Habilitation New Vers 3 [610104]

UNIVERSITY OF BUCHAREST
HABILITATION THESIS
HOMOGENIZATION RESULTS FOR
HETEROGENEOUS MEDIA
CLAUDIA TIMOFTE
Specialization: Applied 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 of paramount importance 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 dierent 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 eort, 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 Capatina,
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
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
3

4 Acknowledgments
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
4

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 represent 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 some of my research works
in this eld, performed after completing my Ph.D., with less details in some proofs and
with some additional hindsights. 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

6
6

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 . . . . . 28
2.1.4 Chemical reactions inside the grains of a porous medium . . . . . . . . 34
2.2 Nonlinear adsorption of chemicals in porous media . . . . . . . . . . . . . . . 36
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 coecients . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . 74
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 . . . . . . . . . . . . . . . . . . . . . . . 91
5 Multiscale modeling of composite media with imperfect interfaces 95
5.1 Multiscale analysis in thermal diusion problems in composite structures . . 96
5.1.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Diusion 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 110
II Career Evolution and Development Plans 113
6 Scientic and academic background and research perspectives 115
6.1 Scientic and academic background . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Further research directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Future plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8

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 c^ ateva 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, fac^ and astfel ca principalele mele
rezultatele 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, mod-
elarea, 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 ^ n limba rom^ an a  si ^ n limba englez a, teza cuprinde dou a
p art i  si o bibliograe cuprinz atoare. Prima parte, structurat a ^ n cinci capitole, este dedicat a
prezent arii principale mele realiz ari  stiint ice obt inute dup a nalizarea studiilor 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 … rezum a principalele
mele contribut ii referitoare la omogenizarea mi sc arilor reactive ^ n medii poroase. Mai precis,
sunt prezentate unele rezultate originale de omogenizare pentru curgerile reactive neliniare
stat ionare ^ n medii poroase  si, de asemenea, sunt prezentate rezultate privind fenomenele de
adsorbt ie neliniar a ^ n medii poroase. Capitolul se bazeaz a pe lucr arile [74], [73], [213]  si [211].
9

10 Rezumat
Al treilea capitol, bazat pe lucr arile [76], [78], [210], [51]  si [54], este dedicat omogeniz arii
unor probleme unilaterale^ n domenii perforate. Sunt cuprinse rezultate originale privitoare la
omogenizarea unor probleme 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^ ncarcate electric, pentru studiul carcinogenezei^ n celulele
vii sau pentru analiza dinamicii calciului ^ n celulele biologice. Rezultatele prezentate ^ n acest
capitol sunt cuprinse ^ n articolele [212], [213], [214], [217], [221], [223], [220]. 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], [215], [216]  si [218]. 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], [149], [156], [162], [163], [207].
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^ ana 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.
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.l. 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 …. amicale pe care le-am avut de-a lungul anilor.
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 neliniare pentru studiul proceselor de carcino
genez 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.
Toate aceste rezultate ar putea deschide perspective noi  si promit  atoare pentru dezvolt ari
ulterioare  si pentru viitoare colabor ari cu institut ii academice  si de cercetare reputate din
t ar a  si din str ain atate.
10

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 some 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-diusion 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 in Romanian and 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 completion of my Ph.D. thesis.
After a brief introductory chapter presenting the state of the art in the eld of homogenization
theory and oering 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 [74], [73], [213], and [211]. The third chapter is devoted to the
homogenization of some relevant unilateral problems in perforated domains. More precisely,
11

12 Abstract
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
papers [76], [78], [210], [51], and [54]. 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 [212],
[213], [214], [217], [221], [223], [220]. 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 based on the articles [47], [48], [49],
[215], [216], and [218]. 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], [149], [156], [162], [163], [207].
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 and 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.
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. Capatina, Dr. I. T  ent ea, and Dr. R. Bunoiu, for a nice
collaboration, for their important contribution, for useful advices and friendly discussions.
My major original contributions contained in this habilitation thesis can be summarized
as follows:
performing a rigorous study of nonlinear reaction-diusion processes in porous media,
including diusion, chemical reactions and dierent 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 diusion processes
in composite structures;
rening the study of diusion problems with dynamical boundary conditions;
obtaining new mathematical models for diusion problems with
ux jump.
I hope that all these results might open new and promising perspectives for further devel-
opments and future collaborations with well-known academic and research institutions from
Romania and from abroad.
12

Part I
Main Scientic Achievements
13

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

16 Introduction
xed domain and they have, in general, simpler or even constant coecients (the eective or
homogenized coecients), while the original equations have rapidly oscillating coecients,
they are dened on a complicated domain and satisfy nonlinear boundary conditions. The
dependence on the real microstructure is given through the homogenized coecients.
The analysis of the macroscopic properties of composite media was initiated by Rayleigh,
Maxwell and Einstein. Around 1970, scientists managed to formulate the physical problems
of composites in such a way that this eld became interesting from a purely mathematical
point of view. This gave rise to a new mathematical discipline, the homogenization theory.
The rst rigorous developments of this theory appeared with the seminal works of Y. Babuka
[28], E. De Giorgi and S. Spagnolo [87], A. Bensoussan, J. L. Lions and G. Papanicolaou
[37], and L. Tartar [207]. De Giorgi's notion of Gamma-convergence marked also an impor-
tant step in the development of this theory. F. Murat and L. Tartar (see [180], [181], [182],
and [208]) 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 [183] and was further developed by Allaire in [1]. An
extension to multiscale problems was obtained by A.I. Ene and J. Saint Jean Paulin [105]
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 oered by the Bloch-wave homogenization
method [77], which is a high frequency method that can provide dispersion relations for wave
propagation in periodic structures. Recently, D. Cioranescu, A. Damlamian, P. Donato, 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 prob-
abilistic 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 [101]). 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-diusive transport in porous media, nonlinear elasticity, the study of composite
polymers, the study of nanocomposite materials, the modeling of interface phenomena in
biology and chemistry, or the problem of obtaining new composite materials with applications
in modern technology. The literature on this subject is vast (see, e.g., [61], [75], [63], [74], [76],
and the references therein). We also mention here some remarkable monographs dedicated
to the mathematical problems of homogenization: [149], [29], [37], [162], [166], [186], [200],
[62], [70], [103].
Multiscale methods oer multiple possibilities for further developments and for useful
applications in many domains of the contemporary science and technology. Their study
is one of the most active and fastest growing areas of modern applied mathematics, and
16

Introduction 17
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.
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-diusion 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 homogenization 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 in Romanian and 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 completion of my Ph.D. thesis. After a brief introductory
chapter presenting the state of the art in the eld of homogenization theory and oering
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 [74], [73], [213], 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 papers [76], [78], [210],
[51], and [54]. 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 [212], [213], [214], [217],
[221], [223], [220]. 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], [215], [216], and [218].
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],
17

18 Introduction
[45], [56], [62], [137], [149], [156], [162], [163], and [207].
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.
The thesis 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 in collaboration with var-
ious 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.
The main original contributions of the author in the eld of homogenization theory, con-
tained 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-diusion processes in porous media,
including diusion, chemical reactions and dierent 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 diusion processes
in composite structures;
rening the study of diusion 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.
18

