1(u") =1u"
1 +1u"; g 2(v") =2v"
1 +2v": (4.25)
4.The concentration ubof the BP molecules on the cell membrane Cis supposed
to be an element of H1=2(C) (see (3.26)) and the initial values u0(x);v0(x)2L2(
),
s0(x);w0(x)2C1(
) are assumed to be nonnegative and bounded independently with
respect to".
Remark 4.3 Let us notice that, as in [131], we can treat in a similar way the case in which
the Lipschitz continuous functions G1andG2are of the form Gi(p;q) =Gi(p;q)(pq),
with 0< Gi;minGi(p;q)Gi;max<1or the case in which Gi(p;q) =Ai(p)Bi(q),
withAiandBiLipschitz continuous and increasing functions, for i= 1;2.
The function g(r) =r=(1 +r) is increasing and one to one from R+to [0;= ].
Despite the fact that gis not dened for r=1=, since we are interested in considering
only non-negative values of the argument r, we can mollify gfor negative values r <0 in
such a way that we get an increasing function on R, growing at most linearly at innity
and having an uniformly bounded derivative (see [7]). Alternatively, since for negative
values of the argument of gsingularities may appear, we can consider, in a rst step, a
modied kinetics g0, obtained by replacing rby its modulusjrjin the denominator of g.
This new function is Lipschitz continuous. Then, proving the existence and uniqueness of
a solution of the problem involving this new kinetics, we show that the solution is non-
negative and, therefore, it is a solution of the initial problem, too. Let us notice that for
a small concentration, i.e. for r1, we are led to a linear function (Henry adsorption
isotherm). We point out that, in fact, from a physical point of view, we can extend the
considered rates by zero for all negative arguments and this would allow a straightforward
proof of the fact that the solution components remain positive if the initial and boundary
data are positive.

MATHEMATICAL MODELS IN BIOLOGY 83
Remark 4.4 Let us notice that for the model analyzed here, as in [80], [227], and [228],
the unknown functions u"ands"can be decoupled in the system and analyzed separately.
Then, the remaining two components of the system can be considered. Of course, this is
due to the particular structure of the model we assume here. If, for instance, we consider
that the nonlinear function G1depends also on v"orw", then such a decoupling is no
longer possible and a complete analysis becomes necessary.
Remark 4.5 We can also deal with the more general case in which the binding-unbinding
processes at the surface S"are given by some rates G"
1(x;u";s")andG"
2(x;v";w")depend-
ing on the concentrations of BP and DE molecules. This setting includes linear, Fre-
undlich, Langmuir or even other isotherms encountered in the literature. The presence
of the variable xin these rates means that we assume that the processes at the surface
of the endoplasmic reticulum are heterogeneous (see [224], [140] and [ ?]). In particular,
this setting includes the case in which G1(x;u";s") =f1(x;u")f2(x;s"); G2(x;v";w") =
f1(x;v")f2(x;w"), withfi, fori= 1;2, increasing Lipschitz continuous functions sat-
isfying conditions which guarantee the positivity and the necessary uniform upper bounds
for the concentration elds. Moreover, by using a regularization procedure, for example
Yosida approximation, as in [80], the hypothesis concerning the smoothness of the nonlin-
ear functions G1andG2can be relaxed. For instance, we can treat the case of single or
multivalued maximal monotone graphs. We point out here that the nonlinearities considered
in the present work are Lipschitz continuous and monotone. For similar nonlinear func-
tions arising in the homogenization of reactive
ows involving adsorption and desorption
at the boundaries of the perforations, see [196].
In order to write the weak formulation of problem (3.19)-(3.22), we need to x the
notation and to introduce some function spaces. In the sequel, the space L2(
") is equipped
with the classical scalar product and norm
(u;v)
"=Z

"u(x)v(x) dx;kuk2

"= (u;u)
";
and the space L2((0;T);L2(
")) is endowed with
(u;v)
";T=ZT
0(u(t);v(t))
dt;kuk2

";T= (u;u)
";T;
whereu(t) =u(t;
);v(t) =v(t;
). Further, following [128] and [129], we set
V(
") =L2((0;T);H1(
"))\H1((0;T);(H1(
"))0);
VN(
") =fv2V(
")jv= 0 on Ng;
VC(
") =fv2V(
")jv=ubon Cg;
V0;C(
") =fv2V(
")jv= 0 on Cg;

84 …..HOMOGENIZATION … heterogeneous structures …media ….
where, for an arbitrary Banach space V, we denote by V0its dual. Similar spaces can be
dened for
and S". We use the notation
hu;vi"=Z
S"g"uvdx;
whereg"is the Riemannian tensor on S". Also, let us dene
VN(
") =fv2H1(
")jv= 0 on Ng;
V0;C(
") =fv2H1(
")jv= 0 on Cg; V (S") =H1(S")
and
V(
;Y) =L2((0;T)
;H1
per(Y));V(
;@F) =L2((0;T)
;H1(@F));
whereH1
per(Y) =f'2H1
loc(Rn
1) :'isYperiodicg. Finally, we assume that ub2
H1=2(C), where, for an arbitrary smooth hypersurface 0Rnand for any 0 < r < 1,
we consider the Sobolev-Slobodeckij space
Hr(0) =fu2L2(0) :juj0;r<1g; (4.26)
where
juj2
0;r=Z
00ju(x)u(y)j2
jxyjn1+2rdxdy:
The spaceHr(0) is endowed with the norm kuk2
Hr(0)=kuk2
L2(0)+juj2
0;r(see [123] and
[127]).
Let us write down the variational formulation of problem (3.19)-(3.22).
Problem 1 : nd (u";v";s";w")2VC(
")VN(
")V(S")V(S"), satisfying the
initial condition
(u"(0);v"(0);s"(0);w"(0)) = (u0;v0;s0;w0);
such that, for a.e. t2(0;T) and for any ( '1;'2;)2VC;0(
")VN(
")V(S"), we have
8
>>>>>>>>>>><
>>>>>>>>>>>:@u"
@t;'1

"+Du(ru";r'1) +"hG1(u";s");'1iS"=(f(u");'1)
";
@v"
@t;'2

"+Dv(rv";r'2) +"hG2(v";w");'2iS"=(g(v");'2)
";
D@s"
@t;E
S"+Dsh"r@Fs";"r@FiS"=hh(s");iS"+hG1(u";s");iS";
D@w"
@t;iS"+Dwh"r@Fw";"r@FiS"=hh(s");iS"+hG2(v";w");iS":(4.27)
In (3.27), to simplify the exposition, we make an abuse of notation, since for the integrals
of the time derivatives we do not use a duality pairing notation. Also, let us mention that
the solution ( u";v";s";w") is continuous in time, which means that the initial condition
makes sense.
Under our hypotheses on the data, there exists a unique weak solution ( u";v";s";w")
of problem (3.27) (see [3, Proposition 2.2] and [129, Theorem 4.4]).

MATHEMATICAL MODELS IN BIOLOGY 85
4.2.2 The macroscopic model
In order to get the homogenized limit for the problem (3.19)-(3.22), we have to pass to
the limit, with "!0, in its variational formulation (3.27). Based on suitable a priori
estimates, we derive compactness results for the solution of the microscopic problem. For
dealing with the nonlinear terms, we need some strong convergence results. To obtain such
strong results, we use the unfolding operators T"andT"
bdened, e. g., in [64], [67], [ ?], [99],
[129], and [130]. The main feature of these operators is that they map functions dened on
the oscillating domains (0 ;T)
"and, respectively, (0 ;T)", into functions dened on
the xed domains (0 ;T)
Yand (0;T)
, respectively. We brie
y recall here the
denitions of these two operators for our geometry. For their general properties, we refer,
for instance, to [ ?, Proposition 2.2 and Proposition 2.6], [99, Proposition 2.10], and [129,
Theorem 2.2, Theorem 2.9, Lemma 2.4 and Lemma 2.5]. For any '2Lp((0;T)
") and
anyp2[1;1], we dene the periodic unfolding operator T":Lp((0;T)
")!Lp((0;T)

Y) by the formula T"(')(t;x;y ) ='
t;hx
"i
+"y
. In a similar manner, for any
function2Lp((0;T)"), the periodic boundary unfolding operator T"
b:Lp((0;T)
")!Lp((0;T)
) is dened byT"
b()(t;x;y ) =
t;"hx
i
+"y
.
Using the unfolding operators T"andT"
b, we are led to the homogenized limit system
(3.28). The strong formulation of the limit problem is given, too.
Theorem 4.6 The solution (u";v";s";w")of system (3.19)-(3.22) converges, as "!0,
in the sense of (3.36), to the unique solution (u;v;s;w )2VC(
)VN(
)V(
;@F)
V(
;@F), with (u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0), of the following macroscopic prob-
lem:
8
>>>>>>>>>>><
>>>>>>>>>>>:jYj@u
@t;'1

+ (Auru;r'1)
+ (G1(u;s);'1)
@F=jYj(f(u);'1)
;
jYj@v
@t;'2

+ (Avrv;r'2)
+ (G2(v;w);'2)
=jYj(g(v);'2)
;
@s
@t;

@F+ (Dsr@F
ys;r@F)
@F(G1(u;s);)
@F=(h(s);)
@F;
@w
@t;

@F+ (Dwr@F
yw;r@F)
@F(G2(v;w);)
@F= (h(s);)
@F;
(4.28)
for('1;'2;)2V0;C(
)VN(
)V(
;@F). Here,AuandAvare the homogenized
matrices, dened by:8
>><
>>:Au
ij=DuZ
Y
ij+@j
@yi
dy;
Av
ij=DvZ
Y
ij+@j
@yi
dy;(4.29)
in terms of the functions j2H1
per(Y)=R, ,j= 1;:::;n; weak solutions of the cell

86 …..HOMOGENIZATION … heterogeneous structures …media ….
problemsry
(ryj+ej) = 0; y2Y;
(ryj+ej)
= 0; y2@F::(4.30)
We also state here the strong form of the limit system (3.28).
Theorem 4.7 The limit function (u;v;s;w )2VC(
)VN(
)V(
;@F)V(
;@F),
dened in Theorem 3.1 and satisfying
(u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0);
is the unique solution of the following problem:
8
>>>><
>>>>:jYj@u
@tr
(Auru) +Z
@FG1(u;s) dy=jYjf(u)in(0;T)
;
u=ubon(0;T)C;
Auru
= 0 on(0;T)N;(4.31)
8
>>>><
>>>>:jYj@v
@tr
(Avrv) +Z
@FG2(v;w) dy=jYjg(v)in(0;T)
;
v= 0 on(0;T)N;
Avrv
= 0 on(0;T)C;(4.32)
8
>><
>>:@s
@tDs
@F
ysG1(u;s) =h(s)on(0;T)
@F;
@w
@tDw
@F
ywG2(v;w) =h(s)on(0;T)
@F::(4.33)
As in [128, Theorem 14] and [129, Theorem 6.1], it follows that the solution of the macro-
scopic problem (3.28) in unique.
Remark 4.8 Let us remark that the in
uence of the properly scaled binding-unbinding pro-
cesses taking place at the surface of the endoplasmic reticulum is re
ected by the appearance
of an extra zero-order term in the equations (3.31)-(3.32). Also, let us point out that the
limit problem involves an additional microvariable y. This local phenomenon yields a more
complicated microstructure of the eective medium; in (3.31)-(3.32), x2
plays the role
of a macroscopic variable and y2@Fis a microscopic one. The limit model consists of two
partial dierential equations, with global diusion (with respect to the macroscopic variable
x), for the limit of the BP and DE molecules in the cytosol (see (3.31)-(3.32)) and two
partial dierential equations, governing the local behavior of the system, with local diusion
(with respect to the microscopic variable y) on@F(see (3.33)). .
Remark 4.9 A nonlinear carcinogenesis model developed in [129] and involving a new
variableR", modeling the concentration of the receptors at the surface of the endoplasmic
reticulum, is discussed in Section 3.2.3, too. .

MATHEMATICAL MODELS IN BIOLOGY 87
Remark 4.10 Let us notice that we can also deal with the case in which the initial condi-
tions are"-dependent, as in [ ?]. Also, we can treat, in a similar manner, the more general
case in which, instead of considering constant diusion coecients, we work with an het-
erogeneous medium represented by periodic symmetric bounded matrices which are assumed
to be uniformly coercive.
Moreover, all the above results can be extended to the situation in which, instead of the
constant diusion coecients DuandDv, we have two matrices A"
uand, respectively, A"
v.
We suppose that A"
uandA"
vare sequences of matrices in M(;;
)such that
T"(A"
u)!Au;T"(A"
v)!Avstrongly in L1(
Y)nn; (4.34)
for some matrices Au=Au(x;y)andAv=Au(x;y)inM(;;
Y)(see [67]). In this
case, since the correctors jdepend also on x, the new homogenized matrices Ahom
u and
Ahom
v are no longer constant, but depend on x. Here, for ;2R, with 0< , we
denote byM(;;
)the set of all the matrices A2(L1(
))nnwith the property that,
for any2Rn,(A(y); )jj2;jA(y)jjj, almost everywhere in
.
Also, for the diusion coecients on the surface S"we can suppose that they are not
constant, but they depend on ". For instance, we can work with the diusion tensors
D"
s(x) =Ds(x=")andD"
w(x) =Dw(x="), whereDsandDware two uniformly coercive
periodic symmetric given tensors Ds(y)andDw(y), with entries belonging to L1(@F).
Moreover, we can also address the case in which we suppose that D"
sandD"
ware such
that there exist Ds=Ds(x;y)andDw=Dw(x;y)with entries in L1(
@F)such that
T"
b(D"
s)!DsandT"
b(D"
w)!Dwstrongly in L1(
@F). .
Remark 4.11 One can also consider a microscopic model in which the diusion coe-
cients in (3.21) and (3.22) vanish and nonlinear (even discontinuous) isotherms are con-
sidered (see [85], [86], [196], and [223]).
In order to prove Theorem 3.7, we start by giving a priori estimates, suitable bounds
and results concerning the existence and uniqueness of a weak solution ( u";v";s";w") of
the problem (3.27).
The following proposition, proven in [218], states that the functions u";v";s"andw"
are nonnegative and bounded from above if the initial data are assumed to be bounded
and nonnegative. The positivity of u";v";s"andw"is a natural requirement, since they
represent concentrations of BP and DE molecules in the cytosol and on the surface of the
endoplasmic reticulum. Also, this property is essential for proving the well-posedness of
our problem. On the other hand, essential boundedness of the solution is necessary from
the point of view of practical applications.
Proposition 4.12 (Proposition …. in [218]) a) The functions u"andv"are nonnegative
for almost every x2
"andt2[0;T]and the functions s"andw"are nonnegative for
almost every x2S"andt2[0;T].

