This work could not have been accomplished without the support of many persons. First of all, I would like to express my deep gratitude to Professor… [610112]

UNIVERSITY OF BUCHAREST
HABILITATION THESIS
HOMOGENIZATION RESULTS FOR
HETEROGENEOUS MEDIA
CLAUDIA TIMOFTE
Specialization: Mathematics
Bucharest, 2016

.

Acknowledgments
This work could not have been accomplished without the support of many persons. First
of all, 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 his continuous support and
professional guidance. His mentorship was very important for the evolution of my academic
and scientic career.
After completing my Ph.D., I beneted, between 2000 and 2003, from four post-doctoral
fellowships at Complutense University of Madrid (Spain), University of Pisa (Italy), and
Center of Mathematical Modelling, University of Chile, Santiago de Chile (Chile). I had the
chance to work, in stimulating environments, under the supervision of top specialists in ap-
plied mathematics: Professor Enrique Zuazua, Professor Giuseppe Buttazzo, and Professor
Carlos Conca. I was really impressed by their remarkable ability to connect different elds of
research and I want to express my deep gratitude to all of them, for their support and generos-
ity, and for the willingness to share their extraordinary knowledge with me. Also, the research
visits performed at Complutense University of Madrid (Spain), Friedrich-Alexander Univer-
sity Erlangen-N urnberg (Germany), Eindhoven University of Technology (Holland), Taras
Shevchenko National University of Kiev (Ukraine), University of Pavia (Italy), Universit e de
Lorraine, Metz (France), or University of Cantabria, Santander (Spain) gave me the chance of
fruitful interactions with well-known specialists in the eld of homogenization theory: Profes-
sor Jes us Ildefonso D az, Professor Iuliu Sorin Pop, Professor Maria Radu-Neuss, Dr. Renata
Bunoiu, Professor Gennady Sandrakov, Professor Giuseppe Savar e, David G omez-Castro,
Professor Mar a Eugenia P erez, Dr. Delna G omez. I am grateful to all of them, for their
warm hospitality, kindness, and for sharing with me their love for mathematics.
The work presented here represents a collective effort, the fruit of many encounters I
had over the years with many persons and I am fully conscious about their importance at
many steps in my career. It is impossible to me to thank now all the people that I met in
this scientic journey. Therefore, I shall mention here only the co-authors of my papers on
which this thesis is based on: Professor Carlos Conca, Professor Fran cois Murat, Professor
Jes us Ildefonso D az, Professor Amable Li~ n an, Professor Horia I. Ene, Dr. Anca C ap at  ^ n a,
Dr. Iulian T ent ea, Dr. Renata Bunoiu. Working together was very important for my
development as a mathematician.
During the last years, I have beneted a lot from inspiring discussions with my colleagues
from the Institute of Mathematics "Simion Stoilow" of the Romanian Academy. I am very
3

4 Acknowledgments
indebted to them for their valuable and constructive suggestions. I would also like to thank to
my colleagues from the Faculty of Physics of the University of Bucharest, who accompanied
me in this transdisciplinary journey, for their support and for the emulating atmosphere they
have always created in our faculty.
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.
Last, but not least, I am grateful to my family and to my friends for their unwavering
support and understanding.
Bucharest, October 2016 Claudia Timofte

Preface
The aim of this manuscript, prepared to defend my Habilitation thesis , is to give an overview
of my research activity in the eld of homogenization theory, which represents the core of my
scientic work done during the last fteen years.
The thesis, written in English, starts by a short summary in Romanian and a brief
overview of the eld of homogenization and then summarizes, with less details in some proofs
and with some additional hindsights, some of my research works in this eld, performed
after completing my Ph.D. studies. The thesis relies on some of my original contributions
to the applications of the homogenization theory, contained in twenty-ve articles already
published or submitted for publication in international journals. Many of the results in
the publications I selected to support my application are closely related to or motivated by
practical applications to real-life problems.
The results included in this thesis have been obtained alone or in collaboration with
several academic and research institutions from Romania or from abroad. I am grateful to all
my co-authors for their important contribution, for useful advices and friendly discussions.
5

Contents
Preface 5
Rezumat 9
Abstract 11
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 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 . . . . . . . . . . . . . 37
2.2.2 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.3 The case of a non-smooth boundary condition . . . . . . . . . . . . . . 42
2.2.4 Laplace-Beltrami model with oscillating coefficients . . . . . . . . . . . 43
3 Homogenization results for unilateral problems 49
3.1 Homogenization results for Signorini's type problems . . . . . . . . . . . . . . 50
3.1.1 The microscopic problem . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.2 The macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Homogenization results for elliptic problems in perforated domains with mixed-
type boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 Setting of the microscopic problem . . . . . . . . . . . . . . . . . . . . 55
3.2.2 The limit problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7

8 Contents
4 Mathematical models in biology 65
4.1 Homogenization results for ionic transport phenomena in periodic charged media 65
4.1.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.2 The homogenized problem . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Multiscale analysis of a carcinogenesis model . . . . . . . . . . . . . . . . . . 73
4.2.1 The microscopic problem . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.2 The macroscopic model . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.3 A nonlinear carcinogenesis model involving free receptors . . . . . . . 84
4.3 Homogenization results for the calcium dynamics in living cells . . . . . . . . 86
4.3.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.2 The main convergence results . . . . . . . . . . . . . . . . . . . . . . . 92
5 Multiscale modeling of composite media with imperfect interfaces 95
5.1 Multiscale analysis in thermal diffusion problems in composite structures . . 96
5.1.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Diffusion problems with dynamical boundary conditions . . . . . . . . . . . . 100
5.3 Homogenization of a thermal problem with
ux jump . . . . . . . . . . . . . 102
5.3.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3.2 The macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4 Other homogenization problems in composite media with imperfect interfaces 112
II Career Evolution and Development Plans 115
6 Scientic and academic background and research perspectives 117
6.1 Scientic and academic background . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Further research directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Future plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Rezumat
Lucrarea de fat  a, preg atit a pentru obt inerea atestatului de abilitare, cuprinde o select ie a
rezultatelor  stiint ice pe care le-am obt inut ^ n domeniul teoriei omogeniz arii dup a dob^ andirea
titlului de doctor ^ n Matematic a. Principala motivat ie din spatele acestui demers o constituie
dorint a de a descrie succint stadiul actual al cunoa sterii ^ n domeniul teoriei omogeniz arii, de
a oferi o imagine de ansamblu asupra contribut iilor mele ^ n acest vast domeniu  si de a discuta
c^ ateva probleme deschise  si c^ ateva posibile perspective de dezvoltare a viitoarei mele cariere
 stiint ice  si academice.
Teza se bazeaz a pe o parte dintre contribut iile mele originale la aplicarea teoriei omoge-
niz arii, contribut ii cont inute^ n dou azeci  si cinci de articole publicate sau trimse spre publicare
^ n reviste internat ionale cu o larg a audient  a, incluz^ and nu doar matematicieni, ci  si zicieni,
ingineri  si cercet atori din diferite domenii aplicative. Multe dintre rezultatele cont inute ^ n
publicat iile pe care le-am selectat pentru a sprijini aceast a aplicat ie sunt str^ ans legate sau
motivate de probleme de interes practic. Voi ^ ncerca s a ofer o imagine de ansamblu, de sine-
st at atoare, asupra contribut iilor mele  si, acolo unde va  necesar, s a dau mai multe detalii
care nu sunt prezente ^ n lucr arile corespunz atoare publicate, f ac^ and astfel ca principalele mele
rezultate s a e accesibile unui public mai larg, cu solide cuno stint e generale de matematic a,
dar nu neap arat expert i ^ n domeniul specic al teoriei omogeniz arii. Teza prezint a c^ ateva
dintre cele mai relevante rezultate pe le-am obt inut pe parcursul ultimilor cincisprezece ani
de cercetare efectuat a, singur a sau ^ n colaborare, ^ n patru arii majore: analiza de multiscar a
a proceselor de react ie-difuzie ^ n medii poroase, omogenizarea problemelor unilaterale, mo-
delarea, cu ajutorul sc arilor multiple, a mediilor compozite cu interfet e imperfecte  si obt inerea
de modele matematice ^ n biologie  si ^ n inginerie.
^In afar a de dou a scurte rezumate, unul ^ n limba rom^ an a  si altul ^ n limba englez a, teza
cont ine dou a p art i  si o bibliograe cuprinz atoare. Prima parte, structurat a ^ n cinci capitole,
este dedicat a prezent arii principalelor mele realiz ari  stiint ice obt inute dup a nalizarea studi-
ilor doctorale. Dup a un capitol introductiv ^ n care este prezentat stadiul actual al cercet arii
^ n domeniul teoriei omogeniz arii  si care ofer a cadrul general  si o motivat ie pentru activitatea
mea de cercetare post-doctoral a ^ n acest domeniu, al doilea capitol cuprinde c^ ateva dintre
contribut iile mele referitoare la omogenizarea mi sc arilor reactive ^ n medii poroase. Mai pre-
cis, sunt prezentate rezultate originale de omogenizare pentru curgerile reactive neliniare
stat ionare ^ n medii poroase  si pentru studiul fenomenelor de adsorbt ie neliniar a ^ n medii
poroase. Capitolul se bazeaz a pe lucr arile [72], [73], [209]  si [211]. Al treilea capitol, bazat
9

10 Rezumat
pe articolele [51], [54], [75], [77]  si [208], este dedicat omogeniz arii unor probleme unilaterale
^ n domenii perforate. Sunt cuprinse rezultate originale privitoare la omogenizarea unor pro-
bleme de tip Signorini  si a unor probleme eliptice cu condit ii mixte pe frontiere ^ n domenii
perforate. Al patrulea capitol cont ine o serie de rezultate recente despre omogenizarea unor
modelele din biologie. Sunt discutate noi modele matematice pentru transportul ionic^ n medii
periodice ^ nc arcate electric, pentru studiul carcinogenezei ^ n celulele vii sau pentru analiza
dinamicii calciului ^ n celulele biologice. Rezultatele prezentate ^ n acest capitol sunt cuprinse
^ n lucr arile [210], [211], [212], [217], [218], [219]  si [221]. Ultimul capitol al acestei prime p art i
sintetizeaz a cele mai importante rezultate pe care le-am obt inut ^ n domeniul transferului de
c aldur a ^ n materiale compozite cu interfet e imperfecte  si se bazeaz a pe lucr arile [47], [48],
[49], [213], [214]  si [215]. Denit iile not iunilor de baz a din teoria omogeniz arii  si rezultatele
generale din analiza funct ional a care vor  folosite pe parcursul acestei lucr ari pot  g asite
^ n [30], [37], [45], [56], [62], [137], [147], [155], [160], [161]  si [205].
A doua parte a acestei teze prezint a c^ ateva planuri de dezvoltare  si de evolut ie ^ n carier a.
Dup a o scurt a trecere ^ n revist a a parcursului meu  stiint ic  si academic de p^ an a acum, sunt
prezentate direct iile viitoare de cercetare  si c^ ateva planuri de dezvoltare pe termen scurt,
mediu  si lung a carierei mele  stiint ice  si academice.
Lucrarea se ^ ncheie cu o bibliograe cuprinz atoare, menit a s a ilustreze stadiul actual al
cunoa sterii ^ n acest domeniu vast al teoriei omogeniz arii  si al aplicat iilor sale.
Principalele mele contribut ii originale cont inute ^ n aceast a tez a pot  sintetizate astfel:
efectuarea unui studiu riguros al proceselor neliniare de react ie-difuzie^ n medii poroase,
care includ difuzie, react ii chimice  si diferite tipuri de rate de adsorbt ie;
obt inerea unor rezultate de omogenizare pentru probleme unilaterale^ n medii perforate;
elaborarea de noi modele matematice pentru fenomenele de transport ionic ^ n medii
periodice ^ nc arcate electric;
derivarea de noi modele matematice pentru studiul proceselor neliniare de carcinogene-
z a ^ n celulele umane  si al dinamicii calciului ^ n celulele vii;
efectuarea unei analize asimptotice riguroase pentru procese de difuzie termic a ^ n
structuri compozite;
ranarea studiului problemelor de difuzie cu condit ii dinamice pe frontier a;
obt inerea de noi modele matematice pentru probleme de difuzie cu salt ^ n
ux.
Rezultatele incluse ^ n aceast a tez a au fost obt inute singur a sau ^ n str^ ans a colaborare cu
mai multe institut ii academice  si de cercetare din Rom^ ania sau din str ain atate. Sunt profund
recunosc atoare tuturor co-autorilor mei, Profesor dr. C. Conca, Profesor dr. F. Murat,
Profesor dr. J. I. D az, Profesor dr. A. Li~ n an, Profesor dr. H. I. Ene, Dr. A. C ap at  ^ n a, Dr.
I. T ent ea, Dr. R. Bunoiu, pentru o frumoas a colaborare, pentru contribut ia lor important a,
pentru sfaturile utile  si pentru discut iile interesante pe care le-am avut de-a lungul anilor.
Toate aceste rezultate ar putea deschide perspective noi  si promit  atoare pentru dezvolt ari
ulterioare  si pentru viitoare colabor ari cu cercet atori reputat i din t ar a  si din str ain atate.

Abstract
This manuscript, prepared to defend my Habilitation thesis, summarizes a selection of my
research results obtained in the eld of homogenization theory after defending my Ph.D.
thesis. The main motivation behind this endeavour is to brie
y describe the state of the art
in the eld of homogenization theory, to give an overview of my contributions in this broad
research area and to discuss some open problems and several perspectives I see for my future
scientic and academic career.
The thesis relies on some of my original contributions to the applications of the homoge-
nization theory, contained in twenty-ve articles already published or submitted for publica-
tion in international journals with a broad audience, including not only mathematicians, but
also physicists, engineers, and scientists from various applied elds. Many of the results in
the publications I selected to support my application are closely related to or motivated by
practical applications to real-life problems. 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, making my main results accessible to an audience with strong, general mathematical
background, but not necessarily experts in the specic eld of homogenization theory.
The thesis is based on some of the most relevant results I obtained during the last fteen
years of research conducted, alone or in collaboration, in four major areas: multiscale analysis
of reaction-diffusion processes in porous media, upscaling in unilateral problems, multiscale
modeling of composite media with imperfect interfaces, and mathematical models in biology
and in engineering. Thus, the homogenization theory and its applications represent the core
of my scientic work done during these last fteen years.
Apart from two short abstracts, one in Romanian and another one in English, the thesis
comprises two parts and a comprehensive bibliography. The rst part, structured into ve
chapters, is devoted to the presentation of my main scientic achievements since the comple-
tion of my Ph.D. thesis. After a brief introductory chapter presenting the state of the art in
the eld of homogenization theory and offering the general framework and a motivation for
my post-doctoral research work in this area, the second chapter is divided in two distinct sec-
tions, summarizing my main contributions to the homogenization of reactive
ows in porous
media. More precisely, some original results for upscaling in stationary nonlinear reactive

ows in porous media and, also, results on nonlinear adsorption phenomena in porous media
are presented. The chapter relies on the papers [72], [73], [209], and [211]. The third chapter
is devoted to the homogenization of some relevant unilateral problems in perforated domains.
11

12 Abstract
More precisely, homogenization results for Signorini's type problems and for elliptic problems
with mixed boundary conditions in perforated media are presented. The chapter is based on
the articles [51], [54], [75], [77], and [208]. The fourth chapter contains some recent results
about homogenized models in biology. New mathematical models for ionic transport phe-
nomena in periodic charged media, for carcinogenesis in living cells or for analyzing calcium
dynamics in biological cells are discussed. The results presented in this chapter are contained
in [210], [211], [212], [217], [218], [219], and [221]. The last chapter of this rst part sum-
marizes the most important results I achieved, alone or in collaboration, in the eld of heat
transfer in composite materials with imperfect interfaces and is based on the papers [47], [48],
[49], [213], [214], and [215]. For the denitions of the basic notions in homogenization theory
and for well-known general results of functional analysis we shall use throughout this thesis,
we refer to [30], [37], [45], [56], [62], [137], [147], [155], [160], [161], and [205].
The second part of this thesis presents some career evolution and development plans.
After a brief review of my scientic and academic background, further research directions
and some future plans on my scientic and academic career are presented. I shall discuss
some short, medium and long term development plans. A brief description of some open
questions I would like to study in the future will be made, as well.
The thesis ends by a comprehensive bibliography, illustrating the state of the art in this
vast eld of homogenization theory and its applications.
My major original contributions contained in this habilitation thesis can be summarized
as follows:
performing a rigorous study of nonlinear reaction-diffusion processes in porous media,
including diffusion, chemical reactions and different types of adsorption rates;
obtaining new homogenization results for unilateral problems in perforated media;
elaborating new mathematical models for ionic transport phenomena in periodic
charged media;
getting original homogenization results for calcium dynamics in living cells;
deriving new nonlinear mathematical models for carcinogenesis in human cells;
performing a rigorous multiscale analysis of some relevant thermal diffusion processes
in composite structures;
rening the study of diffusion problems with dynamical boundary conditions;
obtaining new mathematical models for diffusion problems with
ux jump.
The results included in this thesis have been obtained alone or in close collaboration with
several academic and research institutions from Romania or from abroad. I am grateful to
all my co-authors, Professor C. Conca, Professor F. Murat, Professor J. I. D az, Professor A.
Li~ n an, Professor H. I. Ene, Dr. A. C ap at  ^ n a, Dr. I. T ent ea, and Dr. R. Bunoiu, for a nice
collaboration, for their important contribution, for useful advices and interesting discussions.
I hope that all these results might open new and promising perspectives for further
developments and future collaborations with well-known scientists from Romania and from
abroad.

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 macroscopic
properties of systems having a very complicated microscopic structure. In mechanics, 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 mul-
tiscale techniques represent an essential tool for understanding the macroscopic properties
of such systems having a very complicated microscopic structure. A periodic distribution is
a realistic hypothesis in many situations with practical applications. Typically, in periodic
heterogeneous structures, the physical parameters, such as the electrical or thermal conduc-
tivity or the elastic coefficients, are discontinuous and, moreover, highly oscillating. For
example, in a composite material, constituted by the ne mixing of two or more components,
the physical parameters are obviously discontinuous and they are highly oscillating between
different values characterizing each distinct component. Therefore, the microscopic structure
becomes extremely complicated. If the period of the structure is very small compared to
the domain 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 considered
structure: one which is comparable with the dimension of the period, called the microscopic
scale, and another one which is of the same order of magnitude as 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. The nonhomogeneous system is replaced
by a ctitious homogeneous one, with global characteristics which represent a good approx-
imation of the initial system. In this way, we are led to a general framework for obtaining
these macroscale properties, eliminating the huge difficulties related to the explicit determi-
nation of a solution of the microscopic problem. 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 coeffi-
cients (the so-called effective orhomogenized coefficients ), while the microscopic equations are
15

16 Introduction
dened on a complicated domain, have rapidly oscillating coefficients, and satisfy nonlinear
boundary conditions. Let us remark that the dependence on the real microstructure is given
through the homogenized coefficients.
The study of the macroscopic properties of composite media was initiated by the physicists
Rayleigh, Maxwell, and Einstein. Around 1970, such problems were formulated in such a
way that they became interesting for mathematicians, as well, and this gave rise to a new
mathematical discipline, the homogenization theory . The rst rigorous developments of this
theory appeared with the seminal works of I. Babuka [28], E. De Giorgi and S. Spagnolo [86],
A. Bensoussan, J. L. Lions and G. Papanicolaou [37], and L. Tartar [205]. De Giorgi's notion
of Gamma-convergence marked also an important step in the development of this theory. F.
Murat and L. Tartar (see [178], [179], [180], and [206]) 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 [181] and was
further developed by G. Allaire in [1]. An extension to multiscale problems was obtained
by A.I. Ene and J. Saint Jean Paulin [104] and by G. Allaire and M. Briane [2]. In 1990,
T. Arbogast, J. Douglas, and U. Hornung [21] dened a dilation operator in order to study
homogenization problems in a periodic medium with double porosity. An alternative approach
was offered by the Bloch-wave homogenization method [76], which is a high frequency method
that can provide dispersion relations for wave propagation in periodic structures. Recently,
D. Cioranescu, A. Damlamian, P. Donato, 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 homogenization problems: the periodic unfolding method
(see, e.g., [56]). 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 [100]). It is
important to emphasize that homogenization theory can be applied to non-periodic media,
as well. To this end, one can use G- orH-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, such as
the convective-diffusive transport in porous media, nonlinear elasticity problems, 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 applications in modern technology. The literature on this subject is vast (see, e.g.,
[61], [74], [63], [73], [75], and the references therein). We also mention here some remarkable
monographs dedicated to the mathematical problems of homogenization: [147], [29], [37],
[160], [164], [184], [198], [62], [69], [102].
Multiscale methods offer multiple possibilities for further developments and for useful
applications in many domains of 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.
My interest in this broad eld of homogenization theory started after defending my Ph.D.

Introduction 17
thesis at "Simion Stoilow" Institute of Mathematics of the Romanian Academy under the
supervision of Professor Horia I. Ene. I focused on the applications of the homogenization
theory to a wide category of problems arising in physics, chemistry, biology or engineering.
To summarize, my main research interests have been related to the following areas: mul-
tiscale analysis of reaction-diffusion processes in porous media, homogenization results for
unilateral problems, multiscale modeling of composite media with imperfect interfaces, and
mathematical models in biology and in engineering. My research activity in the eld of ho-
mogenization is interdisciplinary in its nature and in the last years I tried to publish my
results in more application-oriented high quality journals, with a broad audience, including
not only mathematicians, but also physicists, engineers, and scientists from various applied
elds, such as biology or geology.
The aim of this manuscript, prepared to defend my Habilitation thesis, is to give an
overview of my research work in the eld of homogenization theory. As a matter of fact,
the homogenization theory and its applications represent the core of my scientic work done
during the last fteen years. Many of the results presented herein are closely related to or
motivated by practical applications to real-life problems.
Apart from two short abstracts, one in Romanian and another one in English, the thesis
comprises two parts and a comprehensive bibliography. The rst part, structured into ve
chapters, is devoted to the presentation of my main scientic achievements since the comple-
tion of my Ph.D. thesis. After a brief introductory chapter presenting the state of the art
in the eld of homogenization theory and offering the general framework and a motivation
for my post-doctoral research work in this area, the second chapter is divided in two distinct
sections, summarizing my main contributions related to the homogenization of reactive
ows
in porous media. More precisely, some original results for upscaling in stationary nonlinear
reactive
ows in porous media and, also, results on nonlinear adsorption phenomena in porous
media are presented. The chapter relies on the papers [72], [73], [209], and [211]. The third
chapter is devoted to the homogenization of some relevant unilateral problems in perforated
domains. More precisely, some homogenization results for Signorini's type problems and for
elliptic problems with mixed boundary conditions in perforated media are presented. The
chapter is based on the articles [51], [54], [75], [77], and [208]. The fourth chapter contains
some recent results about homogenized models in biology. New mathematical models for
ionic transport phenomena in periodic charged media, for carcinogenesis in living cells or for
analyzing calcium dynamics in biological cells are discussed. The results presented in this
chapter are contained in the papers [210], [211], [212], [217], [218], [219], and [221]. The
last chapter of this rst part summarizes the most important results I achieved, alone or in
collaboration, in the eld of heat transfer in composite materials with imperfect interfaces
and is mainly based on the articles [47], [48], [49], [213], [214], and [215].
For the denitions of the basic notions in homogenization theory and for well-known
general results of functional analysis we shall use throughout this thesis, we refer to [30], [37],
[45], [56], [62], [137], [147], [155], [160], [161], and [205].
The second part of this thesis presents some career evolution and development plans.

18 Introduction
After a brief review of my scientic and academic background, further research directions
and some future plans on my scientic and academic career are presented.
The manuscript end by a comprehensive bibliography, illustrating the state of the art in
this vast eld of homogenization theory and its applications.
The thesis relies on some of my original contributions to the applications of homogeniza-
tion theory, contained in twenty-ve articles. The results included in this thesis have been
obtained during the last fteen years of research studies conducted alone or in collaboration
with various research institutions from Romania and from abroad. Let me emphasize that
most of this work is already published or submitted for publication in international journals.
Thus, in this thesis, I shall explicitly use some parts from my own articles, mentioning each
time the precise references to the corresponding original work.
I am the unique author in twelve of the papers on which this thesis is based. I am co-
author, with equal contribution, for the rest thirteen papers on which this thesis is based
(in Mathematics, the academic norm is to list equally contributed authors in alphabetical
order). I gratefully acknowledge the equal contribution of all my co-authors: Professor C.
Conca, Professor F. Murat, Professor J.I. D az, Professor A. Li~ n an, Professor H.I. Ene, Dr.
A. Capatina, Dr. I. T ent ea, and Dr. R. Bunoiu.
My main original contributions in the eld of homogenization theory, contained in this
habilitation thesis, can be summarized as follows (a more detailed description of a selection of
my results in the eld of homogenization theory will be presented in the following chapters):
performing a rigorous study of nonlinear reaction-diffusion processes in porous media,
including diffusion, chemical reactions, and different types of adsorption rates;
obtaining new homogenization results for unilateral problems in perforated media;
elaborating new mathematical models for ionic transport phenomena in periodic
charged media;
getting original homogenization results for calcium dynamics in living cells;
deriving new nonlinear mathematical models for carcinogenesis in human cells;
performing a rigorous multiscale analysis of some relevant thermal diffusion processes
in composite structures;
rening the study of diffusion problems with dynamical boundary conditions;
obtaining new mathematical models for thermal problems with
ux jump.
All these results might open new and promising perspectives for further developments and
future collaborations with well-known academic and research institutions from Romania and
from abroad.