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

ow, salt water intrusion into coastal aquifers), geology (petroleum reservoir engineering,
geothermal energy), chemical engineering (packed bed rectors, drying of granular 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 eective properties of heterogeneous porous media (see, e.g., [137], [32],
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, diusion, convection, and chemical reactions
is a dicult task. The homogenization theory proves to be a very ecient tool by provid-
ing suitable techniques allowing us to pass from the microscopic scale to the macroscopic
19

20 2.1. Upscaling in stationary reactive
ows in porous media
one and to obtain suitable macroscale models. Since the seminal work of G.I. Taylor [209],
dispersion phenomena in porous media have attracted a lot of attention. There are many
formal or rigorous methods in the literature. We refer to [140] and [138] 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],
[169], [174], [158], [141], [168], [159], [99] 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 [173]. For reactive
ows combined with the mechanics of cells, we refer to [148].
Rigorous homogenization techniques for obtaining the eective model for dissolution and pre-
cipitation in a complex porous medium were successfully applied in [159]. Solute transport in
porous media is also a topic of interest for chemists, geologists and environmental scientists
(see, e.g., [6] and [98]). Related problems, such that single or two-phase
ow or miscible
displacement problems were addressed in various papers (see, for instance, [16], [21], [22],
[172]). For an interesting survey on homogenization techniques applied to problems involving

ow, diusion, 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 [74], [73],
[213], and [211].
2.1 Upscaling in stationary reactive
ows in porous media
We shall discuss now, following [73] and [213], 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 [89], [92], [91], 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 [73], 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
eective 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 [213], where we have analyzed the eective
20

Chapter 2. Homogenization of reactive
ows in porous media 21
behavior of the solution of a nonlinear problem arising in the modeling of enzyme catalyzed
reactions through the exterior of a domain containing periodically distributed reactive solid
obstacles.
2.1.1 The model problem
Let
be an open smooth connected bounded set in Rn(n3) and let us insert in it a set
of periodically distributed reactive obstacles. As a result, we obtain an open set
", 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 setY=YnF. If"is a real parameter taking values in a
sequence of positive numbers converging to zero, for each "and for any integer vector k2Zn,
setF"
kthe translated image of "Fby the 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
, i.e.
F"=[
F"
kjF"
k
; k2Zn
:
Set
"=
nF". Hence,
"is a periodic domain with periodically distributed obstacles
of size of the same order as the period. We remark that the obstacles do not intersect the
boundary@
. LetS"=[f@F"
kjF"
k
; k2Zng. So,@
"=@
[S".
We denote byj!jthe Lebesgue measure of any measurable subset !Rnand, for an
arbitrary function 2L2(
"), we denote by e its extension by zero to the whole of
. Also,
throughout this thesis, by Cwe denote a generic xed strictly positive constant, whose value
can change from line to line.
The rst problem we present in this section concerns the stationary reactive
ow of a
uid
conned in
", of concentration u", reacting on the boundary of the obstacles. A simplied
version of this kind of problem can be written as follows:
8
>>><
>>>:Df
u"=fin
";
Df@u"
@="g(u") onS";
u"= 0 on@
:(2.1)
Here,is the exterior unit normal to
",f2L2(
) andS"is the boundary of our exterior
medium
n
". Moreover, for simplicity, we assume that the
uid is is homogeneous and
isotropic, with a constant diusion coecient Df>0. We can treat in a similar manner the
more general case in which, instead of considering constant diusion coecients, we work
with an heterogeneous medium represented by periodic symmetric bounded matrices which
are assumed to be uniformly coercive.
The semilinear boundary condition imposed on S"in problem (2.1) describes the chemical
reactions which take place locally at the interface between the reactive
uid and the grains.
In fact, from a strictly chemical perspective, such a situation represents, equivalently, the
21

22 2.1. Upscaling in stationary reactive
ows in porous media
eective reaction on the walls of the chemical reactor between the
uid lling
"and a
chemical reactant located in 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 subdierential of a convex lower semicontinuous function G. These two general situations
are well illustrated by the following important practical examples:
a)g(v) =v
1 +v; ; > 0 (Langmuir kinetics) (2.2)
and
b)g(v) =jvjp1v;0<p< 1 (Freundlich kinetics) : (2.3)
The exponent pis called the order of the reaction . We point out that if we assume f0,
one can prove (see, e.g. [91]) that u"0 in
n
"andu">0 in
", although u"is not
uniformly positive except in the case in which gis a monotone smooth function satisfying
the condition g(0) = 0, as, for instance, in example a). In fact, since u"is, in practical
applications, a concentration, we can impose suitable conditions on the data to ensure that
0u"1 (see, e.g., [90]).
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
dierentiable 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(
")jv= 0 on@

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

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

"f'dx8'2V":(2.5)
By classical existence results (see [45]), there exists a unique weak solution u"2V"\H2(
")
of problem (2.1). This solution being dened only on
", we need to extend it to the whole of

to be able to state the convergence result. To this end, let us recall the following well-known
extension result (see [70]):
22

Chapter 2. Homogenization of reactive
ows in porous media 23
Lemma 2.1 There exists a linear continuous extension operator
P"2L(L2(
");L2(
))\L(V";H1
0(
))
and a positive constant C, independent of ", such that, for any v2V",
kP"vkL2(
)CkvkL2(
")
and
krP"vkL2(
)CkrvkL2(
"):
Therefore, we have the following Poincar e's inequality in V":
Lemma 2.2 There exists a positive constant C, independent of ", such that
kvkL2(
")CkrvkL2(
")for anyv2V":
The main convergence result for this case is stated in the following theorem.
Theorem 2.3 ([73]) There exists an extension P"u"of the solution u"of the variational
problem (2.5) such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of
8
><
>:nX
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =f in
;
u= 0 on@
:(2.6)
Here,Q= ((qij))is the homogenized matrix, whose entries are dened as follows:
qij=Df
ij+1
jYjZ
Y@j
@yidy
(2.7)
in terms of the functions i; i= 1;:::;n; solutions of the cell problems
8
>>>><
>>>>:
i= 0 inY;
@(i+yi)
@= 0 on@F;
iYperiodic.(2.8)
The constant matrix Qis symmetric and positive-denite.
Proof. 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 '=u"as a test function
in (2.5), using Schwartz and Poincar e's inequalities, we get
kP"u"kH1
0(
)C
23

24 2.1. Upscaling in stationary reactive
ows in porous media
and, by passing to a subsequence, still denoted by P"u", we can suppose that there exists
u2H1
0(
) such that
P"u"*u weakly inH1
0(
): (2.9)
It remains to determine the limit equation satised by u.
Second step . In order to 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 h2Ls0(@F), 1s01, the linear form "
honW1;s
0(
)
dened by
h"
h;'i="Z
S"hx
"
'd8'2W1;s
0(
);
with 1=s+ 1=s0= 1. Then (see [61]),
"
h!hstrongly in ( W1;s
0(
))0; (2.10)
where
hh;'i=hZ

'dx;
with
h=1
jYjZ
@Fh(y) d:
Ifh2L1(@F) or ifhis constant, we have "
h!hstrongly in W1;1(
) and we denote
by"the above introduced measure in the particular case in which h= 1. Notice that in
this casehbecomes1=j@Fj=jYj. We shall prove now that for any '2D(
) and for any
v"*v weakly inH1
0(
), one has
'g(v")*'g (v) weakly in W1;q
0(
); (2.11)
where
q=2n
q(n2) +n:
To this end, let us remark that
supkrg(v")kLq(
)<1: (2.12)
Indeed, using the growth condition (2.4) imposed to g, we have
Z