88 …..HOMOGENIZATION … heterogeneous structures …media ….
b) The functions u"andv"are bounded independently of "almost everywhere in
[0;T]
"and the functions s"andw"are bounded independently of "almost everywhere
in[0;T]S". .
Hence, the solution ( u";v";s";w") of problem (3.27) is bounded in the L1-norm. Besides,
exactly like in [128, Lemma 2] and [129, Lemma A.2], one can prove the L2-boundedness
of the solution ( u";v";s";w").
Proposition 4.13 There exists a constant C > 0, independent of ", such that
ku"k2

"+kru"k2

";t+kv"k2

"+krv"k2

";tC;
"ks"k2
S"+"3kr@Fs"k2
S";t+"kw"k2
S"+"3kr@Fw"k2
S";tC;
"kG1(u";s")k2
S";t+"kG2(v";w")k2
S";tC;
for almost every t2[0;T]. Also, one gets

@u"
@t

L2((0;T);(H1
0(
"))0)+

@v"
@t

L2((0;T);(H1
0(
"))0)C:: (4.35)
The above a priori estimates will allow us to apply the periodic unfolding method and to get
the needed convergence results for the solution of problem (3.27). Still, the nonlinearity
of the model requires strong compactness results for the sequence of solutions in order
to be able to pass to the limit. We know (see [129]) that u";v"2L2((0;T);H1(
"))\
H1((0;T);(H1
0(
"))0)\L1((0;T)
").
Using suitable extension results (see, for instance, [226], [123] and [184]) and Lemma
5.6 from [131], we know that we can construct two extensions u"andv"that converge
strongly to u;v2L2((0;T);L2(
)). We point out that one can obtain (see e. g. [123] and
[184]) the existence of a linear and bounded extension operator to the whole of
, which
preserves the non-negativity, the essential boundedness and the above priori estimates.
Since the functions g1andg2are Lipschitz, exactly like in Lemma 4.3 in [129], we can
prove thatT"
b(s") andT"
b(w") are Cauchy sequences in L2((0;T)
@F).
Proposition 4.14 For any >0, there exists "0>0such that for any 0< "1;"2< "0
one has
kT"1
b(s"1)T"2
b(s"2)k(0;T)
@F+kT"1
b(w"1)T"2
b(w"2)k(0;T)
@F<::
This implies that the sequences T"
b(s") andT"
b(w") are strongly convergent in L2((0;T)

@F).
As already mentioned, (3.27) is a well-posed problem. Using the above a priori estimates
and the properties of the operators T"andT"
b(see Appendix …), we get immediately the
following compactness result.

MATHEMATICAL MODELS IN BIOLOGY 89
Proposition 4.15 Let(u";v";s";w")be the solution of problem (3.27). Then, there exist
u;v2L2((0;T);H1(
));bu;bv2L2((0;T)
;H1
per(Y)),s;w2L2((0;T)
;H1
per(@F))
such that, up to a subsequence, when "!0, we have
8
>>>>>>>>>><
>>>>>>>>>>:T"(u")*u weakly inL2((0;T)
;H1(Y));
T"(v")*v weakly inL2((0;T)
;H1(Y));
u"!u;v"!vstrongly in L2((0;T)
);
T"(ru")*ru+rybuweakly inL2((0;T)
Y);
T"(rv")*rv+rybvweakly inL2((0;T)
Y);
T"
b(s")*s weakly inL2((0;T)
;H1(@F));
T"
b(w")*w weakly inL2((0;T)
;H1(@F));
T"
b(s")!s;T"
b(w")!wstrongly in L2((0;T)
@F)::(4.36)
In order to pass to the limit in the nonlinear terms involving the functions G1andG2, we
need to prove that T"
b(u")!uandT"
bv")!v, strongly in L2((0;T)
@F). These
strong convergence results follow from the strong convergence of u"andv", respectively,
the trace lemma (see Lemma 3.1 in [123]) and the properties of the unfolding operator T"
b.
More precisely, we have the following result (see [218] and [123]).
Proposition 4.16 Up to a subsequence, we have
T"
b(u")!ustrongly in L2((0;T)
@F)
and
T"
b(v")!vstrongly in L2((0;T)
@F):
Let us notice that using the periodic unfolding operator T", the strong convergence of u"
andv"and the continuity of the Nemytskii operator for fand, respectively, g, which are
bounded and continuous functions, we have
f(T"(u"))!f(u) strongly in L2((0;T)
Y):
g(T"(v"))!g(v) strongly in L2((0;T)
Y):
In a similar manner, we get
h(T"
b(s"))!h(s) strongly in L2((0;T)
@F):
Let us look now at the limit behavior of the terms involving G1andG2. In the rst
situation, i.e. for Henry isotherm with rapidly oscillating coecients given by (2.5), we
can easily pass to the limit since these coecients are uniformly bounded in L1(
) and
converge strongly therein. For the second situation, i.e. isotherm of the form (3.24), we
use the strong convergence of T"
b(u"),T"
b(v"),T"
b(s") andT"
b(w") and the properties of the
functionsg1andg2. Thus, we have
G1(T"
b(u";s"))!G1(u;s) strongly in L2((0;T)
@F)

90 …..HOMOGENIZATION … heterogeneous structures …media ….
and
G2(T"
b(v";w"))!G2(v;w) strongly in L2((0;T)
@F):
We point out that, by classical results (see, for instance, Theorem 2.12 in [64] and Theorem
2.17 in [99] ), uandvare independent of y.
Proof of Theorem 3.7. The above convergence results allow us to pass to the limit in
the variational formulation of problem (3.19)-(3.22) and to obtain the homogenized model
(3.28). More precisely, for obtaining the limit problem (3.28), we take in the rst equation
in (3.27) the admissible test function
'(t;x) ='1(t;x) +"'2
t;x;x
"
; (4.37)
with'12C1
0((0;T);C1(
))'22C1
0((0;T);C1(
;C1
per(Y))).
Integrating with respect to time and applying in each term the corresponding unfolding
operator, we get:
ZT
0Z

Y@
@t
T"(u")
T"(') dxdydt+DuZT
0Z

YT"(ru")
T"(r') dxdydt+
ZT
0Z

@FT"
b(G1(u";s"))(T"
b(')) dxdydt=
ZT
0Z

YT"(f(u"))T"(') dxdydt: (4.38)
Using the above convergence results and Lebesgue's convergence theorem, we can pass to
the limit in (3.38) (see, for details, [67], [80], [129] and ???[ ?]) and we obtain:
ZT
0Z

Y@u
@t'1dxdydt+DuZT
0Z

Y(ru+rybu)
(r'1+ry'2) dxdydt+
ZT
0Z

@FG1(u;s)'1dxdydt=ZT
0Z

Yf(u)'1dxdydt: (4.39)
By standard density arguments, it follows that (3.39) holds true for any '12L2(0;T;H1(
)),
'22L2((0;T)
;H1
per(Y)). In a similar manner, for the limit equation for v", we get
ZT
0Z

Y@v
@t'1dxdydt+DvZT
0Z

Y(rv+rybv)
(r'1+ry'2) dxdydt+
ZT
0Z

@FG2(v;w)'1dxdydt=ZT
0Z

Yg(v)'1dxdydt; (4.40)
for any'12L2((0;T);H1(
)),'22L2((0;T)
;H1
per(Y)).
In order to obtain the limit equations for s"andw", we apply the convergence results
obtained in [129] (see Lemma 2.6 and Theorem 2.9). Indeed, if we apply the boundary
unfolding operator T"
bin (3:27)3, we get
ZT
0Z

@FT"
b@s"
@t
T"
b() dxdydt+DsZT
0Z

@Fr@F
yT"
b(s")
r@F
yT"
b() dxdydt=

MATHEMATICAL MODELS IN BIOLOGY 91
ZT
0Z

@FT"
b(G1(u";s"))T"
b() dxdydtZT
0Z

@FT"
b(h(s"))dxdydt;
which leads to
ZT
0Z

@F@s
@tdxdydt+DsZT
0Z

@Fr@F
ys
r@F
ydxdydt=
ZT
0Z

@FG1(u;s)dxdydtZT
0Z

@Fh(s)dxdydt; (4.41)
for any2C1
0((0;T);C1(
;C1
per(@F))).
In a similar manner, we get
ZT
0Z

@F@w
@tdxdydt+DwZT
0Z

@Fr@F
yw
r@F
ydxdydt=
ZT
0Z

@FG2(v;w)dxdydt+ZT
0Z

@Fh(s)dxdydt; (4.42)
for any2C1
0((0;T);C1(
;C1
per(@F))).
Thus, we get exactly the weak formulation of the limit problem (3.28). Indeed, if we
take'1= 0, we easily get the cell problems (3.30) and
bu=nX
k=1@u
@xkk;bv=nX
k=1@v
@xkk: (4.43)
Then, taking '2= 0 and using (3.43), we obtain (3.28). Moreover, by standard techniques,
we can derive the initial conditions ( u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0). Since the
solution (u;v;s;w ) of problem (3.28) is uniquely determined, the above convergences for
the microscopic solution ( u";v";s";w") hold for the whole sequence and this ends the proof
of Theorem 3.1.
4.2.3 A nonlinear carcinogenesis model involving free receptors
We end this section by brie
y discussing a generalization of a recent nonlinear model
proposed in [129] for the carcinogenesis in human cells, involving a new variable modeling
the free receptors present at the surface of the ER. More precisely, the authors assume that
BP molecules in the cytosol are transformed into BP molecules bound to the surface of the
ERs"only if they nd a free receptor R". Following [129], let R": [0;T]S"![0;1] be
the relative concentration of free receptors on the surface of the ER. The maximal relative
quantity of free receptors is R= 1. At the surface of the ER, BP molecules bind to receptors
and, using the law of mass action, the binding is described by the product kuu"R", with
a constant rate ku>0. DE molecules are assumed to have a similar behavior. When BP
moleculesu"or DE molecules v"bind to the surface of the endoplasmic reticulum, the
amount of free receptors decreases and when the molecules s"andw"unbind from the

92 …..HOMOGENIZATION … heterogeneous structures …media ….
surface of the endoplasmic reticulum the amount of free receptors increases. If receptors
are supposed to be xed on the surface of the ER, then their evolution is described by (see
[129] for details):
@R"
@t=R"jkuu"+kvv"j+ (RR")jkss"+kww"jon (0;T)S":
Here,ks;kw>0 are supposed to be multiples of lsand, respectively, of lw.
Let us write now the variational formulation of this nonlinear problem.
Problem 2: nd (u";v";s";w";R")2VC(
")VN(
")V(S")V(S")VR(S"),
satisfying the initial condition
(u"(0);v"(0);s"(0);w"(0);R"(0)) = (u0;v0;s0;w0;R);
such that, for a.e. t2(0;T) and for all ( '1;'2;)2VC;0(
")VN(
")V(S"), we have
8
>>>>>>>>>>>>><
>>>>>>>>>>>>>:@u"
@t;'1

"+Du(ru";r'1) +"hkuu"R"lss";'1iS"=(f(u");'1)
";
@v"
@t;'2

"+Dv(rv";r'2) +"hkvv"R"lww";'2i"=(g(v");'2)
";
D@s"
@t;E
S"+Dsh"r@Fs";"r@FiS"=hh(s");iS"+hkuu"R"lss";iS";
D@w"
@t;E
S"+Dwh"r@Fw";"r@FiS"=hh(s");iS"+hkvv"R"lww";iS";
h@tR";iS"+hR"jkuu"+kvv"j;iS"=h(RR")jkss"+kww"j;iS":
Here,VR(S") =fu2L2((0;T);L2(S"))j@tu2L2((0;T);L2(S"))g.
The existence of a solution ( u";v";s";w";R")2VC(
")VN(
")V(S")V(S")
VR(S") of the above variational problem is proven in [129, Theorem 4.4]. Also, it is proven
in [129, Lemma 4.1 and Theorem 4.5] that R"is nonnegative and bounded by R> 0 almost
everywhere in [0 ;T]S"andT"
b(R") converges strongly to R2L2((0;T)
@F).
In this case, the homogenized result is stated in the following theorem (see Theorem
5.1 in [129]).
Theorem 4.17 The homogenized limit problem is as follows: nd (u;v;s;w;R )2VC(
)
VN(
)V(
;)V(
;@F)VR(
;@F), with (u(0);v(0);s(0);w(0);R(0)) = (u0;v0;s0;w0;R),
such that8
>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:jYj@u
@t;'1

+ (Auru;r'1)
+ (kuuRlss;'1)
=jYj(f(u);'1)
;
jYj@v
@t;'2

+ (Avrv;r'2)
+ (kvvRlww;' 2)
@F=jYj(g(v);'2)
;
@s
@t;

+ (Dsrs;r)
(kuuRlss;)
@F=(h(s);)
@F;
@w
@t;

@F+ (Dwrw;r)
@F(kvvRlww;)
@F= (h(s);)
@F;
(@tR;)
@F+ (R(kuu+kvv);)
@F= ((RR)(kss+kww);)
@F;

MATHEMATICAL MODELS IN BIOLOGY 93
for('1;'2;)2V0;C(
)VN(
)V(
;@F).
Here, the homogenized matrices AuandAvare given by (3.29). We remark that the
evolution of the receptors is governed by an ordinary dierential equation.
Let us notice that all the above results hold true for the case of highly oscillating
coecients k"
u,k"
vand, respectively, l"
sandl"
w. Moreover, based on the law of mass action,
various other functions G1(R";u") andG2(R";v") can be used to describe the adsorption
phenomena at the surface of the endoplasmic reticulum. As particular situations, we can
mention the case in which G1=R"g1(u") andG2=R"g2(v"), withg1andg2suitable
Lipschitz continuous functions (e.g. the Langmuir kinetics considered above). In such a
case, the equation governing the evolution of the receptors is
@tR"=R"jg1(u") +g2(v")j+ (RR")jkss"+kww"jon (0;T)S":
The case in which the binding processes at the surface of the endoplasmic reticulum is given
by suitable nonlinear functions G1(x=";R";u") and, respectively, G2(x=";R";v"), can be
addressed, too. Such isotherms were proposed in [140]. We can also address the case of
more general adsorption kinetics, such as Freundlich isotherms, and the case of multiple
metabolisms BP!DE.
4.3 Homogenization results for the calcium dynamics in li-
ving cells
In this section, we shall present some results, obtained via the periodic unfolding method
in [217]. More precisely, we shall analyze the eective behavior of a nonlinear system of
coupled reaction-diusion equations arising in the modeling of the dynamics of calcium ions
in living cells is analyzed. We deal, at the microscale, with two reaction-diusion equations
governing the concentration of calcium ions in the endoplasmic reticulum and, respectively,
in the cytosol, coupled through an interfacial exchange term. Depending on the magnitude
of this term, various models arise at the macroscale. In particular, we obtain, at the
limit, a bidomain model. Such a model is widely used for studying the dynamics of the
calcium ions, which are recognized to be important intracellular messengers between the
endoplasmic reticulum and the cytosol inside the biological cells.
Calcium is a very important second messenger in a living cell, participating in many
cellular processes, such as protein synthesis, muscle contraction, cell cycle, metabolism or
apoptosis (see, for instance, [77]). Intracellular free calcium concentrations must be very
well regulated and many buer proteins, pumps or carriers of calcium take part at this
complicated process. The nely structured endoplasmic reticulum, which is surrounded
by the cytosol, is an important multifunctional intracellular organelle involved in calcium
homeostasis and many of its functions depend on the calcium dynamics. The endoplasmic
reticulum plays an important role in the metabolism of human cells. It performs diverse