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 effective properties of
the medium vary on a coarse scale which is suitable for efficient computation and accurately
captures 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 materials),
mechanical engineering (heat exchangers, porous gas burners), the study of industrial ma-
terials (glass ber materials, brick manufacturing). There is an extensive literature on the
determination of the effective properties of heterogeneous porous media (see, e.g., [32] and
[137], and the references therein).
Transport processes in porous media have been extensively studied in last decades by
engineers, geologists, hydrologists, mathematicians, physicists. In particular, mathematical
modeling of chemical reactive
ows through porous media is a topic of huge practical im-
portance in many engineering, physical, chemical, and biological applications. Obtaining
suitable macroscopic laws for the processes in geometrically complex porous media (such as
soil, concrete, rock, or pellets) involving
ow, diffusion, convection, and chemical reactions
is a difficult task. The homogenization theory proves to be a very efficient tool by provid-
ing suitable techniques allowing us to pass from the microscopic scale to the macroscopic
one and to obtain suitable macroscale models. Since the seminal work of G.I. Taylor [207],
19

20 2.1. Upscaling in stationary reactive
ows in porous media
dispersion phenomena in porous media have attracted a lot of attention. There are many
formal or rigorous methods in the literature. We refer to [138] and [139] as one of the rst
works containing rigorous homogenization results for reactive
ows in porous media. By us-
ing 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 [3], [25], [32],
[167], [172], [156], [139], [166], [157], [98] 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
[9], [8], and [171]. For reactive
ows combined with the mechanics of cells, we refer to [146].
Rigorous homogenization techniques for obtaining the effective model for dissolution and pre-
cipitation in a complex porous medium were successfully applied in [157]. Solute transport in
porous media is also a topic of interest for chemists, geologists and environmental scientists
(see, e.g., [6] and [97]). Related problems, such that single or two-phase
ow or miscible
displacement problems were addressed in various papers (see, for instance, [16], [21], [22],
[170]). For an interesting survey on homogenization techniques applied to problems involving

ow, diffusion, convection, and reactions in porous media, we refer to [137].
In this chapter, some applications of the homogenization method to the study of reactive

ows in periodic porous media will be presented. The chapter represents a summary of the
results I obtained in this area, alone or in collaboration, and is based on the papers [72], [73],
[209], and [211].
2.1 Upscaling in stationary reactive
ows in porous media
We shall discuss now, following [72] and [211], some homogenization results for chemical
reactive
ows through porous media. For more details about the chemical aspects involved in
this kind of problems and, also, for some mathematical and historical backgrounds, we refer
to S. N. Antontsev et al. [20], J. Bear [32], J. I. D az [88], [91], [90], and U. Hornung [137]
and the references therein. We shall be concerned 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 [72], 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 [137]. 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
effective model preserves all the relevant 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 [211], where we have analyzed the effective
behavior of the solution of a nonlinear problem arising in the modeling of enzyme catalyzed

Chapter 2. Homogenization of reactive
ows in porous media 21
reactions through the exterior of a domain containing periodically distributed reactive solid
obstacles.
2.1.1 The model problem
We consider an open smooth connected bounded set Ω in Rn(n3) and we insert in it a set
of"-periodically distributed reactive obstacles. In this way, we obtain an open set Ω", called
theexterior domain ;"represents a small parameter related to the characteristic size of the
reactive obstacles. More precisely, let Y= (0;1)nbe the unit cell in Rn. Denote by Fan
open subset of Ywith smooth boundary @Fsuch thatFY. We shall refer to Fas being
the elementary obstacle . We set
Y=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 vector
k,
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 Ω, i.e.
F"=∪{
F"
kjF"
kΩ; k2Zn}
:
Set Ω"= ΩnF". Hence, Ω"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@Ω. Let
S"=[f@F"
kjF"
kΩ; k2Zng:
So,
@Ω"=@Ω[S":
We denote by j!jthe Lebesgue measure of any measurable subset !Rnand, for an
arbitrary function 2L2(Ω"), we denote by e its extension by zero to the whole of Ω. 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 Ω", with concentration u", reacting on the boundary of the obstacles. A
simplied version of this kind of problems can be written as follows:
8
>>><
>>>:Df∆u"=fin Ω";
Df@u"
@="g(u") onS";
u"= 0 on@Ω:(2.1)
Here,is the unit exterior normal to Ω",f2L2(Ω) andS"is the boundary of the exterior
medium Ω nΩ". For simplicity, we assume that the
uid is homogeneous and isotropic,
with a constant diffusion coefficient Df>0. We can treat in a similar manner the more

22 2.1. Upscaling in stationary reactive
ows in porous media
general case in which, instead of considering constant diffusion coefficients, we work with
an heterogeneous medium represented by periodic symmetric bounded matrices which are
assumed to be uniformly coercive.
The semilinear boundary condition imposed on S"in problem (2.1) describes the chemical
reactions which take place locally at the interface between the reactive
uid and the obstacles.
In fact, from a chemical perspective, such a situation represents, equivalently, the effective
reaction on the walls of the chemical reactor between the
uid lling Ω"and a chemical
reactant located inside the rigid solid grains.
For the function g, which is 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 subdifferential 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. [90]) that u"0 in Ω nΩ"andu">0 in Ω", althoughu"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). In fact, since u"is, in practical applications, a
concentration, we can impose suitable conditions on the data to ensure that 0 u"1 (see,
e.g., [89]).
As usual in homogenization, our goal is to obtain a suitable description of the asymptotic
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
differentiable 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 thatdg
dvC(1 +jvjq): (2.4)
We introduce the functional space
V"={
v2H1(Ω")jv= 0 on@Ω}
;
endowed with the norm
∥v∥V"=∥∇v∥L2(Ω"):

Chapter 2. Homogenization of reactive
ows in porous media 23
The weak formulation of problem (2.1) is:
8
><
>:Findu"2V"such that
Df∫
Ω"∇u"
∇φdx+"∫
S"g(u")φd=∫
Ω"fφdx8φ2V":(2.5)
By classical existence results (see [45]), there exists a unique weak solution u"2V"\H2(Ω")
of problem (2.1). This solution being dened only on Ω", we need to extend it to the whole of
Ω to be able to state the convergence result. To this end, let us recall the following well-known
extension result (see [69]):
Lemma 2.1 There exists a linear continuous extension operator
P"2 L(L2(Ω");L2(Ω))\ L(V";H1
0(Ω))
and a positive constant C, independent of ", such that, for any v2V",
∥P"v∥L2(Ω)C∥v∥L2(Ω")
and
∥∇P"v∥L2(Ω)C∥∇v∥L2(Ω"):
Therefore, we have the following Poincar e's inequality in V":
Lemma 2.2 There exists a positive constant C, independent of ", such that
∥v∥L2(Ω")C∥∇v∥L2(Ω")for anyv2V":
The main convergence result for this case is stated in the following theorem.
Theorem 2.3 (Theorem 2.3 in [72]) There exists an extension P"u"of the solution u"of
the variational problem (2.5) such that
P"u"⇀u weakly inH1
0(Ω);
whereuis the unique solution of
8
><
>:n∑
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =finΩ;
u= 0 on@Ω:(2.6)
Here,Q= ((qij))is the homogenized matrix, whose entries are dened as follows:
qij=Df(
ij+1
jYj∫
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 homogenized matrix Qis constant, symmetric, and positive-denite.

24 2.1. Upscaling in stationary reactive
ows in porous media
Proof. The proof of this theorem is divided 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 φ=u"as a test function
in (2.5), using Schwartz and Poincar e's inequalities, we get
∥P"u"∥H1
0(Ω)C
and, by passing to a subsequence, still denoted by P"u", we can suppose that there exists
u2H1
0(Ω) such that
P"u"⇀u weakly inH1
0(Ω): (2.9)
It remains to determine the limit equation satised by u.
Second step . In order to obtain the limit equation satised by u, we need 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 h2Ls′(@F), 1s′ 1 , the linear form "
honW1;s
0(Ω)
dened by
⟨"
h;φ⟩="∫
S"h(x
")
φd8φ2W1;s
0(Ω);
with 1=s+ 1=s′= 1. Then (see [61]),
"
h!hstrongly in ( W1;s
0(Ω))′; (2.10)
where
⟨h;φ⟩=h∫
Ωφdx;
with
h=1
jYj∫
@Fh(y) d:
Ifh2L1(@F) or ifhis constant, we have "
h!hstrongly in W1;1(Ω) and we denote
by"the above introduced measure in the particular case in which h= 1. Notice that in
this casehbecomes1=j@Fj=jYj. We shall prove now that for any φ2 D(Ω) and for any
v"⇀v weakly inH1
0(Ω), one has
φg(v")⇀φg (v) weakly in W1;q
0(Ω); (2.11)
where
q=2n
q(n2) +n:
To this end, let us remark that
sup∥∇g(v")∥Lq(Ω)<1: (2.12)
Indeed, using the growth condition (2.4) imposed to g, we have
∫
Ω@g
@xi(v")q
dxC∫
Ω(
1 +jv"jqq)@v"
@xiq
dx

Chapter 2. Homogenization of reactive
ows in porous media 25
C(
1 +(∫
Ωjv"jqq
dx)1=
)(∫
Ωj∇v"jqdx)1=
;
where we took
andsuch thatq= 2, 1=
+ 1== 1 andqq
= 2n=(n2). Notice that
it is from here that we get
q=2n
q(n2) +n:
Also, due to the fact that 0 q<n= (n2), it follows that q>1. Since
sup∥v"∥
L2n
n2(Ω)<1;
we easily get (2.12). Therefore, to obtain (2.11), it remains only to show that
g(v")!g(v) strongly in Lq(Ω): (2.13)
But this convergence is a direct consequence of the following well-known result (see [80], [155]
and [160]):
Theorem 2.4 LetG: ΩR!Rbe a Carath eodory function, i.e.
a) for every vthe function G(
;v)is measurable with respect to x2Ω:
b) for every (a.e.) x2Ω, 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)j C(
1 +jvjr=t)
;
withr1andt <1, then the map v2Lr(Ω)7!G(x;v(x))2Lt(Ω)is continuous in the
strong topologies.
Indeed, since
jg(v)j C(1 +jvjq+1);
applying the above theorem for G(x;v) =g(v),t=qandr= (2n=(n2))r′, withr′>0
such thatq+ 1< r=t and using the compact injection H1(Ω),!Lr(Ω) we obtain (2.13).
Now, from (2.10), written for h= 1, and (2.11) written for v"=P"u", we get
⟨";φg(P"u")⟩ !j@Fj
jYj∫
Ωφg(u) dx8φ2 D(Ω) (2.14)
and this completes the proof of this step.
Third step . Let"be the gradient of u"in Ω"and let us denote by e"its extension by
zero to the whole of Ω. Then, e"is bounded in ( L2(Ω))nand, as a consequence, there exists
2(L2(Ω))nsuch that
e"⇀ weakly in (L2(Ω))n: (2.15)
Let us identify now the equation satised by . If we take φ2 D(Ω), from (2.5) we get
∫
Ωe"
∇φdx+"∫
S"g(u")φd=∫
ΩΩ"fφdx (2.16)

26 2.1. Upscaling in stationary reactive
ows in porous media
and we can pass to the limit, with "!0, in all the terms of (2.16). For the rst one, we have
lim
"!0∫
Ωe"
∇φdx=∫
Ω
∇φdx: (2.17)
For the second term, using (2.14), we obtain
lim
"!0"∫
S"g(u")φd=j@Fj
jYj∫
Ωg(u)φdx: (2.18)
It is not difficult to pass to the limit in the right-hand side of (2.16). Since
Ω"f ⇀jYj
jYjfweakly inL2(Ω);
we get
lim
"!0∫
ΩΩ"fφdx=jYj
jYj∫
Ωfφdx: (2.19)
Putting together (2.17)-(2.19), we have
∫
Ω
∇φdx+j@Fj
jYj∫
Ωg(u)φdx=jYj
jYj∫
Ωfφdx8φ2 D(Ω):
Thus,satises
div+j@Fj
jYjg(u) =jYj
jYjfin Ω (2.20)
and it remains now only to identify .
Fourth step. For identifying , we shall use the solutions of the local problems (2.8). For
any xedi= 1;:::;n , we dene
i"(x) ="(
i(x
")
+yi)
8×2Ω"; (2.21)
wherey=x=". By periodicity, it follows that
P"i"⇀x iweakly inH1(Ω): (2.22)
Let"
ibe the gradient of  i"in Ω"ande"
ibe the extension by zero of "
iinside the holes.
We have
(
e"
i)
j=(g@i"
@xj)
=(
^@i
@yj(y))
+ijY
and, therefore,
(
e"
i)
j⇀1
jYj(∫
Y@i
@yjdy+jYjij)
=jYj
jYjqijweakly inL2(Ω): (2.23)
We notice that "
isatises{
div"
i= 0 in Ω";
"
i
= 0 onS":(2.24)

Chapter 2. Homogenization of reactive
ows in porous media 27
Letφ2 D(Ω). Multiplying the rst equation in (2.24) by φu"and integrating by parts on
Ω", we obtain∫
Ω""
i
∇φu"dx+∫
Ω""
i
∇u"φdx= 0:
Thus,∫
Ωe"
i
∇φP"u"dx+∫
Ω""
i
∇u"φdx= 0: (2.25)
Takingφi"as a test function in (2.5), we get
∫
Ω"(∇u"
∇φ)i"dx+∫
Ω"(∇u"
∇i")φdx+"∫
S"g(u")φi"d=∫
Ω"fφi"dx;
which, using the denitions of e"ande"
i, leads to
∫
Ωe"
∇φP"i"dx+∫
Ω"∇u"
"
iφdx+"∫
S"g(u")φi"d=∫
ΩfΩ"φP"i"dx:
From (2.25), we obtain
∫
Ωe"
∇φP"i"dx∫
Ωe"
i
∇φP"i"dx+"∫
S"g(u")φi"d=∫
ΩfΩ"φP"i"dx:(2.26)
We pass now to the limit in (2.26). By using (2.15) and (2.22), we get
lim
"!0∫
Ωe"
∇φP"i"dx=∫
Ω
∇φxidx: (2.27)
From (2.9) and (2.23) we obtain
lim
"!0∫
Ωe"
i
∇φP"u"dx=jYj
jYj∫
Ωqi
∇φudx; (2.28)
whereqiis the vector having the j-component equal to qij.
Since the boundary of Fis of classC2,P"i"2W1;1(Ω) andP"i"!xistrongly in
L1(Ω):Then, since g(P"u")P"i"!g(u)xistrongly in Lq(Ω) andg(P"u")P"i"is bounded
inW1;q(Ω), we have g(P"u")P"i"⇀g(u)xiweakly inW1;q(Ω). Thus,
lim
"!0"∫
S"g(u")φi"d=j@Fj
jYj∫
Ωg(u)φxidx: (2.29)
For the limit of the right-hand side of (2.26), since
Ω"f ⇀jYj
jYjfweakly inL2(Ω);
using (2.22), we have
lim
"!0∫
ΩfΩ"φP"i"dx=jYj
jYj∫
Ωfφx idx: (2.30)

28 2.1. Upscaling in stationary reactive
ows in porous media
Thus, we get
∫
Ω
∇φxidxjYj
jYj∫
Ωqi
∇φudx+j@Fj
jYj∫
Ωg(u)φxidx=jYj
jYj∫
Ωfφx idx: (2.31)
Using Green's formula and equation (2.20), we get
∫
Ω
∇xiφdx+jYj
jYj∫
Ωqi
∇uφdx= 0 in Ω:
The above equality is valid for any φ2 D(Ω) and this implies that

∇xi+jYj
jYjqi
∇u= 0 in Ω: (2.32)
Writing (2.32) by components, differentiating with respect to xi;summing after iand using
(2.19), we are led to
jYj
jYjn∑
i;j=1qij@2u
@xi@xj= div=jYj
jYjf+j@Tj
jYjg(u);
which means that uveries
n∑
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =fin Ω:
Sinceu2H1
0(Ω) (i.e.u= 0 on@Ω) 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 Ω. Also, let us emphasize again that if we assume f0,
the function gin example a)is indeed a particular example of our rst model situation.
Remark 2.6 The results in [72] 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 for the nonlinear
functiong(see [209]).
Remark 2.7 In [72], the proof of Theorem 2.3 was done by using the so-called energy method
of L. Tartar (see [205]). 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.,
[66], [56], [57], and [64]), 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.

Chapter 2. Homogenization of reactive
ows in porous media 29
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 [72]. If we denote by
D(g) the domain of g, i.e.D(g) =f2Rjg()̸=∅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)j C(1 +jvjq): (2.33)
Notice that the second important practical example b) mentioned above is a particular ex-
ample of such a single-valued maximal monotone graph.
In this case, there exists a lower semicontinuous convex function GfromRto ]1;+1],
Gproper, i.e. G̸+1, such that gis the subdifferential of G,g=@G. Let
G(v) =∫v
0g(s) ds:
We dene the convex set
K"={
v2V"jG(v)jS"2L1(S")}
:
For a given function f2L2(Ω), the weak solution of the problem (2.1) is also the unique
solution of the following variational inequality:
8
>>><
>>>:ndu"2K"such that
Df∫
Ω"Du"D(v"u") dx∫
Ω"f(v"u") dx+⟨";G(v")G(u")⟩ 0
8v"2K":(2.34)
We start by remarking that there exists a unique weak solution u"2V"\H2(Ω") of the
above variational inequality (see [45]). Moreover, it is well-known that the solution u"of 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
2Df∫
Ω"jDvj2dx+⟨";G(v)⟩ ∫
Ω"fvdx:
If we introduce the following functional dened on H1
0(Ω):
J0(v) =1
2∫
ΩQDvDv dx+j@Fj
jYj∫
ΩG(v) dx∫
Ωfvdx;
the main convergence result for problem (2.34) can be formulated as follows:

30 2.1. Upscaling in stationary reactive
ows in porous media
Theorem 2.8 (Theorem 2.6 in [72]) 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(Ω);
whereuis the unique solution of the minimization problem
8
<
:Findu2H1
0(Ω)such that
J0(u) = inf
v2H1
0(Ω)J0(v):(2.35)
Moreover,G(u)2L1(Ω). 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
>><
>>:n∑
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =finΩ;
u= 0 on@Ω:
Proof of Theorem 2.8. Letu"be the solution of the variational inequality (2.34). Using
the same extension P"u"as in the previous case, it follows immediately that P"u"is bounded
inH1
0(Ω) and, thus, by passing to a subsequence, we have
P"u"⇀u weakly inH1
0(Ω): (2.36)
Letφ2 D(Ω). Using standard regularity results, we know that i2L1. So, from the
boundedness of iandφ, it follows that there exists M0 such that

@φ
@xi

L1

i

L1<M:
Let
v"=φ+∑
i"@φ
@xi(x)i(x
")
: (2.37)
Obviously, v"2K"and this will allow us to take it as a test function in (2.34). Moreover, it
is easy to see that v"!φstrongly in L2(Ω). Further,
Dv"=∑
i@φ
@xi(x)(
ei+Di(x
"))
+"∑
iD@φ
@xi(x)i(x
")
;
where ei, 1in, are the elements of the canonical basis in Rn. Takingv"as a test
function in (2.34), we get
∫
Ω"Du"Dv"dx∫
Ω"f(v"u") dx+∫
Ω"Du"Du"dx ⟨";G(v")G(u")⟩

Chapter 2. Homogenization of reactive
ows in porous media 31
and
∫
ΩDP"u"^(Dv") dx∫
Ω"f(v"u") dx+∫
Ω"Du"Du"dx ⟨";G(v")G(u")⟩:(2.38)
Let us denote
Qej=1
jYj∫
Y(Dj+ej) dy; (2.39)
where=jYj=jYj. It is not difficult 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
∫
ΩDP"u"gDv"dx!∫
ΩQDuDφ dx (2.40)
and ∫
Ω"f(v"u") dx=∫
ΩfΩ"(v"P"u") dx!∫
Ωf(φu) 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(Ω)
and, so
⟨";G(P"u")⟩ !j@Fj
jYj∫
ΩG(u) dx:
In a similar way, we get
⟨";G(v")⟩ !j@Fj
jYj∫
ΩG(φ) dx
and, therefore, we have
⟨";G(v")G(P"u")⟩ !j@Fj
jYj∫
Ω(G(φ)G(u)) dx: (2.42)
Now, it remains only to pass to the limit in the second term of the right-hand side of (2.38).
To this end, let us write down, for any w"2H1
0(Ω), the subdifferential inequality
∫
Ω"Du"Du"dx∫
Ω"Dw"Dw"dx+ 2∫
Ω"Dw"(Du"Dw") dx; (2.43)
Reasoning as before and choosing
w"=φ+∑
i"@φ
@xi(x)i(x
")
;
whereφhas similar properties as the corresponding φ, we can pass to the limit in the right-
hand side of the inequality (2.43) and we get
lim inf
"!0∫
Ω"Du"Du"dx∫
ΩQDφDφdx+ 2∫
ΩQDφ(DuDφ) dx;

32 2.1. Upscaling in stationary reactive
ows in porous media
for anyφ2 D(Ω). Sinceu2H1
0(Ω), taking φ!ustrongly in H1
0(Ω);we obtain
lim inf
"!0∫
Ω"Du"Du"dx∫
ΩQDuDu dx: (2.44)
Putting together (2.40)-(2.42) and (2.44), we get
∫
ΩQDuDφ dx∫
Ωf(φu) dx+∫
ΩQDuDu dxj@Fj
jYj∫
Ω(G(φ)G(u)) dx;
for anyφ2 D(Ω) and, hence, by density, for any v2H1
0(Ω). So, we obtain
∫
ΩQDuD (vu) dx∫
Ωf(vu) dxj@Fj
jYj∫
Ω(G(φ)G(u)) dx;
which leads exactly to the limit problem (2.34). This ends the proof of Theorem 2.8.
Remark 2.10 In fact, the particular form of the test function (2.37) gives 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 diffusion matrix
A"(x) =A(x="), whereA=A(y)is a matrix-valued function on Rnwhich isY-periodic. We
shall assume that
8
<
:A2L1(Ω)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 (see, e.g.
[209] and [211]). In [211], we analyzed the effective 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 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 and, then, the enzyme-substrate complex 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 affect 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 Ω";
Df@u"
@="g(u") onS";
u"= 0 on@Ω:(2.45)

Chapter 2. Homogenization of reactive
ows in porous media 33
Here, the function is continuously differentiable, 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 function g, 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 subdifferential of a convex lower semicontinuous
functionG. More precisely, we shall consider an important practical example, arising in the
diffusion of enzymes, namely the Michaelis-Menten model:
g(v) =8
><
>:v
v+
; v 0;
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 [45] and [99]). Therefore, we know that there exists a
unique weak solution u"2V"∩H2(Ω"). Moreover, u"is also the unique solution of the
following variational problem:
8
>>>>><
>>>>>:Findu"2K"such that
Df∫
Ω"Du"D(v"u") dx+∫
Ω"(u")(v"u") dx
∫
Ω"f(v"u") dx+⟨";G(v")G(u")⟩ 0;8v"2K";(2.46)
where"is the linear form on W1;1
0(Ω) dened by
⟨";φ⟩="∫
S"φd;8φ2W1;1
0(Ω):
The main convergence result in this case, proven in [211], is stated in the following theorem.
Theorem 2.12 The solution u", properly extended to the whole of Ω, converges to the unique
solution of the following variational inequality:
8
>>>>>>><
>>>>>>>:u2H1
0(Ω);
∫
ΩQDuD (vu) dx+∫
Ω(u)(vu) dx∫
Ωf(vu) dx
j@Fj
jYj∫
Ω(G(v)G(u)) dx;8v2H1
0(Ω):(2.47)
Here,Q= ((qij))is the homogenized matrix, dened in (2.7).
Remark 2.13 Notice that ualso satises
8
>><
>>:n∑
i;j=1qij@2u
@xi@xj+(u) +j@Fj
jYjg(u) =finΩ;
u= 0 on@Ω:

34 2.1. Upscaling in stationary reactive
ows in porous media
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
effect of the enzymatic reactions. The effect 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 effect. 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 difference in the limit equation will be the coefficient appearing
in front of this extra zero-order term. So, one can control the effective behavior of such
reactive
ows by choosing different locations for the involved chemical reactions. Moreover,
as we shall see in the next section, we can obtain similar effects 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 Ω", but also inside the grains, instead on their
boundaries. The difference in the limit equation will be the coefficient appearing in front of
this extra zero-order term. Hence, we can control the effective behavior of such reactive
ows
by choosing different 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 [72] 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). A simplied version of this kind of
problems can be formulated as follows:
8
>>>>>>>>><
>>>>>>>>>:Df∆u"=fin Ω";
Dp∆v"+g(v") = 0 in "
Df@u"
@=Dp@v"
@onS";
u"=v"onS";
u"= 0 on@Ω:(2.48)
Here, "= ΩnΩ",is the exterior unit normal to Ω",u"andv"are the concentrations in
Ω"and, respectively, inside the grains ",Df>0,Dp>0,f2L2(Ω) 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 0 q<n= (n2),
such that jg(v)j C(1 +jvjq+1). We remark that the above mentioned examples a) 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
∥w"∥2
H"=∥∇u"∥2
L2(Ω")+∥∇v"∥2
L2("):

Chapter 2. Homogenization of reactive
ows in porous media 35
The variational formulation of problem (2.48) is the following one:
8
>>>><
>>>>:ndw"2H"such that
Df∫
Ω"∇u"
∇φdx+Dp∫
"∇v"
∇ dx+∫
"g(v") dx=∫
Ω"fφdx
8(φ; )2K":(2.49)
Under the above hypotheses and the conditions satised by H", it is well-known (see [45] and
[160]) 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 [72]):
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(Ω);
whereuis the unique solution of
8
><
>:n∑
i;j=1a0
ij@2u
@xi@xj+jFj
jYjg(u) =finΩ;
u= 0 on@Ω:(2.50)
Here,A0= ((a0
ij))is the homogenized matrix, dened by
a0
ij=1
jYj∫
Y(
aij+aik@j
@yk)
dy; (2.51)
wherei; i= 1;:::;n; are the solutions of the 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(Ω)andv2L2(Ω)such that
P"u"⇀u weakly inH1
0(Ω);
ev"⇀v weakly inL2(Ω)
and
v=jFj
jYju:

36 2.1. Upscaling in stationary reactive
ows in porous media
Corollary 2.16 Let"be dened by
"(x) ={u"(x)x2Ω";
v"(x)x2":
Then, there exists 2H1
0(Ω)such that"⇀  weakly inH1
0(Ω), whereis the unique
solution of8
><
>:n∑
i;j=1a0
ij@2
@xi@xj+ajFj
jYjg() =finΩ;
= 0 on@Ω;
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 [72] and [211] is the so-called
energy method orthe oscillating test function method introduced by L. Tartar (see [205]
and [206]) 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 [42]. Also, one can use the periodic unfolding method (see, e.g.,
[57] and [64]).
Remark 2.18 The two reactive
ows studied above, namely (2.1) and (2.48), lead to com-
pletely different effective behaviors. The macroscopic problem (2.1) arises from the homoge-
nization 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 tak-
ing place on the boundaries of the reactive obstacles. On the other hand, the second model is
a boundary-value problem in the whole domain Ω, with discontinuous coefficients. Its macro-
scopic behavior also involves a zero-order term, but of a completely different nature, emerging
from the chemical reactions occurring inside the grains.
Remark 2.19 In (2.48) we considered that the ratio of the diffusion coefficients 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 different orders for the diffusion
in the obstacles and in the pores. More precisely, if one takes the ratio of the diffusion
coefficients 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 effective limit model includes two equations, one in Fand another one in Ω, the
last one containing an extra-term which re
ects the remaining in
uence of the grains (see,
for instance, [21], [41], [42], [73], and [138]).
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