@g
@xi(v")q
dxCZ


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

jv"jqq
dx1=
Z

jrv"jqdx1=
;
24

Chapter 2. Homogenization of reactive
ows in porous media 25
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
supkv"k
L2n
n2(
)<1;
we easily get (2.12). Therefore, to obtain (2.11), it remains only to show that
g(v")!g(v) strongly in Lq(
): (2.13)
But this convergence is a direct consequence of the following well-known result (see [81], [156]
and [162]):
Theorem 2.4 LetG:
R!Rbe a Carath eodory function, i.e.
a) for every vthe function G(
;v)is measurable with respect to x2
:
b) for every (a.e.) x2
, the function G(x;
)is continuous with respect to v.
Moreover, if we assume that there exists a positive constant Csuch that
jG(x;v)jC
1 +jvjr=t
;
withr1andt <1, then the map v2Lr(
)7!G(x;v(x))2Lt(
)is continuous in the
strong topologies.
Indeed, since
jg(v)jC(1 +jvjq+1);
applying the above theorem for G(x;v) =g(v),t=qandr= (2n=(n2))r0, withr0>0
such thatq+ 1< r=t and using the compact injection H1(
),!Lr(
) we obtain (2.13).
Now, from (2.10), written for h= 1, and (2.11) written for v"=P"u", we get
h";'g(P"u")i!j@Fj
jYjZ

'g(u) dx8'2D(
) (2.14)
and this completes the proof of this step.
Third step . Let"be the gradient of u"in
"and let us denote by e"its extension by
zero to the whole of
. Then, e"is bounded in ( L2(
))nand, as a consequence, there exists
2(L2(
))nsuch that
e"* weakly in (L2(
))n: (2.15)
Let us identify now the equation satised by . If we take '2D(
), from (2.5) we get
Z

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


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

e"
r'dx=Z


r'dx: (2.17)
25

26 2.1. Upscaling in stationary reactive
ows in porous media
For the second term, using (2.14), we obtain
lim
"!0"Z
S"g(u")'d=j@Fj
jYjZ

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

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


"f'dx=jYj
jYjZ

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


r'dx+j@Fj
jYjZ

g(u)'dx=jYj
jYjZ

f'dx8'2D(
):
Thus,satises
div+j@Fj
jYjg(u) =jYj
jYjfin
(2.20)
and it remains now only to identify .
Fourth step. In order to identify , we shall use the solutions of the local problems (2.8).
For any xed i= 1;:::;n; let us dene
i"(x) ="
ix
"
+yi
8×2
"; (2.21)
wherey=x=". By periodicity,
P"i"*xiweakly inH1(
): (2.22)
Let"
ibe the gradient of  i"in
"ande"
ibe the extension by zero of "
iinside the holes.
From (2.21), for the j-component of e"
iwe have

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

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

", we obtain Z

""
i
r'u"dx+Z

""
i
ru"'dx= 0:
26

Chapter 2. Homogenization of reactive
ows in porous media 27
Thus, Z

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

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

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

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

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

e"
r'P"i"dx+Z

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

f
"'P"i"dx:
Now, from (2.25), we obtain
Z

e"
r'P"i"dxZ

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

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

e"
r'P"i"dx=Z


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

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

qi
r'udx; (2.28)
whereqiis the vector having the j-component equal to qij.
Since the boundary of Fis of classC2,P"i"2W1;1(
) andP"i"!xistrongly in
L1(
):Then, since g(P"u")P"i"!g(u)xistrongly in Lq(
) andg(P"u")P"i"is bounded
inW1;q(
), we have g(P"u")P"i"*g(u)xiweakly inW1;q(
). Thus,
lim
"!0"Z
S"g(u")'i"d=j@Fj
jYjZ

g(u)'xidx: (2.29)
Finally, for the limit of the right-hand side of (2.26), since

"f *jYj
jYjfweakly inL2(
);
using (2.22), we have
lim
"!0Z

f
"'P"i"dx=jYj
jYjZ

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


r'xidxjYj
jYjZ

qi
r'udx+j@Fj
jYjZ

g(u)'xidx=jYj
jYjZ

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


rxi'dx+jYj
jYjZ

qi
ru'dx= 0 in
:
27

28 2.1. Upscaling in stationary reactive
ows in porous media
The above equality holds true for any '2D(
) and this implies that

rxi+jYj
jYjqi
ru= 0 in
: (2.32)
Writing (2.30) by components, dierentiating with respect to xi;summing after iand using
(2.19), we are led to
jYj
jYjnX
i;j=1qij@2u
@xi@xj= div=jYj
jYjf+j@Tj
jYjg(u);
which means that uveries
nX
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =fin
:
Sinceu2H1
0(
) (i.e.u= 0 on@
) anduis uniquely determined, the whole sequence P"u"
converges to uand Theorem 2.3 is proven.
Remark 2.5 The right scaling "in front of the function gmodeling the contribution of the
nonlinear reactions on the boundary of the grains leads in the limit to the presence of a new
term distributed all over the domain
. 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 [73] 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 [211]).
Remark 2.7 In [73], the proof of Theorem 2.3 was done by using the so-called energy method
of L. Tartar (see [207]). 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.
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 [73]. If we denote by
D(g) the domain of g, i.e.D(g) =f2Rjg()6=?g, then we suppose that D(g) =R.
Moreover, we assume that gis continuous and there exist C0 and an exponent q, with
0q<n= (n2), such that
jg(v)jC(1 +jvjq): (2.33)
Notice that the second important practical example b) mentioned above is a particular ex-
ample of such a single-valued maximal monotone graph.
28

Chapter 2. Homogenization of reactive
ows in porous media 29
In this case, there exists a lower semicontinuous convex function GfromRto ]1;+1],
Gproper, i.e. G6+1, such that gis the subdierential of G,g=@G. Let
G(v) =Zv
0g(s) ds:
We dene the convex set
K"=
v2V"jG(v)jS"2L1(S")
:
For a given function f2L2(
), the weak solution of the problem (2.1) is also the unique
solution of the following variational inequality:
8
>>><
>>>:ndu"2K"such that
DfZ

"Du"D(v"u") dxZ

"f(v"u") dx+h";G(v")G(u")i0
8v"2K":(2.34)
We start by remarking that there exists a unique weak solution u"2V"\H2(
") 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
2DfZ

"jDvj2dx+h";G(v)iZ

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

QDvDv dx+j@Fj
jYjZ

G(v) dxZ

fvdx;
the main convergence result for problem (2.34) can be formulated as follows:
Theorem 2.8 (Theorem 2.6 in [73]) One can construct an extension P"u"of the solution
u"of the variational inequality (2.34) such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of the minimization problem
8
<
:Findu2H1
0(
)such that
J0(u) = inf
v2H1
0(
)J0(v):(2.35)
Moreover,G(u)2L1(
). Here,Q= ((qij))is the classical homogenized matrix, whose entries
were dened in (2.7)-(2.8).
29