94 …..HOMOGENIZATION … heterogeneous structures …media ….
functions, such as protein synthesis, translocation across the membrane, folding, etc. This
complex and highly heterogeneous cellular structure spreads throughout the cytoplasm,
generating various zones with diverse morphology and functions. The study of the dynamics
of calcium ions, acting as messengers between the endoplasmic reticulum and the cytosol
inside living cells, represents a topic of huge interest, which still requires special attention.
Many biological mechanisms involving the functions of the cytosol and of the endoplasmic
reticulum are not yet perfectly understood.
Our goal in [217] was to rigorously analyze, using the periodic unfolding method, the
macroscopic behavior of a nonlinear system of coupled reaction-diusion equations arising
in the modeling of calcium dynamics in living cells. We consider, at the microscale, two
equations governing the concentration of calcium ions in the cytosol and, respectively,
in the endoplasmic reticulum, coupled through an interfacial exchange term. Depending
on the magnitude of this term, dierent models arise at the limit. In a particular case,
we obtain, at the macroscale, a bidomain model, which is largely used for studying the
dynamics of the calcium ions in human cells. The calcium bidomain system consists of two
reaction-diusion equations, one for the concentration of calcium ions in the cytosol and
one for the concentration of calcium ions in the endoplasmic reticulum, coupled through a
reaction term. For details about the physiological background of such a model, the reader
is referred to [151]. Bidomain models arise also in other contexts, such as the modeling of
diusion processes in partially ssured media (see [28], [25] and [106]) or the modeling of
the electrical activity of the heart (see [21], [19] and [191]).
Our models can serve as a tool for biophysicists to analyze the complex mechanisms
involved in the calcium dynamics in living cells, justifying in a rigorous manner some
biological points of view concerning such processes.
The problem of obtaining the calcium bidomain equations using homogenization tech-
niques was addressed by a formal approach in [126] and by a rigorous one, based on the
use of the two-scale convergence method, in [131]. Our results in [217] constitute a gen-
eralization of some of the results contained in [126] and [131]. The proper scaling of the
interfacial exchange term has an important in
uence on the limit problem and, using some
techniques from [99], we extend the analysis from [131] to the case in which the parameter

arising in the exchange term belongs to R.
4.3.1 Setting of the problem
Let us start by describing the geometry of the problem. Let
be a bounded domain in Rn,
withn3, having a Lipschitz boundary @
formed by a nite number of connected com-
ponents. The domain
is supposed to be a periodic structure made up of two connected
parts,
"
1and
"
2, separated by an interface ". We assume that only the phase
"
1reaches
the outer xed boundary @
. Here,"is considered to be a small positive real parameter
related to the characteristic dimension of our two regions. For modeling the dynamics of
the concentration of calcium ions in a biological cell, the phase
"
1represents the cytosol,

MATHEMATICAL MODELS IN BIOLOGY 95
while the phase
"
2is the endoplasmic reticulum. Let Y1be an open connected Lipschitz
subset of the elementary cell Y= (0;1)nandY2=YnY1(see Figure ….). We consider
that the boundary of Y2is locally Lipschitz and that its intersections with the boundary
ofYare reproduced identically on the opposite faces of the elementary cell. Moreover, if
we repeatYin a periodic manner, the union of all the sets Y1is a connected set, with a
locallyC2boundary. Also, we consider that the origin of the coordinate system lies in a
ball contained in the above mentioned union (see [106]).
For any"2(0;1), let
Z"=fk2Znj"k+"Y
g;
K"=fk2Z"j"k"ei+"Y
;8i= 1;:::;ng;
whereeiare the vectors of the canonical basis of Rn. We denote (see Figure 2.)

"
2= int([
k2K"("k+"Y2));
"
1=
n
"
2
and we set =YnY2.
For1;12R, with 0<  1<  1, we denote byM(1;1;Y) the collection of all
the matrices A2(L1(Y))nnwith the property that, for any 2Rn, (A(y); )
1jj2;jA(y)j1jj, almost everywhere in Y. We consider the matrices A"(x) =A(x=")
dened on
, where A2M (1;1;Y) is aY-periodic smooth symmetric matrix and we
denote the matrix AbyA1inY1and, respectively, by A2inY2.
If (0;T) is the time interval under consideration, we shall be concerned with the macro-
scopic behavior of the solutions of the following microscopic system:
8
>>>>>>>>>>>>><
>>>>>>>>>>>>>:@u"
1
@tdiv (A1"ru"
1) =f(u"
1) in (0;T)
"
1;
@u"
2
@tdiv (A2"ru"
2) =g(u"
2) in (0;T)
"
2;
A1"ru"
1
=A2"ru"
2
on (0;T)";
A1"ru"
1
="
h(u"
1;u"
2) on (0;T)";
u"
1= 0 on (0 ;T)@
;
u"
1(0;x) =u0
1(x) in
"
1; u"
2(0;x) =u0
2(x) in
"
2;(4.44)
whereis the unit outward normal to
"
1and the scaling exponent
is a given real
number, related to the speed of the interfacial exchange. As we shall see, three important
cases will arise at the limit, i.e.
= 1,
= 0 and
=1 (see, also, Remark 3.21). We
assume that the initial conditions are non-negative and that the functions fandgare
Lipschitz-continuous, with f(0) =g(0) = 0. We also suppose that
h(u"
1;u"
2) =h"
0(x)(u"
2u"
1); (4.45)

96 …..HOMOGENIZATION … heterogeneous structures …media ….
whereh"
0(x) =h0(x=") andh0=h0(y) is a real Y-periodic function in L1(), with
h0(y)>0. Besides, we consider that
H=Z
h0(y) dy6= 0:
As in [131], we can treat in a similar way the case in which the function his Lipschitz-
continuous in both arguments and is given by:
h(r;s) =h(r;s)(sr); (4.46)
with 0<hminh(r;s)hmax<1.
Since it is not easy to nd an explicit solution of the well-posed microscopic problem
(3.44), we need to apply an homogenization procedure for obtaining a suitable model that
describes the averaged properties of the complicated microstructure. Using the periodic
unfolding method, we can nd the asymptotic behavior of the solution of our problem.
For the case
= 1, this behavior is described by a new nonlinear system (see (3.48)), a
bidomain model . So, in this case, at a macroscopic scale, our medium can be represented
by a continuous model, i.e. the superimposition of two interpenetrating continuous media,
the cytosol and the endoplasmic reticulum, which coexist at any point. For the other
two relevant cases, see (3.49) and (3.50). For the particular geometry considered in this
section, we use two unfolding operators, mapping functions dened on oscillating domains
into functions given on xed domains (see Appendix …).
It might seem that these simplied assumptions about the complex calcium dynamics
inside a cell are quite strong. However, the homogenized solution ts well with experimental
data (see [151]). Also, one could argue that the periodicity of the microstructure is not
a realistic assumption and it would be interesting to work with a random microstructure.
Still, such a periodic structure provides a very good description, in agreement with all the
experimental ndings (see [135]).
We can deal, in a similar manner, with the more general case of an heterogeneous
medium represented by a matrix A"
0=A0(x;x=" ) or by a matrix D"=D(t;x=" ), under
reasonable assumptions on the matrices A0andD. For instance, we can suppose that
Dis a symmetric matrix, with D;@D
@t2L1(0;T;L1
per(Y))nnand such that, for any
2Rn, (D(t;x);)2jj2andjD(t;x)j2jj, almost everywhere in (0 ;T)Y, for
0< 2< 2.
In order to prove the main convergence results for our problem, obtained via the periodic
unfolding method, let us introduce now the function spaces and norms we shall work with
in the sequel. Let
H1
@
(
"
1) =fv2H1(
"
1)jv= 0 on@
\@
"
1g;
V(
"
1) =L2(0;T;H1
@
(
"
1));V(
"
1) =n
v2V(
"
1)j@v
@t2L2((0;T)
"
1)o
;

MATHEMATICAL MODELS IN BIOLOGY 97
V(
"
2) =L2(0;T;H1(
"
2));V(
"
2) =n
v2V(
"
2)j@v
@t2L2((0;T)
"
2)o
;
with
(u(t);v(t))
"=Z

"u(t;x)v(t;x) dx;ku(t)k2

"= (u(t);u(t))
";
(u;v)
";t=Zt
0(u(t);v(t))
"dt;kuk2

";t= (u;u)
";t;
for= 1;2. Also, let
V(
) =L2(0;T;H1(
));V(
) =n
v2V(
)j@v
@t2L2((0;T)
)o
;
with
(u(t);v(t))
=Z

u(t;x)v(t;x) dx;ku(t)k2

= (u(t);u(t))
;
(u;v)
;t=Zt
0(u(t);v(t))
dt;kuk2

;t= (u;u)
;t
and
V0(
) =fv2V(
)jv= 0 on@
a.e. on (0 ;T)g;V0(
) =V0(
)\V(
):
The variational formulation of problem (3.44) is as follows: nd ( u"
1;u"
2)2V(
"
1)
V(
"
2), with (u"
1(0;x);u"
2(0;x)) = (u0
1(x);u0
2(x))2(L2(
))2and
@u"
1
@t(t);'(t)

"
1+@u"
2
@t(t); (t)

"
2+
(A"
1(t)ru"
1;r'(t))
"
1+ (A"
2(t)ru"
2;r (t))
"
2
"
(h(u"
1;u"
2);'(t) (t))"= (f(u"
1(t));'(t))
"
1+ (g(u"
2(t)); (t))
"
2; (4.47)
for a.e.t2(0;T) and any ( '; )2V(
"
1)V(
"
2).
Following the same techniques used in [131], it is not dicult to prove that (3.47) is a
well-posed problem and that u"andv"are non-negative and bounded almost everywhere.
Taking (u"
1;u"
2) as test function in (3.47), integrating with respect to time and taking
into account that u"
1andu"
2are bounded and non-negative, it follows that there exists a
constantC0, independent of ", such that
ku"
1(t)k2

"
1+ku"
2(t)k2

"
2+kru"
1k2

"
1;t+kru"
2k2

"
2;t+"
(h(u"
1;u"
2);u"
1u"
2)";tC;
for a.e.t2(0;T). Also, as in [131] or ???[ ?], we can see that there exists a positive
constantC0, independent of ", such that

@u"
1
@t(t)

2

"
1+

@u"
2
@t(t)

2

"
2C;
for
1 and

@u"
1
@t

L2(0;T;H1(
"
1))+

@u"
2
@t

L2(0;T;H1(
"
2))C;
for
<1. These a priori estimates allow us to use the periodic unfolding method and to
obtain the needed convergence results in all the above mentioned relevant cases.

98 …..HOMOGENIZATION … heterogeneous structures …media ….
4.3.2 The periodic unfolding method for a two-component domain
For retrieving the macroscopic behavior of the solution of problem (3.47), we use two unfol-
ding operators,T"
1andT"
2, which transform functions dened on oscillating domains into
functions dened on xed domains (see [64], [67] and [99]).
We brie
y recall here the denitions and the main properties of these unfolding oper-
ators.
Forx2Rn, we denote by [ x]Yits integer part k2Zn, such that x[x]Y2Y
and we setfxgY=x[x]Yfor a.e.x2Rn. So, for almost every x2Rn, we have
x="x
"
Y+x
"
Y

. For dening the above mentioned periodic unfolding operators, we
consider the following sets (see [99]):
bZ"=n
k2Znj"Yk
o
;b
"= int[
k2bZ"
"Yk
;"=
nb
";
b
"
=[
k2bZ"
"Yk

;"
=
"
nb
"
;b"=@b
"
2:
Denition 4.18 For any Lebesgue measurable function 'on
"
,2f1;2g, we dene
the periodic unfolding operators by the formula
T"
(')(x;y) =8
<
:'
"x
"
Y+"y

for a.e. (x;y)2b
"Y
0 for a.e. (x;y)2"Y
If'is a function dened in
, for simplicity, we write T"
(')instead ofT"
('j
").
For any function which is Lebesgue-measurable on ", the periodic boundary unfolding
operatorT"
bis dened by
T"
b()(x;y) =8
<
:
"x
"
Y+"y

for a.e. (x;y)2b
"
0 for a.e. (x;y)2"
Remark 4.19 We notice that if '2H1(
"
), thenT"
b(') =T"
(')jb
".
We recall here some useful properties of these operators (see, for instance, [ ?], [101],
and [99]).
Proposition 4.20 Forp2[1;1)and= 1;2, the operatorsT"
are linear and continuous
fromLp(
"
)toLp(
Y)and
(i) if'and are two Lebesgue measurable functions on
"
, one hasT"
(' ) =
T"
(')T"
( );
(ii) for every '2L1(
"
), one has
1
jYjZ

YT"
(')(x;y) dxdy=Z
b
"'(x) dx=Z

"'(x) dxZ
"'(x) dx;

MATHEMATICAL MODELS IN BIOLOGY 99
(iii) iff'"g"Lp(
) is a sequence such that '"!'strongly in Lp(
), then
T"
('")!'
strongly in Lp(
Y);
(iv) if'2Lp(Y)isY-periodic and '"(x) ='(x="), thenT"
('")!'strongly in
Lp(
Y);
(v) if'2W1;p(
"
), thenry(T"
(')) ="T"
(r')andT"
(')belongs toL2

;W1;p(Y)

.
Moreover, for every '2L1("), one has
Z
b"'(x) dx=1
"jYjZ

T"
b(')(x;y) dxdy:
For
= 1, using the obtained a priori estimates and the properties of the operators
T"
1andT"
2, it follows that there exist u12L2(0;T;H1
0(
)),u22L2(0;T;H1(
)),cu12
L2((0;T)
;H1
per(Y1)),cu22L2((0;T)
;H1
per(Y2)) such that, passing to a subsequence,
for"!0, we have:
8
>><
>>:T"
1(u"
1)!u1strongly in L2((0;T)
;H1(Y1));
T"
1(ru"
1)*ru1+rycu1weakly inL2((0;T)
Y1);
T"
2(u"
2)*u 2weakly inL2((0;T)
;H1(Y2));
T"
2(ru"
2)*ru2+rycu2weakly inL2((0;T)
Y2):(4.48)
Moreover, as in [131] and ???[ ?],@u1
@t2L2(0;T;L2(
));@u2
@t2L2(0;T;L2(
)) andu12
C0([0;T];H1
0(
)); u22C0([0;T];H1(
)). So,u12V0(
) andu22V(
).
Let us mention that, in fact, under our hypotheses, passing to a subsequence, T"
1(u"
1)
converges strongly to u1inLp((0;T)
Y1), for 1p <1. As a consequence, since
the Nemytskii operator corresponding to the nonlinear function fis continuous, it follows
thatf(T"
1(u"
1)) converges to f(u1). A similar result holds true for u"
2.
Since
kT"
1(u"
1)T"
2(u"
2)kL2((0;T)
)C"1