Chapter 2. Homogenization of reactive
ows in porous media 37
Robin problems, Signorini's problems, problems arising in chemistry (see [61], [73], [75], and
[209]). Also, for the case of a different geometry of the perforated domain and different
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 [73], concerning
the effective behavior of some chemical reactive
ows involving diffusion, different 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, [137], 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 [137], was treated in [73]. Two well-known examples of such nonlinear models, namely
the so-called Freundlich and Langmuir kinetics, were studied. We brie
y describe here these
results. 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 a surface diffusion modeled by a Laplace-Beltrami operator
to take place on this surface. In this last case, we show that the effective 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 effects are present in this limit model.
2.2.1 The microscopic model and its weak solvability
Our main goal in [73] 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 Ω"and a solid skeleton
(grains or pores), Ω nΩ". We assume that chemical substances are dissolved in the
uid
part. They are transported by diffusion and also, by adsorption, they can change from being
dissolved in the
uid to residing on the surface of the pores, where chemical reactions take
place. Thus, the model consists of a diffusion system in the
uid phase, a reaction system on
the pore surface and a boundary condition coupling them (see (2.54)):
(V")8
>>><
>>>:@v"
@t(t;x)Df∆v"(t;x) =h(t;x); x2Ω"; t> 0;
v"(t;x) = 0; x2@Ω; t> 0;
v"(t;x) =v1(x); x2Ω"; 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)

38 2.2. Nonlinear adsorption of chemicals in porous media
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 concentration
of the solute on the surface of the skeleton Ω nΩ",v12H1
0(Ω) is the initial concentration of
the solute and w12H1
0(Ω) 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
diffusion coefficient Df>0,a;
> 0 are the reaction and, respectively, the adsorption factor
andhis 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 analyzed: the case of a monotone
smooth function gwithg(0) = 0 and, respectively, the case of a maximal monotone graph
withg(0) = 0, i.e. the case in which gis the subdifferential of a convex lower semicontinuous
functionG. These two general situations are well illustrated by the two important practical
examplesa) andb) (see (2.2) and (2.3)) mentioned in Section 2.1.1, namely the Langmuir
and, respectively, the Freundlich kinetics.
Let us notice that if v"0 in Ω"andv">0 in Ω", then the function gin example a)
is a particular example of our rst model situation, i.e. gis a monotone smooth function
satisfying the condition g(0) = 0. In fact, instead of (2.56), we could consider a slightly more
general boundary condition, given by
f"(t;x) =
1g(v"(t;x))
2w"(t;x);
with
1>0 and
2>0 being the adsorption factor and the desorption factor , respectively
(see [138]).
The existence and uniqueness of a weak solution u"= (v";w") of the system (2.53)-(2.56)
can be established by using the classical theory of semilinear monotone problems (see, for
instance, [45] and [160]).
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(Ω), with the classical scalar product
(u;v)Ω=∫
Ωu(x)v(x) dx;
and let H=L2(0;T;H), with the scalar product
(u;v)Ω;T=∫T
0(u(t);v(t))Ωdt;
whereu(t) =u(t;
);v(t) =v(t;
). Also, let V=H1(Ω), with
(u;v)V= (u;v)Ω+ (∇u;∇v)Ω

Chapter 2. Homogenization of reactive
ows in porous media 39
andV=L2(0;T;V), with
(u;v)V=∫T
0(u(t);v(t))Vdt:
We set
W={
v2 Vdv
dt2 V′}
;
where V′is the dual space of V,
V0={
v2 Vv= 0 on@Ω a.e. on (0 ;T)}
;
W0=V0∩
W:
In a similar manner, we dene the spaces V(Ω"),V(Ω"),V(S") and V(S"). For the latter we
write
⟨u;v⟩S"=∫
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"by∇"and the Laplace-Beltrami operator on S"by ∆", we have
(∆"u;v)S"=⟨∇"u;∇"v⟩S":
For the space of test functions we use the notation
D=C1
0((0;T)Ω)):
We shall start our analysis with the case in which gis a continuously differentiable func-
tion, monotonously non-decreasing and such that g(v) = 0 if and only if v= 0 (for the
non+mooth case, see Section 2.2.3). Moreover, we shall suppose that there exist a positive
constantCand an exponent q, with 0 q<n= (n2), such that
dg
dvC(1 +jvjq): (2.57)
The weak formulation of problem (2.53)-(2.56) is as follows:
8
>>>><
>>>>:ndv"2 W 0(Ω");v"(0) =v1jΩ"such that
(
v";dφ
dt)
Ω";T+"(f";φ)Ω";T=Df(∇v";∇φ)Ω";T+ (h;φ)Ω";T;
8φ2 W 0(Ω");(2.58)
and8
>><
>>:ndw"2 W(S");w"(0) =w1jS"such that
(
w";dφ
dt)
S";T+a(w";φ)S";T= (f";φ)S";T;8φ2 W(S"):(2.59)
Proposition 2.20 There exists a unique weak solution u"= (v"; w")of the system (2.58)-
(2.59).

40 2.2. Nonlinear adsorption of chemicals in porous media
Remark 2.21 The solution of (2.59) can be written as
w"(t;x) =w1(x)e(a+
)t+
∫t
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 Ω", we need to extend it to the
whole of Ω 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 [71]):
Lemma 2.22 There exists a positive constant C, independent of ", such that, for any v2V",
∥v∥2
L2(S")C("1∥v∥2
L2(Ω")+"∥∇v∥2
L2(Ω")):
2.2.2 The main result
Theorem 2.23 (Theorem 2.5 in [73]) One can construct an extension P"v"of the solution
v"of the problem (V")such that
P"v"⇀v weakly in V;
wherevis the unique solution of the following limit problem:
8
>>>>><
>>>>>:@v
@t(t;x) +F0(t;x)n∑
i;j=1qij@2v
@xi@xj(t;x) =h(t;x); t> 0; x2Ω;
v(t;x) = 0; t> 0; x2@Ω;
v(t;x) =v1(x); t = 0; x2Ω;(2.60)
with
F0(t;x) =j@Fj
jY⋆j
[
g(v(t;x))w1(x)e(a+
)t
r(
)⋆g(v(
;x))(t)]
:
Here,Q= ((qij))is the 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; x2Ω;
w(t;x) =w1(x); t = 0; x2Ω(2.61)
andwcan be written as
w(t;x) =w1(x)e(a+
)t+
r(t)⋆g(v(t;x)):

Chapter 2. Homogenization of reactive
ows in porous media 41
Remark 2.24 The weak formulation of problem (2.60) is:
8
>>>><
>>>>:ndv2 W 0(Ω);v(0) =v1such that
(
v;dφ
dt)
Ω;T+ (F0;φ)Ω;T=(Q∇v;∇φ)Ω;T+ (h;φ)Ω;T
8φ2 W 0(Ω):(2.62)
Proof of Theorem 2.23. The proof is divided into several steps (see [73]). The rst step
consists in proving the uniqueness of the limit problem (2.62). This is stated in the following
proposition, proven in [73]:
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 in describing the macroscopic
behavior of the solution u"= (v";w"), as"!0. To achieve this goal, some a priori estimates
on this solution are required (for a detailed proof, see [73]).
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
∥w"(t)∥2
S"(∥w"(0)∥2
S"+

∥g(v")∥2
S"; t)e
t;8t0;8>0;

@w"
@t

2
S"; tC(∥w"(0)∥2
S"+∥g(v")∥2
S"; t);8t0;
∥v"(t)∥2
Ω"C;∥∇v"(t)∥2
Ω"; tC
and

@v"
@t(t)

2
Ω"C:
The last step is the limit passage and the identication of the homogenized problem. Let
v"2 W 0(Ω") 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
∥P"v"(t)∥Ω+∥∇P"v"∥Ω;t+∥@t(P"v")(t)∥ΩC;
for alltT. Therefore, by passing to a subsequence, still denoted by P"v", we can assume
that there exists v2 Vsuch that the following convergences hold:
P"v"⇀v weakly in V;
@t(P"v")⇀@ tvweakly in H;
P"v"!vstrongly in H:

42 2.2. Nonlinear adsorption of chemicals in porous media
It remains now 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 difficult 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
⟨";φg(P"v"(t))⟩ !j@Fj
jYj∫
Ωφg(v(t))dx8φ2 D:
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");φ)S";T=j@Fj
jYj
(g(v);φ)Ω; T:
For the rest of the terms, the proof is standard and we obtain immediately (2.60). Since
v2 W 0(Ω) (i.e.v= 0 on@Ω) andvis uniquely determined, the whole sequence P"v"
converges to vand Theorem 2.23 is proven.
2.2.3 The case of a non-smooth boundary condition
In this subsection, we address 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 [137], 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)j C(1 +jvj). As in Section 2.1.3, let Gbe such that g=@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
>>>><
>>>>:ndv"
2 W 0(Ω");v"
(0) =v1jΩ"such that
(
v"
;dφ
dt)Ω";T+"(f"
;φ)
Ω";T=Df(∇v"
;∇φ)Ω";T+ (h;φ)Ω";T;
8φ2 W 0(Ω")(2.63)
and8
><
>:ndw"
2 W(S");w"
(0) =w1jS"such that
(
w"
;dφ
dt)
S";T+a(w"
;φ)S";T= (f"
;φ)S";T;8φ2 W(S");(2.64)
where
f"
=
(g(v"
)w"
)
and
g=IJ
;

Chapter 2. Homogenization of reactive
ows in porous media 43
with
J= (I+@G)1:
We remark that gis a non-decreasing Lipschitz function, with g(0) = 0.
Problem (2.63)-(2.64) has a unique solution ( v"
;w"
), for every >0 (see [45] and [160]).
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"
.
Mollifyinggto make it a smooth function (see [34]) and using the results of the previous
subsection, for any >0, we get
P"v"
!vstrongly in H(Ω):
Then, it is not difficult to see that, proving suitable a priori estimates (classical energy
estimates) on the solutions v, we can ensure, via compactness arguments (see [30]), the
strong convergence of v, as!0, tov, the unique solution of problem (2.60). Hence
v!vstrongly in H(Ω):
Finally, since
∥P"v"v∥Ω;T ∥P"v"P"v"
∥Ω;T+∥P"v"
v∥Ω;T+∥vv∥Ω;T;
we get the strong convergence of P"v"tovinH(Ω).
Remark 2.27 The conclusion of the above theorem remains true for more general situations.
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. For the correct mathematical
treatment, one needs to reformulate the problem by using the maximal monotone graph of
R2associated to the Heaviside function (v) =f0gifv <0,(0) = [0;1]and(v) =f1g
ifv > 0. The existence and uniqueness of a solution can be found, for instance, in [45].
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 coefficients
In problem (2.53)-(2.56), the rate aof chemical reactions on S"and the adsorption coefficient

were assumed to be constant. A more realistic model implies to assume that the surface @F
is chemically and physically heterogeneous, which means that aand
are rapidly oscillating
functions, i.e.
a"(x) =a(x
")
;
"(x) =
(x
")
;

44 2.2. Nonlinear adsorption of chemicals in porous media
withaand
positive functions in W1;1(Ω) which are Y-periodic (for linear adsorption rates,
see [138]). In this case, v"andw"satisfy the following system of equations:
(V")8
><
>:@v"
@t(t;x)Df∆v"(t;x) =h(t;x); x2Ω"; t> 0;
v"(t;x) = 0; x2@Ω; t> 0;
v"(t;x) =v1(x); x2Ω"; 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:
Theorem 2.28 (Theorem 4.1 in [73]) The effective behavior of vandwis governed by the
following system:
8
>>><
>>>:@v
@t(t;x) +G0(t;x)n∑
i;j=1qij@2v
@xi@xj=h(t;x); t> 0; x2Ω;
v(t;x) = 0t>0; x2@Ω;
v(t;x) =v1(x)t= 0; x2Ω;(2.69)
and
{@w
@t(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(v(t;x)); t> 0; x2Ω; y2@F
w(t;x;y ) =w1(x)t= 0; x2Ω; y2@F;(2.70)
where
G0(t;x) =1
jY⋆j∫
@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)∫t
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);

Chapter 2. Homogenization of reactive
ows in porous media 45
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)n∑
i;j=1qij@2v
@xi@xj(t;x) +F0(t;x) =h(t;x); t> 0; x2Ω; (2.71)
with
F0(t;x) =1
jY⋆j∫
@F{

(y)[g(v(t;x))w1(x)e(a(y)+
(y))t
(y)r(
;y)⋆g(v(
;x))(t)]}
d:
Remark 2.29 The above adsorption model can be slightly generalized by allowing surface
diffusion on S". In fact, the chemical substances can creep on the surface and this effect is
similar to a surface-like diffusion. From a mathematical point of view, we can model this
phenomenon by introducing a diffusion 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 diffusion 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; x2Ω; 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-diffusion system with respect to an
additional microvariable. Also, notice that the local behavior is no longer governed by an
ordinary differential equation, but by a partial differential 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 ef-
fective 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 diffusion-reaction equation (2.71).
The above results can be extended 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 diffusion matrix D"(x).

46 2.2. Nonlinear adsorption of chemicals in porous media
More precisely, we analyze the asymptotic behavior, as "!0, of the following coupled system
of equations:
8
>>><
>>>:@v"
@t(t;x)div(D"(x)∇v"(t;x)) +(x;v") =h(t;x); x2Ω"; t> 0;
v"(t;x) = 0; x2@Ω; t> 0;
v"(t;x) =v1(x); x2Ω"; t= 0;(2.72)
D"(x)∇v"(t;x)
="f"(t;x); x2S"; t> 0; (2.73)
and8
<
:@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)
where
f"(t;x) =
"(x)(g(x;v"(t;x))w"(t;x)): (2.75)
To deal with this case, we shall make some new assumptions on the data.
1) The diffusion matrix is dened as being
D"(x) =D(x
")
;
whereD=D(y) is a matrix-valued function on Rnwhich isY-periodic. We shall assume
that8
<
:D2L1(Ω)nn;
Dis a symmetric matrix,
for some 0<
<;
jj2D(y)
jj28; y2Rn:
2) The function =(x;v) is continuous, monotonously non-decreasing with respect to
vfor anyxand such that (x;0) = 0.
3) The function g=g(x;v) is continuously differentiable, monotonously non-decreasing
with respect to vfor anyxand withg(x;0) = 0. We suppose that there exist C0 and two
exponentsqandrsuch that
j(x;v)j C(1 +jvjq) (2.76)
and 8
>><
>>:@g
@vC(1 +jvjq);
@g
@xiC(1 +jvjr) 1in;(2.77)
with 0 q<n= (n2) and with 0 r<n= (n2) +q.
Using the theory of semilinear monotone problems (see [45] and [160]), 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 can obtain the following result:

Chapter 2. Homogenization of reactive
ows in porous media 47
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(Ω));
wherevis the unique solution of the following limit problem:
8
><
>:@v
@t(t;x)div(D0∇v) +(x;v) +F0(t;x) =h(t;x); t> 0; x2Ω;
v(t;x) = 0; t> 0; x2@Ω;
v(t;x) =v1(x); t = 0; x2Ω;(2.78)
with
F0(t;x) =
1
jY⋆j∫
@F{

(y)[g(x;v(t;x))w1(x)e(a(y)+
(y))t
(y)r(
;y)⋆g(x;v(
;x))(t)]}
d:
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; x2Ω; y2@F
w(t;x;y ) =w1(x)t= 0; x2Ω; y2@F:(2.79)
Here,D0= ((d0
ij))is the homogenized matrix, dened by:
d0
ij=1
jYj∫
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)(∇yj+ej) = 0 inY;
D(y)(∇j+ej)
= 0 on@F;
j2H1
#Y(Y⋆);∫
Y⋆j= 0;
where ei,1in, are the elements of the canonical basis in Rn.
The constant matrix D0is symmetric and positive-denite.
Remark 2.32 The solution of (2.79) can be found using the method of variation of para-
meters. We have
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)∫t
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);
with
r(;y) =e(a(y)+
(y)):

48 2.2. Nonlinear adsorption of chemicals in porous media
Remark 2.33 If we consider the case in which we have diffusion 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 diffusion constant on the surface S"and∆"is the Laplace-Beltrami
operator on S", then instead of (2.79) we get the following local partial differential 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; x2Ω; y2@F.
As already mentioned, related problems were addressed in the literature by many authors
(see, for instance, U. Hornung and W. J ager [138], U. Hornung [137], D. Cioranescu, P.
Donato and R. Zaki [65], C. Conca, J.I. D az, and C. Timofte [73], or G. Allaire and H.
Hutridurga [6]). The results presented in this section constitute a generalization of some of
the results obtained in [73] and [137], 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 would be to investigate, in multi-component porous media with
imperfect interfaces, the case of systems of reaction-diffusion equations involving nonlinear
reaction-terms and nonlinear boundary conditions. Also, it would be of interest to deal with
the case of other geometries of the porous media under consideration or with the case of more
general nonlinear, even discontinuous, kinetics.

Chapter 3
Homogenization results for
unilateral problems
The study of variational inequalities has attracted a lot of interest in the last decades due to
their applications to a large class of nonlinear boundary-value problems arising in mechanics,
biology, geology, or engineering. The theory of variational inequalities provides us with the
natural framework for analyzing the classical problem of
ow through porous media. Also,
the approximation and the numerical analysis of variational inequalities is a modern topic,
with a wide range of applications to important and difficult free boundary problems arising
in the study of
ow through porous media. We can mention here the representative papers
of G. Fichera [111], G. Signorini [202], G. Stampacchia [204], G. Stampacchia and J. L.
Lions [162], L Boccardo and P. Marcellini [39], L. Boccardo and F. Murat [40], H. Br ezis,
U. Mosco [176], R. Glowinski, J.L. Lions and R. Tr emoli eres [119], Duvaut and Lions [99],
J.T. Oden and N. Kikuchi [183], D. Kinderlehrer and G. Stampacchia [154]. For a nice and
comprehensive presentation of the theory of variational inequalities and its applications, we
refer to the monographs [45], [50], [99], [154], [119], [183], and [132]. For homogenization
results for variational inequalities, we refer, e.g., to D. Cioranescu and F. Murat [68], Yu. A.
Kazmerchuk and T. A. Mel'nyk [151], or G.A. Yosian [222].
Our goal in this chapter is to discuss some homogenization results for a class of unilateral
problems in perforated media. In a series of papers (see [51], [54], [75], [77], and [208]), we
analyzed the limit behavior of the solutions of some Signorini's type-like problems in pe-
riodically perforated domains. The classical weak formulation of such unilateral problems
involves a standard variational inequality (in the sense of [162]), corresponding to a nonlinear
free boundary-value problem. Such a model was introduced in the earliest '30 by A. Sig-
norini [202] (see also G. Fichera [111]) 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 [45] and [99].
The chapter is based on the papers [51], [54], [75], [77], and [208].
49

50 3.1. Homogenization results for Signorini's type problems
3.1 Homogenization results for Signorini's type problems
In this section, we shall present some homogenization results, obtained in [75] by using Tar-
tar's oscillating test function method, for the solutions of some Signorini's type-like problems
in"-periodically perforated domains. The main feature of these kind of unilateral problems
is the existence of a critical size of the inclusions that separates the emerging phenomena at
the macroscale. In the critical case, it is shown in [75] 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. The limit problem captures the two sources of oscillations involved in this kind
of free boundary-value problems, namely, those arising from the special 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 [75] constitute a generalization of those obtained in the the well-known
pioneering work of D. Cioranescu and F. Murat [68]. 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 considered the case in which the constraint u"0 acts
on the holes (which includes the classical Dirichlet condition u"= 0 onS"). In [75], 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".
3.1.1 The microscopic problem
Let Ω 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 to F
and thatFis star-shaped with respect to 0. Since Fis bounded, without loss of generality,
we shall assume that FY, whereY= (1
2;1
2)nis the representative cell in Rn. We set
Y=YnF. Letr:R+!R+be a continuous map, characterizing 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, let
F"
k="k+r(")F:
Also, let us denote by F"the set of all the holes contained in Ω, i.e.
F"=∪{
F"
kjF"
kΩ; k2Zn}
:

Chapter 3. Homogenization results for unilateral problems 51
Set
Ω"= ΩnF"
and
S"=[f@F"
kjF"
kΩ;k2Zng:
So,
@Ω"=@Ω[S":
Let us consider a family of inhomogeneous media occupying the region Ω, parameter-
ized by"and represented by nnmatricesA"(x) of real-valued coefficients dened on Ω.
Therefore, the positive parameter "will also dene a length scale measuring how densely the
inhomogeneities are distributed in Ω. One of our main goals in this paper was to understand
the interactions between these two sources of oscillations represented by the parameter ",
namely those coming from the geometry (and more precisely, from the size of the holes) and
those due to the inhomogeneity of the medium (the matrix A"(x) involves rapidly oscillating
coefficients).
We dene the following nonempty closed convex subset of H1(Ω"):
K"={
v2H1(Ω")jv= 0 on@Ω; v0 onS"}
: (3.3)
Our main motivation is to study the asymptotic behavior of the solution of the following
variational problem in Ω":
8
<
:ndu"2K"such that∫
Ω"A"Du"D(v"u") dx∫
Ω"f(v"u") dx8v"2K";(3.4)
wherefis a given function in L2(Ω).
The solution u"of (3.4) is also known to be characterized as being the solution of the
following nonlinear free boundary-value problem: nd a smooth function u"and two subsets
S"
0andS"
+such thatS"
0[S"
+=S"; S"
0\S"
+=∅, and
8
<
:div(A"Du") =fin Ω";
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". Thus, on S", there are two a 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) =A(x
")
;

52 3.1. Homogenization results for Signorini's type problems
whereA=A(y) is a matrix-valued function on Rnwhich isY-periodic and satises the
following conditions:
8
<
:A2L1(Ω)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 exis-
tence and uniqueness result of J. L. Lions and G. Stampacchia [162] that (3.4) is a well-posed
problem.
3.1.2 The macroscopic models
Several situations can occur depending on the asymptotic behavior of the size of the holes
and there exists a critical size that separates different behaviors of the solution u"as"!0.
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 [75] that the solution u"of problem (3.4), properly
extended to the whole of Ω, converges to the unique solution of a Dirichlet problem in Ω,
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 [75]) 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(Ω);
whereuis the unique solution of
8
<
:u2H1
0(Ω);∫
ΩA0DuDv dx⟨
0u;v⟩
H1(Ω);H1
0(Ω)=∫
Ωfvdx8v2H1
0(Ω):(3.7)
Here,A0is the homogenized matrix, whose entries are dened as follows:
a0
ij=1
jYj∫
Y(
aij(y) +aik(y)@j
@yk)
dy;
in terms of the functions j; j= 1;:::;n; solutions of the cell problems
8
<
:divy(A(y)Dy(yj+j)) = 0 inY;
jYperiodic

Chapter 3. Homogenization results for unilateral problems 53
and0is given by
0= inf
w2H1(Rn){∫
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-
deed, in (3.4) 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 effect of the unilateral condition u"0imposed on S", which can be seen by
the fact that the strange term charges only the negative part of the homogenized solution u;
it is only its negative part uthat is penalized at the limit.
The proof of Theorem 3.1, given in [75], is based on the use of a technical result of E. De
Giorgi [85] for matching boundary conditions for minimizing sequences. This result allows
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 coming from the periodic oscillating structure of the
medium and those arising from the oscillations of the boundaries of the holes.
Besides this critical case , there are three other situations that are considered in [75].
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 Ω associated
to the matrix A. 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"becomes 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 (stated 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 Ω,
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 Ω and, on the other one,
their special size affects the homogenized medium. More precisely, for the case of holes of
the same size as the period, we have the following result (see [75]):
Theorem 3.3 (Theorem 4.6 in [75]) 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(Ω);

54 3.2. Elliptic problems in perforated domains with mixed-type boundary conditions
whereuis the unique solution of
8
>>>><
>>>>:u2H1
0(Ω); u0inΩ;
∫
ΩA0DuDu dx2∫
Ωfudx∫
ΩA0DvDv dx2∫
Ωfvdx;
8v2H1
0(Ω); v0inΩ:(3.9)
Here,A0= (a0
ij)is the homogenized matrix, dened by
a0
ij=1
jYj∫
Y(
aij(y) +aik(y)@j
@yk)
dy;
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⋆);∫
Y⋆jdy= 0;
where ei,1in, are the elements of the canonical basis in Rn. The constant matrix A0
is symmetric and positive-denite.
Let us notice that the variational inequality in (3.9) can be written as
∫
ΩA0DuD (vu)dx∫
Ωf(vu)dx:
As mentioned before, the method we followed in [75] is the energy method of L. Tar-
tar [205]. 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 [80] and
of A. Braides and A. Defranceschi [43]. Also, as we shall see in the next section, another
way to obtain convergence results for such problems is to use the recently developed periodic
unfolding method. This method was introduced, for xed domains, by D. Cioranescu, A.
Damlamian, and G. Griso in [57], [58] and by A. Damlamian in [83]. Their results were
extended to perforated domains by D. Cioranescu, P. Donato, and R. Zaki [64], [66] and,
further, by D. Cioranescu, A. Damlamian, G. Griso, andD. Onofrei in [59], by D. Onofrei in
[185] and by A. Damlamian and N. Meunier in [84] 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 perforated
domains with mixed-type boundary conditions
In this section, we present some results obtained, via the periodic unfolding method, in [51]
and generalizing the corresponding results in [75]. More precisely, the asymptotic behavior

Chapter 3. Homogenization results for unilateral problems 55
of a class of elliptic equations with highly oscillating coefficients, in an "-periodic perforated
structure, with two holes of different sizes in each period, will be analyzed. Two distinct
conditions, one of Signorini's type and another one of Neumann type, are imposed on the
corresponding boundaries of the holes, while on the exterior xed boundary of the perforated
domain, an homogeneous Dirichlet condition is prescribed. As mentioned in [51], the main
characteristic of this type of problems is represented by the existence of a critical size of the
perforations that separates different phenomena arising at the macroscale. In this critical
case, it was proven in [51] that the homogenized problem contains two additional terms
generated by the particular geometry of the domain. These new terms, a right-hand side
term and, respectively, a strange one, capture the two sources of oscillations involved in this
problem, namely 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 and F. Murat [68], putting into evidence, in the case of critical holes,
the appearance of a strange term. Their results were extended, using different techniques, to
heterogeneous media by N. Ansini, A. Braides [19], G. Dal Maso and F. Murat [82] and D.
Cioranescu, A. Damlamian, G. Griso, and D. Onofrei in [59]. Recently, A. Damlamian and
N. Meunier [84] 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 and P. Donato [74] and by D. Onofrei [185]. For
problems with Robin or nonlinear boundary conditions we refer, for instance, to D. Cioranescu
and P. Donato [61], D. Cioranescu, P. Donato, and H. Ene [63], D. Cioranescu, P. Donato,
and R. Zaki [64] and A. Capatina and H. Ene [52]. Also, for Signorini's type problems we
mention Yu. A. Kazmerchuk and T. A. Mel'nyk [151]. The homogenization of problems
involving perforated domains with two kinds of holes of various sizes was recently considered
by D. Cioranescu and O. Hammouda [67].
The non-standard feature of the problem we present here is given by the presence, in
each period, of two holes of different sizes and with different conditions (3.10) 2;3imposed on
their boundaries. More precisely, we consider the case of Signorini and, respectively, criti-
cal Neumann holes. The Signorini condition (3.10) 2(see [202]) implies that the variational
formulation (3.11) of our problem is expressed as an inequality, which creates further difficul-
ties. 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 and J.L. Lions [99] and U. Hornung [137].
3.2.1 Setting of the microscopic problem
Let us brie
y describe now the new geometry of the problem. Let Ω Rn,n3, be a
bounded open set with j@Ωj= 0 and let Y=(
1
2;1
2)nbe the reference cell. We deal with
an"Y-periodic perforated structure with two types of holes: some of size "1and the other