30 2.1. Upscaling in stationary reactive
ows in porous media
Remark 2.9 We notice that ualso satises
8
>><
>>:nX
i;j=1qij@2u
@xi@xj+j@Fj
jYjg(u) =f in
;
u= 0 on@
:
Proof 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(
) and, thus, by passing to a subsequence, we have
P"u"*u weakly inH1
0(
): (2.36)
Let'2D(
). By classical regularity results, i2L1. So, from the boundedness of iand
', it follows that there exists M0 such that

@'
@xi

L1

i

L1<M:
Let
v"='+X
i"@'
@xi(x)ix
"
: (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"!'strongly in L2(
). Further,
Dv"=X
i@'
@xi(x)
ei+Dix
"
+"X
iD@'
@xi(x)ix
"
;
where ei, 1in, are the elements of the canonical basis in Rn. Takingv"as a test
function in (2.34), we get
Z

"Du"Dv"dxZ

"f(v"u") dx+Z

"Du"Du"dxh";G(v")G(u")i
and
Z

DP"u"^(Dv") dxZ

"f(v"u") dx+Z

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

DP"u"gDv"dx!Z

QDuD' dx (2.40)
and Z

"f(v"u") dx=Z

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

f('u) dx: (2.41)
30

Chapter 2. Homogenization of reactive
ows in porous media 31
For the third term of the right-hand side of (2.38), assuming the growth condition (2.33)
for the single-valued maximal monotone graph gand reasoning exactly like in the previous
subsection, we get
G(P"u")*G(u) weakly in W1;q
0(
)
and, so
h";G(P"u")i!j@Fj
jYjZ

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

G(') dx
and, therefore, we have
h";G(v")G(P"u")i!j@Fj
jYjZ

(G(')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(
), the subdierential inequality
Z

"Du"Du"dxZ

"Dw"Dw"dx+ 2Z

"Dw"(Du"Dw") dx; (2.43)
Reasoning as before and choosing
w"='+X
i"@'
@xi(x)ix
"
;
where'has similar properties as the corresponding ', we can pass to the limit in the right-
hand side of the inequality (2.43) and we get
lim inf
"!0Z

"Du"Du"dxZ

QD'D'dx+ 2Z

QD'(DuD') dx;
for any'2D(
). Sinceu2H1
0(
), taking '!ustrongly in H1
0(
);we obtain
lim inf
"!0Z

"Du"Du"dxZ

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

QDuD' dxZ

f('u) dx+Z

QDuDu dxj@Fj
jYjZ

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

QDuD (vu) dxZ

f(vu) dxj@Fj
jYjZ

(G(')G(u)) dx;
which leads exactly to the limit problem (2.34). This ends the proof of Theorem 2.8.
31

32 2.1. Upscaling in stationary reactive
ows in porous media
Remark 2.10 The choice of the test function (2.37) gives, in fact, a rst-corrector term for
the weak convergence of P"u"tou.
Remark 2.11 All the results of this section can be obtained for a general diusion matrix
A"(x) =A(x="), whereA=A(y)is a matrix-valued function on Rnwhich isY-periodic. We
shall assume that
8
<
:A2L1(
)nn;
Ais a symmetric matrix,
for some 0<
<;
jj2A(y)
jj28; y2Rn:
Problems similar to the one presented here may arise in various other contexts (see, e.g.
[211] and [213]). In [213], we analyzed the eective behavior of the solution of a nonlinear
problem arising in the modeling of enzyme catalyzed reactions through the exterior of a
domain containing periodically distributed reactive solid obstacles, with period ". Enzymes
are proteins that speed up the rate of a chemical reaction without being used up. They
are specic to particular substrates. The substrates in the reaction bind to active sites on
the surface of the enzyme. The enzyme-substrate complex then undergoes a reaction to
form a product along with the original enzyme. The rate of chemical reactions increases
with the substrate concentration. However, enzymes become saturated when the substrate
concentration is high. Additionally, the reaction rate depends on the properties of the enzyme
and the enzyme concentration. We can describe the reaction rate with a simple equation
to understand how enzymes aect chemical reactions. Michaelis-Menten equation remains
the most generally applicable equation for describing enzymatic reactions. In this case, we
consider the following elliptic problem:
8
>>><
>>>:Df
u"+(u") =fin
";
Df@u"
@="g(u") onS";
u"= 0 on@
:(2.45)
Here, the function is continuously dierentiable, monotonously non-decreasing and such
that(0) = 0. For example, we can take to be a linear function, i.e. (v) =v, or we
can consider the nonlinear case in which is given by (2.2) (Langmuir kinetics). For the
given 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 subdierential of a convex lower semicontinuous
functionG. More precisely, we shall consider an important practical example, arising in the
diusion of enzymes (the Michaelis-Menten model):
g(v) =8
><
>:v
v+
; v0;
0; v< 0;
for;
> 0.
32

Chapter 2. Homogenization of reactive
ows in porous media 33
The existence and uniqueness of a weak solution of (2.45) is ensured by the classical theory
of monotone problems (see [45] and [100]). Therefore, we know that there exists a unique
weak solution u"2V"TH2(
"). Moreover, u"is also the unique solution of the following
variational problem:
8
>>>>><
>>>>>:Findu"2K"such that
DfZ

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

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

"f(v"u") dx+h";G(v")G(u")i0;8v"2K";(2.46)
where"is the linear form on W1;1
0(
) dened by
h";'i="Z
S"'d;8'2W1;1
0(
):
The main convergence result in this case, proven in [213], is stated in the following theorem.
Theorem 2.12 The solution u", properly extended to the whole of
, converges to the unique
solution of the following variational inequality:
8
>>>>>>><
>>>>>>>:u2H1
0(
);
Z

QDuD (vu) dx+Z

(u)(vu) dxZ

f(vu) dx
j@Fj
jYjZ

(G(v)G(u)) dx;8v2H1
0(
):(2.47)
Here,Q= ((qij))is the homogenized matrix, dened in (2.7).
Remark 2.13 Notice that ualso satises
8
>><
>>:nX
i;j=1qij@2u
@xi@xj+(u) +j@Fj
jYjg(u) =f in
;
u= 0 on@
:
Thus, the asymptotic behavior of the solution of the microscopic problem (2.45) is governed
by a new elliptic boundary-value problem, with an extra zero-order term that captures the
eect of the enzymatic reactions. The eect of the enzymatic reactions initially situated
on the boundaries of the grains spread out in the limit all over the domain, giving the extra
zero-order term which captures this boundary eect. In fact, one could obtain a similar result
by considering interior enzymatic nonlinear chemical reactions given by the same well-known
nonlinear function g. The only dierence in the limit equation will be the coecient appearing
in front of this extra zero-order term. So, one can control the eective behavior of such
reactive
ows by choosing dierent locations for the involved chemical reactions. Moreover,
as we shall see in the next section, we can obtain similar eects by considering transmission
33

34 2.1. Upscaling in stationary reactive
ows in porous media
problems, with an unknown
ux on the boundary of each grain, i.e. we can consider the
case in which we have chemical reactions in
", but also inside the grains, instead on their
boundaries. The dierence in the limit equation will be the coecient appearing in front of
this extra zero-order term. Hence, we can control the eective behavior of such reactive
ows
by choosing dierent locations for the involved chemical reactions.
2.1.4 Chemical reactions inside the grains of a porous medium
We shall brie
y present now some results obtained in [73] 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
";
Dp
v"+g(v") = 0 in "
Df@u"
@=Dp@v"
@onS";
u"=v"onS";
u"= 0 on@
:(2.48)
Here, "=
n
",is the exterior unit normal to
",u"andv"are the concentrations in