2
it follows that for the case
= 0 and
=1 we have, at the macroscale, u1=u2=u02
V0(
). Moreover, for
=1, following the techniques from [99], one can prove that
T"
1(u"
1)T"
2(u"
2)
"*cu1cu2weakly inL2((0;T)
):
4.3.3 The main convergence results
We present now the main convergence results obtained in [217].
Theorem 4.21 (Theorem …. in [217]) If
= 1, the solution (u"
1; u"
2)of system (3.44)
converges in the sense of (3.48), as "!0, to the unique solution (u1; u2)of the following

100 …..HOMOGENIZATION … heterogeneous structures …media ….
macroscopic problem:
8
>>>><
>>>>:@u1
@tdiv(A1ru1)H(u2u1) =f(u1)in(0;T)
;
(1)@u2
@tdiv(A2ru2) +H(u2u1) = (1)g(u2)in(0;T)
;
u1(0;x) =u0
1(x); u 2(0;x) =u0
2(x)in
:(4.49)
Here,
H=Z
h0(y) dy
andA1andA2are the homogenized matrices, given by:
A1
ij=Z
Y1
a1
ij+nX
k=1a1
ik@1j
@yk!
dy;
A2
ij=Z
Y2
a2
ij+nX
k=1a2
ik@2j
@yk!
dy;
wherea1
ij=A1
ij,a2
ij=A2
ijand1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n , are the
weak solutions of the cell problems
8
<
:divy(A1(y)(ry1k+ek)) = 0; y2Y1;
A1(y)(ry1k+ek)
= 0; y2;
8
<
:divy(A2(y)(ry2k+ek)) = 0; y2Y2;
A2(y)(ry2k+ek)
= 0; y2:
At a macroscopic scale, we obtain a continuous model, a so-called bidomain model , similar
to those arising in the context of the modeling of diusion processes in partially ssured
media (see [28] and [106]) or in the case of the modeling of the electrical activity of the
heart (see [21], [19] and [191]). If we assume that his given by (3.46), then, at the limit,
the exchange term appearing in (3.49) is of the form jjh(u1;u2).
Theorem 4.22 (Theorem …. in [217]) For
= 0, i.e. for high contact resistance, we
obtain, at the macroscale, only one concentration eld. So, u1=u2=u0andu0is the
unique solution of the following problem:
8
><
>:@u0
@tdiv(A0ru0) =f(u0) + (1)g(u0)in(0;T)
;
u0(0;x) =u0
1(x) +u0
2(x)in
:(4.50)

MATHEMATICAL MODELS IN BIOLOGY 101
Here, the eective matrix A0is given by:
A0
ij=Z
Y1
a1
ij+nX
k=1a1
ik@1j
@yk!
dy+Z
Y2
a2
ij+nX
k=1a2
ik@2j
@yk!
dy;
in terms of the functions 1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n; weak solutions
of the local problems
8
<
:divy(A1(y)(ry1k+ek)) = 0; y2Y1;
A1(y)(ry1k+ek)
= 0; y2;
8
<
:divy(A2(y)(ry2k+ek)) = 0; y2Y2;
A2(y)(ry2k+ek)
= 0; y2:
In this case, the exchange at the interface leads to the modication of the limiting diusion
matrix, but the insulation is not enough strong to impose the existence of two dierent
limit concentrations.
Theorem 4.23 (Theorem …. in [217]) For the case
=1, i.e. for very fast interfacial
exchange of calcium between the cytosol and the endoplasmic reticulum (i.e. for weak
contact resistance), at the limit, we also obtain u1=u2=u0and, in this case, the
eective concentration eld u0satises:
8
<
:@u0
@tdiv(A0ru0) =f(u0) + (1)g(u0)in(0;T)
;
u0(0;x) =u0
1(x) +u0
2(x)in
:(4.51)
The eective coecients are given by:
A0;ij=Z
Y1
a1
ij+nX
k=1a1
ik@w1j
@yk!
dy+Z
Y2
a2
ij+nX
k=1a2
ik@w2j
@yk!
dy;
wherew1k2H1
per(Y1)=R; w2k2H1
per(Y2)=R,k= 1;:::;n; are the weak solutions of the cell
problems
8
>><
>>:divy(A1(y)(ryw1k+ek)) = 0; y2Y1;
divy(A2(y)(ryw2k+ek)) = 0; y2Y2;
(A1(y)ryw1k)
= (A2(y)ryw2k)
; y2;
(A1(y)ryw1k)
+h0(y)(w1kw2k) =A1(y)ek
; y2:
It is important to notice that the diusion coecients depend now on h0. A similar result
holds true for the case in which his given by (3.46). Let us notice that in this case the
homogenized matrix is no longer constant, but it depends on the solution u0. A similar
eect was noticed in [7].

Remark 4.24 For simplicity, we address here only the relevant cases
=1;0;1. For
the case
2(1;1), we get, at the limit, the macroscopic problem (3.50), while for
>
1, we obtain a problem similar to (3.49), but without the exchange term H(u2u1)or
jjh(u1;u2), respectively. Finally, for the case
<1, we obtain, at the limit, a standard
composite medium without any barrier resistance. It is worth mentioning that in this case
we getw1k=w2kon, fork= 1;:::;n .
Remark 4.25 The conditions imposed on the nonlinear functions f;g, andhcan be re-
laxed. For instance, we can consider that fandgare maximal monotone graphs, verifying
suitable growth conditions (see [80]). Also, as in [231], [223] and [ ?], we can work with
more general functions h.
102

Chapter 5
Multiscale modeling of heat
transfer in composite materials
with imperfect interfaces
In the last two decades, the study of the macroscopic properties of heterogeneous composite
materials with imperfect contact between their constituents has been a subject of major
interest for engineers, mathematicians, physicists (see [24] and [149]). In particular, the
problem of thermal transfer in such heterogeneous media has attracted the attention of a
broad category of researchers, due to the fact that the macroscopic properties of a composite
can be strongly in
uenced by the imperfect bonding between its components (for a review
of the literature on imperfect interfaces in heterogeneous media, we refer to [158] and [ ?]).
This imperfect contact can be generated by various causes: the presence of impurities at
the boundaries, the presence of a thin interphase, the interface damage, chemical processes.
A large variety of multiscale problems are homogenization problems. Microstructural
features in heterogeneous materials strongly aect what happens at a larger scale. Making
connections among scales is a dicult task. Fundamental physical and mathematical prin-
ciples are applied to the modeling at each scale, and data are then passed to the next scale
up. Combining a multiscale-modeling strategy with computational techniques, scientists
try to bridge the dierent length scales, shedding light on the fundamental mechanisms
that determine how such materials behave. We shall try to understand the interplay of
eects on dierent scales and to nd the relevant quantities at microscale needed for an
accurate description of the macroscopic behaviour. A major goal today is to develop sys-
tematic and reliable multiscale techniques, allowing to deal with a variety of applications.
The basic strategy is to use as much as possible the microscopic model information for
extracting the macroscale behaviour of the analyzed systems. The coupling mechanisms
between scales is not an easy task and it is far from being a completely solved problem.
Multiscale techniques proved to be very useful for the multiscale analysis of a large
class of problems arising in the modeling of physical and chemical phenomena of theo-
103

104 …..HOMOGENIZATION … heterogeneous structures …media ….
retical and practical interest. Progress in the problems we plan to address could have a
considerable impact on practical applications, such as the designing and the optimization
of new composite materials, which are nowadays widely exploited for the realization of in-
novative devices useful in various elds. Multiscale methods still oer multiple possibilities
for further developments, the study of multiscale problems being one of the most active
and fastest growing areas of modern applied mathematics, and denitely one of the most
interdisciplinary eld of mathematics.
The behavior of heterogeneous materials, with inhomogeneities at a length scale which
is much smaller than the characteristic dimensions of the system, is of huge interest in the
theory of composite materials. The homogenization theory was successfully applied for
modeling the behavior of such materials, leading to appropriate macroscopic continuum
models, obtained by averaging the rapid oscillations of the material properties. Besides,
such eective models have the advantage of avoiding extensive numerical computations
arising when dealing with the small scale behavior of the system.
The homogenization of a thermal problem in a two-component composite with inter-
facial barrier, with jump of the temperature and continuity of the
ux, was studied for
the rst time in the pioneering work [106], where the asymptotic expansion method was
used. Many mathematical studies were performed since then, in order to rigorously justify
the convergence results. Various mathematical methods were used: the energy method in
[102] and [185], the two-scale convergence method in [109], and more recently the unfolding
method for periodic homogenization in [99] and [215], to cite just a few of them. The main
common point of all these studies is the fact that at the interface between the two compo-
nents the
ux of the temperature is continuous, the temperature eld has a jump and the

ux is proportional to this jump. Several cases are studied, following the order of magni-
tude with respect to the small parameter "of the resistance generated by the imperfect
contact between the constituents, leading to completely dierent macroscopic problems.
Here,"is a small real parameter related to the characteristic size of the two constituents.
In some cases, an eect of the imperfect conditions is observed in the coecients of the
homogenized matrix, via the local problems; in other cases, there is no eect at all in the
homogenized problem.
Such problems were addressed mainly in two geometrical settings. For the case when
both components of the composite material are connected, we refer to [181], [193], [194],
[195], [212] and [213]. The case in which only one phase is connected, while the other one
is disconnected was considered, for instance, in [102], [185], and [99].
For similar homogenization problems of parabolic or hyperbolic type, we refer the
reader to [100] and [178]. Also, for problems involving jumps in the solution in other
contexts, such as heat transfer in polycrystals with interfacial resistance, linear elasticity
or thermoelasticity problems or problems modeling the electrical conduction in biological
tissues, see [21], [19], [103], [110], [111], [112], [143], [191], and [217].
In this chapter, we shall present some homogenization results for …. We start by

HEAT TRANSFER IN COMPOSITE MATERIALS 105
presenting the results obtained in … for the … In the same geometry, we also ….We notice
that … results for elasticity or thermoelasticity … both geometries … We end this chapter
by presenting some recent results for … involving jumps both in the solution and in the

ux .. This chapter is based on ..[215], [46], [47], [48], [212], [213] ……………
5.1 Multiscale analysis in nonlinear thermal diusion prob-
lems in composite structures
Multiscale analysis in nonlinear diusion problems in composite structures
In [215], we have analyzed, using the periodic unfolding method, the eective thermal
transfer in a periodic composite material formed by two constituents, separated by an
imperfect interface. Our results were set in the framework of thermal transfer, but they
remain true for more general reaction-diusion processes. We assumed that we have non-
linear sources acting in each media and that at the interface between the two constituents
the
ux is continuous, but the temperature eld has a jump. We were interested in de-
scribing the asymptotic behavior, as the small parameter which characterizes the sizes of
the two constituents tends to zero, of the temperature eld in the periodic composite. The
imperfect contact between the constituents generates a contact resistance and, depending
on the magnitude of this resistance, a threshold phenomenon arises. So, depending on the
rate exchange between the two phases, three important cases are considered and three dif-
ferent types of limit problems are obtained from the same type of micromodel. The results
in [215] constitute a generalization of those obtained in [106], [192], ???[ ?] and ???[ ?]. For
heat conduction problems in a periodic material with a dierent geometry, we refer to [99]
and [185] and the references therein.
In [215], for simplicity, we dealt only with the stationary case, but the dynamic one can
be treated in a similar manner (see [80] and [192]). Similar problems have been addressed,
using dierent techniques, formal or not, in [24], [25], [158] and [106].
5.1.1 Problem setting
We place ourselves in the same setting as in Section 3.3. So, we consider an open bounded
material body
in Rn, withn3, with a Lipschitz-continuous boundary @
. We assume
that
is formed by two constituents,
"
1and
"
2, representing two materials with dierent
thermal characteristics, separated by an imperfect interface ". We assume that both
phases
"
1and
"
2=
n
"
1are connected, but only
"
1reaches the external xed boundary
@
. Here,"represents a small parameter related to the characteristic size of the two
constituents.
Our goal in [215] was to describe the eective behavior of the solution ( u";v") of the

106 …..HOMOGENIZATION … heterogeneous structures …media ….
following coupled system of equations:
8
>>>><
>>>>:div (A"
1ru"
1) +(u"
1) =fin
"
1;
div (A"
2ru"
2) +(u"
2) =fin
"
2;
A"
1ru"
1
=A"
2ru"
2
on ";
A"
1ru"
1
="
h(u"
1;u"
2) on ";
u"
1= 0 on@
:(5.1)
Here,is the unit outward normal to
"
1andf2L2(
).
Thus, we consider that the
ux is continuous across the boundary ", but, since the
interface between the two phases is not perfect, the continuity of temperatures is replaced
by a Biot boundary condition. We assume that the functions =(r) and=(r)
are continuous, monotonously non-decreasing with respect to rand such that (0) = 0
and(0) = 0. Moreover, we suppose that there exist C0 and an exponent q, with
0q<n= (n2), such that
j(r)jC(1 +jrjq) (5.2)
and
j(r)jC(1 +jrjq): (5.3)
We also assume, as in Section 3.3, that
h(u"
1;u"
2) =h"
0(x)(u"
2u"
1); (5.4)
whereh"
0(x) =h0x
"
andh0(y) is aY-periodic, smooth real function with h0(y)>0.
Moreover, we consider that
H=Z
h0(y) d6= 0:
Let us notice that we can deal with the more general case in which the nonlinear functions
andare multi-valued maximal monotone graphs, as in [80].
For results concerning the well posedness of problem (4.1), we refer to [ ?], [192], [191],
???[?] and ???[ ?]. Since it is impossible to solve this system, the microstructure must be
homogenized in order to obtain a new model that describes its macroscopic properties.
Using the periodic unfolding method, we can describe the asymptotic behavior of the
solution of system (4.1). This behavior depends on the values of the parameter
, i.e. on
the contact resistance between the two constituents. There are three interesting cases to
be considered:
= 1,
= 0, and
=1.
In the most interesting case,
= 1, we obtain at the limit a new nonlinear system (see
(4.5)). At a macroscopic scale, the composite medium can be represented by a continuous
model, which conceives it as the superimposition of two interpenetrating continuous media,
coexisting at every point of the domain. For the other two cases, we obtain at the limit
only one equation (see (4.6) and (4.7)).
Using similar techniques as those developed in …, i.e. obtaining suitable a priori
estimates and using … compactness results .. unfolding method, we get
… convergences …(3.48) …to write them here …notatation u;v,u1;u2.