56 3.2. Elliptic problems in perforated domains with mixed-type boundary conditions
ones of size "2, with1and2depending on the small parameter "and going to zero as "
tends to zero. More precisely, we consider two smooth open sets BandFsuch thatBY,
FYandB\F=  and we denote the above mentioned holes by
B"1=∪
2Zn"(+1B);
F"2=∪
2Zn"(+2F):
We assume that
Y12=Yn(1B[2F);
the region occupied by the material in the unit cell, is connected. The perforated domain
Ω";12is dened by
Ω";12= Ωn(B"1[F"2) ={
x2Ωj{x
"}
Y2Y12}
:
As in Section 3.1.1, let A2L1(Ω)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(Ω), we consider the
following microscopic problem:
8
>>>>>>><
>>>>>>>:div (A"∇u";12) =fin Ω ";12;
u";120; A"∇u";12
B0; u ";12A"∇u";12
B= 0 on@B"1;
A"∇u";12
F=g"2on@F"2;
u";12= 0 on@extΩ";12;(3.10)
where
A"(x) =A(x
")
and
g"2(x) =g(1
2{x
"}
Y)
a.e.x2@F"2:
In (3.10),BandFare the unit exterior normals to the sets B"1and, respectively, F"2.
We introduce the space
V"
12=fv2H1(Ω";12)jv= 0 on@extΩ";12g
and the convex set
K"
12=fv2V"
12jv0 on@B"1g:

Chapter 3. Homogenization results for unilateral problems 57
Then, the variational formulation of (3.10) is given by the following variational inequality:
(P";12)8
>>>>>>>><
>>>>>>>>:ndu";122K"
12such that
∫
Ω";12A"∇u";12
(∇v ∇u";12) dx∫
Ω";12f"(vu";12) dx
+∫
@F"2g"2(vu";12) ds8v2K"
12:(3.11)
Standard results for variational inequalities (see, for example, [44], [204], and [119]) ensure
the existence and the uniqueness of a weak solution of the problem (3 :11).
Our main goal is to obtain the asymptotic behavior of the solution of problem (3.11) when
"; 1; 2!0. Following [51], 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 signies that we deal with the case of critical sizes both for the Signorini and for the
Neumann holes. Due to (3.12), we can write "!0 instead of writing ( ";1;2)!(0;0;0).
3.2.2 The limit problem
For stating the main convergence result for this problem, we introduce the functional space
KB=fv2L2(Rn) ;∇v2L2(Rn); v= constant on Bg
where 2is the Sobolev exponent2n
n2associated to 2. Also, for i= 1;:::;n , letibe the
solution of the cell problem
8
><
>:i2H1
per(Y);
∫
YA∇(iyi)
∇ϕdy= 08ϕ2H1
per(Y)(3.13)
andbe the solution of the problem
8
>><
>>:2KB; (B) = 1;
∫
RnnBA(0)∇
∇vdz= 08v2KBwithv(B) = 0:(3.14)
For the special geometry of this problem, we need to introduce, following [56] and [59], 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,

58 3.2. Elliptic problems in perforated domains with mixed-type boundary conditions
we have to introduce some notation. For x2Rn, we denote by [ x]Yits integer part k2Zn,
such thatx[x]Y2Yand we set fxgY=x[x]Yforx2Rn. So, forx2Rn, we have
x="([x
"]
+{x
"})
:
LetYk=Y+k, fork2Zn. We consider the following sets:
bZ"={
k2Znj"YkΩ}
;bΩ"= int∪
k2bZ"(
"Yk)
;"= ΩnbΩ":
Denition 3.4 For any function φ2Lp(Ω), withp2[1;1), we dene the periodic unfolding
operator T":Lp(Ω)!Lp(ΩY)by the formula
T"(φ)(x;y) =8
<
:φ(
"[x
"]
Y+"y)
for a.e. (x;y)2bΩ"Y;
0 for a.e. (x;y)2"Y:
The operator T"is linear and continuous from Lp(Ω) toLp(ΩY). We recall here some
useful properties of this operator (see, for instance, [56]):
(i) ifφand are two Lebesgue measurable functions on Ω, one has
T"(φ ) =T"(φ)T"( );
(ii) for every φ2L1(Ω), one has
1
jYj∫
ΩYT"(φ)(x;y) dxdy=∫
bΩ"φ(x) dx=∫
Ωφ(x) dx∫
"φ(x) dx;
(iii) if fφ"g L2(Ω) is a sequence such that φ"!φstrongly in L2(Ω), then
T"(φ")!φstrongly in Lp(ΩY);
(iv) ifφ2L2(Y) isY-periodic and φ"(x) =φ(x="), then
T"(φ")!φstrongly in L2(ΩY);
(v) ifφ"⇀φ weakly inH1(Ω), then there exists a subsequence and bφ2L2(Ω;H1
per(Y))
such that
T"(∇φ")⇀∇xφ+∇ybφweakly inL2(ΩY):
For domains with small holes, we need to introduce an unfolding operator depending on
two parameters "and. We recall now its denition (for details, see [59]). To this end, let
us consider domains with "Y-periodically distributed holes of size ", for>0 going to zero
with"tending to zero. More precisely, for a given open set BY, we denote Y
=YnB.
The perforated domain Ω
"is dened by
Ω
"={
x2Ω{x
"}
2Y
}
:
If we consider functions which vanish on the whole boundary of the perforated domain, i.e.
functions belonging to H1
0(Ω
"), then we can extend them by zero to the whole of Ω. In this
case, we shall not distinguish between functions in H1
0(Ω
") and their extensions in H1
0(Ω).

Chapter 3. Homogenization results for unilateral problems 59
Denition 3.5 For anyφ2Lp(Ω), withp2[1;1), we dene the periodic unfolding operator
T"by the formula
T"(φ)(x;z) =8
<
:T"(φ)(z;z)if(x;z)2bΩ"1
Y;
0 otherwise:
By using the change of variable z= (1=)y, one can obtain similar properties for the operator
T"to those stated for T"(see [59]).
Further, following [67] and [185], we brie
y recall the denition of the boundary unfolding
operator Tb
".
Denition 3.6 For anyφ2Lp(@B"), withp2[1;1), we dene the boundary unfolding
operator Tb
"by the formula
Tb
"(φ)(x;z) =φ(
"[x
"]
Y+"z)
a.e. forx2Rn;z2@B:
The main convergence result obtained in [51] is stated in the following theorem.
Theorem 3.7 (Theorem 3.1 in [51]) Let u";12be the solution of the variational inequality
(3.11). Under the above hypotheses, there exists u2H1
0(Ω)such that
T"(u";12)⇀u weakly inL2(Ω;H1(Y)); (3.15)
whereu2H1
0(Ω)is the unique solution of the homogenized problem
8
>>>>>><
>>>>>>:u2H1
0(Ω);
∫
ΩAhom∇u
∇φdxk2
1∫
Ωuφdx =∫
Ωfφdx
+k2j@FjM@F(g)∫
Ωφdx8φ2H1
0(Ω):(3.16)
In (3.16),Ahomis the homogenized matrix, dened, in terms of the solution iof (3.13), by
Ahom
ij=∫
Y(
aij(y)n∑
k=1aik(y)@j
@yk(y))
dy
andis the capacity of the set B, given by
=∫
RnnBA(0)∇z
∇zdz;
whereveries (3.14).
Remark 3.8 The limit problem (3.16) contains two extra terms, generated by the suitable
sizes of our holes. Also, let us remark in (3.16) the spreading effect 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.

60 3.2. Elliptic problems in perforated domains with mixed-type boundary conditions
Remark 3.9 Fork1= 0, the extra term generated by the Signorini holes disappears at the
limit, while for k2= 0the contribution of the Neumann holes vanishes at the macroscale.
Proof of Theorem 3.7. The variational inequality (3.11) is equivalent to the following
minimization problem:
{ndu";122K"
12such that
J"
12(u";12)J"
12(v)8v2K"
12;(3.17)
where
J"
12(v) =1
2∫
Ω";12A"∇v
∇vdx∫
Ω";12fvdx∫
@F"2g"2vds: (3.18)
Let us prove that
lim sup
"!0J"
12(u";12)J0(φ)8φ2 D(Ω); (3.19)
where
J0(φ) =1
2∫
ΩAhom∇φ
∇φdx+1
2k2
1∫
Ω(φ)2dx
∫
Ωfφdx+k2j@FjM@F(g)∫
Ωφdx:(3.20)
Ifφ2 D(Ω) andiis the solution of the problem (3.13), we set
h"(x) =φ(x)"n∑
i=1@φ
@xi(x)i(x
")
:
If we takev"1=h+
"w"1h
", where
w"1(x) = 1(1
1{x
"}
Y)
8x2Rn;
andis given by (3.14), we obtain
J"
12(v"1) =I1
"I2
";
where
I1
"=1
2∫
Ω";12A"(∇h+
"w"1∇h
"h
"∇w"1)
(∇h+
"w"1∇h
"h
"∇w"1) dx;
I2
"=∫
Ω";12f"(h+
"w"1h
")dx+∫
@F"2g"(h+
"w"1h
") ds:
Using the periodic unfolding operators T"andT"1(see [58] and [59]), we get
T"(A")(x;y) =A(y) in Ω Y;
T"1(w"1)(x;z) =T"(w"1)(x;1z) = 1(z) in Ω Rn;
T"1(∇w"1)(x;z) =1
"1∇z(z) in Ω Rn:

Chapter 3. Homogenization results for unilateral problems 61
We also have {
w"1⇀1 weakly in H1(Ω);
h"!φstrongly in H1(Ω):(3.21)
Taking into account the properties of the unfolding operator T"andT"1, we get the following
convergences 8
>><
>>:T"(h")!φstrongly in L2(ΩY);
T"(∇h")! ∇ xφ+∇yφ1strongly in L2(ΩY);
T"1(∇h")! ∇ xφ+∇yφ1strongly in L2(ΩRn);(3.22)
where
φ1=n∑
i=1@φ
@xii:
By unfolding and by using the fact that f∇h
"g"is bounded in ( L2(Ω))n, we can pass to the
limit in the unfolded form of I1
"and we get
lim
"!0I1
"=1
2∫
ΩYA(∇φ+∇yφ1)
(∇φ+∇yφ1) dxdy+
1
2k2
1∫
Ω(RnnB)A(0)(φ)2∇z
∇zdxdz;
which, together with (3.13), yields
lim
"!0I1
"=1
2∫
ΩYAhom∇φ
∇φdxdy +1
2k2
1∫
Ω(RnnB)A(0)(φ)2∇z
∇zdxdz:(3.23)
Exactly like in [67], i.e. using the boundary unfolding operator Tb
"2, we can pass to the limit
inI2
"and we obtain
lim
"!0I2
"=∫
Ωfφdx+k2j@FjM@F(g)∫
Ωφdx: (3.24)
Putting together (3.23) and (3.24), we are led to
lim
"!0J"
12(v"1) =J0(φ)8φ2 D(Ω) (3.25)
and, thus, we get (3.19).
Let us show 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
∥u";12∥H1(Ω";12)C: (3.27)

62 3.2. Elliptic problems in perforated domains with mixed-type boundary conditions
Sinceu";122V"
12, we can suppose that, up to a subsequence, there exists u2H1
0(Ω) such
that 8
<
:T"(u";12)⇀u weakly inL2(Ω;H1(Y));
∥u
";12u∥L2(Ω";12)!0;
T"(u
";12)!ustrongly in L2(ΩY):(3.28)
It is not difficult to check that we have
lim inf
"!0∫
Ω";12A"∇u+
";12
∇u+
";12dx∫
ΩAhom∇u+
∇u+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
"!0∫
Ω";12A"∇u
";12
∇u
";12dx∫
ΩAhom∇u
∇udx+k2
1∫
Ω(u)2dx:(3.30)
Since ∫
Ω";12A"∇(u
";12h"w"1)
∇(u
";12h"w"1) dx0;
we have
∫
Ω";12A"∇u
";12
∇u
";12dx ∫
Ω";12A"(h")2∇w"1
∇w"1dx
∫
Ω";12A"(w"1)2∇h"
∇h"dx+ 2∫
Ω";12A"h"∇u
";12
∇w"1dx
+2∫
Ω";12A"w"1∇u
";12
∇h"dx2∫
Ω";12A"h"w"1∇w"1
∇h"dx
(
n
21
1
")2∫
ΩRnT"1(A")(T"1(h"))2∇z
∇zdxdz
∫
ΩYT"(A")T"(∇h")
T"(∇h")(T"(w"1))2dxdy
2n
21
1
"∫
ΩRnT"1(A")T"1(h")[
n=2
1T"1(∇u
";12)]

∇z(z) dxdz
+2∫
ΩYT"(A")T"(w"1)T"(∇u
";12)
T"(∇h") dx
+2n
21
1
"n=2
1∫
ΩRnT"1(A")T"1(h")(1(z))∇z
T"1(∇h") dxdz:
From (3.27) and (3.28) and the properties of the unfolding operators T"andT"1(see [59]),
it follows that there exist u12L2(Ω;H1
per(Y)) and U12L2(Ω;L2
loc(Rn)) such that
8
<
:T"(∇u
";12)⇀∇u+∇yu1 weakly inL2(ΩY);
n
2
1T"1(∇u
";12)⇀∇zU1 weakly inL2(ΩRn):(3.31)

Chapter 3. Homogenization results for unilateral problems 63
Therefore, we get
lim inf
"!0∫
Ω";12A"∇u
";12
∇u
";12dx k2
1∫
Ωφ2dx
∫
ΩYA(∇φ+∇yφ1)
(∇φ+∇yφ1) dxdy2k1∫
Ω(RnnB)A0φ∇zU1
∇zdxdz
+2∫
ΩYA(∇u+∇yu1)
(∇φ+∇yφ1)dxdy 8φ2H1
0(Ω):
(3.32)
SinceT"1(u
";12) = 0 on Ω B, we have U1= 0 on Ω B. Thus,W1=U1k1u2L2(Ω;KB).
On the other hand, from the cell problem (3.14), we obtain
divz(A(0)∇z) = 0 in D′(Ω(RnnB))
which, by Stokes formula, leads to
∫
RnnBA(0)∇z
∇z dz= (B)∫
@BA(0)∇z
Bds8 2KB: (3.33)
For almost every x2Ω,W1(x;
)2KBand, so, (3.33) gives
∫
RnnBA(0)∇z
∇zW1dz=W1(x;B)∫
@BA(0)∇z
Bds:
Since ∇zW1=∇zU1andU1(x;B) = 0, we obtain
∫
RnnBA(0)∇z
∇zU1dz=k1u∫
@BA(0)∇z
Bds=k1u
which implies that
2k1∫
Ω(RnnB)A(0)φ∇zU1
∇zdx= 2k2
1∫
Ωuφdx:
Takingφ=uin (3.32) and using the fact thatn∑
i=1@u
@xii=u1, we obtain (3.30).
Finally, using (3.19) and (3.26), by standard density arguments, we deduce
lim
"!0J"
12(u";12) =J0(u)J0(φ)8φ2H1
0(Ω): (3.34)
Sinceis non-negative, by using Lax-Milgram theorem, it follows that the minimum point
of the functional J0is unique and this implies that the whole sequence T"(u";12) converges
tou.
Using a classical technique (see, for instance, [183] and [68]), one can prove that the
functional
P(v) =1
2∫
Ω(v)2dx8v2H1
0(Ω)

64 3.2. Elliptic problems in perforated domains with mixed-type boundary conditions
is Fr echet (and, hence, G^ ateaux) differentiable and its gradient is given by
P′(u)
v=∫
Ωuvdx8u;v2H1
0(Ω):
Therefore, the functional J0is G^ ateaux differentiable on H1
0(Ω), which ensures the equivalence
of the minimization problem
J0(u) = min
φ2H1
0(Ω)J0(φ) (3.35)
with the problem (3.16). This completes the proof of Theorem 3.7.
Remark 3.10 From (3.27), it follows that there exists an extension bu";12of our solution
to the whole of Ω, positive on the Signorini holes (see [75]), such that
bu";12⇀u weakly inH1
0(Ω): (3.36)
For example, 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 further extend it in a standard way (see, e.g., [74]) inside the Neumann holes.
In fact, the use of the periodic unfolding method allows us to work without extending our
solution to the whole of the domain Ω.
Remark 3.11 We can deal in a similar manner with the case in which in (3.10) Ais a
general matrix satisfying the usual conditions of boundedness and coercivity. In this case, we
have to suppose, like in [59] or [84], that there exist two matrix elds AandA0such that
T"(A")(x;y)!A(x;y)a.e. in ΩY
and
T"1(A")(x;z)!A0(x;z)a.e. in Ω(RnnB):
The only difference consists in the fact that now 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 5.4 we shall brie
y mention some
related models, obtained via the periodic unfolding method in [54]. More precisely, we shall
be concerned with the derivation of macroscopic models for some elasticity problems in pe-
riodically perforated domains with rigid inclusions of the same size as the period.

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 media exhibit-
ing multiple scales. It is widely recognized that multiscale techniques represent an essential
tool for understanding the macroscopic properties of such complex systems. A lot of efforts
have been made in the last years to obtain suitable mathematical models in biology. Still,
despite the all these efforts, many rigorous mathematical models can be viewed as toy models ,
being far from capturing the complexity of the phenomena involved in the biological processes.
Much work needs to be done to develop a unied view of the various mechanisms governing
biological systems and to understand the causal relationships among different parts of the
analyzed systems. Using multiscale techniques, it is necessary to develop suitable models at
various scales and, more important, to bridge the gaps between different methodologies and
models applied at different scales.
In this chapter, we shall present some homogenization results for a series of problems
arising in the mathematical modeling of various reaction-diffusion processes in biological
tissues. This chapter is based on the papers [210], [211], [212], [217], [219], [221], and [218].
4.1 Homogenization results for ionic transport phenomena in
periodic charged media
We start this chapter by presenting some homogenization results for ionic transport pheno-
mena in periodic charged porous media. These results were obtained, via the periodic unfold-
ing method, in [217]. More precisely, we shall describe the effective behavior of the solution
of a system of coupled partial differential equations arising in the modeling of ionic transfer
phenomena, coupled with electrocapillary effects, in periodic charged porous media. The so-
called Nernst-Planck-Poisson system was proposed by W. Nernst and M. Planck (see [197])
for describing the potential difference in galvanic cells. Such a model 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
65

66 4.1. Ionic transport phenomena in periodic charged media
the physical aspects behind these models and for a review of the recent relevant literature,
we refer to [114], [143], [197], [199], and [200].
We shall 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. Due to the complexity of the
geometry and of the governing equations, an asymptotic approach becomes necessary for
a reasonable description of the solution of such a problem. Using the periodic unfolding
method, we show that the effective behavior of the solution of our problem is governed by
a new coupled system of equations (see (4.11)-(4.14)). In particular, the evolution of the
macroscopic electrostatic potential is described by a new law, which is similar to the well-
known Grahame's law (see [114] and [130]). An advantage of using such an approach based
on unfolding operators 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).
Besides, the homogenized equations being dened on a xed domain Ω and having simpler
coefficients will be easier to be handled numerically than the original equations.
Related problems have been addressed, using different techniques, in [114], [143] or [199].
As already mentioned, our approach in [217] 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 brie
y 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 2.1, but we shall use a specic terminology for this case. Thus,
we consider a bounded connected smooth open set Ω in Rn, with j@Ωj= 0 and with n2.
We shall deal here only with the physically relevant cases n= 2 orn= 3. The reference
cellY= (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 Ω"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 @Ω, it follows that S"\@Ω =∅.
In this periodic microstructure, we consider the Poisson-Nernst-Planck system, with suit-
able boundary and initial conditions. The diffusion in the
uid phase is described by the
Nernst-Planck equations, while the electric potential is governed by the Poisson equation.
Also, we include electrocapillary effects in our analysis.
More precisely, if we denote by [0 ;T], withT > 0, the time interval of interest, we shall
analyze the effective behavior, as the small parameter "!0, of the solution of the following

Chapter 4. Mathematical models in biology 67
system:
8
>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>:∆  "=c+
"c
"+D in (0;T)Ω";
∇"
=""G(x;") on (0;T)S";
∇"
= 0 in (0 ;T)@Ω;
@c
"
@t ∇
(∇c
"c
"∇") =F(c+
";c
") in (0;T)Ω";
(∇c
"c
"∇")
= 0 on (0 ;T)S";
(∇c
"c
"∇")
= 0 on (0 ;T)@Ω;
c
"=c
0inft= 0g Ω":(4.1)
Here,is the unit exterior normal to Ω", "is the electrostatic potential, c
"are the con-
centrations of the ions (or the density of electrons and holes in the case of van Roosbroeck
model),D2L1(Ω) is the given doping prole, Gis a nonlinear function describing the effect
of the electrical double layer phenomenon arising at the interface S"andFrepresents the
reaction term.
We remark 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
different scalings in (4.1), see [143] and [220].
We assume 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 differentiable, monotonously increasing with respect to sfor anyxand with
G(x;0) = 0. Also, we assume that, for n3, there exist C0 and two exponents pandm
such that 8
>>>><
>>>>:@G
@sC(1 +jsjp);
@G
@xiC(1 +jsjm) 1in;(4.2)
with 0 pn=(n2) and 0 m<n= (n2)+p. By using a regularization procedure, such as
Yosida approximation (see [73]), the hypothesis concerning the smoothness of the nonlinearity
Gcan be relaxed. For instance, we can deal with the case of single or multivalued maximal
monotone graphs, as in [73]. Also, our results can be obtained, under our assumptions,
without imposing any growth condition (see [209]).
In practical applications, based on the Gouy-Chapman theory, one can use Grahame
equation (see [114] and [130]) in which
G(s) =K1sinh(K2s); K 1;K2>0:

68 4.1. Ionic transport phenomena in periodic charged media
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!.
For the reaction terms, in [216] we addressed the case in which
F(c+
";c
") =∓(c+
"c
"):
Of course, the case in which
F(c+
";c
") =∓(a"c+
"b"c
");
with
a"(x) =a(x
")
; b"(x) =b(x
")
;
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 [217], we were concerned 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 particular,
this setting includes the case treated in [108], 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 [73], [109] or [220], we can study other relevant types of reaction
rates, such as those appearing in the so-called Auger generation-recombination model or in
theShockley-Read-Hall model (see [134]). More precisely, we can deal with the case in which
F=F(u;v) is a continuously differentiable 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 [137], [139] and [220].
We suppose that the initial data are non-negative and bounded independently with respect
to"and ∫
Ω"(c+
0c
0+D) dx="∫
S""G(x;") ds:
Moreover, we suppose that the mean value in Ω"of the potential  "is zero.
From the Nernst-Planck equation, it is not difficult to see that the the total mass
M=∫
Ω"(c+
"+c
") dx
is conserved and suitable physical equilibrium conditions are veried, both at the microscale
and at the macroscale (see, for details, [114] and [220]). Let us mention that, for simplifying
the notation, we have eliminated in system (4.1) some constant physical relevant parameters.

Chapter 4. Mathematical models in biology 69
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
diffusivity 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 Ω, with
suitable transmission conditions at the interface S", as in [114] or [200].
The weak formulation of problem (4.1) is as follows: nd ( "; c+
"; c
"), with
8
>>>><
>>>>:"2L1(0;T;H1(Ω"));
c
"2L1(0;T;L2(Ω"))\L2(0;T;H1(Ω"));
@c
"
@t2L2(0;T; (H1(Ω"))′)(4.3)
such that, for any t>0 and for any φ1; φ22H1(Ω"), the triple ( "; c+
"; c
") satises:
∫
Ω"∇"
∇φ1dx∫
S"∇"
φ1d=∫
Ω"(c+
"c
"+D)φ1dx; (4.4)
⟨@c
"
@t; φ2⟩
(H1)′;H1+∫
Ω"(∇c
"c
"∇")
∇φ2dx=∫
Ω"F(c+
";c
")φ2dx (4.5)
and
c
"(0;x) =c
0(x) in Ω": (4.6)
The variational problem (4.3)-(4.6) has a unique weak solution ( "; c+
"; c
") (see [114], [143]
or [220]). Moreover, like in [143], we can prove that the concentrations are non-negative. As
a matter of fact, they are bounded from below and above, uniformly in ".
Under the above hypotheses, it follows that there exists a constant C2R+, independent
of", such that the following a priori estimates hold true:
∥"∥L2((0;T)Ω")+∥∇"∥L2((0;T)Ω")C
max
0tT∥c
"∥L2(Ω")+ max
0tT∥c+
"∥L2(Ω")+∥∇c
"∥L2((0;T)Ω")+∥∇c+
"∥L2((0;T)Ω")+

@c
"
@t

L2(0;T;(H1(Ω"))′)+

@c
"
@t

L2(0;T;(H1(Ω"))′)C:
Our goal is to obtain, via the periodic unfolding method, the effective behavior, as "!0, of
the solution ( "; c+
"; c
") of problem (4.3)-(4.6).
Let us brie
y recall here the denition of the unfolding operator T"introduced in [58]
and [64] 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 [57], [56], [64], and
[66].