"and, respectively, inside the grains ",Df>0,Dp>0,f2L2(
) andgis a continuous
function, monotonously non-decreasing and such that g(v) = 0 if and only if v= 0. Moreover,
we suppose that there exist a positive constant Cand an exponent q, with 0q<n= (n2),
such thatjg(v)jC(1 +jvjq+1). 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
kw"k2
H"=kru"k2
L2(
")+krv"k2
L2("):
The variational formulation of problem (2.48) is the following one:
8
>>>><
>>>>:ndw"2H"such that
DfZ

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

"f'dx
8('; )2K":(2.49)
Under the above hypotheses and the conditions satised by H", it is well-known (see [45] and
[162]) that (2.49) is a well-posed problem.
34

Chapter 2. Homogenization of reactive
ows in porous media 35
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 [73]):
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(
);
whereuis the unique solution of
8
><
>:nX
i;j=1a0
ij@2u
@xi@xj+jFj
jYjg(u) =f in
;
u= 0 on@
:(2.50)
Here,A0= ((a0
ij))is the homogenized matrix, dened by
a0
ij=1
jYjZ
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(
)andv2L2(
)such that
P"u"*u weakly inH1
0(
);
ev"*v weakly inL2(
)
and
v=jFj
jYju:
Corollary 2.16 Let"be dened by
"(x) =u"(x)x2
";
v"(x)x2":
Then, there exists 2H1
0(
) such that"*  weakly inH1
0(
), whereis the unique
solution of8
><
>:nX
i;j=1a0
ij@2
@xi@xj+ajFj
jYjg() =f in
;
= 0 on@
;
andA0is given by (2.51)-(2.52), i.e. =u, due to the well-posedness of problem (2.49).
35

36 2.2. Nonlinear adsorption of chemicals in porous media
Remark 2.17 As already mentioned, the approach used in [73] and [213] is the so-called
energy method or the oscillating test function method introduced by L. Tartar [207], [208]
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 dierent eective 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
, with discontinuous coecients. Its macro-
scopic behavior also involves a zero-order term, but of a completely dierent nature, emerging
from the chemical reactions occurring inside the grains.
Remark 2.19 In (2.48) we considered that the ratio of the diusion coecients in the two
media is of order one in order to compare the case in which the chemical reactions take place
on the boundary of the grains with the case in which the chemical reactions occur inside them.
However, a more interesting problem arises if we consider dierent orders for the diusion
in the obstacles and in the pores. More precisely, if one takes the ratio of the diusion
coecients to be of order "2, then the limit model will be the so-called double-porosity model .
This scaling preserves the physics of the
ow inside the grains, as "!0. The less permeable
part of our medium (the grains) contributes in the limit as a nonlinear memory term. In
fact, the eective limit model includes two equations, one in Fand another one in
, the
last one containing an extra-term which re
ects the remaining in
uence of the grains (see,
for instance, [21], [41], [42], [74], [140]).
We can treat in a similar manner the case of multi-valued maximal monotone graphs,
which includes various semilinear boundary-value problems, such as Dirichlet, Neumann or
Robin problems, Signorini's problems, problems arising in chemistry (see [61], [74], [76], and
[211]). Also, for the case of a dierent geometry of the perforated domain and dierent
transmission conditions, see Chapter 5.
2.2 Nonlinear adsorption of chemicals in porous media
In this section, we shall present some homogenization results, obtained in [74], concerning
the eective behavior of some chemical reactive
ows involving diusion, dierent types of
adsorption rates and chemical reactions which take place at the boundary of the grains of a
porous material. Such problems arise in many domains, such as chemical engineering or soil
sciences (see, for instance, [137], where the asymptotic behavior of such chemical processes
was analyzed and rigorous convergence results were given for the case of linear adsorption
36

Chapter 2. Homogenization of reactive
ows in porous media 37
rates and linear chemical reactions). The case of nonlinear adsorption rates, left as open
in [137], was treated in [74]. 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 diusion modeled by a Laplace-Beltrami operator
to take place on this surface. In this last case, we show that the eective behavior of our
system is governed by a new boundary-value problem, with an additional microvariable and
a zero-order extra term proving that memory eects are present in this limit model.
2.2.1 The microscopic model and its weak solvability
Our main goal in [74] was to obtain the asymptotic behavior, as "!0, of the microscopic
models (2.53)-(2.55) below. The geometry of this problem is the same as the one in Section
2.1. More precisely, the domain consists of two parts: a
uid phase
"and a solid skeleton
(grains or pores),
n
". We assume that chemical substances are dissolved in the
uid
part. They are transported by diusion and also, by adsorption, they can change from being
dissolved in the
uid to residing on the surface of the pores, where chemical reactions take
place. Thus, the model consists of a diusion system in the
uid phase, a reaction system on
the pore surface and a boundary condition coupling them (see (2.54)):
(V")8
>>><
>>>:@v"
@t(t;x)Df
v"(t;x) =h(t;x); x2
"; t> 0;
v"(t;x) = 0; x2@
; t> 0;
v"(t;x) =v1(x); x2
"; t= 0;(2.53)
Df@v"
@(t;x) ="f"(t;x); x2S"; t> 0; (2.54)
and
(W")8
<
:@w"
@t(t;x) +aw"(t;x) =f"(t;x); x2S"; t> 0;
w"(t;x) =w1(x); x2S"; t= 0;(2.55)
where
f"(t;x) =
(g(v"(t;x))w"(t;x)): (2.56)
Here,v"represents the concentration of the solute in the
uid region, w"is the concentration
of the solute on the surface of the skeleton
n
",v12H1
0(
) is the initial concentration of
the solute and w12H1
0(
) is the initial concentration of the reactants on the surface S"of
the skeleton. Also, the
uid is assumed to be homogeneous and isotropic, with a constant
diusion coecient Df>0,a;
> 0 are the reaction and, respectively, the adsorption factor
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
37

38 2.2. Nonlinear adsorption of chemicals in porous media
by the nonlinear function g. Two model situations are considered: the case of a monotone
smooth function gwithg(0) = 0 and, respectively, the case of a maximal monotone graph
withg(0) = 0, i.e. the case in which gis the subdierential 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
"andv">0 in
", then the function gin example a)
is indeed a particular example of our rst model situation ( gis a monotone smooth function
satisfying the condition g(0) = 0). In fact, instead of (2.56), we could consider a 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 [140]).
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 [162]).
In order to write down the variational formulation of problem (2.53)-(2.56), let us dene
some suitable function spaces. Let H=L2(
), with the classical scalar product
(u;v)
=Z

u(x)v(x) dx;
and letH=L2(0;T;H), with the scalar product
(u;v)
;T=ZT
0(u(t);v(t))
dt;whereu(t) =u(t;
);v(t) =v(t;
):
Also, letV=H1(
), with (u;v)V= (u;v)
+ (ru;rv)
andV=L2(0;T;V), with
(u;v)V=ZT
0(u(t);v(t))Vdt:
We set
W=
v2Vdv
dt2V0
whereV0is the dual space of V;
V0=
v2Vv= 0 on@
a.e. on (0 ;T)
;W0=V0\
W:
In a similar manner, we dene the spaces V(
"),V(
"),V(S") andV(S"). For the latter we
write
hu;viS"=Z
S"g"uvd;
whereg"is the metric tensor on S"; the rule of partial integration on S"applies and, if we
denote the gradient on S"byr"and the Laplace-Beltrami operator on S"by
", we have
(
"u;v)S"=hr"u;r"viS":
38