HEAT TRANSFER IN COMPOSITE MATERIALS 107
5.1.2 The main results
We describe now the eective behavior of the solutions of the microscopic model (4.1) for
the above mentioned three cases.
Theorem 5.1 For
= 1, the solution (u"
1; u"
2)of system (4.1) converges, as "!0, in
the sense of (…), to the unique solution (u1; u2), withu1;u22H1
0(
), of the following
macroscopic problem:
(
div(A1ru1) +(u1)H(u2u1) =fin
;
div(A2ru2) + (1)(u2) +H(u2u1) = (1)fin
:(5.5)
In (4.5),A1andA2are the homogenized matrices, dened by:
A1
ij=Z
Y1
aij+aik@1j
@yk
dy;
A2
ij=Z
Y2
aij+aik@2j
@yk
dy
and1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n , are the weak solutions of the cell
problems8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
So, at a macroscopic scale, the composite medium, despite of its discrete structure, can
be represented by a continuous model, which is similar to the so-called bidomain model ,
arising in the context of diusion in partially ssured media (see [28] and [106]) or in the
case of electrical activity of the heart (see [ ?] and [191]).
Theorem 5.2 For
= 0, i.e. for high contact resistance, we get, at the macroscale, only
one temperature eld. So, u1=u2=u02H1
0(
)andu0satises:
div(A0ru0) +(u0) + (1)(u0) =fin
: (5.6)
Here, the eective matrix A0is given by:
A0
ij=Z
Y1
aij+aik@1j
@yk
dy+Z
Y2
aij+aik@2j
@yk
dy;

108 …..HOMOGENIZATION … heterogeneous structures …media ….
in terms of the functions 1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n; weak solutions
of the cell problems
8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
Let us notice that in this case, the insulation provided by the interface is sucient to
modify the limiting diusion matrix, but it is not strong enough to force the existence of
two dierent limit phases.
Theorem 5.3 For the case
=1, i.e. for weak contact resistance, we also get, at the
limit,u1=u2=u02H1
0(
)and, in this case, the eective temperature eld u0satises:
div(A0ru0) +(u0) + (1)(u0) =fin
: (5.7)
The macroscopic coecients are given by:
A0;ij=Z
Y1
aij+aik@w1j
@yk
dy+Z
Y2
aij+aik@w2j
@yk
dy;
wherew1k2H1
per(Y1)=R; w2k2H1
per(Y2)=R,k= 1;:::;n; are the weak solutions of the cell
problems
8
>><
>>:ry
(A1(y)ryw1k) =ryA1(y)ek; y2Y1;
ry
(A2(y)ryw2k) =ryA2(y)ek; y2Y2;
(A1(y)ryw1k)
= (A2(y)ryw2k)
 y2;
(A1(y)ryw1k)
+h0(y)(w1kw2k) =A1(y)ek
 y2:
In this case, as expected, the eective coecients depend on h0.
5.2 Diusion problems with dynamical boundary conditions
Recent theoretical and advanced computational studies investigating the eective behavior
of composite materials are based on a model which considers the composite material,
despite of its discrete structure, as the coupling of two continuous superimposed domains.
Homogenization theory will allow us to justify such a model and to give a meaning to the
eective properties of highly heterogeneous materials, modeled by equations with rapidly
oscillating periodic coecients.
In this section, we shall present some homogenization results obtained by using the …
on [212] and generalized later, using the periodic unfolding method, in [213]. The aim of

HEAT TRANSFER IN COMPOSITE MATERIALS 109
these papers was to analyze the asymptotic behavior of the solution of a nonlinear problem
arising in the modeling of thermal diusion in a two-component composite material. We
consider, at the microscale, a periodic structure formed by two materials with dierent
thermal properties. We assume that we have nonlinear sources and that at the interface
between the two materials the
ux is continuous and depends in a dynamical nonlinear
way on the jump of the temperature eld. As usual, we are interested in describing the
asymptotic behavior of the temperature eld in the periodic composite as the small pa-
rameter which characterizes the sizes of our two regions tends to zero. We prove that the
eective behavior of the solution of this system is governed by a new system, similar to
Barenblatt's model, with additional terms capturing the eect of the interfacial barrier, of
the dynamical boundary condition, and of the nonlinear sources.
More precisely, in the same geometry as the one described in Section 4.1, we shall be
interested in analyzing the asymptotic behavior of the solutions of the following nonlinear
system:8
>>>>>>>><
>>>>>>>>:div (A"
1ru"
1) +(u"
1) =fin
"
1(0;T);
div (A"
2ru"
2) =f; in
"
2(0;T);
A"
1ru"
1
=A"
2ru"
2
on "(0;T);
A"
1ru"
1
+a"@
@t(u"
1u"
2) ="g(u"
2u"
1) on "(0;T);
u"
1= 0 on@
(0;T);
u"
1(0;x)u"
2(0;x) =c0(x);on ":(5.8)
Here,is the exterior unit normal to
"
1,f2L2(0;T;L2(
)),c02H1
0(
) anda>0. The
functionis continuous, monotonously non-decreasing and such that (0) = 0 and the
functiongis continuously dierentiable, monotonously non-decreasing and with g(0) = 0.
We shall suppose that there exist a positive constant Cand an exponent q, with 0q <
n=(n2), such that
j(v)jC(1 +jvjq);dg
dvC(1 +jvjq): (5.9)
As particular examples of such functions we can consider, for instance, the following impor-
tant practical ones: (v) =v
1 +
v; ;
> 0 (Langmuir kinetics), (v) =jvjp1v;0<
p<1 (Freundlich kinetics), g(v) =avorg(v) =av3, witha>0.
Well posedness of problem (4.8) in suitable function spaces and proper energy estimates
have been obtained in [21], [43] and [191]. Our goal is to obtain the asymptotic behavior,
when"!0, of the solution of problem (4.8). Using Tartar's method of oscillating test
functions (see [ ?]), coupled with monotonicity methods and results from the theory of
semilinear problems (see [43] and [80]), we can prove that the asymptotic behavior of the
solution of our problem is governed by a new nonlinear system, similar to the famous
Barenblatt's model (see [28] and [106]), with additional terms capturing the eect of the
interfacial barrier, of the dynamical boundary condition and of the nonlinear sources. Our

110 …..HOMOGENIZATION … heterogeneous structures …media ….
results constitute a generalization of those obtained in [28] and [106], by considering nonli-
near sources, nonlinear dynamical transmission conditions and dierent techniques in the
proofs. Similar problems have been considered, using dierent techniques, in [19] and [191],
for studying electrical conduction in biological tissues.
Using well-known extension results (see, for instance, [73], [ ?] and [191]) and suitable
test functions, we can take the limit in the variational formulation of problem (4.8) and
obtain the eective behavior of the solution of our microscopic model. Therefore, the main
result can be formulated as follows:
Theorem 5.4 One can construct two extensions P"u"
1andP"u"
2of the solutions u"
1and
u"
2of problem (4.8) such that P"u"
1*u 1; P"u"
2*v, weakly in L2(0;T;H1
0(
)), where
8
>>>>>>><
>>>>>>>:ajj@
@t(u1u2)div(A1ru1) +(u1)jjg(u2u1) =
=fin
(0;T);
ajj@
@t(u2u1)div(A2ru2)+jjg(u2u1) =
= (1)fin
(0;T);
u1(0;x)u2(0;x) =c0(x)on
:(5.10)
Here,A1andA2are the homogenized matrices, dened by:
A1
ij=Z
Y1
aij+aik@1j
@yk
dy;
A2
ij=Z
Y2
aij+aik@2j
@yk
dy;
in terms of the functions 1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n , weak solutions
of the cell problems
8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
Thus, in the limit, we obtain a system similar to the so-called Barenblatt model . Alterna-
tively, such a system is similar to the so-called bidomain model , appearing in the context
of electrical activity of the heart. At a macroscopic level, despite of the discrete cellular
structure, the composite material can be represented by a continuous model, describ-
ing the averaged properties of the complex structured composite material. The resulting
macroscopic model describes the composite material as the superimposition of two interpe-
netrating continuous media, coexisting at every point of the domain. Also, note that the

HEAT TRANSFER IN COMPOSITE MATERIALS 111
above model is a degenerate parabolic system, as the time derivatives involve the unknown
vu, instead of the unknowns uandvoccurring in the second-order conduction term.
Similar problems to that described herein were considered by D. Cioranescu and P.
Donato [ ?], D. Cioranescu, P. Donato and H.I. Ene [61], C. Conca and P. Donato [ ?], C.
Conca, J.I. D az and C. Timofte [80], C. Conca, F. Murat and C. Timofte [83], C. Timofte
???[?], ????[ ?] and ????[ ?], H. Ene and D. Polisevski [106], C. Timofte ???? [ ?], for the
deterministic case, W. Wang and J. Duan [ ?] for the stochastic one.
These results were generalized in [213] by using the periodic unfolding method, which
allows us to avoid the use of extension operators and, hence, to deal with more general
media and by considering more general nonlinearities (x;u") andg(x;v"u")).
5.3 Homogenization of a thermal problem with
ux jump
In [46], our goal was to analyze, through homogenization techniques, the eective thermal
transfer in a periodic composite material formed by two constituents, one connected and the
other one disconnected, separated by an imperfect interface where both the temperature
and the
ux exhibit jumps. This mathematical model is not restricted to the thermal
transfer, but can be used in other contexts, too. Transmission problems involving jumps in
the solutions or in the
uxes are encountered in various domains, such as linear elasticity,
theory of semiconductors, the study of photovoltaic systems or problems in media with
cracks (see, for instance, [ ?], [41], [45], [138] and [153]). Formal methods of averaging were
widely used in the literature to deal with such imperfect transmission problems. Still,
obtaining rigorous results based on the homogenization theory is a dicult task in many
cases. Some results were nevertheless obtained for problems with
ux jump, by using
homogenization techniques. We mention here the results obtained in [198] for problems
arising in the combustion theory and in [119] for a problem corresponding to the Gouy-
Chapman-Stern model for an electric double layer.
The main novelty brought by us in [46] consists in allowing the presence, apart from the
discontinuity in the temperature eld, of a jump in the thermal
ux across the imperfect
interface ", given by the function G". Two dierent representative cases were studied,
following the conditions imposed on G"(stated explicitly in Section 4.3…. Case 1 and Case
2). Let us mention that such functions were already encountered in a dierent context,
more precisely in [64] and [66] for the case of the perforated domains with non homogeneous
Neumann boundary conditions on the perforations. After passage to the limit with the
unfolding method, we obtain here two dierent unfolded problems (stated in Theorem 4.1
and Theorem 4.7), corresponding to the above mentioned cases for the
ux jump function
G". In Corollary 4.2 ??? and Theorem 4.9 ???, we then give the corresponding homogenized
problems. In both situations, the homogenized matrix Ahomis constant and it depends on
the function describing the jump of the solution. This phenomenon was already observed
in some cases without
ux jump. Moreover, for the rst case studied here, we notice in the

112 …..HOMOGENIZATION … heterogeneous structures …media ….
right-hand side the presence of a new source term distributed all over the domain
and
depending on the
ux jump function. For the second case, we notice that the in
uence of
the jump in the
ux is captured by the correctors only and so this jump plays no role in
the homogenized problem; nevertheless, in Remark … 4.10 we mention a case when the
homogenized problem depends on this jump, too. This type of result is to be compared
with the Neumann problem in perforated domains (see [64], [66]), where similar phenomena
occur.
5.3.1 Setting of the problem
We consider a composite material occupying an open bounded set
in Rn, withn2, with
a Lipschitz-continuous boundary @
. We assume that
is formed by two parts denoted

"
1and
"
2, occupied by two materials with dierent thermal characteristics, separated by
an imperfect interface ". We also assume that the phase
"
1is connected and reaches
the external xed boundary @
and that
"
2is disconnected: it is the union of domains of
size", periodically distributed in
with periodicity ". More precisely, let Y= (0;1)nbe
the reference cell in Rn. We assume that Y1andY2are two non-empty disjoint connected
open subsets of Ysuch thatY2YandY=Y1[Y2. We also suppose that = @Y2
is Lipschitz continuous and that Y2is connected. In fact, our results can be extended to
the case in which the set Y2has a nite number of connected components, as in [99]. For
eachk2ZN, we denote Yk=k+YandYk
=k+Y, for= 1;2. For each ", we
dene,Z"=n
k2ZN:"Yk
2
o
and we set
"
2=S
k2Z"
"Yk
2

and
"
1=
n
"
2. The
boundary of
"
2is denoted by "andis the unit outward normal to
"
1.
Our goal is to describe the asymptotic behavior, as "!0, of the solution u"= (u"
1;u"
2)
of the following problem:
8
>>>>>>><
>>>>>>>:div (A"ru"
1) =fin
"
1;
div (A"ru"
2) =fin
"
2;
A"ru"
1
=h"
"(u"
1u"
2)G"on ";
A"ru"
2
=h"
"(u"
1u"
2) on ";
u"
1= 0 on@
:(5.11)
Remark 5.5 We notice that
A"ru"
2
A"ru"
1
=G"; (5.12)
which clearly shows that the
ux of the solution exhibits a jump across ".
The function f2L2(
) is given. Let hbe aY{periodic function in L1() such that
there exists h02Rwith 0<h 0<h(y) a.e. on . We set
h"(x) =hx
"
a.e. on ":

HEAT TRANSFER IN COMPOSITE MATERIALS 113
For;2R, with 0< , letM(;;Y ) be the set of all the matrices A2
(L1(Y))NNwith the property that, for any 2RN,jj2(A(y); )jj2, al-
most everywhere in Y. For aY-periodic smooth symmetric matrix A2M (;;Y ), we
set
A"(x) =Ax
"
a.e. in
:
Letgbe aY-periodic function that belongs to L2(). We dene
g"(x) =gx
"
a.e. on ":
For the given function G"in (5:11), we consider the following two relevant forms (see [66]):
Case 1 :G"="gx
"
, ifM(g)6= 0.
Case 2 :G"=gx
"
, ifM(g) = 0.
Here,M(g) =1
jjZ
g(y) dydenotes the mean value of the function gon .
In order to write the variational formulation of problem (5.11), we introduce, for every
positive"<1, the Hilbert space
H"=V"H1(
"
2): (5.13)
The spaceV"=
v2H1(
"
1); v= 0 on@

is endowed with the norm kvkV"=krvkL2(
"
1),
for anyv2V"and the space H1(
"
2) is equipped with the usual norm. On the space H",
we consider the scalar product
(u;v)H"=Z