70 4.1. Ionic transport phenomena in periodic charged media
For any Lebesgue measurable function φon Ω", the periodic unfolding operator T"is the
linear operator dened by
T"(φ)(x;y) =8
<
:φ(
"[x
"]
Y+"y)
for a.e. (x;y)2bΩ"Y;
0 for a.e. ( x;y)2"Y:
The periodic unfolding operator T"has similar properties as the corresponding operator T"
dened for xed domains in Section 3.2.2.
Using the properties of the unfolding operator T"and the above a priori estimates, it
is not difficult to see that there exist  2L2(0;T;H1(Ω)),b2L2((0;T)Ω;H1
per(Y)),
c2L2(0;T;H1(Ω)),bc2L2((0;T)Ω;H1
per(Y)), such that, up to a subsequence,
T"(")⇀ weakly in L2((0;T)Ω;H1(Y)); (4.7)
T"(∇")⇀∇ +∇yb weakly in L2((0;T)ΩY); (4.8)
T"(c
")!cstrongly in L2((0;T)Ω;H1(Y)); (4.9)
T"(∇c
")⇀∇c+∇ybcweakly inL2((0;T)ΩY): (4.10)
4.1.2 The homogenized problem
In order to obtain the needed asymptotic behavior of the solution of our microscopic model,
we shall pass to the limit, with "!0, in the variational formulation of problem (4.1). The
convergence result is stated in the next theorem.
Theorem 4.1 (Theorem 1 in [217]) Under the above hypotheses, the solution ("; c+
";c
")
of system (4.1) converges, in the sense of (4.7)-(4.10), as "!0, to the unique solution
(; c+;c)of the following macroscopic problem in (0;T)Ω:
8
>><
>>:div(D0∇) +1
jYj0G=c+c+D;
@c
@tdiv(D0∇cD0c∇) =F
0;(4.11)
with the boundary conditions on (0;T)@Ω:
{
D0∇
= 0;
(D0∇cD0c∇)
= 0(4.12)
and the initial conditions
c(0;x) =c
0(x);8×2Ω: (4.13)
Here,
0=∫
@F(y) ds;
F
0=F(c+;c) =∓g(c+c)

Chapter 4. Mathematical models in biology 71
andD0=(
d0
ij)
is the homogenized matrix, dened as follows:
d0
ij=1
jYj∫
Y(
ij+@j
@yi(y))
dy;
withj; j= 1;:::;n; solutions of the cell problems
8
>>>><
>>>>:j2H1
per(Y);∫
Yj= 0;
∆j= 0 inY;
(∇j+ej)
= 0 on@F(4.14)
andei,1in, the vectors in the canonical basis of Rn.
Proof. In order to prove Theorem 4.1, let us rst take in the Poisson equation (4.4) the
test function
φ1(t;x) = 0(t;x) +" 1(
t;x;x
")
;
with 02 D((0;T);C1(Ω)) and 12 D((0;T)Ω;H1
per(Y)). By unfolding, we obtain
∫T
0∫
ΩYT"(∇")T"(∇( 0+" 1)) dxdydt+
∫T
0∫
Ω@FT"()T"(G("))T"( 0+" 1)) dxdsdt=
∫T
0∫
ΩYT"(c+
"c
"+D)T"( 0+" 1)) dxdydt: (4.15)
Using the above convergence results, we can easily compute the limit of the linear terms in
(4.15) dened on Ω Y(see, for instance, [58], [73] and [220]). For the term containing
the nonlinear function G, let us notice that, exactly like in [73], one can show that if Ris a
continuously differentiable function, monotonously increasing, with R(x;v) = 0 if and only if
v= 0 and fullling the assumption (4.2), then, for any w"⇀w weakly inH1
0(Ω), we have
R(x;w")⇀R(x;w);
weakly inW1;p
0(Ω), where
p=2n
q(n2) +n:
Using the properties of the unfolding operator T"and Lebesgue's convergence theorem, we
get
∫T
0∫
Ω@FT"()T"(G(x;"))T"( 0+" 1)) dxdsdt!0∫T
0∫
ΩG(x;) 0dxdt:
Therefore, for "!0, we obtain:
∫T
0∫
ΩY(∇(t;x) +∇yb(t;x;y )) (∇ 0(t;x) +∇y 1(t;x;y )) dxdydt+

72 4.1. Ionic transport phenomena in periodic charged media
0∫T
0∫
ΩG(x;(t;x)) 0(t;x) dxdt=
∫T
0∫
ΩY(c+(t;x)c(t;x) +D(x)) 0(t;x) dxdydt: (4.16)
By density, (4.16) is valid for any 02L2(0;T;H1(Ω)) and 12L2((0;T)Ω;H1
per(Y)).
Taking 0(t;x) = 0, we get
8
><
>:∆yb(t;x;y ) = 0 in (0 ;T)ΩY;
∇yb
=∇ x(t;x)
on (0;T)Ω@F;
b(t;x;y ) periodic in y:
By linearity, we obtain
b(t; x; y ) =n∑
j=1j(y)@
@xj(t; x); (4.17)
wherej; j=1; n, are the solutions of the local problems (4.14).
Taking 1(t;x;y ) = 0, integrating with respect to the variable xand using (4.17), we
easily get the homogenized problem for the electrostatic potential .
In a second step, taking in the Nernst-Planck equation the test function
φ2(t;x) = 0(t;x) +" 1(
t;x;x
")
;
with 02 D((0;T);C1(Ω)) and 12 D((0;T)Ω;H1
per(Y)), we have:
∫T
0∫
ΩYT"(c
")T"(@
@t( 0+" 1))
dxdydt+
∫T
0∫
ΩYT"(∇c
"c
"∇")T"( 0+" 1)) dxdydt=
∫T
0∫
ΩYT"(F(c+
";c
"))T"( 0+" 1) dxdydt:
Passing to the limit with "!0, we obtain
∫T
0∫
ΩYc(t;x)@
@t 0(t;x) dxdydt+
∫T
0∫
ΩY(∇c(t;x) +∇ybc(t;x;y ))(∇ 0(t;x) +∇y 1(t;x;y )) dxdydt=
∫T
0∫
ΩYF
0(c+;c) 0(t;x) dxdydt: (4.18)
Using again standard density arguments, (4.18) can be written for any 02L2(0;T;H1(Ω))
and 12L2((0;T)Ω;H1
per(Y)).
Then, 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 (4.3)-(4.6) (see [143] and [220]),
the whole sequences of microscopic solutions converge to the solution of the homogenized
problem and this completes the proof of Theorem 4.1.

Chapter 4. Mathematical models in biology 73
4.2 Multiscale analysis of a carcinogenesis model
In this section, we shall focus on the results obtained in [221], where our goal was to analyze,
using homogenization techniques, the effective behavior of a coupled system of reaction-
diffusion equations arising in the modeling of some biochemical processes contributing to
carcinogenesis in living cells. We shall be concerned with the carcinogenic effects produced
in the human cells by Benzo-[a]-pyrene (BP) molecules. Such molecules can be found in
coal tar, cigarette smoke, charbroiled food, etc. To understand the complex behavior of
these molecules, mathematical models including reaction-diffusion processes and binding and
cleaning mechanisms have been developed. As in [126], we consider a simplied setting in
which BP molecules enter in the cytosol inside of a human cell. There, they are allowed
to diffuse freely, but they cannot enter inside the nucleus. Moreover, they can bind to
the surface of the endoplasmic reticulum (ER), where chemical reactions, produced by the
enzyme system called MFO (microsomal mixed-function oxidases), take place. As a result,
the BP molecules are chemically activated to a diol epoxide molecule, Benzo-[a]-pyrene-7,8-
diol-9,10-epoxide (DE). These new molecules can unbind from the surface of the endoplasmic
reticulum, they can diffuse again in the cytosol, and they may enter inside the nucleus. Thus,
by such a mechanism, these new molecules can bind to DNA, DNA damage being known as
a main cause of cancer. We also include in our model natural cleaning mechanisms occurring
in the cytosol and making the carcinogenic molecules harmless. The slow diffusion process
taking place at the surface of the endoplasmic reticulum is modeled with the aid of the
Laplace-Beltrami operator, properly scaled. The binding-unbinding process at the surface of
the endoplasmic reticulum is described by various functions, leading to different homogenized
models. Another carcinogenesis model, introduced in [127], will be brie
y discussed in Section
4.2.3. For more details about the complex mechanisms governing carcinogenesis in human
cells, we refer to [116] and [189].
Related problems to the one we treat here were addressed in [6], [78], [126], [127], and
[129]. For results about the upscaling of reactive transport in porous media, we refer, among
others, to [8], [109], [137], [138], [139], [177], and [193]. Also, for reaction-diffusion problems
involving adsorption and desorption, we refer to [6], [72], [73], [97], [137], and [157].
The method we follow to prove our main convergence results is the periodic unfolding
method (see [56], [58], [64] and [94]), extended in [127] and [128] for dealing with gradients
of functions dened on smooth periodic manifolds. Our results in [221], announced in [218],
generalized some of those obtained in [126] and [127]. More precisely, we addressed the case
in which the surface of the endoplasmic reticulum is supposed to be heterogeneous and, also,
the case in which the adsorption is modeled with the aid of a nonlinear isotherm of Langmuir
type.
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 2.1. More precisely, we consider a bounded connected open set Ω in Rn, with a

74 4.2. Multiscale analysis of a carcinogenesis model
Lipschitz boundary @Ω and with n2. The domain Ω, which, as in [127], is supposed to
be representable by a nite union of axis-parallel cuboids with corner coordinates belonging
toQn, represents a human cell with the region occupied by the nucleus removed. Following
[127] and [221], we denote by Cthe cell membrane and by Nthe boundary of the nucleus.
Thus,@Ω = C[N. LetY= (0;1)nbe the unit 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 is Y=YnF.
If we repeat Yby periodicity, then the union of all Yis a connected set in Rn, denoted by
Rn
1.
Let"2(0;1) be a small parameter related to the periodicity length and taking values in
a positive real sequence tending to zero and such that Ω is a nite union of cuboids which
are homothetic to the unit cell with the same ratio ". We denote
Ω"=∪
k2Zn"(k+Y)\Ω;
S"=∪
k2Zn"(k+@F)\Ω
and we suppose that S"\@Ω =∅. In such a way, Ω is a nite union of cuboids which are
homothetic to the reference cell and the inclusions do not touch the exterior boundary @Ω.
If we denote by [0 ;T], with 0< T < 1, the time interval of interest, we shall analyze
the effective behavior, as the small parameter "!0, of the solution of the following coupled
system of equations:
8
>>>>>>>>>><
>>>>>>>>>>:@u"
@tDu∆u"=f(uϵ) in (0;T)Ω";
u"=ubon (0;T)C;
Du∇u"
= 0 on (0;T)N;
Du∇u"
="G1(u";s") on (0;T)S";
u"(0;x) =u0(x) in Ω";(4.19)
8
>>>>>>>>><
>>>>>>>>>:@v"
@tDv∆v"=g(v") in (0;T)Ω";
v"= 0 on (0;T)N;
Dv∇v"
= 0 on (0;T)C;
Du∇u"
="G2(v";w") on (0;T)S";
v"(0;x) =v0(x) in Ω";(4.20)
8
<
:@s"
@t"2Ds∆"s"=h(s") +G1(u";s") on (0;T)S";
s"(0;x) =s0(x) onS":(4.21)

Chapter 4. Mathematical models in biology 75
8
<
:@w"
@t"2Dw∆"w"=h(s") +G2(v";w") on (0;T)S";
w"(0;x) =w0(x) onS":(4.22)
In (4.19)-(4.22), is the outward unit normal to Ω", ∆"denotes the Laplace-Beltrami operator
onS",u": [0;T]Ω"!Randv": [0;T]Ω"!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.
We notice that the diffusion on the surface of the endoplasmic reticulum is scaled with "2,
in order to keep the in
uence of the slow surface diffusion term at the macroscale. Besides,
the scaling in the right-hand side of the boundary conditions (4.19) and (4.20) on S"ensures
that we keep the in
uence of the binding processes at the macroscale. We can deal in a
similar way with 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) (see, for the linear case, [126]).
We make several assumptions on the data.
1.The diffusion coefficients Du;Dv;Ds;Dw>0 are supposed to be, to simplify the
presentation, constant.
2.f,g, andhare nonlinear functions describing the cleaning mechanisms in Ω"and,
respectively, the transformation of the BP molecules to DE molecules bound to the surface of
the endoplasmic reticulum. As in [127], we suppose that the cleaning mechanism is given 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 assumed to be of the same form as f, but with
different parameters. Let us mention that we consider here Michaelis-Menten functions, but
we can also treat 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 [137], [193], and [217]).
3.The binding-unbinding phenomena taking place at the surface of the endoplasmic
reticulum are modeled with the aid of two given functions G1andG2. Various types of such
functions can be considered, provided that one imposes suitable conditions guarantying the
positivity and the boundedness of the solution ( u";v";s";w"). In [126], the authors deal with
the standard linear case in which
G1(u";s") =ls(u"s"); G 2(v";w") =lw(v"w");
wherels;lw>0 are the constant binding and unbinding rates to the endoplasmic reticulum.
We deal here with two cases: the linear Henry isotherm with highly oscillating coefficients

76 4.2. Multiscale analysis of a carcinogenesis model
and, respectively, the case of a Langmuir isotherm. Thus, we rstly consider that
G1(u";s") =l"
uu"l"
ss"; G 2(v";w") =l"
vv"l"
ww"; (4.23)
with
l"
u(x) =lu(x
")
; l"
s(x) =ls(x
")
; l"
v(x) =lv(x
")
; l"
w(x) =lw(x
")
;
wherelu(y),ls(y),lv(y) andlw(y) areY-periodic, real, smooth, bounded functions with
lu(y)l0
u>0; l s(y)l0
s>0; l v(y)l0
v>0; l w(y)l0
w>0:
Such a model, involving coefficients which depend on the surface variable is physically
justied (for examples where the processes on the membrane are inhomogeneous, see [79]).
As a result, in the homogenized limit, new integral terms are present, re
ecting the effect of
the heterogeneity of the cell on the effective behavior of the solution of system (4.19)-(4.22).
To simplify the presentation, we suppose that all the parameters involved in our model are
time independent, but the case in which they depend on time can be also addressed.
In the second case we shall address here, we consider that G1andG2are nonlinear
functions given in terms of isotherms of Langmuir type:
G1(u";s") =l"
s(1u"
1 +1u"s")
; G 2(v";w") =l"
w(2v"
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 (4.26)) and the initial values u0(x);v0(x)2L2(Ω),s0(x);w0(x)2
C1(Ω) are supposed to be nonnegative and bounded independently with respect to ".
Remark 4.2 We point out that, exactly like in [129], we can deal also with the case in which
the Lipschitz continuous functions G1andG2are of the form Gi(p;q) =Gi(p;q)(pq), with
0<G i;minGi(p;q)Gi;max<1or with the case in which Gi(p;q) =Ai(p)Bi(q), with
AiandBiLipschitz continuous and increasing functions, for i= 1;2.
The function g(r) =r=(1 +r) is one to one and increasing from R+to [0;= ].gis
not dened for r=1=, but, since we plan to consider only non-negative values for the
argumentr, we can mollify gfor negative values r<0 to obtain an increasing function on R,
growing at most linearly at innity and having an uniformly bounded derivative (see [6]). We
point out that, as a matter of fact, from a strictly physical point of view, one could extend
the considered rates by zero for all negative arguments and this would allow a direct proof
of the fact that the solution components remain positive as long as the initial and boundary
data are positive.

Chapter 4. Mathematical models in biology 77
Remark 4.3 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")depending
on the concentrations of BP and DE molecules. This setting includes linear, Freundlich,
Langmuir or other isotherms encountered in the literature (see [98], [137], and [220]).
For giving the weak formulation of problem (4.19)-(4.22), we introduce some function
spaces. In what follows, the space L2(Ω") is endowed with the classical scalar product and
norm
(u;v)Ω"=∫
Ω"u(x)v(x) dx;∥u∥2
Ω"= (u;u)Ω";
and the space L2((0;T);L2(Ω")) is equipped with
(u;v)Ω";T=∫T
0(u(t);v(t))Ωdt;∥u∥2
Ω";T= (u;u)Ω";T;
whereu(t) =u(t;
);v(t) =v(t;
).
Following [126] and [127], we introduce the spaces
V(Ω") =L2((0;T);H1(Ω"))\H1((0;T);(H1(Ω"))′);
VN(Ω") =fv2 V(Ω")jv= 0 on Ng;
VC(Ω") =fv2 V(Ω")jv=ubon Cg;
V0;C(Ω") =fv2 V(Ω")jv= 0 on Cg;
where, for an arbitrary Banach space V, we denote by V′its dual. Similar spaces can be
dened for Ω and S". We use the notation
⟨u;v⟩"=∫
S"g"uvdx;
whereg"is the Riemannian tensor on S". Further, we dene
VN(Ω") =fv2H1(Ω")jv= 0 on Ng;
V0;C(Ω") =fv2H1(Ω")jv= 0 on Cg; V (S") =H1(S")
and
V(Ω;Y) =L2((0;T)Ω;H1
per(Y));V(Ω;@F) =L2((0;T)Ω;H1(@F));
where
H1
per(Y) =fφ2H1
loc(Rn
1) :φisYperiodic g:
Finally, we suppose that ub2H1=2(C), where, for an arbitrary smooth hypersurface 0Rn
and for any 0 <r< 1, we consider the Sobolev-Slobodeckij space
Hr(0) =fu2L2(0) :juj0;r<1g; (4.26)

78 4.2. Multiscale analysis of a carcinogenesis model
where
juj2
0;r=∫
00ju(x)u(y)j2
jxyjn1+2rdxdy:
The spaceHr(0) is endowed with the norm (see [115] and [125])
∥u∥2
Hr(0)=∥u∥2
L2(0)+juj2
0;r:
Let us give now the variational formulation of problem (4.19)-(4.22).
Problem 1 : nd (u";v";s";w")2 VC(Ω")VN(Ω")V(S")V(S"), verifying 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 ( φ1;φ2;ϕ)2VC;0(Ω")VN(Ω")V(S"), we have
8
>>>>>>>>>>><
>>>>>>>>>>>:(@u"
@t;φ1)
Ω"+Du(∇u";∇φ1) +"⟨G1(u";s");φ1⟩S"=(f(u");φ1)Ω";
(@v"
@t;φ2)
Ω"+Dv(∇v";∇φ2) +"⟨G2(v";w");φ2⟩S"=(g(v");φ2)Ω";
⟨@s"
@t;ϕ⟩
S"+Ds⟨"∇@Fs";"∇@Fϕ⟩S"=⟨h(s");ϕ⟩S"+⟨G1(u";s");ϕ⟩S";
⟨@w"
@t;ϕ⟩S"+Dw⟨"∇@Fw";"∇@Fϕ⟩S"=⟨h(s");ϕ⟩S"+⟨G2(v";w");ϕ⟩S":(4.27)
Let us remark that in (4.27), to simplify the presentation, we made a slight 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 the hypotheses we imposed on the data, one can prove the existence of a unique
weak solution ( u";v";s";w") of problem (4.27) (see [7, Proposition 2.2] and [127, Theorem
4.4]).
4.2.2 The macroscopic model
Our goal now is to obtain the homogenized limit for the problem (4.19)-(4.22). Thus, we
have to pass to the limit, with "!0, in its variational formulation (4.27). For dealing with
the nonlinear terms, we need to prove some strong convergence results, obtained by using
the unfolding operators T"andT"
bdened, e.g., in [56], [58], [64], [94], [127], and [128].
The main feature of these operators is that they map functions dened on the oscillating
domains (0;T)Ω"and, respectively, (0 ;T)", into functions dened on the xed domains
(0;T)ΩYand (0;T)Ω, respectively. We brie
y recall here the denitions of these
two operators for our particular geometry. For any φ2Lp((0;T)Ω") and anyp2[1;1],
we dene the periodic unfolding operator
T":Lp((0;T)Ω")!Lp((0;T)ΩY)

Chapter 4. Mathematical models in biology 79
by the formula
T"(φ)(t;x;y ) =φ(
t;ϵ[x
"]
+"y)
:
In a similar manner, for any function ϕ2Lp((0;T)"), the periodic boundary unfolding
operator
T"
b:Lp((0;T)")!Lp((0;T)Ω)
is dened by
T"
b(ϕ)(t;x;y ) =ϕ(
t;"[x
ϵ]
+"y)
:
Using these unfolding operators, we can deduce the homogenized limit system.
Theorem 4.4 (Theorem 3.1 in [221]) The solution (u";v";s";w")of system (4.19)-(4.22)
converges, as "!0, in the sense of (3.36), to the unique solution (u;v;s;w )2 V C(Ω)
VN(Ω) V(Ω;@F) V(Ω;@F), with
(u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0);
of the following macroscopic problem:
8
>>>>>>>>>>><
>>>>>>>>>>>:jYj(@u
@t;φ1)
Ω+ (Au∇u;∇φ1)Ω+ (G1(u;s);φ1)Ω@F=jYj(f(u);φ1)Ω;
jYj(@v
@t;φ2)
Ω+ (Av∇v;∇φ2)Ω+ (G2(v;w);φ2)Ω=jYj(g(v);φ2)Ω;
(@s
@t;ϕ)
Ω@F+ (Ds∇@F
ys;∇@Fϕ)Ω@F(G1(u;s);ϕ)Ω@F=(h(s);ϕ)Ω@F;
(@w
@t;ϕ)
Ω@F+ (Dw∇@F
yw;∇@Fϕ)Ω@F(G2(v;w);ϕ)Ω@F= (h(s);ϕ)Ω@F;(4.28)
for(φ1;φ2;ϕ)2V0;C(Ω)VN(Ω)V(Ω;@F). Here,AuandAvare the homogenized matrices,
dened by:8
>><
>>:Au
ij=Du∫
Y(
ij+@j
@yi)
dy;
Av
ij=Dv∫
Y(
ij+@j
@yi)
dy;(4.29)
in terms of the functions j2H1
per(Y)=R, ,j= 1;:::;n; weak solutions of the cell problems
{∇ y
(∇yj+ej) = 0; y2Y;
(∇yj+ej)
= 0; y2@F::(4.30)
We also state here the strong form of the limit system (4.28).
Theorem 4.5 (Theorem 3.2 in [221]) The limit function (u;v;s;w )2 V C(Ω) V N(Ω)
V(Ω;@F) V(Ω;@F), dened in Theorem 4.1 and satisfying
(u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0);

80 4.2. Multiscale analysis of a carcinogenesis model
is the unique solution of the following problem:
8
>>>><
>>>>:jYj@u
@t ∇
(Au∇u) +∫
@FG1(u;s) dy=jYjf(u)in(0;T)Ω;
u=ubon(0;T)C;
Au∇u
= 0 on(0;T)N;(4.31)
8
>>>><
>>>>:jYj@v
@t ∇
(Av∇v) +∫
@FG2(v;w) dy=jYjg(v)in(0;T)Ω;
v= 0 on(0;T)N;
Av∇v
= 0 on(0;T)C;(4.32)
8
>><
>>:@s
@tDs∆@F
ysG1(u;s) =h(s)on(0;T)Ω@F;
@w
@tDw∆@F
ywG2(v;w) =h(s)on(0;T)Ω@F::(4.33)
As in [126, Theorem 14] and [127, Theorem 6.1], it follows that the solution of the macroscopic
problem (4.28) in unique.
Remark 4.6 We notice that the in
uence of well-balanced binding-unbinding processes tak-
ing place at the surface of the endoplasmic reticulum is re
ected by the appearance of an
extra zero-order term in the equations (4.31)-(4.32). Also, it is worth remarking that the
limit problem contains an additional microvariable y. This local phenomenon yields a more
complicated microstructure of the effective medium; in (4.31)-(4.32), x2Ωplays the role
of a macroscopic variable and y2@Fcan be seen as a microscopic one. The limit model
consists of two partial differential equations, with global diffusion (with respect to the macro-
scopic variable x), for the limit of the BP and DE molecules in the cytosol (see (4.31)-(4.32))
and two partial differential equations, governing the local behavior of the system, with local
diffusion (with respect to the microscopic variable y) on@F(see (4.33)). .
Remark 4.7 We can deal, in a similar manner, with the more general case in which, in-
stead of considering constant diffusion coefficients, we work with an heterogeneous medium
represented by periodic symmetric bounded matrices which are assumed to be uniformly coer-
cive. Moreover, all the above results can be extended to the situation in which, instead of the
constant diffusion coefficients DuandDv, we have two matrices A"
uand, respectively, A"
v.
We suppose that A"
uandA"
vare sequences of matrices in M(;; Ω)such that
T"(A"
u)!Au;T"(A"
v)!Avstrongly in L1(ΩY)nn; (4.34)
for some matrices Au=Au(x;y)andAv=Au(x;y)inM(;; ΩY)(see [58]). In this
case, since the correctors jdepend also on x, the new homogenized matrices Ahom
uandAhom
v
are no longer constant, but depend on x. Here, for ;2R, with 0< , we denote

Chapter 4. Mathematical models in biology 81
byM(;; Ω)the set of all the matrices A2(L1(Ω))nnwith the property that, for any
2Rn,(A(y); )jj2;jA(y)j jj, almost everywhere in Ω.
Also, for the diffusion coefficients on the surface S"we can suppose that they are not con-
stant, but they depend on ". For instance, we can work with the diffusion tensors D"
s(x) =
Ds(x=")andD"
w(x) =Dw(x="), whereDsandDware two uniformly coercive periodic sym-
metric 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(Ω@F)such that T"
b(D"
s)!Dsand
T"
b(D"
w)!Dwstrongly in L1(Ω@F). .
In order to prove Theorem 4.7, we need to derive a priori estimates, suitable bounds
and results concerning the existence and uniqueness of a weak solution ( u";v";s";w") of the
problem (4.27).
In the following proposition, proven in [221], we state that the functions u";v";s"and
w"are nonnegative and bounded from above if the initial data are assumed to be bounded
and nonnegative. Such a positivity condition is a natural requirement, since u";v";s"andw"
represent the concentrations of BP and DE molecules in the cytosol and on the surface of the
endoplasmic reticulum. Besides, this property is important for proving the well-posedness of
our problem. Moreover, the essential boundedness of the solution is necessary from the point
of view of practical applications.
Proposition 4.8 (Proposition 4.1 in [221]) The functions u"andv"are nonnegative for
almost every x2Ω"andt2[0;T]and the functions s"andw"are nonnegative for almost
everyx2S"andt2[0;T]. Also, the functions u"andv"are bounded independently of "
almost everywhere in [0;T]Ω"and the functions s"andw"are bounded independently of "
almost everywhere in [0;T]S".
Hence, the solution ( u";v";s";w") of problem (4.27) is bounded in the L1-norm. Additionally,
like in [126, Lemma 2] and [127, Lemma A.2], we can prove the L2-boundedness of the solution
(u";v";s";w").
Proposition 4.9 (Proposition 4.2 in [221]) There exists a constant C > 0, independent of
", such that
∥u"∥2
Ω"+∥∇u"∥2
Ω";t+∥v"∥2
Ω"+∥∇v"∥2
Ω";tC;
"∥s"∥2
S"+"3∥∇@Fs"∥2
S";t+"∥w"∥2
S"+"3∥∇@Fw"∥2
S";tC;
"∥G1(u";s")∥2
S";t+"∥G2(v";w")∥2
S";tC;
for almost every t2[0;T]. Also, one gets

@u"
@t

L2((0;T);(H1
0(Ω"))′)+

@v"
@t

L2((0;T);(H1
0(Ω"))′)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 (4.27). Still, the nonlinearity of