Chapter 2. Homogenization of reactive
ows in porous media 39
Also, for the space of test functions we use the notation D=C1
0((0;T)
)).
We shall start our analysis with the case in which gis a continuously dierentiable 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 0q<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"2W 0(
");v"(0) =v1j
"such that

v";d'
dt

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

w";d'
dt
S";T+a(w";')S";T= (f";')S";T;8'2W(S"):(2.59)
Proposition 2.20 There exists a unique weak solution u"= (v"; w")of the system (2.58)-
(2.59).
Remark 2.21 The solution of (2.59) can be written as
w"(t;x) =w1(x)e(a+
)t+
Zt
0e(a+
)(ts)g(v"(s;x)) ds
or, if we denote by ?the convolution with respect to time, as
w"(
;x) =w1(x)e(a+
)t+
r(
)?g(v"(
;x));
where
r() =e(a+
):
The solution v"of problem ( V") being dened only on
", we need to extend it to the
whole of
to be able to state the convergence result. In order to do that, we use Lemma
2.1. We also recall the following well-known result (see [72]):
Lemma 2.22 There exists a positive constant C, independent of ", such that, for any v2V",
kvk2
L2(S")C("1kvk2
L2(
")+"krvk2
L2(
")):
39

40 2.2. Nonlinear adsorption of chemicals in porous media
2.2.2 The main result
Theorem 2.23 (Theorem 2.5 in [74]) One can construct an extension P"v"of the solution
v"of the problem (V")such that
P"v"*v weakly inV;
wherevis the unique solution of the following limit problem:
8
>>>>><
>>>>>:@v
@t(t;x) +F0(t;x)nX
i;j=1qij@2v
@xi@xj(t;x) =h(t;x); t> 0; x2
;
v(t;x) = 0; t> 0; x2@
;
v(t;x) =v1(x); t = 0; x2
;(2.60)
with
F0(t;x) =j@Fj
jY?j
h
g(v(t;x))w1(x)e(a+
)t
r(
)?g(v(
;x))(t)i
:
Here,Q= ((qij))is the homogenized matrix, whose entries are dened in (2.7). Moreover,
the limit problem for the surface concentration is:
8
><
>:@w
@t(t;x) + (a+
)w(t;x) =
g(v(t;x)); t> 0; x2
;
w(t;x) =w1(x); t = 0; x2
(2.61)
andwcan be written as
w(t;x) =w1(x)e(a+
)t+
r(t)?g(v(t;x)):
Remark 2.24 The weak formulation of problem (2.60) is:
8
>>>><
>>>>:ndv2W 0(
);v(0) =v1such that

v;d'
dt

;T+ (F0;')
;T=(Qrv;r')
;T+ (h;')
;T
8'2W 0(
):(2.62)
Proof of Theorem 2.23. The proof is divided into several steps (see [74]). The rst step
consists in proving the uniqueness of the limit problem (2.62). This is stated in the following
proposition, proven in [74]:
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 [74]).
40

Chapter 2. Homogenization of reactive
ows in porous media 41
Proposition 2.26 Letv"andw"be the solutions of the problem (2.53)-(2.56). There exists
a positive constant C, independent of ", such that
kw"(t)k2
S"(kw"(0)k2
S"+

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

@w"
@t

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

"C;krv"(t)k2

";tC
and

@v"
@t(t)

2

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

'g(v(t))dx8'2D:
We are now in a position to use Lebesgue's convergence theorem. Using the above pointwise
convergence, the a priori estimates stated in Proposition 2.26 and the growth condition (2.57),
we get
lim
"!0"
(g(v");')S";T=j@Fj
jYj
(g(v);')
;T:
For the rest of the terms, the proof is standard and we obtain immediately (2.60). Since
v2W 0(
) (i.e.v= 0 on@
) andvis uniquely determined, the whole sequence P"v"
converges to vand Theorem 2.23 is proven.
41

42 2.2. Nonlinear adsorption of chemicals in porous media
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)jC(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"
2W 0(
");v"
(0) =v1j
"such that

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

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

w"
;d'
dt
S";T+a(w"
;')S";T= (f"
;')S";T;8'2W(S");(2.64)
where
f"
=
(g(v"
)w"
)
and
g=IJ
;
with
J= (I+@G)1:
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 [162]).
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 inH(
):
Then, it is not dicult to see that, proving suitable a priori estimates (classical energy
estimates) on the solutions v, we can ensure, via compactness arguments (see [30]), the
strong convergence of v, as!0, tov, the unique solution of problem (2.60). Hence
v!vstrongly inH(
):
42

Chapter 2. Homogenization of reactive
ows in porous media 43
Finally, since
kP"v"vk
;TkP"v"P"v"
k
;T+kP"v"
vk
;T+kvvk
;T;
we get the strong convergence of P"v"tovinH(
).
Remark 2.27 The conclusion of the above theorem remains true for more general 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 coecients
In problem (2.53)-(2.56), the rate aof chemical reactions on S"and the adsorption coecient

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) =ax
"
;
"(x) =
x
"
;
withaand
positive functions in W1;1(
) which are Y-periodic (for linear adsorption rates,
see [140]). In this case, v"andw"satisfy the following system of equations:
(V")8
><
>:@v"
@t(t;x)Df
v"(t;x) =h(t;x); x2
"; t> 0;
v"(t;x) = 0; x2@
; t> 0;
v"(t;x) =v1(x); x2
"; t= 0;(2.65)
Df@v"
@(t;x) ="f"(t;x); x2S"; t> 0; (2.66)
and
(W")(@w"
@t(t;x) +a"(x)w"(t;x) =f"(t;x); x2S"; t> 0;
w"(t;x) =w1(x); x2S"; t= 0;(2.67)
where
f"(t;x) =
"(x)(g(v"(t;x))w"(t;x)): (2.68)
If we denote y=x=", then the main result in this case is the following one:
Theorem 2.28 (Theorem 4.1 in [74]) The eective behavior of vandwis governed by the
following system:
8
>>><
>>>:@v
@t(t;x) +G0(t;x)nX
i;j=1qij@2v
@xi@xj=h(t;x); t> 0; x2
;
v(t;x) = 0t>0; x2@
;
v(t;x) =v1(x)t= 0; x2
;(2.69)
43

44 2.2. Nonlinear adsorption of chemicals in porous media
and
(@w
@t(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(v(t;x)); t> 0; x2
; y2@F
w(t;x;y ) =w1(x)t= 0; x2
; y2@F;(2.70)
where
G0(t;x) =1
jY?jZ
@Ff0(t;x;y ) d
and
f0=
(y)(g(v(t;x))w(t;x;y )):
Here,Q= ((qij))is the classical homogenized matrix, dened by (2.7).
Obviously, the solution of (2.70) can be found using the method of variation of parameters.
Hence, we get
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)Zt
0e(a(y)+
(y))(ts)g(v(s;x)) ds;
or, using the convolution notation
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)r(
;y)?g(v(
;x))(t);
with
r(;y) =e(a(y)+
(y)):
Moreover, let us notice that (2.69)-(2.70) imply that v(t;x) satises the following equation
@v
@t(t;x)nX
i;j=1qij@2v
@xi@xj(t;x) +F0(t;x) =h(t;x); t> 0; x2
; (2.71)
with
F0(t;x) =1
jY?jZ
@Fn