"
1ru1rv1dx+Z

"
2ru2rv2dx+1
"Z
"(u1u2)(v1v2) dx (5.14)
whereu= (u1;u2) andv= (v1;v2) belong to H". The norm generated by the scalar
product (5.14) is given by
kvk2
H"=krv1k2
L2(
"
1)+krv2k2
L2(
"
2)+1
"kv1v2k2
L2("): (5.15)
The variational formulation of problem (5.11) is the following one: nd u"2H"such
that
a(u";v) =l(v);8v2H"; (5.16)
where the bilinear form a:H"H"!Rand the linear form l:H"!Rare given by
a(u;v) =Z

"
1A"ru1rv1dx+Z

"
2A"ru2rv2dx+Z
"h"
"(u1u2)(v1v2) dx

114 …..HOMOGENIZATION … heterogeneous structures …media ….
and
l(v) =Z

"
1fv1dx+Z

"
2fv2dx+Z
"G"v1dx;
respectively.
We state now an existence and uniqueness result and some necessary a priori estimates
for the solution of the variational problem (5.16).
Theorem 5.6 For any"2(0;1), the variational problem (5:16)has a unique solution
u"2H". Moreover, there exists a constant C > 0, independent of ", such that
kru"
1kL2(
"
1)C;kru"
2kL2(
"
2)C (5.17)
and
ku"
1u"
2kL2(")C"1=2: (5.18)
5.3.2 The periodic unfolding method for a two-component domain
In this section, we shall brie
y recall the denitions and the main properties of the unfolding
operatorsT"
1andT"
2, introduced, for a two-component domain, by P. Donato et al. in [99]
(see, also, [ ?], [?], [?] and [101]) and of the boundary unfolding operator T"
b, introduced
in [?] and [ ?]. The main feature of these operators is that they map functions dened on
the oscillating domains
"
1,
"
2and, respectively, ", into functions dened on the xed
domains
Y1,
Y2and
, respectively.
Forx2RN, we denote by [ x]Yits integer part k2ZN, such that x[x]Y2Y
and we setfxgY=x[x]Yfor a.e.x2RN. So, for almost every x2RN, we have
x="hx
"i
+nx
"o
. For dening the above mentioned periodic unfolding operators, we
consider the following sets (see [99]):
bZ"=n
k2ZNj"Yk
o
;b
"= int[
k2bZ"
"Yk
;"=
nb
";
b
"
=[
k2bZ"
"Yk

;"
=
"
nb
"
;b"=@b
"
2:
Denition 5.7 For any Lebesgue measurable function 'on
"
,2f1;2g, we dene the
periodic unfolding operators by the formula
T"
(')(x;y) =8
<
:'
"hx
"i
+"y
for a.e. (x;y)2b
"Y
0 for a.e. (x;y)2"Y
If'is a function dened in
, for simplicity, we write T"
(')instead ofT"
('j
").

HEAT TRANSFER IN COMPOSITE MATERIALS 115
For any function which is Lebesgue-measurable on ", the periodic boundary unfolding
operatorT"
bis dened by
T"
b()(x;y) =8
<
:
"hx
"i
+"y
for a.e. (x;y)2b
"
0 for a.e. (x;y)2"
Remark 5.8 We notice that if '2H1(
"
), thenT"
b(') =T"
(')jb
".
We recall here some useful properties of these operators (see, for instance, [ ?], [101]
and [99]).
Proposition 5.9 Forp2[1;1)and= 1;2, the operatorsT"
are linear and continuous
fromLp(
"
)toLp(
Y)and
(i) if'and are two Lebesgue measurable functions on
"
, one has
T"
(' ) =T"
(')T"
( );
(ii) for every '2L1(
"
), one has
1
jYjZ

YT"
(')(x;y) dxdy=Z
b
"'(x) dx=Z

"'(x) dxZ
"'(x) dx;
(iii) iff'"g"Lp(
)is a sequence such that '"!'strongly in Lp(
), then
T"
('")!'strongly in Lp(
Y);
(iv) if'2Lp(Y)isY-periodic and '"(x) ='(x="), then
T"
('")!'strongly in Lp(
Y);
(v) if'2W1;p(
"
), thenry(T"
(')) ="T"
(r')andT"
(')belongs toL2

;W1;p(Y)

.
Moreover, for every '2L1("), one has
Z
b"'(x) dx=1
"jYjZ

T"
b(')(x;y) dxdy:
The following result was proven, for our geometry, in [99].
Lemma 5.10 Ifu"= (u"
1;u"
2)is a sequence in H", then
1
"jYjZ

jT"
1(u"
1)T"
2(u"
2)j2dxdyZ
"ju"
1u"
2j2dx:
Moreover, if '2D(
), then, for"small enough, we have
"Z
"h"(u"
1u"
2)'dx=Z

h(y) (T"
1(u"
1)T"
2(u"
2))T"
(') dxdy;
with= 1or= 2:

116 …..HOMOGENIZATION … heterogeneous structures …media ….
We also recall here some general compactness results obtained in [99] for bounded sequences
inH".
Lemma 5.11 Letu"= (u"
1;u"
2)be a bounded sequence in H". Then, there exists a constant
C > 0, independent of ", such that
kT"
1(ru"
1)kL2(
Y1)C;
kT"
2(ru"
2)kL2(
Y2)C;
kT"
2(u"
1)T"
1(u"
2)kL2(
)C":
Theorem 5.12 Letu"= (u"
1;u"
2)be a bounded sequence in H". Then, up to a subsequence,
still denoted by ", there exist u12H1
0(
),u22L2(
),bu12L2

;H1
per(Y1)

andbu22
L2

;H1(Y2)

such that
T"
1(u"
1)!u1strongly in L2

;H1(Y1)

;
T"
1(ru"
1)*ru1+rybu1weakly inL2(
Y1);
T"
2(u"
2)*u 2weakly inL2(
;H1(Y2));
T"
2(ru"
2)*rybu2weakly inL2(
Y2);
eu"
*jYj
jYjuweakly inL2(
);  = 1;2;
whereM(bu1) = 0 for almost every x2
and~
denotes the extension by zero of a function
to the whole of the domain
. Moreover, we have u1=u2and
1
"[T"
1(u"
1)M (T"
1(u"
1))]*y ru1+bu1weakly inL2

;H1(Y1)

;
withy=yM (y)and
1
"[T"
2(u"
2)M (T"
2(u"
2))]*bu2weakly inL2

;H1(Y2)

:
5.3.3 The macroscopic models
Our goal now is to pass to the limit, with "!0, in the variational formulation (5.16) of
the problem (5.11). To this end, we use the periodic unfolding method and the general
compactness results given in Appendix ….
We start by emphasizing again that by applying the general results stated in Theorem
… to the solution u"= (u"
1;u"
2) of the variational problem (5.16), which is bounded in H",
we obtain, at the macroscale, u1=u2. In what follows, we shall denote their common
value byu. We notice that ubelongs toH1
0(
).

HEAT TRANSFER IN COMPOSITE MATERIALS 117
Moreover, using the priori estimates (4.17)-(4.18) and the general compactness results
(…)-(…) in Appendix …, we know that there exist u2H1
0(
),bu12L2(
;H1
per(Y1)),
bu22L2(
;H1(Y2)) such thatM(bu1) = 0 and up to a subsequence, for "!0, we have:
T"
1(u"
1)!ustrongly in L2(
;H1(Y1));
T"
1(ru"
1)*ru+rybu1weakly inL2(
Y1);
T"
2(u"
2)*u weakly inL2(
;H1(Y2));
T"
2(ru"
2)*rybu2weakly inL2(
Y2);
eu"
*jYj
jYjuweakly inL2(
);  = 1;2:(5.19)
Moreover, one has
T"
1(u"
1)T"
2(u"
2)
"*bu1u2weakly inL2(
); (5.20)
where u22L2(
;H1(Y2)) is dened by
u2=bu2yru;
for some2L2(
).
LetWper(Y1) =fv2H1
per(Y1)jM (v) = 0g. We consider the space
V=H1
0(
)L2(
;Wper(Y1))L2

;H1(Y2)

;
endowed with the norm
kVk2
V=krv+rybv1k2
L2(
Y1)+krv+ryv2k2
L2(
Y2)+kbv1v2k2
L2(
);
for allV= (v;bv1;v2)2V.
For the passage to the limit, we have to distinguish between two cases, following the
form of the function G".
Case 1 :G"="gx
"
, ifM(g)6= 0.
Theorem 5.13 The unique solution u"= (u"
1;u"
2)of the variational problem (5.16) con-
verges, in the sense of (4.19), to the unique solution (u;bu1;u2)2Vof the following unfolded
limit problem:
1
jYjZ

Y1A(y)(ru+rybu1)(r'+ry1) dxdy+
1
jYjZ

Y2A(y)(ru+ryu2)(r'+ry2) dxdy+
1
jYjZ

h(y)(bu1u2)(12) dxdy=Z

f(x)'(x) dx+jj
jYjM(g)Z

'(x) dx;(5.21)
for all'2H1
0(
),12L2(
;H1
per(Y1))and22L2(
;H1(Y2)).

118 …..HOMOGENIZATION … heterogeneous structures …media ….
Proof. In order to obtain the limit problem (5.21), we rst unfold the variational
formulation (5.16) and by using Lemma … we get
1
jYjZ

Y1T"
1(A")T"
1(ru"
1)T"
1(rv1) dx+1
jYjZ

Y2T"
2(A")T"
2(ru"
2)T"
2(rv2) dx+
1
jYjZ

h(y)T"
1(u"
1)T"
2(u"
2)
"T"
1(v1)T"
2(v2)
"dx=
1
jYjZ

Y1T"
1(f)T"
1(v1) dx+1
jYjZ

Y2T"
2(f)T"
2(v2) dx+1
"1
jYjZ

T"
b(G")T"
b(v1) dx:
Then, for= 1;2, we choose in this unfolded problem the admissible test functions
v='(x) +"!(x) x
"
; (5.22)
with';!2D(
), 12H1
per(Y1), 22H1(Y2) and for which we obviously have
T"
(v)!'(x) strongly in L2(
Y) (5.23)
and
T"
(rv)!r'(x) +rystrongly in L2(
Y); (5.24)
where (x;y) =!(x) (y).
Then, the passage to the limit with "!0 is classical, by using the above convergences
and the ideas in [101]. The only term which requires more attention is in the right-hand
side, the integral term involving the function G". For this term, we have:
1
"1
jYjZ

T"
b(G")T"
b(v1) dx=1
jYjZ

T"
b
gx
"
T"
b
'(x) +"!1(x) 1x
"
dx=
1
jYjZ

g(y)T"
b(')(x;y) dxdy+"1
jYjZ

g(y)T"
b(!1)(x;y)T"
b( 1)(x;y) dxdy!
jj
jYjM(g)Z

'(x) dx:
By the density ofD(
)
H1
per(Y1) inL2(
;H1
per(Y1)) and ofD(
)
H1(Y2) inL2(
;H1(Y2)),
we get (4.21).
We notice that our limit problem (4.21) is similar with the one obtained in [99] (see
relation (3.43)), the only dierence being the right-hand side, in which an extra term
involving the function garises. More precisely, our right-hand side writes
Z

F(x)'(x) dx;

HEAT TRANSFER IN COMPOSITE MATERIALS 119
with
F(x) =f(x) +jj
jYjM(g):
The extra term is, in fact, just a real constant and this allows us to prove the uniqueness
of the solution of problem (4.21) exactly as in [99], since the presence of this constant term
does not change the linearity nor the continuity of its right-hand side. Thus, due to the
uniqueness of ( u;bu1;u2)2V, all the above convergences hold true for the whole sequence,
which ends the proof of the theorem. 
Corollary 5.14 The function u2H1
0(
)dened by (4.19) is the unique solution of the
following homogenized equation:
div(Ahomru) =f+jj
jYjM(g)in
; (5.25)
whereAhomis the homogenized matrix whose entries are given, for i;j= 1;:::;n , by
Ahom
ij=1
jYjZ
Y1
aijnX
k=1aik@j
1
@yk!
dy+1
jYjZ
Y2
aijnX
k=1aik@j
2
@yk!
dy; (5.26)
in terms of j
12H1
per(Y1)andj
22H1(Y2),j= 1;:::;n , the weak solutions of the
following cell problems:
8
>>>>><
>>>>>:divy(A(y)(ryj
1+ej)) = 0; y2Y1;
divy(A(y)(ryj
2+ej)) = 0; y2Y2;
(A(y)ryj
1)
= (A(y)ryj
2)
; y2;
(A(y)(ryj
1+ej))
+h(y)(j
1j
2) = 0; y2:
M(j
1) = 0:(5.27)
wheredenotes the outward normal to Y1.
The proof of this result is classical. Indeed, by choosing successively '= 0 and  1=
2= 0 in (4.21), we obtain:
1
jYjZ

Y1A(y)(ru+rybu1)ry1dxdy+1
jYjZ

Y2A(y)(ru+ryu2)ry2dxdy+
1
jYjZ

h(y)(bu1u2)(12) dxdy= 0 (5.28)
and
1
jYjZ

Y1A(y)(ru+rybu1)r'dxdy+1
jYjZ

Y2A(y)(ru+ryu2)r'dxdy=
Z

f(x)'(x) dx+jj
jYjM(g)Z

'(x) dx: (5.29)

120 …..HOMOGENIZATION … heterogeneous structures …media ….
We search now bu1and u2in the usual form
bu1(x;y) =nX
j=1@u
@xj(x)j
1(y); (5.30)
u2(x;y) =nX
j=1@u
@xj(x)j
2(y): (5.31)
Standard computations lead to the homogenized limit problem, in which the term contain-
ingggives a contribution only to the right-hand side, without aecting the cell problems
and the homogenized matrix. 
Remark 5.15 The right scaling "in front of the function g"prescribed at the interface "
leads in the limit to the presence of a new source term distributed all over the domain
.
Remark 5.16 It is possible to study our initial problem (5.11) also for a nonzero function
gwith mean-valueM(g)equal to zero. But, in this situation, there is no contribution of
gin the right-hand side of the homogenized equation and, thus, the limit problem is the
same as in the case with no gat all in the microscopic problem.
Remark 5.17 We remark that the homogenized matrix Ahomdepends on the function h.
So, the eect of the two jumps involved in our microscopic problem is recovered in the
homogenized problem, in the right-hand side and also in the left-hand side (through the
homogenized coecients).
Case 2 :G"(x) =gx
"
, ifM(g) = 0.
Theorem 5.18 The unique solution u"= (u"
1;u"
2)of the variational problem (5.16) con-
verges, in the sense of (4.19), to the unique solution (u;bu1;u2)2Vof the following unfolded
limit problem:
1
jYjZ

Y1A(y)(ru+rybu1)(r'+ry1) dxdy+
1
jYjZ

Y2A(y)(ru+ryu2)(r'+ry2) dxdy+
1
jYjZ

h(y)(bu1u2)(12) dxdy=
Z

f(x)'(x) dx+1
jYjZ

g(y)1(x;y) dxdy; (5.32)
for all'2H1
0(
),12L2(
;H1
per(Y1)),22L2(
;H1(Y2)).