82 4.2. Multiscale analysis of a carcinogenesis model
the model requires strong compactness results for the sequence of solutions in order to be
able to pass to the limit. We know (see [127]) that
u";v"2L2((0;T);H1(Ω"))\H1((0;T);(H1
0(Ω"))′)\L1((0;T)Ω"):
Using suitable extension results (see, e.g., [136], [115] and [169]) and Lemma 5.6 from [129],
we can construct two extensions u"andv"that converge strongly to u;v2L2((0;T);L2(Ω)).
We remark that one can obtain (see [115] and [169]) the existence of a linear and bounded
extension operator to the whole of Ω, which preserves the non-negativity, the essential bound-
edness and the above priori estimates.
Due to the fact that the functions g1andg2are Lipschitz, following the lines of Lemma 4.3
in [127], we can prove that T"
b(s") and T"
b(w") are Cauchy sequences in L2((0;T)Ω@F).
Proposition 4.10 (Proposition 4.3 in [221]) For any  >0, there exists "0>0such that
for any 0<"1;"2<"0one has
∥T"1
b(s"1) T"2
b(s"2)∥(0;T)Ω@F+∥T"1
b(w"1) T"2
b(w"2)∥(0;T)Ω@F<::
Thus, the sequences T"
b(s") and T"
b(w") are strongly convergent in L2((0;T)Ω@F).
We point out again here that (4.27) is a well-posed problem. Using the above a priori
estimates and the properties of the operators T"andT"
b(see [56], [58], [64], [94], [127], and
[128]), we easily obtain the following compactness result.
Proposition 4.11 (Proposition 4.4 in [221]) Let (u";v";s";w")be the solution of problem
(4.27). Then, there exist u;v2L2((0;T);H1(Ω));bu;bv2L2((0;T)Ω;H1
per(Y)),s;w2
L2((0;T)Ω;H1
per(@F))such that, up to a subsequence, when "!0, we have
8
>>>>>>>>>><
>>>>>>>>>>:T"(u")⇀u weakly inL2((0;T)Ω;H1(Y));
T"(v")⇀v weakly inL2((0;T)Ω;H1(Y));
u"!u;v"!vstrongly in L2((0;T)Ω);
T"(∇u")⇀∇u+∇ybuweakly inL2((0;T)ΩY);
T"(∇v")⇀∇v+∇ybvweakly inL2((0;T)ΩY);
T"
b(s")⇀s weakly inL2((0;T)Ω;H1(@F));
T"
b(w")⇀w weakly inL2((0;T)Ω;H1(@F));
T"
b(s")!s;T"
b(w")!wstrongly in L2((0;T)Ω@F)::(4.36)
For passing to the limit in the nonlinear terms containing the functions G1andG2, we have
to show that T"
b(u")!uandT"
bv")!v, strongly in L2((0;T)Ω@F). These strong
convergence results follow from the strong convergence of u"andv", respectively, the trace
lemma (see Lemma 3.1 in [115]) and the properties of the unfolding operator T"
b. So, we have
the following result (see [221] and [115]).
Proposition 4.12 (Proposition 4.5 in [221]) Up to a subsequence, one has
T"
b(u")!ustrongly in L2((0;T)Ω@F)
and
T"
b(v")!vstrongly in L2((0;T)Ω@F):

Chapter 4. Mathematical models in biology 83
We remark that from 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)ΩY):
g(T"(v"))!g(v) strongly in L2((0;T)ΩY):
In a similar way, we obtain
h(T"
b(s"))!h(s) strongly in L2((0;T)Ω@F):
For getting the limit behavior of the terms involving G1andG2, in the rst situation, i.e. for
Henry isotherm with rapidly oscillating coefficients given by (2.5), we can easily pass to the
limit since these coefficients are uniformly bounded in L1(Ω) and converge strongly therein,
while for the second situation, i.e. isotherm of the form (3.24), we need to use the strong
convergence of T"
b(u"),T"
b(v"),T"
b(s") and T"
b(w") and the properties of the functions g1and
g2. Therefore, we obtain
G1(T"
b(u";s"))!G1(u;s) strongly in L2((0;T)Ω@F)
and
G2(T"
b(v";w"))!G2(v;w) strongly in L2((0;T)Ω@F):
By classical results (see, for instance, Theorem 2.12 in [56] and Theorem 2.17 in [94] ), uand
vare independent of y.
Proof of Theorem 4.7. For getting the limit problem (4.28), we take in the rst equation
in (4.27) the admissible test function
φ(t;x) =φ1(t;x) +"φ2(
t;x;x
")
; (4.37)
withφ12C1
0((0;T);C1(Ω))φ22C1
0((0;T);C1(Ω;C1
per(Y))).
Integrating with respect to time, applying in each term the corresponding unfolding op-
erator and passing to the limit, by using the above convergence results and Lebesgue's con-
vergence theorem (see, for details, [58], [73], [127] and [217]), we get:
∫T
0∫
ΩY@u
@tφ1dxdydt+Du∫T
0∫
ΩY(∇u+∇ybu)
(∇φ1+∇yφ2) dxdydt+
∫T
0∫
Ω@FG1(u;s)φ1dxdydt=∫T
0∫
ΩYf(u)φ1dxdydt: (4.38)
Using standard density arguments, it follows that (4.38) is valid for any φ12L2(0;T;H1(Ω)),
φ22L2((0;T)Ω;H1
per(Y)). In a similar manner, for the limit equation for v", we obtain
∫T
0∫
ΩY@v
@tφ1dxdydt+Dv∫T
0∫
ΩY(∇v+∇ybv)
(∇φ1+∇yφ2) dxdydt+
∫T
0∫
Ω@FG2(v;w)φ1dxdydt=∫T
0∫
ΩYg(v)φ1dxdydt; (4.39)

84 4.2. Multiscale analysis of a carcinogenesis model
for anyφ12L2((0;T);H1(Ω)),φ22L2((0;T)Ω;H1
per(Y)).
In order to obtain the limit equations for s"andw", we apply the convergence results
obtained in [127] (see Lemma 2.6 and Theorem 2.9). Indeed, using the boundary unfolding
operator T"
bin (4:27)3, by passing to the limit we get
∫T
0∫
Ω@F@s
@tϕdxdydt+Ds∫T
0∫
Ω@F∇@F
ys
∇@F
yϕdxdydt=
∫T
0∫
Ω@FG1(u;s)ϕdxdydt∫T
0∫
Ω@Fh(s)ϕdxdydt; (4.40)
for anyϕ2C1
0((0;T);C1(Ω;C1
per(@F))).
In a similar manner, we get
∫T
0∫
Ω@F@w
@tϕdxdydt+Dw∫T
0∫
Ω@F∇@F
yw
∇@F
yϕdxdydt=
∫T
0∫
Ω@FG2(v;w)ϕdxdydt+∫T
0∫
Ω@Fh(s)ϕdxdydt; (4.41)
for anyϕ2C1
0((0;T);C1(Ω;C1
per(@F))).
Hence, we obtain the weak formulation of the limit problem (4.28). Indeed, if we take
φ1= 0, we easily get the cell problems (4.30) and
bu=n∑
k=1@u
@xkk;bv=n∑
k=1@v
@xkk: (4.42)
Then, taking φ2= 0 and using (4.42), we obtain (4.28). Also, by classical techniques, we
can obtain the initial conditions ( u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0). Since the solution
(u;v;s;w ) of problem (4.28) is unique, the above convergences for the microscopic solution
(u";v";s";w") hold for the whole sequence and this completes the proof of Theorem 4.7.
4.2.3 A nonlinear carcinogenesis model involving free receptors
We end this section by brie
y discussing a generalization of a nonlinear model proposed in
[127] (see, also, [221]) for the carcinogenesis in living cells, involving a new variable describing
the free receptors which are present at the surface of the endoplasmic reticulum. In this new
model, the BP molecules present in the cytosol are transformed into BP molecules bound to
the surface of the ER only if they nd a free receptor R". As in [127], 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, following the law of mass action, the binding is given by the product kuu"R",
with a constant rate ku>0. For the DE molecules, we assume a similar behavior. It is
natural to consider that when BP molecules u"or DE molecules v"bind to the surface of the
endoplasmic reticulum, the quantity of free receptors decreases and when the molecules s"
andw"unbind from the surface of the endoplasmic reticulum the amount of free receptors

Chapter 4. Mathematical models in biology 85
increases. Assuming that the receptors are xed on the surface of the endoplasmic reticulum,
their evolution is governed by (see [127] 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.
The variational formulation of this new nonlinear problem is stated below.
Problem 2: nd (u";v";s";w";R")2 V C(Ω") V N(Ω") V(S") V(S") V R(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 ( φ1;φ2;ϕ)2VC;0(Ω")VN(Ω")V(S"), we have
8
>>>>>>>>>>>>><
>>>>>>>>>>>>>:(@u"
@t;φ1)
Ω"+Du(∇u";∇φ1) +"⟨kuu"R"lss";φ1⟩S"=(f(u");φ1)Ω";
(@v"
@t;φ2)
Ω"+Dv(∇v";∇φ2) +"⟨kvv"R"lww";φ2⟩"=(g(v");φ2)Ω";
⟨@s"
@t;ϕ⟩
S"+Ds⟨"∇@Fs";"∇@Fϕ⟩S"=⟨h(s");ϕ⟩S"+⟨kuu"R"lss";ϕ⟩S";
⟨@w"
@t;ϕ⟩
S"+Dw⟨"∇@Fw";"∇@Fϕ⟩S"=⟨h(s");ϕ⟩S"+⟨kvv"R"lww";ϕ⟩S";
⟨@tR";ϕ⟩S"+⟨R"jkuu"+kvv"j;ϕ⟩S"=⟨(RR")jkss"+kww"j;ϕ⟩S":(4.43)
In (4.43),
VR(S") =fu2L2((0;T);L2(S"))j@tu2L2((0;T);L2(S"))g:
The existence of a solution
(u";v";s";w";R")2 VC(Ω") V N(Ω") V(S") V(S") V R(S")
of this variational problem is proven in [127, Theorem 4.4]. Moreover, it is shown in [127,
Lemma 4.1 and Theorem 4.5] that R"is nonnegative and bounded by R > 0 almost every-
where in [0;T]S"andT"
b(R") converges strongly to R2L2((0;T)Ω@F).
The homogenized result for this case is stated in the following theorem (see Theorem 5.1
in [127]).
Theorem 4.13 The homogenized problem corresponding to (4.43) reads as follows: nd
(u;v;s;w;R )2 VC(Ω) V N(Ω) V(Ω;) V(Ω;@F) V R(Ω;@F);
with
(u(0);v(0);s(0);w(0);R(0)) = (u0;v0;s0;w0;R);

86 4.3. Homogenization results for the calcium dynamics in living cells
such that
8
>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:jYj(@u
@t;φ1)
Ω+ (Au∇u;∇φ1)Ω+ (kuuRlss;φ1)Ω=jYj(f(u);φ1)Ω;
jYj(@v
@t;φ2)
Ω+ (Av∇v;∇φ2)Ω+ (kvvRlww;φ 2)Ω@F=jYj(g(v);φ2)Ω;
(@s
@t;ϕ)
Ω+ (Ds∇s;∇ϕ)Ω(kuuRlss;ϕ)Ω@F=(h(s);ϕ)Ω@F;
(@w
@t;ϕ)
Ω@F+ (Dw∇w;∇ϕ)Ω@F(kvvRlww;ϕ)Ω@F= (h(s);ϕ)Ω@F;
(@tR;ϕ)Ω@F+ (R(kuu+kvv);ϕ)Ω@F= ((RR)(kss+kww);ϕ)Ω@F;
for(φ1;φ2;ϕ)2V0;C(Ω)VN(Ω)V(Ω;@F).
Notice that the homogenized matrices AuandAvare given by (4.29). We point out that the
evolution of the receptors is governed by an ordinary differential equation.
All the above results are still valid for the case of highly oscillating coefficients 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.
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 ad-
dressed, as well.
4.3 Homogenization results for the calcium dynamics in living
cells
In the last section of this chapter, we shall present some results, obtained via the periodic
unfolding method in [219]. More precisely, our goal was to analyze the effective behavior
of a nonlinear system of coupled reaction-diffusion equations arising in the modeling of the
dynamics of calcium ions in living cells. It is well-known that calcium is a very important
second messenger in a living cell, contributing to many cellular processes, such as protein
synthesis, muscle contraction, or metabolism (see, for instance, [70]). Controlling the in-
tracellular free calcium concentrations is a very complicated process. The nely structured
endoplasmic reticulum is an important multifunctional intracellular organelle involved in cal-
cium homeostasis and many of its functions depend on the calcium dynamics. This highly
heterogeneous complex cellular structure spreads throughout the cytoplasm, creating 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,

Chapter 4. Mathematical models in biology 87
is a modern topic which needs further investigations, since many biological mechanisms in-
volving the functions of the cytosol and of the endoplasmic reticulum are not yet perfectly
understood.
The model considered in [219] consists, at the microscale, of two equations governing the
concentration of calcium ions in the cytosol and, respectively, in the endoplasmic reticulum,
coupled through an interfacial exchange term (for details about the physiological background
of such a model, see [152]). Depending on the magnitude of this exchange term, various
models are obtained at the limit. An interesting situation is that in which we obtain, at the
macroscale, a bidomain model, i.e. a system consisting of two reaction-diffusion 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. Our results
in [219] constitute a generalization of some of the results contained in [121] and [129]. The
scaling of the interfacial exchange term has an important in
uence on the limit problem and,
using some techniques from [94], we can extend the analysis from [129] to the case in which
the parameter
arising in the exchange term belongs to R.
Let us point out that bidomain models can be encountered also in other contexts, such
as the modeling of diffusion processes in partially ssured media (see [31], [27], and [101]) or
the modeling of the electrical activity of the heart (see [15], [13], and [187]).
4.3.1 Setting of the problem
Let us start by describing the geometry of the problem. Let Ω be a bounded domain in
Rn, withn3, having a Lipschitz boundary @Ω made up of a nite number of connected
components. The domain Ω is assumed to be a periodic structure formed by two connected
parts, Ω"
1and Ω"
2, separated by an interface ". We suppose that only the phase Ω"
1reaches the
exterior xed boundary @Ω. The small positive real parameter "is related to the characteristic
dimension of these two regions. For dealing with the dynamics of the concentration of calcium
ions in a biological cell, the phase Ω"
1models the cytosol, while the phase Ω"
2represents
the endoplasmic reticulum. Let Y1be an open connected Lipschitz subset of the unit cell
Y= (0;1)nandY2=YnY1. We set=YnY2. We consider that the boundary of Y2is
locally Lipschitz and that its intersections with the boundary of Yare identically reproduced
on the opposite faces of the elementary cell. Besides, if we repeat Yin a periodic manner,
the union of all the sets Y1is a connected set, with a locally C2boundary. Also, we consider
that the origin of the coordinate system lies in a ball contained in the above mentioned union
(see [101]).
For any"2(0;1), let
Z"=fk2Znj"k+"YΩg;
K"=fk2Z"j"k"ei+"YΩ;8i= 1;:::;n g;
whereeiare the vectors of the canonical basis of Rn. We set
Ω"
2= int(∪
k2K"("k+"Y2)) Ω"
1= ΩnΩ"
2:

88 4.3. Homogenization results for the calcium dynamics in living cells
For1;12R, with 0< 1<1, we denote by M(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 Ω, where A2 M (1;1;Y) is aY-periodic symmetric matrix and we denote the
matrixAbyA1inY1and, respectively, by A2inY2.
If we denote by (0 ;T) the time interval of interest, we shall be concerned with the effective
behavior of the solutions of the following microscopic system:
8
>>>>>>>>>>>>><
>>>>>>>>>>>>>:@u"
1
@tdiv (A1"∇u"
1) =f(u"
1) in (0;T)Ω"
1;
@u"
2
@tdiv (A2"∇u"
2) =g(u"
2) in (0;T)Ω"
2;
A1"∇u"
1
=A2"∇u"
2
on (0;T)";
A1"∇u"
1
="
h(u"
1;u"
2) on (0;T)";
u"
1= 0 on (0 ;T)@Ω;
u"
1(0;x) =u0
1(x) in Ω"
1; u"
2(0;x) =u0
2(x) in Ω"
2;(4.44)
whereis the outward unit normal to Ω"
1and the parameter
is supposed to be a given real
number, related to the speed of the interfacial exchange. Three important cases arise at the
limit, i.e.
= 1,
= 0 and
=1 (see, also, Remark 4.20).
We suppose that the initial conditions are non-negative and that the functions fandg
are Lipschitz-continuous, with f(0) =g(0) = 0. Further, we assume that
h(u"
1;u"
2) =h"
0(x)(u"
2u"
1); (4.45)
where
h"
0(x) =h0(x
")
andh0=h0(y) is a realY-periodic function in L1(), withh0(y)>0. Let
H=∫
h0(y) dy̸= 0:
Exactly like in [129], we can deal in a similar manner with the case in which the function h
is Lipschitz-continuous in both arguments and is given by
h(r;s) =h(r;s)(sr); (4.46)
with 0<h minh(r;s)hmax<1.
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

Chapter 4. Mathematical models in biology 89
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.
The well-posed microscopic problem (4.44) can be homogenized via the periodic unfolding
method and the homogenized solution ts well with experimental data (see [152] and [135]).
In order to write the weak form of our microscopic problem, let us introduce the needed
function spaces and norms. Let
H1
@Ω(Ω"
1) =fv2H1(Ω"
1)jv= 0 on@Ω\@Ω"
1g;
V(Ω"
1) =L2(0;T;H1
@Ω(Ω"
1));V(Ω"
1) ={
v2V(Ω"
1)j@v
@t2L2((0;T)Ω"
1)}
;
V(Ω"
2) =L2(0;T;H1(Ω"
2));V(Ω"
2) ={
v2V(Ω"
2)j@v
@t2L2((0;T)Ω"
2)}
;
with
(u(t);v(t))Ω"=∫
Ω"u(t;x)v(t;x) dx;∥u(t)∥2
Ω"= (u(t);u(t))Ω";
(u;v)Ω";t=∫t
0(u(t);v(t))Ω"dt;∥u∥2
Ω";t= (u;u)Ω";t;
for= 1;2. Also, let
V(Ω) =L2(0;T;H1(Ω));V(Ω) ={
v2V(Ω)j@v
@t2L2((0;T)Ω)}
;
with
(u(t);v(t))Ω=∫
Ωu(t;x)v(t;x) dx;∥u(t)∥2
Ω= (u(t);u(t))Ω;
(u;v)Ω;t=∫t
0(u(t);v(t))Ωdt;∥u∥2
Ω;t= (u;u)Ω;t
and
V0(Ω) = fv2V(Ω)jv= 0 on@Ω a.e. on (0 ;T)g;V0(Ω) =V0(Ω)\ V(Ω):
The variational formulation of problem (4.44) is the following one: nd ( u"
1;u"
2)2 V(Ω"
1)
V(Ω"
2), with
(u"
1(0;x);u"
2(0;x)) = (u0
1(x);u0
2(x))2(L2(Ω))2
and(@u"
1
@t(t);φ(t))
Ω"
1+(@u"
2
@t(t); (t))
Ω"
2+
(A"
1(t)∇u"
1;∇φ(t))Ω"
1+ (A"
2(t)∇u"
2;∇ (t))Ω"
2
"
(h(u"
1;u"
2);φ(t) (t))"= (f(u"
1(t));φ(t))Ω"
1+ (g(u"
2(t)); (t))Ω"
2; (4.47)
for a.e.t2(0;T) and any ( φ; )2V(Ω"
1)V(Ω"
2).
As in [129], one can prove that (4.47) is a well-posed problem and that u"andv"are
non-negative and bounded almost everywhere. Moreover, taking ( u"
1;u"
2) as test function in

90 4.3. Homogenization results for the calcium dynamics in living cells
(4.47), integrating with respect to time and using the fact that u"
1andu"
2are bounded and
non-negative, it follows that there exists a constant C0, independent of ", such that
∥u"
1(t)∥2
Ω"
1+∥u"
2(t)∥2
Ω"
2+∥∇u"
1∥2
Ω"
1;t+∥∇u"
2∥2
Ω"
2;t+"
(h(u"
1;u"
2);u"
1u"
2)";tC;
for a.e.t2(0;T). Also, as in [129] or [213], we can see that there exists a positive constant
C0, independent of ", such that

@u"
1
@t(t)

2
Ω"
1+

@u"
2
@t(t)

2
Ω"
2C;
for
1 and

@u"
1
@t

L2(0;T;H1(Ω"
1))+

@u"
2
@t

L2(0;T;H1(Ω"
2))C;
for
<1.
For getting the macroscopic behavior of the solution of problem (4.47), 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 [56], [58] and [94]). We brie
y recall here the deni-
tions and the main properties of these unfolding operators.
For dening the above mentioned periodic unfolding operators, we consider the following
sets (see [94]):
bZ"={
k2Znj"YkΩ}
;bΩ"= int∪
k2bZ"(
"Yk)
;"= ΩnbΩ";
bΩ"
=∪
k2bZ"(
"Yk
)
;"
= Ω"
nbΩ"
;
and
b"=@bΩ"
2:
Denition 4.14 For any Lebesgue measurable function φonΩ"
,2 f1;2g, we dene the
periodic unfolding operators by the formula
T"
(φ)(x;y) =8
<
:φ(
"[x
"]
Y+"y)
for a.e. (x;y)2bΩ"Y;
0 for a.e. (x;y)2"Y:
Ifφis a function dened in Ω, for simplicity, we write T"
(φ)instead of T"
(φjΩ").
For any function ϕwhich is Lebesgue-measurable on ", the periodic boundary unfolding
operator T"
bis dened by
T"
b(ϕ)(x;y) =8
<
:ϕ(
"[x
"]
Y+"y)
for a.e. (x;y)2bΩ";
0 for a.e. (x;y)2":
Remark 4.15 We notice that if φ2H1(Ω"
), then T"
b(φ) =T"
(φ)jbΩ".

Chapter 4. Mathematical models in biology 91
We recall here some useful properties of these operators (for detailed proofs, see, for
instance, [56], [93], and [94]).
Proposition 4.16 Forp2[1;1)and= 1;2, the operators T"
are linear and continuous
fromLp(Ω"
)toLp(ΩY)and
(i) ifφand are two Lebesgue measurable functions on Ω"
, one has
T"
(φ ) =T"
(φ)T"
( );
(ii) for every φ2L1(Ω"
), one has
1
jYj∫
ΩYT"
(φ)(x;y) dxdy=∫
bΩ"φ(x) dx=∫
Ω"φ(x) dx∫
"φ(x) dx;
(iii) if fφ"g"Lp(Ω)is a sequence such that φ"!φstrongly in Lp(Ω), then
T"
(φ")!φstrongly in Lp(ΩY);
(iv) ifφ2Lp(Y)isY-periodic and φ"(x) =φ(x="), then
T"
(φ")!φstrongly in Lp(ΩY);
(v) ifφ2W1;p(Ω"
), then ∇y(T"
(φ)) ="T"
(∇φ)andT"
(φ)belongs toL2(
Ω;W1;p(Y))
.
Moreover, for every φ2L1("), one has
∫
b"φ(x) dx=1
"jYj∫
ΩT"
b(φ)(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(Ω)),u22L2(0;T;H1(Ω)),bu12
L2((0;T)Ω;H1
per(Y1)),bu22L2((0;T)Ω;H1
per(Y2)) such that, passing to a subsequence,
for"!0, we have:
8
>><
>>:T"
1(u"
1)!u1strongly in L2((0;T)Ω;H1(Y1));
T"
1(∇u"
1)⇀∇u1+∇ybu1weakly inL2((0;T)ΩY1);
T"
2(u"
2)⇀u 2weakly inL2((0;T)Ω;H1(Y2));
T"
2(∇u"
2)⇀∇u2+∇ybu2weakly inL2((0;T)ΩY2):(4.48)
Moreover, as in [129] and [213],
@u1
@t2L2(0;T;L2(Ω));@u2
@t2L2(0;T;L2(Ω))
and
u12C0([0;T];H1
0(Ω)); u 22C0([0;T];H1(Ω)):
So,u12 V0(Ω) andu22 V(Ω).
Let us mention that, in fact, under our hypotheses, passing to a subsequence, T"
1(u"
1)
converges strongly to u1inLp((0;T)ΩY1), for 1 p<1. As a consequence, since the

92 4.3. Homogenization results for the calcium dynamics in living cells
Nemytskii operator corresponding to the nonlinear function fis continuous, it follows that
f(T"
1(u"
1)) converges to f(u1). A similar result holds true for u"
2.
Since
∥T"
1(u"
1) T"
2(u"
2)∥L2((0;T)Ω)C"1

2
it follows that for the case
= 0 and
=1 we have, at the macroscale,
u1=u2=u02 V0(Ω):
Moreover, for
=1, following the techniques from [94], one can prove that
T"
1(u"
1) T"
2(u"
2)
"⇀bu1bu2weakly inL2((0;T)Ω):
4.3.2 The main convergence results
We present now, without proofs, the main convergence results obtained in [219].
Theorem 4.17 (Theorem 1 in [219]) If
= 1, the solution (u"
1; u"
2)of system (4.44) con-
verges in the sense of (4.48), as "!0, to the unique solution (u1; u2)of the following
macroscopic problem:
8
>>>><
>>>>:@u1
@tdiv(A1∇u1)H(u2u1) =f(u1)in(0;T)Ω;
(1)@u2
@tdiv(A2∇u2) +H(u2u1) = (1 )g(u2)in(0;T)Ω;
u1(0;x) =u0
1(x); u 2(0;x) =u0
2(x)inΩ:(4.49)
Here,
H=∫
h0(y) dy
andA1andA2are the homogenized matrices, given by:
A1
ij=∫
Y1(
a1
ij+n∑
k=1a1
ik@1j
@yk)
dy;
A2
ij=∫
Y2(
a2
ij+n∑
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)(∇y1k+ek)) = 0; y2Y1;
A1(y)(∇y1k+ek)
= 0; y2;(4.50)
8
<
:divy(A2(y)(∇y2k+ek)) = 0; y2Y2;
A2(y)(∇y2k+ek)
= 0; y2:(4.51)

Chapter 4. Mathematical models in biology 93
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 diffusion processes in partially ssured media
(see [31] and [101]) or in the case of the modeling of the electrical activity of the heart (see
[15], [13] and [187]). 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.
If we assume that his given by (4.46), then, at the limit, the exchange term appearing
in (4.49) is of the form jjh(u1;u2).
Theorem 4.18 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(A0∇u0) =f(u0) + (1 )g(u0)in(0;T)Ω;
u0(0;x) =u0
1(x) +u0
2(x)inΩ:(4.52)
Here, the effective matrix A0is given by:
A0
ij=∫
Y1(
a1
ij+n∑
k=1a1
ik@1j
@yk)
dy+∫
Y2(
a2
ij+n∑
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)(∇y1k+ek)) = 0; y2Y1;
A1(y)(∇y1k+ek)
= 0; y2;(4.53)
8
<
:divy(A2(y)(∇y2k+ek)) = 0; y2Y2;
A2(y)(∇y2k+ek)
= 0; y2:(4.54)
In this case, the exchange at the interface leads to the modication of the limiting diffusion
matrix, but the insulation is not enough strong to impose the existence of two different limit
concentrations.
Theorem 4.19 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 effective concentration eld u0
satises:8
<
:@u0
@tdiv(A0∇u0) =f(u0) + (1 )g(u0)in(0;T)Ω;
u0(0;x) =u0
1(x) +u0
2(x)inΩ:(4.55)

94 4.3. Homogenization results for the calcium dynamics in living cells
The effective coefficients are given by:
A0;ij=∫
Y1(
a1
ij+n∑
k=1a1
ik@w1j
@yk)
dy+∫
Y2(
a2
ij+n∑
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)(∇yw1k+ek)) = 0; y2Y1;
divy(A2(y)(∇yw2k+ek)) = 0; y2Y2;
(A1(y)∇yw1k)
= (A2(y)∇yw2k)
; y 2;
(A1(y)∇yw1k)
+h0(y)(w1kw2k) =A1(y)ek
; y 2:(4.56)
It is important to notice that the diffusion coefficients depend now on h0. A similar result
holds true for the case in which his given by (4.46). Let us notice that in this case the
homogenized matrix is no longer constant, but it depends on the solution u0. A similar effect
was noticed in [6].
Remark 4.20 To simplify the presentation, we present here only the relevant cases
=
1;0;1. If
2(1;1), we obtain, at the limit, the macroscopic problem (4.50) and if

> 1, we obtain a problem similar to (4.49), but without the exchange term H(u2u1)
orjjh(u1;u2), respectively. For the case
<1, we get, at the macroscale, a standard
composite medium without any barrier resistance. Let us mention that in this case we have
w1k=w2kon, fork= 1;:::;n .
Remark 4.21 The conditions imposed for the nonlinear functions f;g, andhcan be relaxed.
For example, we can deal with the case in which fandgare maximal monotone graphs,
satisfying suitable growth conditions (see, e.g., [73]). Also, as in [177], [193], and [221], we
can consider more general nonlinear functions h.