(y)[g(v(t;x))w1(x)e(a(y)+
(y))t
(y)r(
;y)?g(v(
;x))(t)]o
d:
Remark 2.29 The above adsorption model can be slightly generalized by allowing surface
diusion on S". In fact, the chemical substances can creep on the surface and this eect is
similar to a surface-like diusion. From a mathematical point of view, we can model this
phenomenon by introducing a diusion term in the law governing the evolution of the surface
concentration w". This new term is the properly rescaled Laplace-Beltrami operator. This
implies that the rst equation in (2.67) has to be replaced by
@w"
@t(t;x)"2E
"w"(t;x) +a"(x)w"(t;x) =f"(t;x)x2S"; t> 0;
whereE > 0is the diusion constant on the surface S"and
"is the Laplace-Beltrami
operator on S".
44

Chapter 2. Homogenization of reactive
ows in porous media 45
In this case, the homogenized limit is the following one:
@w
@t(t;x;y )E
@F
yw(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(v(t;x));
fort>0; x2
; y2@F. Here,
@Fdenotes the Laplace-Beltrami operator on @Fand the
subscriptyindicates the fact that the derivatives are taken with respect to the local variable
y. The limit problem involves the solution of a reaction-diusion system with respect to an
additional microvariable. Also, notice that the local behavior is no longer governed by an
ordinary dierential equation, but by a partial dierential one.
Remark 2.30 We notice that the bulk behavior of system (V")-(W")involves an additional
microvariable y. This local phenomena yields a more complicated microstructure of the 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 diusion-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 diusion matrix D"(x).
More precisely, we analyze the asymptotic behavior, as "!0, of the following coupled system
of equations:
8
>>><
>>>:@v"
@t(t;x)div(D"(x)rv"(t;x)) +(x;v") =h(t;x); x2
"; t> 0;
v"(t;x) = 0; x2@
; t> 0;
v"(t;x) =v1(x); x2
"; t= 0;(2.72)
D"(x)rv"(t;x)
="f"(t;x); x2S"; t> 0; (2.73)
and 8
<
:@w"
@t(t;x) +a"(x)w"(t;x) =f"(t;x); x2S"; t> 0;
w"(t;x) =w1(x); x2S"; t= 0;(2.74)
where
f"(t;x) =
"(x)(g(x;v"(t;x))w"(t;x)): (2.75)
Assumptions.
1) The diusion matrix is dened as being D"(x) =D(x="), whereD=D(y) is a
matrix-valued function on Rnwhich isY-periodic. We shall assume that
8
<
:D2L1(
)nn;
Dis a symmetric matrix,
for some 0<
<;
jj2D(y)
jj28; y2Rn:
45

46 2.2. Nonlinear adsorption of chemicals in porous media
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 dierentiable, 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)jC(1 +jvjq) (2.76)
and 8
>><
>>:@g
@vC(1 +jvjq);
@g
@xiC(1 +jvjr) 1in;(2.77)
with 0q<n= (n2) and with 0r<n= (n2) +q.
Using the theory of semilinear monotone problems (see [45] and [162]), 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:
Theorem 2.31 One can construct an extension P"v"of the solution v"of the problem (2.72)-
(2.75) such that
P"v"*v weakly inL2(0;T;H1(
));
wherevis the unique solution of the following limit problem:
8
><
>:@v
@t(t;x)div(D0rv) +(x;v) +F0(t;x) =h(t;x); t> 0; x2
;
v(t;x) = 0; t> 0; x2@
;
v(t;x) =v1(x); t = 0; x2
;(2.78)
with
F0(t;x) =
1
jY?jZ
@Fn

(y)[g(x;v(t;x))w1(x)e(a(y)+
(y))t
(y)r(
;y)?g(x;v(
;x))(t)]o
d:
The limit problem for the surface concentration is:
(@w
@t(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(x;v(t;x)); t> 0; x2
; y2@F
w(t;x;y ) =w1(x)t= 0; x2
; y2@F:(2.79)
Here,D0= ((d0
ij))is the homogenized matrix, dened by:
d0
ij=1
jYjZ
Y
dij(y) +dik(y)@j
@yk
dy;
in terms of the functions j; j= 1;:::;n; solutions of the cell problems
8
>>><
>>>:divyD(y)(ryj+ej) = 0 inY;
D(y)(rj+ej)
= 0 on@F;
j2H1
#Y(Y?);Z
Y?j= 0;
46

Chapter 2. Homogenization of reactive
ows in porous media 47
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 param-
eters. We have
w(t;x;y ) =w1(x)e(a(y)+
(y))t+
(y)Zt
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)):
Remark 2.33 If we consider the case in which we have diusion of the chemical species on
the surface S", i.e.
@w"
@t(t;x)"2E
"w"(t;x) +a"(x)w"(t;x) =f"(t;x)x2S"; t> 0;
whereE > 0is the diusion constant on the surface S"and
"is the Laplace-Beltrami
operator on S", then instead of (2.79) we get the following local partial dierential equation:
@w
@t(t;x;y )E
@F
yw(t;x;y ) + (a(y) +
(y))w(t;x;y ) =
(y)g(x;v(t;x));(2.80)
fort>0; x2
; y2@F.
As already mentioned, related problems were addressed in the literature by many authors
(see, for instance, U. Hornung and W. J ager [140], U. Hornung [137], D. Cioranescu, P.
Donato and R. Zaki [65], C. Conca, J.I. D az and C. Timofte [74], or G. Allaire and H.
Hutridurga [6]). The results presented in this section constitute a generalization of some of
the results obtained in [74] 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 is to investigate, in multi-component porous media with im-
perfect interfaces, the case of systems of reaction-diusion equations involving nonlinear
reaction-terms and nonlinear boundary conditions. Also, it would be of interest to deal with
the case of other geometries of the porous media under consideration or with the case of more
general nonlinear, even discontinuous, kinetics.
47