HEAT TRANSFER IN COMPOSITE MATERIALS 121
In order to get the problem (5.32), we pass to the limit in the unfolded form of the
variational formulation (5.16) with the same test functions (5 :22) as in Theorem 4.7, which
verify (5:23) and (5:24). The only dierence is that now the limit of the term involving
the function G"is dierent. More precisely, we have:
1
"1
jYjZ

T"
b(G")T"
b(v1) dx=1
"1
jYjZ

T"
b
gx
"
T"
b
'(x) +"!1(x) 1x
"
dx=
1
"1
jYjZ

g(y)T"
b(')(x;y) dxdy+1
jYjZ

g(y)T"
b(!1)(x;y)T"
b( 1)(x;y) dxdy=
1
"jj
jYjM(g)Z

'(x) dx+1
jYjZ

g(y)!1(x) 1(y) dxdy:
Then, sinceM(g) = 0, by using the density of D(
)
H1
per(Y1) inL2(
;H1
per(Y1)) and
ofD(
)
H1(Y2) inL2(
;H1(Y2)), we get the unfolded limit problem (5.32).
Due to the uniqueness of ( u;bu1;u2)2V, which is proven by the Lax-Milgram theorem,
all the above convergences hold true for the whole sequence, which ends the proof of the
theorem.

Remark 5.19 Let us point out that the term1
jYjZ

g(y)1(x;y) dxdyin (5.32) rep-
resents the main dierence with respect to the unfolded equation (5.21), where the term
involvinggis a nonzero constant, recovered explicitly in the right-hand side of the homoge-
nized equation (4….). This cannot be the case here, since this term involves now explicitly
both variables xandy. We have to understand the contribution in the homogenized problem
of this nonstandard term generated by the discontinuity of the
ux in the initial problem.
Actually, it will be seen in the next theorem that, apart from the classical solutions j
1and
j
2of the cell problems (5.27), we are led to introduce in (4…)-(4…) two additional scalar
terms1and2, verifying a new imperfect transmission cell problem (see (4….)).
Theorem 5.20 The solution (u;bu1;u2)2V of (4…. ) is such that:
bu1(x;y) =NX
j=1@u
@xj(x)j
1(y) +1(y);
u2(x;y) =NX
j=1@u
@xj(x)j
2(y) +2(y);
wherej
1andj
2are dened by (4….) and the function (1;2)is the unique solution of
the cell problem8
>>>><
>>>>:divy(A(y)r1) = 0 inY1;
divy(A(y)r2) = 0 inY2;
A(y)r1
=h(y)(12)g(y)on;
A(y)r2
=h(y)(12)on;
M(1) = 0:

122 …..HOMOGENIZATION … heterogeneous structures …media ….
The function u2H1
0(
)is the unique solution of the following homogenized equation
div(Ahomru) =fin
; (5.33)
whereAhomis the homogenized matrix whose entries are given in (4…).
Proof. By choosing '= 0 in (5.32), we obtain:
1
jYjZ

Y1A(y)(ru+rybu1)ry1dxdy+1
jYjZ

Y2A(y)(ru+ryu2)ry2dxdy+
1
jYjZ

h(y)(bu1u2)(12) dxdy=1
jYjZ

g(y)1(x;y) dxdy: (5.34)
Let us point out again that the presence of the term1
jYjZ

g(y)1(x;y) dxdyin
this equation represents the main dierence with respect to the previous case.
By choosing now suitable test functions  1and  2in (5.34), we obtain
divy(A(y)rybu1) = divy(A(y)ru) a.e. in
Y1; (5.35)
divy(A(y)ryu2) = divy(A(y)ru) a.e. in
Y2; (5.36)
A(y)(ru+ryu2)
=h(y)(bu1u2) a.e. on
; (5.37)
A(y)(ru+rybu1)
=h(y)(bu1u2)g(y) a.e. on
: (5.38)
We point out here that we also have a discontinuity type condition:
A(y)(ru+ryu2)
A(y)(ru+rybu1)
=g(y) a.e. on
: (5.39)
In the classical case with jump in the solution and with continuity of the
ux, the use
of the standard correctors j
1andj
2dened in (4…) is enough in order to express the
functionsbu1and u2in terms of the function ru. The presence of the function gin relations
(4….) and (4….) suggests us to search bu1and u2in the following nonstandard form:
bu1(x;y) =nX
j=1@u
@xj(x)j
1(y) +1(y); (5.40)
u2(x;y) =nX
j=1@u
@xj(x)j
2(y) +2(y); (5.41)
wherej
1andj
2are dened by (4….) and the functions 1,2have to be found. To this
end, we introduce (5.40) and (5.41) in (…)-(…) and we obtain:
8
>>>><
>>>>:divy(A(y)r1) = 0 inY1;
divy(A(y)r2) = 0 inY2;
A(y)r1
=h(y)(12)g(y) on ;
A(y)r2
=h(y)(12) on ;
M(1) = 0:(5.42)

HEAT TRANSFER IN COMPOSITE MATERIALS 123
We obviously have
A(y)r2
A(y)r1
=g(y) (5.43)
and then we notice that the new local problem (4.42) is an imperfect transmission problem,
involving both the discontinuities in the solution and in the
ux, given in terms of hand
g, respectively.
By the Lax-Milgram theorem, the problem (4.42) has a unique solution in the space
H=Wper(Y1)H1(Y2);
endowed with the scalar product
(;)H= (r1;r1)L2(Y1)+ (r2;r2)L2(Y2)+ (12;12)L2():
By choosing now  1=  2= 0 in (4. ..), we get:
1
jYjZ

Y1A(y)(ru+rybu1)r'dxdy+1
jYjZ

Y2A(y)(ru+ryu2)r'dxdy=
Z

f(x)'(x) dx: (5.44)
Integrating it by parts with respect to x, we obtain:
divx1
jYjZ
Y1A(y)(ru+rybu1) dy+1
jYjZ
Y2A(y)(ru+ryu2) dy
=f(x) in
:
By using here the particular form (5.40) and (5.41) of the functions bu1and u2and the
denition (4….) of the matrix Ahom, we get:
divx(Ahomru) =f+ divx0
@1
jYjZ
Y1A(y)r1(y)dy+1
jYjZ
Y2A(y)r2(y)dy1
Ain
;
(5.45)
which leads immediately to the homogenized problem (4….). We notice that this problem
does not involve the function g, because the second term of the right-hand side in (4. …)
actually vanishes.

Remark 5.21 All the above results can be extended to the case in which A"is a sequence
of matrices inM(;;
)such that
T"
(A")!Astrongly in L1(
Y); (5.46)
for some matrix A=A(x;y)inM(;;
Y). The heterogeneity of the medium modeled
by such a matrix induces dierent eects in our limit problems (4.3) and (4.14) respectively.
In both cases, since the correctors j
depend also on x, the new homogenized matrix Ahom
x

124 …..HOMOGENIZATION … heterogeneous structures …media ….
is no longer constant, but it depends on x. A more interesting eect arises in the second
case. As we have seen in Theorem 4.9, if the matrix Adepends only on the variable y,
the functions are independent of xand there is no contribution of the term containing
gin the decoupled form of the limit problem. So, the limit equation is the same as that
corresponding to the case with no jump on the
ux in the microscopic problem. Now, the
dependence of Aonxprevails this phenomenon to occur, and, hence, the function gbrings
an explicit contribution in the homogenized problem, which becomes
divx(Ahom
xru) =f+divx0
@1
jYjZ
Y1A(x;y)r1(x;y)dy+1
jYjZ
Y2A(x;y)r2(x;y)dy1
Ain
:
A similar eect was observed in the homogenization of the Neumann problem in perforated
domains (see [64]).
Problems involving jumps in the solution can be encountered in various other situations.
For instance, our goal in [110] was to rigorously obtain, via the periodic unfolding method, a
macroscopic model for a periodic elastic composite formed by two interwoven and connected
components with imperfect contact at the interface. In the last decades, the study of the
macroscopic properties of composite materials which exhibit imperfect contact between
their constituents has been a subject of major interest for mathematicians, physicists,
engineers, etc. In particular, the problem of modeling the contact between two elastic
media which represent the components of a periodic composite material is of considerable
interest for people working in the eld of material and structural engineering. For the case
of perfect contact between the two elastic media, the continuity of the displacements and
the tractions across their common boundary is assumed. This idealized contact condition
can be relaxed by allowing a discontinuity in the displacement elds across the imperfect
interface between the two elastic media, the jump in displacements being proportional to
the traction vector. In such a model, called a spring type interface model in the literature,
the imperfect interface conditions are equivalent to the eect produced by a very thin and
soft (i.e. very compliant) elastic interphase between the two media. Another interesting
imperfect interface condition arises in the case of a thin and sti interphase, characterized
by a jump of the traction vector across the interface between the two media (see [ ?] and
[138]). In both cases, the imperfect contact can be generated by various causes (the presence
of a thin interphase, chemical processes, the presence of impurities at the boundaries, the
interface damage, etc.). Let us notice that by imposing such imperfect interface conditions,
we are allowed to deal with only two-phases media, instead of considering a threephase
model, consisting of two constituents and an interphase formed by a third material, with
perfect interface conditions between them (see [190]). Imperfect contact problems are of
huge importance for studying composite materials, which might contain coated particles
or bers and cannot be modeled with the aid of continuous displacements and tractions
across the boundaries. For more details concerning the corresponding mechanical models,
the interested reader is referred to [ ?], [138], [158], [166], and [190].

In [110], we assumed that on the interface between the two media there is a jump
in the displacement vector. The order of magnitude of this jump with respect to the
small parameter "denes the macroscopic elastostatic equations and our analysis reveals
three dierent important cases. More precisely, we obtain, at the macroscale, one or two
equations, with dierent stiness tensors: (i) if the intensity of the jump is of order "1,
we obtain only one equation at the macroscale, with the stiness tensor depending on the
jump coecient; (ii) if the intensity of the jump is of order ", we get a system of two
coupled equations with classical stiness tensors; (iii) if the intensity of the jump is of
order one, we obtain at the macroscale only one equation, with no in
uence of the jump
in the macroscopic tensor. The convergence of the homogenization process is proven in
all the cases. Our setting is relevant for dealing with contact problems for multiphase
composites with an interfacial resistance that may arise from the presence of impurities at
the boundaries between the phases or from chemical reactions between the constituents.
An example of such a material is represented by a concrete structure. Also, our techniques
can be applied for dealing with other geomaterials, such as mortar, soils or rocks. Similar
problems have been considered, using dierent techniques, formal or not, in [107], [ ?],
[158], and [169]. Recently, using the periodic unfolding method, some elasticity problems
for media with open and closed cracks were studied in [54]. For other related elasticity
problems, see [125] and [205]. The dynamic case can be addressed in a similar manner.
Homogenization results for a dynamic coupled thermoelasticity problem in a periodic
composite material made up of two connected constituents with imperfect contact at their
interface were obtained in [111]. The homogenized problem, derived via the periodic un-
folding method, comprises new coupling terms involving the macroscopic displacement and
temperature elds, generated by the imperfect bonding at the interface between the two
phases of the composite. Related problems have been studied, with various methods, over
the last years. For a nice presentation of the classical theory of thermoelasticity, the reader
is referred to [144]. Also, for some interesting thermoelasticity models, we refer to [173],
[106], [107] and [120].
In [112], a similar model was considered, but in a dierent geometry and with dierent
scalings of the temperature-displacement tensors of the two constituents, leading to dier-
ent homogenized results. More precisely, the domain
was considered to be the union of
a connected part
"
1and a disconnected one
"
2and the temperature-displacement tensor
was supposed to be of order of unity in the connected part of the medium and, respectively,
of order"in the disconnected one. As a consequence, the macroscopic elasticity tensor, the
temperature-displacement tensor and the thermic-conductivity tensor corresponding to the
disconnected part canceled at the limit. In our case, we keep these tensors in the macro-
scopic system and, in addition, we get a dierent specic heat coecient in the equation
for the macroscopic temperature eld coming from the disconnected part. Moreover, let us
mention the presence of new coupling terms in the macroscopic system and the dierent
functional setting.
125

126 …..HOMOGENIZATION … heterogeneous structures …media ….

Part II
Career Evolution and
Development Plans
127

Chapter 6
..???.
Career Development Plan ???
Research perspectives
CAREER EVOLUTION AND DEVELOPMENT PLANS
– Further research directions – Further plans On the scientic and professional career .
. . . . . . . . . . . . . . . . . . . . . On the academical career . . . . .
6.1 Academic Background
I graduated in 1988 from the Faculty of Mathematics of the University of Bucharest, with
a specialization in Fluid Mechanics. The advisor of my thesis, entitled …., was Professor
Horia I. Ene. I have defended my PhD thesis in 1996, at the Institute of Mathematics
"Simion Stoilow" of the Romanian Academy. My advisor was Professor Horia I. Ene and
the jury consisted of …(was formed by …)…Professor …. The thesis …"Applications of
stochastic processes in
uid mechanics" ….
I had the chance to study in one of the best Romanian universities and to … my PhD
in a leading research institution in Romania. I had … with some of the best Romanian
mathematicians of our times ???… I am indebted to all of them for ………
After a few years, I had the chance to benet
After defending my PhD thesis, my main research interests were … related to the
following areas:
……
Here is a short summary of the main results that I have obtained after the completion
of my PhD thesis. For a more detailed description of some of these results, see …..
Research Grants
Fellowships
Research visits. Collaborations ….
Autonomy and Visibility of the Scientic Activity. I have given invited talks
at many prestigious international conferences such as: …….
129