Chapter 5
Multiscale modeling of composite
media with imperfect interfaces
In the last decades, the study of the effective properties of heterogeneous composite materials
with imperfect contact between their constituents has been a subject of huge interest for a
broad category of researchers, such as engineers, mathematicians, or physicists (see [26] and
[150]). In particular, the problem of analyzing the thermal transfer in media with imperfect
interfaces has attracted a lot of attention (for a nice review of the literature on imperfect
interfaces in heterogeneous media, we refer to [163] and [174]). The imperfect contact between
the constituents of a composite material can be produced by various causes: the existence of
some impurities at the boundaries, the presence of a thin interphase, the interface damage,
or chemical processes. The homogenization theory was successfully applied for describing
the behavior of such heterogeneous materials, with inhomogeneities at a length scale which
is much smaller than the characteristic dimensions of the system, leading to appropriate
macroscopic continuum models, obtained by averaging the rapid oscillations of the material
properties. Besides, such effective 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 material with
interfacial barrier, in the case in which the
ux is continuous and there exists a jump of the
temperature across the interface, was studied for the rst time, with the aid of the asymptotic
expansion method, in [27]. The convergence results in [ ?] were rigorously justied later,
in [95], [175], [103], [94], and [215], to cite just a few reference. In all these studies, at
the interface between the two components the
ux of the temperature was supposed to be
continuous, the temperature eld had a jump and the
ux was proportional to this jump.
Several cases were studied, depending on the order of magnitude with respect to the small
parameter"characterizing the size of the two constituents of the resistance generated by the
imperfect contact between the constituents, leading to different macroscopic problems.
Problems similar to the ones we are discussing here were addressed in the literature mainly
in two geometrical settings. For the case when both components of the two-composite material
are connected, we refer to [159], [192], [191], [190], [213], and [214]. The case in which only one
95

96 5.1. Multiscale analysis in thermal diffusion problems in composite structures
phase is connected, while the other one is disconnected was considered, for instance, in [95],
[175], and [94]. For similar homogenization problems of parabolic or hyperbolic type, we refer
to [92] and [148]. 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 [15], [13],
[96], [105], [107], [106], [140], [187], and [221].
In this chapter, we shall present some recent homogenization results for diffusion problems
in composite media with imperfect imperfect interfaces. We start by brie
y describing the
results obtained in [215] for a thermal diffusion problem in a so-called bi-connected structure .
In the same geometry, we shall present some homogenization results contained in [213] and
[214] for diffusion problems with dynamical boundary conditions. We remark that, using
similar techniques, we can analyze the asymptotic behavior of the solution of a system of
coupled partial differential equations appearing in the modeling of an elasticity problem in
a periodic structure formed by two interwoven and connected components with imperfect
contact at the interface (see [105]) or of a dynamic coupled thermoelasticity problem in
composite media with imperfect interfaces and various geometries (see [107] and [106]). We
end this chapter by presenting some recent results obtained in [47], [48], and [49] for diffusion
problems involving jumps both in the solution and in the
ux.
This chapter is based on the papers [47], [48], [49], [213], [214], and [215].
5.1 Multiscale analysis in thermal diffusion problems in com-
posite structures
In [215], we have analyzed, using the periodic unfolding method, the effective 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 are true for
more general reaction-diffusion processes. We considered that nonlinear sources are acting
in each phase and that at the interface between the two constituents the
ux is continuous,
while the temperature eld has a jump. Our goal was to describe 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 medium. Depending on the magnitude
of the resistance generated by the imperfect contact between the constituents, a threshold
phenomenon arises. More precisely, depending on the rate exchange between the two phases,
three important cases are addressed, leading to different limit problems. The results in
[215] constitute a generalization of those obtained in [101], [188], [213], and [214]. For heat
conduction problems in periodic materials with a different geometry, we refer to [94] and [175]
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 [73] and [188]). Similar problems have been addressed,
using different techniques, formal or not, in [26], [27], [163], and [101].

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 97
5.1.1 Problem setting
We place ourselves in the same setting as in Section 4.3. So, we consider an open bounded
material body Ω in Rn, withn3, with a Lipschitz-continuous boundary @Ω. We assume
that Ω is formed by two constituents, Ω"
1and Ω"
2, representing two materials with different
thermal features, separated by an imperfect interface ". We assume that both phases Ω"
1
and Ω"
2= ΩnΩ"
1are connected, but only Ω"
1touches the outer boundary @Ω.
Our main goal in [215] was to analyze the macroscopic behavior of the solution ( u"
1;u"
2)
of the following coupled system of equations:
8
>>>><
>>>>:div (A"
1∇u"
1) +(u"
1) =fin Ω"
1;
div (A"
2∇u"
2) +(u"
2) =fin Ω"
2;
A"
1∇u"
1
=A"
2∇u"
2
on ";
A"
1∇u"
1
="
h(u"
1;u"
2) on ";
u"
1= 0 on@Ω:(5.1)
In (5.1),is the exterior unit normal to Ω"
1,A"=A(x="), whereA2 M (1;1;Y) is a
Y-periodic symmetric matrix (as in Section 4.3.1), and f2L2(Ω). We remark that the

ux is continuous across the boundary ", but instead of the continuity of temperature, we
prescribe a Biot boundary condition. The functions =(r) and=(r) are supposed
to be continuous, non-decreasing with respect to rand such that (0) = 0 and (0) = 0.
Further, we suppose that there exist C0 andq, with 0 q<n= (n2), such that
j(r)j C(1 +jrjq) (5.2)
and
j(r)j C(1 +jrjq): (5.3)
We also assume, as in Section 4.3, that
h(u"
1;u"
2) =h"
0(x)(u"
2u"
1);
where
h"
0(x) =h0(x
")
andh0(y) is aY-periodic, smooth real function with h0(y)>0. Let let
H=∫
h0(y) d̸= 0:
We point out here that we can deal with the more general case in which the nonlinear functions
andare multi-valued maximal monotone graphs, as in [73].
For the proof of the well-posedness of problem (5.1), we refer to [15], [13], [188], [187],
[213], and [214].
Using the periodic unfolding method, we can describe the asymptotic behavior of the
solution of system (5.1), in terms of the values of the parameter
. Three important cases
need to be considered:
= 1,
= 0, and
=1.

98 5.1. Multiscale analysis in thermal diffusion problems in composite structures
5.1.2 The main results
We give now the effective behavior of the solutions of the microscopic model (5.1) in the
above mentioned cases.
For
= 1, using similar techniques as those developed in Section 4.3, i.e. obtaining
suitable a priori estimates and compactness results and using the periodic unfolding method,
we get similar convergence results as those stated in (4.48). More precisely, it follows that
there exist u12H1
0(Ω),u22H1(Ω),bu12L2(Ω;H1
per(Y1)),bu22L2(Ω;H1
per(Y2)) such that,
passing to a subsequence, for "!0, we have:
8
>><
>>:T"
1(u"
1)!u1strongly in L2(Ω;H1(Y1));
T"
1(∇u"
1)⇀∇u1+∇ybu1weakly inL2(ΩY1);
T"
2(u"
2)⇀u 2weakly inL2(Ω;H1(Y2));
T"
2(∇u"
2)⇀∇u2+∇ybu2weakly inL2(ΩY2):(5.4)
The main convergence result in this case is stated in the following theorem, proven in [215].
Theorem 5.1 (Theorem 1 in [215]) The solution (u"
1; u"
2)of system (5.1) converges, as
"!0, in the sense of (5.4), to the unique solution (u1; u2), withu1;u22H1
0(Ω), of the
following macroscopic problem:
{
div(A1∇u1) +(u1)H(u2u1) =finΩ;
div(A2∇u2) + (1 )(u2) +H(u2u1) = (1 )finΩ:(5.5)
Here,A1andA2are the homogenized matrices, dened by:
A1
ij=∫
Y1(
aij+aik@1j
@yk)
dy;
A2
ij=∫
Y2(
aij+aik@2j
@yk)
dy
and1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n , are the weak solutions of the local
problems8
<
:∇ y
((A1(y)∇y1k) =∇yA1(y)ek; y2Y1;
(A1(y)∇y1k)
=A1(y)ek
; y 2
and 8
<
:∇ y
((A2(y)∇y2k) =∇yA2(y)ek; y2Y2;
(A2(y)∇y2k)
=A2(y)ek
; y 2:
Thus, at macroscale, the composite medium, regardless of its discrete structure, is represented
by a continuous model, which is similar to the standard bidomain model , arising in the
context of diffusion in partially ssured media (see [31] and [101]) or in the case of electrical
activity of the heart (see [15], [13], and [187]). The composite medium is conceived as the

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 99
superimposition of two interpenetrating continuous media, coexisting at every point of the
domain.
Following the same techniques as those used in Section 4.3, we can treat the other two
relevant cases, namely
= 0 and
=1.
Theorem 5.2 For high contact resistance, i.e. for
= 0, we obtain, at the macroscale, only
one temperature eld. Thus,
u1=u2=u02H1
0(Ω)
andu0satises:
div(A0∇u0) +(u0) + (1 )(u0) =finΩ: (5.6)
Here, the effective matrix A0is given by:
A0
ij=∫
Y1(
aij+aik@1j
@yk)
dy+∫
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
<
:∇ y
((A1(y)∇y1k) =∇yA1(y)ek; y2Y1;
(A1(y)∇y1k)
=A1(y)ek
; y 2
and 8
<
:∇ y
((A2(y)∇y2k) =∇yA2(y)ek; y2Y2;
(A2(y)∇y2k)
=A2(y)ek
; y 2:
In this case, the insulation provided by the interface is sufficient to affect the homogenized
diffusion matrix, but it is not strong enough to ensure the existence of two different limit
phases.
Theorem 5.3 For the case
=1, i.e. for weak contact resistance, we still get, at the
limit,
u1=u2=u02H1
0(Ω)
and the effective temperature eld u0veries:
div(A0∇u0) +(u0) + (1 )(u0) =finΩ: (5.7)
The macroscopic coefficients are given by:
A0;ij=∫
Y1(
aij+aik@w1j
@yk)
dy+∫
Y2(
aij+aik@w2j
@yk)
dy;
wherew1k2H1
per(Y1)=R; w2k2H1
per(Y2)=R,k= 1;:::;n; are the weak solutions of the local
problems
8
>><
>>:∇ y
(A1(y)∇yw1k) =∇yA1(y)ek; y2Y1;
∇ y
(A2(y)∇yw2k) =∇yA2(y)ek; y2Y2;
(A1(y)∇yw1k)
= (A2(y)∇yw2k)
 y2;
(A1(y)∇yw1k)
+h0(y)(w1kw2k) =A1(y)ek
 y2:

100 5.2. Diffusion problems with dynamical boundary conditions
In this case, as expected, the effective coefficients depend on h0.
5.2 Diffusion problems with dynamical boundary conditions
As already mentioned in the previous section, recent computational and theoretical studies
investigating the macroscopic behavior of composite materials are based on a model which
considers the composite material, despite of its discrete structure, as a bidomain , i.e. the
coupling of two continuous superimposed domains. In this section, we shall present some
homogenization results obtained by using the oscillating test function method of L. Tartar
in [213] and generalized later, via the periodic unfolding method, in [214]. The aim of these
papers was to analyze the asymptotic behavior of the solution of a nonlinear problem arising
in the modeling of thermal diffusion in a two-component composite material. We assume
that we have nonlinear sources acting in one phase 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. More precisely, in the same geometry as the one described in Section 5.1,
we shall be interested in analyzing the asymptotic behavior of the solutions of the following
nonlinear system:
8
>>>>>>>><
>>>>>>>>:div (A"
1∇u"
1) +(u"
1) =fin Ω"
1(0;T);
div (A"
2∇u"
2) =f; in Ω"
2(0;T);
A"
1∇u"
1
=A"
2∇u"
2
on "(0;T);
A"
1∇u"
1
+a"@
@t(u"
1u"
2) ="g(u"
2u"
1) on "(0;T);
u"
1= 0 on@Ω(0;T);
u"
1(0;x)u"
2(0;x) =c0(x);on ":(5.8)
Here,f2L2(0;T;L2(Ω)),c02H1
0(Ω), anda>0. The function is continuous, monotonously
non-decreasing and such that (0) = 0 and the function gis continuously differentiable,
monotonously non-decreasing, and with g(0) = 0. We shall assume that there exist an expo-
nentq, with 0 q<n= (n2), and a positive constant Csuch that
j(v)j C(1 +jvjq);dg
dvC(1 +jvjq): (5.9)
For relevant particular examples of such functions, we can mention, e.g., the following ones:
(v) =v
1 +
v; ;
> 0 (Langmuir kinetics)
and
(v) =jvjp1v;0<p< 1 (Freundlich kinetics) :
Also, we can deal with the cases in which g(v) =kvorg(v) =kv3, withk>0.
For results about the well-posedness of problem (5.8) in suitable function spaces and for
proper energy estimates, we refer to [15], [45], and [187]. Using Tartar's method of oscillating
test functions (see [205]) and results from the classical theory of semilinear problems (see

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 101
[45] and [73]), we can show that the effective behavior of the solution of problem (5.8) is
described by a nonlinear system, similar to Barenblatt's model (see [31] and [101]), with new
terms capturing the effect of the interfacial barrier, of the dynamical boundary condition, and
of the presence of the given nonlinear sources. The results in [213] represent a generalization
of the corresponding ones obtained in [31] and [101], by dealing with nonlinear sources and
nonlinear dynamical transmission conditions. For related problems, see in [13] and [187].
By applying well-known extension results (see [69], [61], and [187]) and by using suitable
test functions, we can pass to the limit, with "!0, in the weak formulation of problem
and we obtain the effective behavior of the solution of our microscopic model. The main
convergence result in this case is stated in the next theorem.
Theorem 5.4 (Theorem 2.1 in [213]) There exist two extensions P"u"
1andP"u"
2of the solu-
tionsu"
1andu"
2of problem (5.8) such that P"u"
1⇀u 1; P"u"
2⇀v, weakly in L2(0;T;H1
0(Ω)),
where8
>>><
>>>:ajj@
@t(u1u2)div(A1∇u1) +(u1) jjg(u2u1) =finΩ(0;T);
ajj@
@t(u2u1)div(A2∇u2)+jjg(u2u1) = (1 )finΩ(0;T);
u1(0;x)u2(0;x) =c0(x)onΩ:(5.10)
Here,A1andA2are the homogenized matrices, given by
A1
ij=∫
Y1(
aij+aik@1j
@yk)
dy;
A2
ij=∫
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 local problems
8
<
:∇ y
((A1(y)∇y1k) =∇yA1(y)ek; y2Y1;
(A1(y)∇y1k)
=A1(y)ek
; y 2;
8
<
:∇ y
((A2(y)∇y2k) =∇yA2(y)ek; y2Y2;
(A2(y)∇y2k)
=A2(y)ek
; y 2:
Thus, at the macroscale, we are led to a Barenblatt model, which is similar to the bidomain
model arising in the context of electrical activity of the heart. As already mentioned, in such
a model, at a macroscopic level, regardless of its discrete cellular structure, the composite
medium can be represented by a continuous model, describing its averaged properties.
We point out that the above model is a degenerate parabolic system, as the time deriva-
tives involve the unknown vuand not the unknowns uandvoccurring in the second-order
conduction terms.
These results were generalized in [214] where we used the periodic unfolding method and
we considered more general nonlinearities (x;u") andg(x;v"u").

102 5.3. Homogenization of a thermal problem with
ux jump
5.3 Homogenization of a thermal problem with
ux jump
In [47], our aim was to study, via homogenization techniques, the macroscopic thermal trans-
fer in a periodic composite material made up of two constituents, one connected and the other
one disconnected, separated by an interface where both the temperature and its
ux present
jumps. Such a mathematical model can be used in other contexts, as well. For instance,
transmission problems involving jumps in the solutions or in their
uxes arise in linear elas-
ticity, in the theory of semiconductors, in the study of photovoltaic systems, or in problems
in media with cracks (see, e.g., [26], [38], [46], [133], and [153]). With few exceptions, only
formal methods of averaging were used in the literature to deal with such imperfect transmis-
sion problems and obtaining rigorous homogenization results is still a difficult task. However,
we can mention here the rigorous results obtained in [143], [115], or [110].
The major novelty brought by us in [47] resides in assuming, apart from the discontinuity
of the temperature eld, of a jump in the thermal
ux across the imperfect interface ", given
by suitable functions G". Two representative cases for such functions were analyzed, leading,
at the limit, to two different homogenized problems (stated in Theorem 5.10 and Theorem
5.15 below).
5.3.1 Setting of the problem
Following [47], let us consider a composite material occupying an open bounded set Ω in
Rn, withn2, with a Lipschitz-continuous boundary @Ω. We assume that Ω is made up
of two parts denoted Ω"
1and Ω"
2, separated by an imperfect interface ". We suppose that
the phase Ω"
1is connected and touches the exterior boundary @Ω and that the second phase,
Ω"
2, is disconnected, being the union of domains of size ", periodically distributed in Ω, with
periodicity ". More precisely, we assume that Y1andY2are two non-empty disjoint connected
open subsets of the reference cell Y= (0;1)ninRn, such that Y2YandY=Y1[Y2.
We also assume that = @Y2is Lipschitz continuous and that the set Y2is connected.
For eachk2ZN, we setYk=k+YandYk
=k+Y, for= 1;2. For each ", we
deneZ"={
k2ZN:"Yk
2Ω}
and we take Ω"
2=∪
k2Z"(
"Yk
2)
and Ω"
1= ΩnΩ"
2. The
boundary of Ω"
2is denoted by "andis the outward unit normal to Ω"
1.
We aim at describing the asymptotic behavior, as "!0, of the solution u"= (u"
1;u"
2) of
the following problem:
8
>>>>>>><
>>>>>>>:div (A"∇u"
1) =fin Ω"
1;
div (A"∇u"
2) =fin Ω"
2;
A"∇u"
1
=h"
"(u"
1u"
2)G"on ";
A"∇u"
2
=h"
"(u"
1u"
2) on ";
u"
1= 0 on@Ω:(5.11)
Remark 5.5 We point out that
A"∇u"
2
A"∇u"
1
=G"; (5.12)

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 103
which means that the thermal
ux indeed exhibits a jump across the interface ".
The function f2L2(Ω) is given. If his aY{periodic function in L1() such that there
existsh02Rwith 0<h0<h(y) a.e. on , then we take
h"(x) =h(x
")
a.e. on ":
For aY-periodic symmetric matrix A2 M (1;1;Y) (see Section 4.3.1), we set
A"(x) =A(x
")
a.e. in Ω:
Letgbe aY-periodic function that belongs to L2(). We dene
g"(x) =g(x
")
a.e. on ":
As already mentioned, for the given function G"in (5:11) we consider the following relevant
forms (see, also, [55]):
Case 1 :G"="g(x
")
, ifM(g)̸= 0.
Case 2 :G"=g(x
")
, ifM(g) = 0.
Here,
M(g) =1
jj∫
g(y) dy
denotes the mean value of the function gon .
For writing the weak formulation of problem (5.11), we consider, for every positive "<1,
the Hilbert space
H"=V"H1(Ω"
2); (5.13)
where the space
V"={
v2H1(Ω"
1); v= 0 on@Ω}
is equipped with the norm
∥v∥V"=∥∇v∥L2(Ω"
1);
for anyv2V", and the space H1(Ω"
2) is endowed with its standard norm. On H", we dene
the scalar product
(u;v)H"=∫
Ω"
1∇u1∇v1dx+∫
Ω"
2∇u2∇v2dx+1
"∫
"(u1u2)(v1v2) dx (5.14)
whereu= (u1;u2) andv= (v1;v2) belong to H". The norm induced by the scalar product
(5.14) is given by
∥v∥2
H"=∥∇v1∥2
L2(Ω"
1)+∥∇v2∥2
L2(Ω"
2)+1
"∥v1v2∥2
L2("): (5.15)

104 5.3. Homogenization of a thermal problem with
ux jump
The weak 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 dened by
a(u;v) =∫
Ω"
1A"∇u1∇v1dx+∫
Ω"
2A"∇u2∇v2dx+∫
"h"
"(u1u2)(v1v2) dx
and
l(v) =∫
Ω"
1fv1dx+∫
Ω"
2fv2dx+∫
"G"v1dx;
respectively.
In the next theorem, we state an existence and uniqueness result and some a priori
estimates for the solution of the weak problem (5.16).
Theorem 5.6 For any"2(0;1), the variational problem (5:16)has a unique solution u"2
H"and there exists a constant C > 0, independent of ", such that
∥∇u"
1∥L2(Ω"
1)C;∥∇u"
2∥L2(Ω"
2)C (5.17)
and
∥u"
1u"
2∥L2(")C"1=2: (5.18)
In order to obtain the macroscopic behavior of the solution of problem (5.16), we shall use
the unfolding operators T"
1andT"
2and the boundary unfolding operator T"
bdened in Section
4.3.1. As already mentioned, the main feature of these operators is that they map functions
dened on the oscillating domains Ω"
1, Ω"
2and, respectively, ", into functions dened on the
xed domains Ω Y1, ΩY2and Ω , respectively.
From [94], we know that if u"= (u"
1;u"
2) is a sequence in H", then
1
"jYj∫
ΩjT"
1(u"
1) T"
2(u"
2)j2dxdy∫
"ju"
1u"
2j2dx:
Moreover, if φ2 D(Ω), then, for "small enough, one has, for = 1 or= 2,
"∫
"h"(u"
1u"
2)φdx=∫
Ωh(y) (T"
1(u"
1) T"
2(u"
2))T"
(φ) dxdy:
Besides, the following general compactness results were obtained in [94].
Lemma 5.7 Letu"= (u"
1;u"
2)be a bounded sequence in H". Then, there exists a constant
C > 0, independent of ", such that
∥T"
1(∇u"
1)∥L2(ΩY1)C;
∥T"
2(∇u"
2)∥L2(ΩY2)C;
∥T"
2(u"
1) T"
1(u"
2)∥L2(Ω)C":

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 105
Theorem 5.8 Letu"= (u"
1;u"
2)be a bounded sequence in H". Then, up to a subsequence,
still denoted by ", there exist u12H1
0(Ω),u22L2(Ω),bu12L2(
Ω;H1
per(Y1))
andbu22
L2(
Ω;H1(Y2))
such that
T"
1(u"
1)!u1strongly in L2(
Ω;H1(Y1))
;
T"
1(∇u"
1)⇀∇u1+∇ybu1weakly inL2(ΩY1);
T"
2(u"
2)⇀u 2weakly inL2(Ω;H1(Y2));
T"
2(∇u"
2)⇀∇ybu2weakly inL2(ΩY2);
eu"
⇀jYj
jYjuweakly inL2(Ω);  = 1;2;
where M(bu1) = 0 for almost every x2Ωand~
is the extension by zero of a function to the
whole of the domain Ω. Further, we get u1=u2and
1
"[T"
1(u"
1) M (T"
1(u"
1))]⇀y ∇u1+bu1weakly inL2(
Ω;H1(Y1))
;
withy=y M (y), and
1
"[T"
2(u"
2) M (T"
2(u"
2))]⇀bu2weakly inL2(
Ω;H1(Y2))
:
5.3.2 The macroscopic models
We aim now at passing to the limit, with "!0, in the variational formulation (5.16) of the
problem (5.11). We point out again that by applying the general results stated in Theorem
5.8 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 the sequel, their common value will be denoted by
u. We remark that ubelongs toH1
0(Ω).
From the priori estimates (5.17)-(5.18) and the above mentioned general compactness
results, it follows that there exist u2H1
0(Ω),bu12L2(Ω;H1
per(Y1)),bu22L2(Ω;H1(Y2)) such
thatM(bu1) = 0 and, up to a subsequence, for "!0, we obtain:
T"
1(u"
1)!ustrongly in L2(Ω;H1(Y1));
T"
1(∇u"
1)⇀∇u+∇ybu1weakly inL2(ΩY1);
T"
2(u"
2)⇀u weakly inL2(Ω;H1(Y2));
T"
2(∇u"
2)⇀∇ybu2weakly inL2(ΩY2);
eu"
⇀jYj
jYjuweakly inL2(Ω);  = 1;2:(5.19)
Besides, one gets
T"
1(u"
1) T"
2(u"
2)
"⇀bu1u2weakly inL2(Ω); (5.20)

106 5.3. Homogenization of a thermal problem with
ux jump
where, for some 2L2(Ω), u22L2(Ω;H1(Y2)) is given by
u2=bu2y∇u:
Let
Wper(Y1) =fv2H1
per(Y1)jM(v) = 0g:
We consider the space
V=H1
0(Ω)L2(Ω;Wper(Y1))L2(
Ω;H1(Y2))
;
endowed with the norm
∥V∥2
V=∥∇v+∇ybv1∥2
L2(ΩY1)+∥∇v+∇yv2∥2
L2(ΩY2)+∥bv1v2∥2
L2(Ω);
for all
V= (v;bv1;v2)2 V:
In order to pass to the limit, we have to treat separately the above mentioned two cases
for the function G".
Case 1 :G"="g(x
")
, ifM(g)̸= 0.
For this case, via the periodic unfolding method, we proved in [47] the following convergence
result:
Theorem 5.9 (Theorem 4.1 in [47]) The unique solution u"= (u"
1;u"
2)of the variational
problem (5.16) converges, in the sense of (5.19), to the unique solution (u;bu1;u2)2 Vof the
following unfolded limit problem:
1
jYj∫
ΩY1A(y)(∇u+∇ybu1)(∇φ+∇y1) dxdy+
1
jYj∫
ΩY2A(y)(∇u+∇yu2)(∇φ+∇y2) dxdy+
1
jYj∫
Ωh(y)(bu1u2)(12) dxdy=∫
Ωf(x)φ(x) dx+jj
jYjM(g)∫
Ωφ(x) dx;(5.21)
for allφ2H1
0(Ω),12L2(Ω;H1
per(Y1))and22L2(Ω;H1(Y2)).
We remark that in the limit problem (5.21) the right-hand side contains an extra term
involving the function g. More precisely, our right-hand side writes
∫
ΩF(x)φ(x) dx;
with
F(x) =f(x) +jj
jYjM(g):

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 107
Corollary 5.10 (Corollary 1 in [47]) The function u2H1
0(Ω)dened by (5.19) is the unique
solution of the following homogenized equation:
div(Ahom∇u) =f+jj
jYjM(g)inΩ; (5.22)
whereAhomis the homogenized matrix, given, for i;j= 1;:::;n , by
Ahom
ij=1
jYj∫
Y1(
aijn∑
k=1aik@j
1
@yk)
dy+1
jYj∫
Y2(
aijn∑
k=1aik@j
2
@yk)
dy; (5.23)
in terms of j
12H1
per(Y1)andj
22H1(Y2),j= 1;:::;n , the weak solutions of the local
problems8
>>>>><
>>>>>:divy(A(y)(∇yj
1+ej)) = 0; y2Y1;
divy(A(y)(∇yj
2+ej)) = 0; y2Y2;
(A(y)∇yj
1)
= (A(y)∇yj
2)
; y 2;
(A(y)(∇yj
1+ej))
+h(y)(j
1j
2) = 0; y2:
M(j
1) = 0;(5.24)
whereis the exterior unit normal to Y1.
Remark 5.11 We notice that the homogenized matrix Ahomdepends on the function h.
Thus, the effect of the two jump functions involved in the microscopic problem is recovered in
the limit problem, both in the right-hand side and in the left-hand side, through the homoge-
nized coefficients.
Case 2 :G"(x) =g(x
")
, ifM(g) = 0.
For this case, the convergence result, proven in [47], is stated in the next theorem.
Theorem 5.12 (Theorem 4.2 in [47]) The unique solution u"= (u"
1;u"
2)of the variational
problem (5.16) converges, in the sense of (5.19), to the unique solution (u;bu1;u2)2 Vof the
following unfolded limit problem:
1
jYj∫
ΩY1A(y)(∇u+∇ybu1)(∇φ+∇y1) dxdy+
1
jYj∫
ΩY2A(y)(∇u+∇yu2)(∇φ+∇y2) dxdy+
1
jYj∫
Ωh(y)(bu1u2)(12) dxdy=
∫
Ωf(x)φ(x) dx+1
jYj∫
Ωg(y)1(x;y) dxdy; (5.25)
for allφ2H1
0(Ω),12L2(Ω;H1
per(Y1)),22L2(Ω;H1(Y2)).