130
I beneted from a number of research stages and I have given several talks at universities
and research institutions from abroad, such as University of ….
I am a member of the American Mathematical Society, … Also, I am a reviewer for
Mathematical Reviews and for more than 30 scientic journals (such as ….).
6.2 Further Research Directions
….litere se tipar? Cuvinte cu litere mari in titlu?
Research perspectives. I shall brie
y describe here the perspectives I see for my
research. A few of them are already ongoing works.
I plan to perform a rigorous multiscale analysis of some relevant nonlinear phenomena
in heterogeneous media, with applications in biology and engineering. More precisely, I aim
at obtaining new mathematical models for electrically coupled excitable tissues and for skin
electropermeabilization, at developing new multiscale techniques for studying carcinogene-
sis in living cells and at performing a rigorous homogenization study for periodic structured
materials with imperfect interfaces. Also, I think at elaborating new mathematical models
for electromagnetic periodic composites .and at analyzing nonlinear transmission problems
in composites with various other geometries.
Ongoing projects …… The two main projects I am actually involved in are the …. The
goal of these two projects is to develop …..
I.1. Mathematical models for electrochemically coupled excitable biologi-
cal tissues. We plan to rigorously justify and generalize some existing homogenized
models for the description of excitable biological tissues electrochemically coupled through
gap junctions. We shall rigorously prove, via the periodic unfolding method, the formal
results obtained in […20] for doughball gap junction model. Homogenization results were
obtained in the literature mainly for the syncytial model. In the doughball gap junction
model, gap junctions are considered to be thin conductors between cells, coupling them
electrically and chemically. We think that this model is well suited to tackle more general
gating laws than the syncytial model. Our study is motivated by the need to fully un-
derstand wave propagation and failure experimentally observed in the pancreatic islets of
Langerhans. Recent theoretical and experimental facts suggested that calcium is capable of
gating control over gap junction permeability in islets. We shall treat the case of nonlinear
calcium-dependent conductive
uxes across junctions. There are very few results for junc-
tional nonlinearities in islets and many aspects of such models need further investigations.
A realistic comparison of the syncytial and doughball models will be made, as well.
I.2. Homogenization results for skin electropermeabilization. In an ongoing
project, which is a collaboration with Professor Daniele Andreucci and Professor Micol
Amar from Sapienza University of Rome, Italy, we aim at studying, via homogenization
techniques, some suitable mathematical models for the evolution of thermal and electrical
elds in biological tissues. As a rst goal, we plan to analyze alternating interface condi-

131
tions on cell membranes in biological tissues. We aim at a deeper understanding of the
mathematical theory of parabolic equations with boundary conditions strongly oscillating
in time. The specic problem we have in mind couples such a behavior with strong oscilla-
tions in time of the leading coecients (the diusivity) in the diusion partial dierential
equation, in order to model anity. The model should shed light on the role played by
diusion in the selective transport mechanism through the gating pores.
Transdermal drug delivery represents an alternative to standard drug delivery methods
of injection or oral administration. The outermost layer of the epidermis acts as a barrier,
limiting the penetration of drugs through the skin. To overcome this barrier, innovative
technologies were developed. In particular, electropermeabilization, i.e. the application
of high voltage pulses to the skin, increases its permeability and enables the delivery of
various substances through it (see [6]). We need to control the electric pulse parameters
in order to maximize the amount of electropermeabilized tissue in the targeted area and
to minimize the damage produced to the surrounding tissue. Apart from the amount of
electropermeabilized tissue, it is important to take into account the thermal eects pro-
duced in the skin by the electrical pulses in order to design useful electropermeabilization
protocols. The problem is complex, involving a very complicated geometry and the non-
linear coupling of a diusion equation for the drug molecules, of a heat equation, and of
an equation for the electric potential. We shall make simplifying assumptions in order to
capture the essential features of the model, while making it tractable. Modeling the skin
as a a composite medium, our goal is to analyze the eective behavior, as the period of the
microstructure tends to zero, of the solutions of this coupled system of partial dierential
equations. We shall analytically investigate the eect of various parameters on the eective
temperature eld in the tissue exposed to permeabilizing electric pulses. The results can
be used for designing skin electropermeabilization protocols for cancer treatment planning.
I.3. Mathematical models for carcinogenesis in living cells. We shall be con-
cerned with the carcinogenic eects produced in the human cells by Benzo-[a]-pyrene
molecules (BP), which are reactive toxic molecules found in coal tar, cigarette smoke,
charbroiled food, etc. We shall generalize the results obtained in [21] and [36]. The
microscopic mathematical model, including reaction-diusion processes and binding and
cleaning mechanisms, will be homogenized in order to reduce its complexity and to make
it numerically treatable and not so computationally expensive. We shall consider that BP
molecules enter in the cytosol inside of a human cell. There, they diuse freely, but they
cannot enter in the nucleus. Also, they bind to the surface of the endoplasmic reticulum
(ER), where chemical reactions take place, BP molecules being chemically activated to
Benzo-[a]-pyrene-7,8-diol-9,10-epoxide molecules (DE). These molecules can unbind from
the surface of the ER and they can diuse again in the cytosol, entering in the nucleus.
Natural cleaning mechanisms occurring in the cytosol are taken into account, too. For
describing the binding-unbinding process at the surface of the ER, we shall consider var-
ious nonlinear functions, with various scalings, leading to dierent homogenized models.

132
We shall deal with the case of general nonlinear (even discontinuous) isotherms, similar
to those used in [9], [24], and [31], and of multiple metabolisms BP !DE. We shal also
generalize a carcinogenesis model, introduced in [21], involving free receptors on the surface
of the ER.
II.1. Homogenization of a two-conductivity problem with
ux jump. This
is an ongoing joint work with Renata Bunoiu from the University of Lorraine – Metz,
France. We shall continue our study on the homogenization of a thermal diusion problem
in a highly heterogeneous medium formed by two constituents, separated by an imperfect
interface (see …). In this case, the order of magnitude of the thermal conductivity of the
material occupying the domain
"
2is of order "2, while the conductivity of the material
occupying the domain
"
1is supposed to be of order one. Our problem presents various
sources of singularities: the geometric one related to the interspersed periodic distribution
of the components, the material one related to the conductivities and the ones generated
by the presence of an imperfect interface between the two materials. The case G"= 0,
which corresponds to a continuous
ux, proportional to the jump of the temperature eld
across the imperfect interface, has attracted, in the last two decades, the interest of a
broad category of researchers (see, e.g., [193], [194], [195], [103], [133], [ ?], and [ ?]). After
passing to the limit with respect to the small parameter ", a regularised model of diusion
is obtained, which in fact is a special case of the double-porosity model, introduced in [199]
in the frame of the heat transfer and in [37] in the context of the
ow in porous media.
We shall consider the case G"6= 0, which corresponds to a discontinous
ux as well. We
shall study the two representative cases for the jump function G", stated in Section …. ,
which both lead to dierent modied reguralized models of diusion. More precisely, in the
rst case, a new global source term, macroscopically distributed over the entire equivalent
domain, appears in the right-hand side of the homogenized equation, while in the second
case, the novelty brought by the presence of the
ux jump is the emergence of the new non
homogeneous Neumann cell problem and the presence of its solution in the corrector. We
also point out here the eect of the jump of the solution, which is recovered in the corrector
and in the weak limit, via the solution of a local Robin problem. This phenomenon was
already noticed in the case in which G"= 0.
Our primary objective is to obtain new homogenization results for reaction-diusion
problems in periodic composite media which exhibit at the interfaces between their compo-
nents jumps of the solution and of the
ux. Such problems are relevant in the the context
of thermal diusion in composites, in the theory of semiconductors, in linear elasticity or
in reaction-diusion problems in biological tissues. Almost all the existing studies in the
literature deal with the case in which at the interface between the components the
ux of
the solution is continuous, while the solution has a jump (see [34]). There are very few
rigorous mathematical results for the case in which the
ux is discontinuous, as well (see
[7], [18], [19] and [23]). Obtaining rigorous results based on the homogenization theory is
still a dicult task. We plan to generalize the results contained in [7] and to apply them

133
to the study of calcium dynamics in biological tissues modeled as media with imperfect
interfaces. The main novelty brought by our research will consist in allowing the presence,
apart from the discontinuity in the concentration eld, of a jump in the
ux across the
surface of the endoplasmic reticulum modeled as an imperfect interface. We shall study
several representative cases for the functions modeling the jumps and we shall consider
various scalings for the concentration and the
ux jumps, leading to dierent macroscopic
problems. We shall extend our analysis to nonlinear problems, this being a largely open
case in the literature.
II.2. …………. Using the periodic unfolding method or Gamma-convergence method,
as an alternative plan, we shall address the problem of nding the eective parameters
for electromagnetic periodic composite materials in the quasi-static case. The developed
strategy will allow us to deal with quite general microscopic geometries and can be applied
to other heterogeneous materials in which the scale of the period is much smaller than the
wavelength of the electromagnetic eld. We shall generalize some of the results obtained
in [29]. Also, using some improvements of the oscillating test function method and the
periodic unfolding method, we shall analyze the macroscopic properties of nanocomposite
materials, with complicated microstructures, which make impossible the application of
conventional methods. Through this multiscale approach, we can understand how the
small-scale material structure controls the macroscopic behavior of such materials. Our
model is based on imposing suitable nonlinear interface conditions, in order to capture the
microstructural features of such materials and to contribute to a better understanding of
their eective properties.
Elements of originality and of diculty. We aim at developing new ideas and
methods in the eld of multiscale analysis, the focus being on obtaining a better under-
standing of some aspects of the modeling of heterogeneous media, with applications in
biology and in material science. There are very few results for junctional nonlinearities
in islets and many aspects of such models need further investigations. Rigorous proofs
of convergence results for homogenization models are still missing in the literature. Also,
there are not many rigorous studies for skin electropermeabilization and mathematical re-
sults of existence, uniqueness, regularity and asymptotic behavior of the solutions of such
models are still not present in the literature. Another novelty brought by our project
is the development of new mathematical models for calcium dynamics in biological tis-
sues, involving interfacial jumps not only in the concentration, as previously considered
in the literature, but also in its
ux. By assuming
ux jumps or nonlinear jumps at the
microscale we are led to new macroscopic features of the biological systems under con-
sideration compared to the ones previously known in the literature. For instance, we can
get at the macroscale a modied bidomain model. Also, understanding complex interface
phenomena in composites could lead to the design of innovative materials. Besides, in the
homogenization of our nonlinear systems, non-local eects will appear. To summarize, the
original elements brought by our research are the general nonlinear interface conditions

134
to be considered, the nonlinear sources introduced in the models, the involved coupled
phenomena, the techniques we use and the non-standard applications.
The diculties behind our models come, e.g., from the involved coupled phenomena,
from the complex geometry, from the nonlinearity of the models or from the non-standard
applications. A major challenge resides in developing suitable multiscale models that en-
able us to understand the coupling mechanisms between scales and to establish realistic
models for media exhibiting multiple scales, with applications in biology and engineering.
On the other hand, addressing problems with interfacial
ux jump in composite media cre-
ates many mathematical diculties. Also, when analyzing skin electropermeabilization,
diculties come, e.g., from the heterogeneity of the tissues and from the changes in con-
ductivity due to electropermeabilization. We plan to address some of the limitations of the
proposed models in the literature. To this end, we shall use some multiscale methods and
various variational techniques. Each method has its own advantages and limitations, but
we hope that combining them we shall obtain results contributing to a better understanding
of the macroscopic behavior of the complex heterogeneous systems under consideration.
….
In another ongoing project, with …. and …, we plan to ….
small holes Kaizu, Goncharenko, Maria Neuss, Maria Eugenia, Shaposhnikova …..z
The study of … could represent a ??? fruitful thesis subject.
…The case in which the strain-stress law is viscoelastic and the case in which we consider
thermal eects in the history of a composite material will be treated in a forthcoming paper.
….
The prediction of the macroscopic behavior of thermoelastic microstructured materials
is a subject of topical interest for a broad category of researchers. The growing interest in
such a problem is justied by the increased need of designing advanced composite materials,
with useful mechanical and thermodynamical properties. In particular, the problem of
multiscale modeling of thermoelastic composites with imperfect interfaces has attracted a
lot of interest in the last years, due to the great importance of such heterogeneous materials
in many engineering applications. For instance, there are important applications of the
interphase eects on the thermoelastic response of polymer nanocomposite materials.
…..
All these subjects could lead to relevant PhD thesis subjects, …??? connected to the
main stream of applied mathematical research. …….
Supervision of PhD candidates
List of papers of this habilitation thesis
Main contributions of this habilitation
State of the art
Perspectives
Ongoing collaborations
Intended works

135
………..
(a) Rezumatul (minimum 4.000 de caractere, maximum 6.000 de caractere) prezint
sinteza tezei de abilitare. (b) Realizri tiinice i profesionale i planuri de evoluie i dezvoltare
a carierei. Aceasta parte are trei seciuni. (i) In prima seciune, de minimum 150.000
de caractere si de maximum 300.000 de caractere (inclusiv formule), se prezint realizrile
tiinice, profesionale i academice, pe direcii tematice disciplinare sau interdisciplinare. Cele
mai importante lucrri (maxim 10) vor  incluse in dosarul de abilitare. Realizrile personale
sunt prezentate n contextul stadiului actual al cercetrii tiinice din domeniul tematic al
specialitii, pe plan internaional. (ii) n a doua seciune, de maximum 25.000 de caractere, se
prezint planuri de evoluie i dezvoltare a propriei cariere profesionale, tiinice i academice,
respectiv direcii de cercetare/predare/aplicaii practice.
(iii) A treia seciune prezint referine bibliograce asociate coninutului primelor dou
seciuni.
Rezumatul tezei de abilitare se redacteaz n dou versiuni, n limbile romn i englez.

136

List of publications
This habilitation thesis is based on the following publications:
137

138

Bibliography
[1] G. Allaire, ….1992
[2] G. Allaire, Z. Habibi, Second order corrector in the homogenization of a conductive-
radiative heat transfer problem, Discrete Contin. Dyn. Syst. Ser. B 18(1) (2013) 1-36.
[3] G. Allaire, A. Mikeli c and A. Piatnitski, Homogenization of the linearized ionic trans-
port equations in rigid periodic porous media, Journal of Mathematical Physics, 51
(12) (2010), pp. 123103.1-123103.18.
[4] G. Allaire, A. Piatnitski, Homogenization of nonlinear reaction-diusion equation with
a large reaction term, Ann. Univ. Ferrara Sez. VII Sci. Mat., 56 (1), 141-161, 2010.
[5] G. Allaire, A. Mikeli c, A. Piatnitski, Homogenization approach to the dispersion the-
ory for reactive transport through porous media, SIAM J. Math. Anal. 42 (1) (2010)
125-144.
[6] G. Allaire, H. Hutridurga IMA J Appl Math (2012) 77 (6): 788-815 "Homogenization
of reactive
ows in porous media and competition between bulk and surface diusion"
……..
[7] G. Allaire, H. Hutridurga, Upscaling nonlinear adsorption in periodic porous media-
homogenization approach, arXiv:1412.8301v1 (2014), to appear in Appl. Anal., 2015.
[8]S.N. Antontsev, A.V. Kazhikhov and V.N. Monakhov ,Boundary Value Prob-
lems in Mechanics of Nonhomogeneous Fluids , North-Holland, Amsterdam, 1990.
[9]R. Aris ,The Mathematical Theory of Diusion and Reaction in Permeable Catalysis ,
Clarendon Press, Oxford, 1975.
[10] S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, Boundary Value Problems in
Mechanics of Nonhomogeneous Fluids , North-Holland, Amsterdam, 1990.
[11] T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of a
single phase
ow via homogenization theory , SIAM J. Math. Anal. 21(1990), 823{836.
[12] T. Arbogast, …………..
139

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