108 5.3. Homogenization of a thermal problem with
ux jump
Let us note that the term1
jYj∫
Ωg(y)1(x;y) dxdyin (5.25) involves explicitly both
variablesxandyand its contribution in the homogenized problem will be nonstandard. In
fact, as we shall see in the next theorem, apart from the standard solutions j
1andj
2of
the cell problems (5.24), we are forced to introduce two additional scalar terms 1and2,
verifying a new imperfect transmission cell problem (see (5.35)).
Theorem 5.13 (Theorem 4.3 in [47]) The function u2H1
0(Ω)dened in (5.19) is the
unique solution of the homogenized equation
div(Ahom∇u) =finΩ; (5.26)
whereAhomis the homogenized matrix whose entries are given in (5.26). Besides, we have
bu1(x;y) =N∑
j=1@u
@xj(x)j
1(y) +1(y);
u2(x;y) =N∑
j=1@u
@xj(x)j
2(y) +2(y);
wherej
1andj
2are dened by (5.27) and the function (1;2)is the unique solution of the
cell problem8
>>>><
>>>>:divy(A(y)∇1) = 0 inY1;
divy(A(y)∇2) = 0 inY2;
A(y)∇1
=h(y)(12)g(y)on;
A(y)∇2
=h(y)(12)on;
M(1) = 0:
Proof. If we takeφ= 0 in (5.25), we get
1
jYj∫
ΩY1A(y)(∇u+∇ybu1)∇y1dxdy+1
jYj∫
ΩY2A(y)(∇u+∇yu2)∇y2dxdy+
1
jYj∫
Ωh(y)(bu1u2)(12) dxdy=1
jYj∫
Ωg(y)1(x;y) dxdy: (5.27)
If we choose now suitable test functions  1and  2in (5.27), we obtain
divy(A(y)∇ybu1) = div y(A(y)∇u) a.e. in Ω Y1; (5.28)
divy(A(y)∇yu2) = div y(A(y)∇u) a.e. in Ω Y2; (5.29)
A(y)(∇u+∇yu2)
=h(y)(bu1u2) a.e. on Ω ; (5.30)
A(y)(∇u+∇ybu1)
=h(y)(bu1u2)g(y) a.e. on Ω : (5.31)
We remark that we have a discontinuity type condition:
A(y)(∇u+∇yu2)
A(y)(∇u+∇ybu1)
=g(y) a.e. on Ω : (5.32)

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 109
As in [47], we search bu1and u2in the following nonstandard form:
bu1(x;y) =n∑
j=1@u
@xj(x)j
1(y) +1(y); (5.33)
u2(x;y) =n∑
j=1@u
@xj(x)j
2(y) +2(y); (5.34)
wherej
1andj
2are dened by (5.24) and the functions 1,2have to be found. In this way,
we obtain 8
>>>><
>>>>:divy(A(y)∇1) = 0 inY1;
divy(A(y)∇2) = 0 inY2;
A(y)∇1
=h(y)(12)g(y) on ;
A(y)∇2
=h(y)(12) on ;
M(1) = 0:(5.35)
Since
A(y)∇2
A(y)∇1
=g(y); (5.36)
we note that the new local problem (5.35) is an imperfect transmission problem, involving
both the discontinuities in the solution and in the
ux, given in terms of handg, respectively.
Using Lax-Milgram theorem, we can prove this problem has a unique solution in the space
H=Wper(Y1)H1(Y2);
equipped with the scalar product
(;)H= (∇1;∇1)L2(Y1)+ (∇2;∇2)L2(Y2)+ (12;12)L2():
By taking now  1=  2= 0 in (5.25), we obtain:
1
jYj∫
ΩY1A(y)(∇u+∇ybu1)∇φdxdy+1
jYj∫
ΩY2A(y)(∇u+∇yu2)∇φdxdy=
∫
Ωf(x)φ(x) dx: (5.37)
Integrating it by parts with respect to x, we have:
divx(1
jYj∫
Y1A(y)(∇u+∇ybu1) dy+1
jYj∫
Y2A(y)(∇u+∇yu2) dy)
=f(x) in Ω:
By using the formulas (5.33) and (5.34) for the functions bu1and u2and the denition of the
matrixAhom, we obtain
divx(Ahom∇u) =f+ div x(1
jYj∫
Y1A(y)∇1(y)dy+1
jYj∫
Y2A(y)∇2(y)dy)
in Ω;
which gives exactly the homogenized problem (5.26). Due to the fact that the second term of
the right-hand side in the above equation vanishes, the limit problem (5.26) does not involve
the function g.

110 5.3. Homogenization of a thermal problem with
ux jump
Remark 5.14 As mentioned in [47], all the above results remain true in the case in which
A"is a sequence of matrices in M(1;1;Ω)such that
T"
(A")!Astrongly in L1(ΩY);
for some matrix A=A(x;y)inM(1;1;ΩY). The heterogeneity of the medium given
by such a matrix generates different effects in the limit problems. More precisely, in both
cases, due to the fact that the correctors j
depend also on x, the homogenized matrix Ahom
x
depends on x. Moreover, an interesting effect arises in the second case, since, due to the fact
that the functions and the matrix Aare depending on x, the function gbrings an explicit
contribution in the homogenized problem, which becomes
divx(Ahom
x∇u) =
f+divx(1
jYj∫
Y1A(x;y)∇1(x;y)dy+1
jYj∫
Y2A(x;y)∇2(x;y)dy)
inΩ:
A similar effect was observed in the homogenization of the Neumann problem in perforated
domains (see [56]).
Remark 5.15 Corrector results for these problems can be obtained, too (see [47]).
The homogenization of a thermal diffusion problem in a highly heterogeneous medium
formed by two constituents, Ω"
1and Ω"
2, separated by an imperfect interface was addressed in
[48], as well. We assume the discontinuity of the thermal conductivity over the domain as we
go from one constituent to another one and the presence of an imperfect interface between
the two constituents, where both the temperature and its
ux exhibit jumps. The order of
magnitude of the thermal conductivity of the material occupying the domain Ω"
2is"2, while
the conductivity of the material occupying the domain Ω"
1is supposed to be of order one.
Our problem presents various sources of singularities, described in terms of ": 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.
More precisely, our goal in [48] was to study the asymptotic behavior, as "!0, of the
solutionu"= (u"
1;u"
2) of the following problem:
8
>>>>><
>>>>>:div (A"∇u"
1) =fin Ω"
1;
div ("2A"∇u"
2) =fin Ω"
2;
A"∇u"
1
n"="h"(u"
1u"
2)G"on ";
"2A"∇u"
2
n"="h"(u"
1u"
2) on ";
u"
1= 0 on@Ω:(5.38)
For the given function G"in (5.38), we consider the two relevant situations mentioned
above. In the rst case, using similar techniques as before, we are led to the following result.

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 111
Theorem 5.16 The unique solution u"= (u"
1;u"
2)of the problem (5.38) converges to
(u1;bu1;bu2)2 V;
whereu1is the unique solution of the homogenized problem
8
<
:div(Ahom∇u1(x)) =f(x) +jj
jYjM(g)inΩ;
u1= 0 on@Ω(5.39)
and
bu1(x;y) =n∑
j=1@u1
@xj(x)j
1(y)inΩY1; (5.40)
bu2(x;y) =u1(x) +f(x)2(y)inΩY2: (5.41)
Here,Ahomis the constant homogenized matrix whose entries are dened, for i;j= 1;:::;n
by
Ahom
ij=1
jYj∫
Y1(
aijN∑
k=1aik@j
1
@yk)
dy: (5.42)
The vectorial function j
12H1
per(Y1)(j= 1;:::;n ) and the scalar function 22H1(Y2)are
the weak solutions of the following cell problems:
8
><
>:divy(A(y)(∇yj
1ej)) = 0 inY1;
(A(y)(∇yj
1ej))
n= 0 on;
M(j
1) = 0(5.43)
and {divy(A(y)∇y2) = 1 inY2;
A(y)∇y2
n+h2= 0 on;(5.44)
wherendenotes the unit outward normal to Y2.
For the second case, the main convergence result in [48] is stated in the following theorem.
Theorem 5.17 The unique solution u"= (u"
1;u"
2)of the problem (5.38) converges to
(u1;bu1;bu2)2 V;
whereu1is the unique solution of the homogenized problem
{div(Ahom∇u1(x)) =f(x)inΩ;
u1= 0 on@Ω(5.45)
and
bu1(x;y) =n∑
j=1@u1
@xj(x)j
1(y) +(y); (5.46)
bu2(x;y) =u1(x) +f(x)2(y):

112 5.4. Other homogenization problems in media with imperfect interfaces
Here,Ahomis the homogenized matrix whose entries are given by (5.42) and the functions
j
1and2are dened by (5.43) and (5.44), respectively. The Y-periodic function is the
unique solution of the following non homogeneous Neumann cell problem:
8
<
:divy(A(y)∇y) = 0 inY1;
A(y)∇y
n=g(y)on;
M() = 0:(5.47)
Thus, the limit problems, obtained via the periodic unfolding method, capture the in-

uence of the jumps in the limit temperature eld, in an additional source term, or in the
correctors.
5.4 Other homogenization problems in composite media with
imperfect interfaces
Problems involving jumps in the solution can be encountered in various other situations. We
shall brie
y mention here some results we obtained recently for such problems.
For instance, our goal in [105] 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. 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
and many models have been proposed in the literature. 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 effect 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 stiff interphase, characterized by a jump of the traction
vector across the interface between the two media (see [38] and [133]). 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
[184]). For more details concerning the corresponding mechanical models, we refer to [38],
[133], [163], [168], and [184].
In [105], we assumed that on the interface between the constituents of the two-composite
medium 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 different important cases. More precisely, we obtain, at the macroscale,
one or two equations, with different stiffness tensors:

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 113
if the jump is of order "1, we obtain only one equation at the macroscale, with the
stiffness tensor depending on the jump coefficient;
if the jump is of order ", we get a system of two coupled equations with classical stiffness
tensors;
if 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. Such a setting
is relevant for dealing with contact problems for multiphase composites with an interfacial
resistance generated by 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 used for dealing with other
geomaterials, such as mortar, soils or rocks. Similar problems have been considered, using
different techniques, formal or not, in [102], [158], [163], and [173]. Recently, using the
periodic unfolding method, some elasticity problems for media with open and closed cracks
were studied in [60]. For other related elasticity problems, see [117] and [203]. 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 [106]. 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 [142]. Also, for some interesting thermoelasticity models, we refer to [10], [101],
[102], and [112]. In [107], a similar model was considered, but in a different geometry and with
different scalings of the temperature-displacement tensors of the two constituents, leading to
different homogenized results. More precisely, the domain Ω was considered to be the union
of a connected part Ω"
1and a disconnected one Ω"
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. Also, let us remark that the functional setting is
different compared to the one used in [106].
Recently, in [54], we have been 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. We considered a periodic structure, occupied by a linearly elastic body,
clamped along a part of its outer boundary. On the rest of the exterior boundary, surface
tractions were given. The body was subjected to the action of given volume forces and several
nonlinear conditions on the boundary of the rigid inclusions were considered. More precisely,

114 5.4. Other homogenization problems in media with imperfect interfaces
we treated the case in which a nonlinear Robin condition is imposed and, respectively, the
case in which unilateral contact with given friction is taken into consideration. For the Robin
problem, we extended, via the periodic unfolding method, some of the results contained
in [118] and [144], by considering general nonlinearities in the condition imposed on the
boundary of the inclusions. Also, we established an homogenization result for a Signorini
problem with Tresca friction. The difficulties of this problem came from the fact that the
unilateral condition generates a convex cone of admissible displacements, and, especially, from
the fact that the friction condition involves a nonlinear functional containing the norm of the
tangential displacement on the boundary of the rigid inclusions. The macroscopic problem
is different from the one addressed in [60]. In particular, for the frictionless contact case,
we regained a result obtained, under more restrictive assumptions, in [144]. This frictionless
problem was also addressed in [145], by the two-scale convergence method, for more general
geometric structures of the inclusions on which the Signorini conditions act.

Part II
Career Evolution and Development
Plans
115

Chapter 6
Scientic and academic background
and research perspectives
In this chapter, I shall brie
y present my scientic and academic career, emphasizing the au-
tonomy and the visibility of my research activity performed after obtaining my Ph.D. in 1996.
Also, some further research directions and some future plans on my scientic and academic
career will be presented. I shall discuss some short, medium and long term development
plans and a brief description of some open questions I would like to study in the future will
be made, as well.
6.1 Scientic and 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 "The atmospheric

uid
ow in the presence of orographic obstacles", was Professor Horia I. Ene. In 1992, I
started a Ph.D. program at the Institute of Mathematics "Simion Stoilow" of the Romanian
Academy, under the supervision of Professor Horia I. Ene. The title of my thesis, defended
in 1996, was "Applications of stochastic processes in
uid mechanics". During these years, I
had the chance to study in one of the best Romanian universities and to prepare my Ph.D.
in a leading research institution in Romania, where I had the opportunity to interact with
some of the best Romanian mathematicians of our times.
After completing my Ph.D., between 2000 and 2003, I beneted from four post-doctoral
fellowships at University Complutense of Madrid (Spain), University of Pisa (Italy), and
Center of Mathematical Modelling, University of Chile, Santiago de Chile (Chile). I had
the chance to work, in a stimulating environment, under the supervision of top specialists in
applied mathematics: Professor Enrique Zuazua, Professor Giuseppe Buttazzo, and Professor
Carlos Conca.
I was invited to perform several research visits and I gave several talks at universities and
research institutions from abroad, where I had the chance to establish fruitful collaborations
with well-known specialists in the eld of homogenization theory: University of Cantabria,
117

118 6.1. Scientic and academic background
Santander, Spain (2016); University of Lorraine, Metz, France (2015, 2016); Complutense
University, Madrid, Spain (2005, 2014, 2015); Humboldt University, Berlin, Germany (2010);
RWTH Aachen, Germany (2007, 2008, 2009); Taras Sevcenko University, Kiev, Ukraine
(2008); University of Pavia, Italy (2008); Technical University of Eindhoven, Holland (2005).
All these visits helped me to enlarge my horizon and to establish solid international colla-
borations with outstanding researchers: C. Conca (Chile), J. I. D az (Spain), F. Murat
(France), A. Li~ n an (Spain), G. Buttazzo (Italy), E. Zuazua (Spain), M. E. P erez (Spain), D.
G omez (Spain), D. G omez-Castro (Spain), and R. Bunoiu (France).
I gave invited talks at several prestigious international conferences and various seminars at
foreign universities in countries like Italy, Spain, France, Germany, Holland, Chile, Bulgaria,
Ukraine, or Turkey.
My post-doctoral scientic research has been mainly devoted to the following elds, in
which I published more than 80 papers (see the list of publications): homogenization theory;
multiscale modeling, reaction-diffusion processes in porous media, and mathematical models
in biology and in engineering.
I was director for several research grants, such as Grant 3046GR/1997 of the Romanian
Academy, Grant 4064GR/1998 of the Romanian Academy, CNCSIS 1059, 2006-2007, PN II
– IDEAS, 2007 – 2010, Bilateral project LEA Math Mode /2015 (co-director with dr. R.
Bunoiu, Metz, France). I was also member in several other national or international projects.
Concerning my didactic activity, between 1991 and 2008, I was assistant professor, lec-
turer, and then associate professor at the Faculty of Physics of the University of Bucharest.
Since 2008, I am professor at the same faculty. During this period, I was involved in teaching
various courses and seminars, both at undergraduate and at graduate level: Real Analysis,
Complex Analysis, Ordinary Differential Equations, Complements of Mathematics, etc. I
published ten books or chapters in books dedicated to my students.
Theautonomy and the visibility of my research activity performed after the
completion of my Ph.D. studies is supported by the following arguments:
I published, as main author, more than sixty papers in peer-reviewed journals;
I published more than twenty papers in proceedings of national and international confe-
rences;
I am single author for more than fty papers;
I published more than seventy papers after completing my Ph.D. thesis;
I gave more than forty talks at international conferences;
I gave sixteen invited talks at international conferences;
I gave eleven invited seminars abroad;
I obtained four post-doctoral fellowships;

Chapter 6. Scientic and academic background and research perspectives 119
I was director of ve research projects and member of seven other national or interna-
tional projects;
I am a member of the American Mathematical Society, the Society for Industrial and
Applied Mathematics, and of Romanian Mathematical Society;
I am reviewer for Mathematical Reviews and for more than thirty international journals
(SIAM Journal of Applied Mathematics, Networks and Heterogeneous Media, etc);
I am member of editorial board of international journals (Biomath Communications,
Abstract and Applied Analysis);
I was member of the scientic committee for several international conferences (MMSC
2016, BIOMATH 2016, SVCS 2014, 2015, 2016);
I was evaluator for international research projects (Tubitak, NWO);
I was evaluator for Ph.D. theses and member of several academic promotion or recruit-
ment committees.
6.2 Further research directions
I shall brie
y describe here the perspectives I see for my research in the next years. A
few of them are, in fact, already ongoing works. Basically, I plan to continue my work in
the broad eld of homogenization theory and 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 carcinogenesis in living cells and at performing a rigorous homoge-
nization study for periodic structured materials with imperfect interfaces. Also, I think at
elaborating new mathematical models for electromagnetic periodic composites and at study-
ing nonlinear transmission problems in composites with various other geometries than those
already considered in the literature.
I.1. Mathematical models for electrochemically coupled excitable biological
tissues. I plan to rigorously justify and generalize some existing homogenized models for
the description of excitable biological tissues electrochemically coupled through gap junctions.
The formal results obtained in [120] for doughball gap junction model will be rigorously
proven, via the periodic unfolding method. 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
and I think that this model is well suited to tackle more general gating laws than the syncytial
model. Such a study is motivated by the need to fully understand wave propagation and
failure experimentally observed in the pancreatic islets of Langerhans. Recent theoretical

120 6.2. Further research directions
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 junctional 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 skin electropermeabilization. Transdermal drug de-
livery is an alternative to standard drug delivery methods of injection or oral administration.
The exterior 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 [36]). 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 effects produced in the skin by the electrical pulses in order to design useful electro-
permeabilization protocols. The problem is complex, involving a very complicated geometry
and the nonlinear coupling of a diffusion 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 composite medium, our goal is to analyze the effective behavior, as the period of the
microstructure tends to zero, of the solutions of this coupled system of partial differential
equations. We shall analytically investigate the effect of various parameters on the effective
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. I shall be concerned
with the carcinogenic effects 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.
I plan to generalize the results obtained in [127] and [221]. The microscopic mathematical
model, including reaction-diffusion 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. I shall consider that BP molecules enter in the cytosol inside
of a human cell. There, they diffuse 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 diffuse 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

Chapter 6. Scientic and academic background and research perspectives 121
ER, I shall consider various nonlinear functions, with various scalings, leading to different
homogenized models. I shall deal with the case of general nonlinear (even discontinuous)
isotherms, similar to those used in [78], [157], and [193], and of multiple metabolisms BP !
DE. I shall also generalize a carcinogenesis model, introduced in [127], involving free receptors
on the surface of the ER (see, also, Section 4.2.3).
II.1. Homogenization of reaction-diffusion problems with
ux jump. This is
an ongoing joint work with Dr. Renata Bunoiu from the University of Lorraine-Metz, France.
We shall continue our study on the homogenization of a thermal diffusion problem in a highly
heterogeneous medium formed by two constituents, separated by an imperfect interface (see
Section 5.3). We shall be interested in dealing with other geometrical settings and with
some nonlinear reaction-diffusion problems in periodic composite media which exhibit at the
interfaces between their components jumps of the solution and of the
ux. Such problems are
relevant in the the context of thermal diffusion in composites, in the theory of semiconductors,
in linear elasticity, or in reaction-diffusion problems in biological tissues. We plan to apply
our results to the study of calcium dynamics in biological tissues modeled as media with
imperfect interfaces. We also think at extending our analysis to nonlinear problems, this
being a largely open case in the literature.
II.2. Homogenization results for electromagnetic composite materials. Using
the periodic unfolding method or Gamma-convergence method, as an alternative plan, I shall
address the problem of nding the effective parameters for electromagnetic periodic composite
materials in the quasi-static case. The developed strategy will allow one 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. I
shall generalize some of the results obtained in [186].
II.3. Multiscale analysis of nanocomposite materials. In a collaboration with M.
E. P erez and D. G omez, we plan to obtain new homogenization results for the case of per-
forated domains with critical inclusions (see [124], [149], [122], and [123]). 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 mi-
crostructures, 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 effective properties.
II.4. Multiscale modeling of thermoelastic microstructured materials. The
prediction of the effective 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 me-
chanical and thermodynamical properties. In particular, the problem of multiscale modeling

122 6.2. Further research directions
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 effects on the
thermoelastic response of polymer nanocomposite materials. We shall try to generalize our
results in [105], [106], and [107] to include more general interface effects. The case in which
the strain-stress law is viscoelastic and the case in which we consider thermal effects in the
history of the composite material will be treated, as well.
6.3 Future plans
In the next years, in order to disseminate my results, I plan to publish them in well-known
international journals and to attend several prestigious international conferences. 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 application-oriented high quality journals, with a broad
audience, including not only mathematicians, but also physicists, engineers, and scientists
from various applied elds, such as biology or geology. I would like to give talks at foreign
universities, to take part in the organization of scientic events in the eld of homogenization
and to extend the editorial activities for applied mathematics scientic journals. Also, I wish
to continue and strengthen the already established collaborations and to establish new ones.
I plan to apply for national and international interdisciplinary research projects, as project
director or as a member. I aim at attracting young researchers to the eld of homogenization
theory, by including them in solid research teams of national and international grants. All
the subjects mentioned in Section 6.2 could lead to relevant Ph.D. thesis subjects, connected
to the main stream of applied mathematical research. Obtaining the habilitation would give
me the chance to supervise Ph.D. candidates in the dynamic eld of applied mathematics.
I intend to carry on with the teaching activities at the Faculty of Physics of the Uni-
versity of Bucharest. Meanwhile, in the near future, I plan to publish new lecture notes
and monographs for students or researchers. In particular, I would like to write a textbook
on Functional Analysis for my students and a monograph about homogenization results for
interface phenomena in composite media. I would like to introduce in the curriculum for
graduate students at the Faculty of Physics of the University of Bucharest a course about
homogenization techniques, with applications in material science.

List of publications
This habilitation thesis is based on the following publications:
[1]C. Conca, J. I. D az, C. Timofte ,Effective chemical processes in porous media , Math.
Models Methods Appl. Sci. (M3AS), 13(10), 1437-1462, 2003.
[2]C. Conca, F. Murat, C. Timofte ,A generalized strange term in Signorini's type problems ,
ESAIM: Mod el. Math. Anal. Num er. (M2AN), 37(5), 773-806, 2003.
[3]C. Conca, J. I. D az, A. Li~ n an, C. Timofte ,Homogenization in chemical reactive
ows ,
Electronic Journal of Differential Equations, 40, 1-22, 2004.
[4]C. Conca, C. Timofte ,Interactive oscillation sources in Signorini's type problems ,
Contemporary Mathematics, 362, 381-392, American Mathematical Society Book Se
ries, Providence, Rhode Island, 2004.
[5] C. Timofte ,Upscaling of variational inequalities arising in nonlinear problems with
unilateral constraints , Z. Angew. Math. Mech., 87(6), 406-412, 2007.
[6] C. Timofte ,Homogenization results for climatization problems , Annali dell'Universita
di Ferrara Sez. VII (N.S.), 53(2), 437-448, 2007.
[7] C. Timofte ,Upscaling in dynamical heat transfer problems in biological tissues , Acta
Physica Polonica B, 39(11), 2811-2822, 2008.
[8] C. Timofte ,Homogenization results for enzyme catalyzed reactions through porous media ,
Acta Mathematica Scientia, 29B (1), 74-82, 2009.
[9] C. Timofte ,Homogenization results for dynamical heat transfer problems in heteroge-
neous biological tissues , Bulletin of the Transilvania University of Bra sov, 2(51), 143-148,
2009.
[10] C. Timofte ,Multiscale analysis in nonlinear thermal diffusion problems in composite
structures , Cent. Eur. J. Phys., 8, 555-561, 2010.
[11] A. Capatina, H.I. Ene, C. Timofte ,Homogenization results for elliptic problems in
periodically perforated domains with mixed-type boundary conditions , Asymptotic Anal.,
80(1-2), 45-56, 2012.
[12] C. Timofte ,Multiscale analysis of diffusion processes in composite media , Comp. Math.
Appl., 66(9), 1573-1580, 2013.
123

124 List of publications
[13] C. Timofte ,Multiscale modeling of heat transfer in composite materials , Romanian
Journal of Physics, 58 (9-10), 1418-1427, 2013.
[14] C. Timofte ,Homogenization results for ionic transport in periodic porous media , Comp.
Math. Appl. 68 (9) (2014) 1024-1031.
[15] A. Capatina, C. Timofte ,Boundary optimal control for quasistatic bilateral frictional
contact problems , Nonlinear Analysis: Theory, Methods and Applications, 94, 84-99,
2014.
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