48 2.2. Nonlinear adsorption of chemicals in porous media
48

Chapter 3
Homogenization results for
unilateral problems
The study of variational inequalities has attracted a lot of interest in the last decades due to
its 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 numerical analysis of variational inequalities is a modern topic, with
a wide range of applications to important and dicult free boundary problems arising in
the study of
ow through porous media. We can mention here the representative papers
of G. Fichera [112], G. Signorini [204], G. Stampacchia [206], G. Stampacchia and J. L.
Lions [164], L Boccardo and P. Marcellini [39], L. Boccardo and F. Murat [40], H. Br ezis,
U. Mosco [178], R. Glowinski, J.L. Lions and R. Tr emoli eres [120], Duvaut and Lions [100],
J.T. Oden and N. Kikuchi [185], D. Kinderlehrer and G. Stampacchia [155]. For a nice and
a comprehensive presentation of the theory of variational inequalities and its applications,
we refer to the monographs [45], [50], [100], [155], [120], [185] and [133]. 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 [153], or G.A. Yosian [224].
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], [76], [78], and [210]),
the limit behavior of the solutions of some Signorini's type-like problems in periodically per-
forated domains was analyzed. The classical weak formulation of such unilateral problems
involves a standard variational inequality (in the sense of [164]), corresponding to a nonlinear
free boundary-value problem. Such a model was introduced in the earliest '30 by A. Sig-
norini [204] (see also G. Fichera [112]) 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 [100].
The chapter is based on the papers [51], [54], [76], [78], and [210].
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 [76] by using Tar-
tar's oscillating test function method, for the solutions of some Signorini's type-like problems
in periodically perforated domains with period ". The main feature of these kind of problems
is the existence of a critical size of the perforations that separates dierent emerging phe-
nomena as "!0. In the critical case, it is shown in [76] 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 [76] 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 [76], 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
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 Fand
thatFis star-shaped with respect to 0. Since Fis bounded, to simplify matters, without
loss of generality, we shall assume that FY, whereY= (1
2;1
2)nis the representative cell
inRn. We setY=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, letF"
k="k+r(")F. Also, let us denote by
F"the set of all the holes contained in
, i.e.
F"=[
F"
kjF"
k
; k2Zn
:
50

Chapter 3. Homogenization results for unilateral problems 51
Set
"=
nF"andS"=[f@F"
kjF"
k
;k2Zng. So,@
"=@
[S".
Let us consider a family of inhomogeneous media occupying the region
, parameter-
ized by"and represented by nnmatricesA"(x) of real-valued coecients dened on
.
Therefore, the positive parameter "will also dene a length scale measuring how densely
the inhomogeneities are distributed in
. One of our main goals in this paper was to get a
reasonable understanding of the interactions between these two sources of oscillations rep-
resented 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 coecients).
We dene the following nonempty closed convex subset of H1(
"):
K"=
v2H1(
")jv= 0 on@
; v0 onS"
: (3.3)
Our main motivation is to study the asymptotic behavior of the solution of the following
variational problem in
":
8
<
:Findu"2K"such thatZ

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

"f(v"u") dx8v"2K";(3.4)
wherefis a given function in L2(
).
The solution u"of (3.4) is also known to be characterized as being the solution of the
following non-linear free boundary-value problem: Find a smooth function u"and two subsets
S"
0andS"
+such thatS"
0[S"
+=S"; S"
0\S"
+=;, and
8
<
:div(A"Du") =fin
";
u"= 0 onS"
0; A"Du"
0 onS"
0;
u">0 onS"
+; A"Du"
= 0 onS"
+;(3.5)
whereis the exterior unit normal to the surface S". 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) =Ax
"
;
whereA=A(y) is a matrix-valued function on Rnwhich isY-periodic and satises the
following conditions:
8
<
:A2L1(
)nn;
Ais a symmetric matrix,
for some 0<<; jj2A(y)
jj28; y2I Rn:
For simplicity, we further assume that Ais continuous with respect to y. Under the above
structural hypotheses and the conditions fullled by K", it is well-known by a classical exis-
tence and uniqueness result of J. L. Lions and G. Stampacchia [164] that (3.4) is a well-posed
problem.
51

52 3.1. Homogenization results for Signorini's type problems
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 dierent 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 [76] that the solution u"of problem (3.4), properly
extended to the whole of
, converges to the unique solution of a Dirichlet problem in
,
associated with a new operator which is the sum of the standard homogenized one and an
extra term strange term that comes in from the special geometry (the size of the holes). More
precisely, we have the following result:
Theorem 3.1 (Theorem 4.1 in [76]) There exists an extension P"u"of the solution u"of
the variational inequality (3.4), positive inside the holes, such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of
8
<
:u2H1
0(
);Z

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

fvdx8v2H1
0(
):(3.7)
Here,A0is the homogenized matrix, whose entries are dened as follows:
a0
ij=1
jYjZ
Y
aij(y) +aik(y)@j
@yk
dy;
in terms of the functions j; j= 1;:::;n; solutions of the cell problems
8
<
:divy(A(y)Dy(yj+j)) = 0 inY;
jYperiodic
and0is given by
0= inf
w2H1(Rn)Z
RnA(0)DwDw dxjw1q.e. onF
: (3.8)
The constant matrix A0is symmetric and positive-denite.
Remark 3.2 The limit problem takes into account all the ingredients involved in (3.4). In
(3.4) 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
eect 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 just its
negative part uthat is penalized at the limit.
52

Chapter 3. Homogenization results for unilateral problems 53
The proof of Theorem 3.1, given in [76], is based on the use of a technical result of E. De
Giorgi [86] for matching boundary conditions for minimizing sequences. This result allowed
us to modify sequences of functions near the holes Br(")("i) and to separate the contribution
of the gradient of our solution close and far from the holes. By doing this, we are able to
capture, simultaneously, the oscillations 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 cases that are considered in [76]. The rst
one is the case in which the holes are much smaller than the critical ones. In this case, they are
too small to produce any visible contribution at the limit and the solution u"converges to the
solution of a classical homogenized Dirichlet problem in
associated to the 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"will become a positivity condition, u0, all over the domain. The
limit problem is an obstacle problem associated to the corresponding homogenized medium.
In this case, the holes only spread the positivity condition all over the domain. The last
case that we state explicitly below is characterized by the fact that the size of the holes is
exactly of order ". The solution u", properly extended to the whole of
, converges in this
case to an obstacle problem, associated to the homogenization of a periodic heterogeneous
and perforated medium. The in
uence of the holes comes twofold: on one hand, they spread
the positivity condition on S"to the whole of
and, on the other one, their particular size
aects the homogenized medium. More precisely, for the case of holes of the same size as the
period, we have the following result (see [76]):
Theorem 3.3 (Theorem 4.6 in [76]) There exists an extension P"u"of the solution u"of
the variational inequality (3.4), positive inside the holes, such that
P"u"*u weakly inH1
0(
);
whereuis the unique solution of
8
>>>><
>>>>:u2H1
0(
); u0in
;
Z

A0DuDu dx2Z

fudxZ

A0DvDv dx2Z

fvdx;
8v2H1
0(
); v0in
:(3.9)
Here,A0= (a0
ij)is the homogenized matrix, dened by
a0
ij=1
jYjZ
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?);R
Y?jdy= 0;
53

54 3.1. Elliptic problems in perforated domains with mixed-type boundary conditions
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
Z

A0DuD (vu)dxZ

f(vu)dx:
As mentioned before, the method we followed in [76] is the energy method of L. Tar-
tar [207]. 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 [81] 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, G. Griso in [58], [57] and by A. Damlamian in [84]. Their results were extended
to perforated domains by D. Cioranescu, P. Donato, R. Zaki [64], [66] and, further, by D.
Cioranescu, A. Damlamian, G. Griso, D. Onofrei in [59], by D. Onofrei in [187] and by A.
Damlamian, N. Meunier in [85] 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 [76]. More precisely, the asymptotic behavior
of a class of elliptic equations with highly oscillating coecients, in an "-periodic perforated
structure, with two holes of dierent sizes in each period, 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 perfo-
rated domain, an homogeneous Dirichlet condition is prescribed. As mentioned in [51], the
main feature of this type of problems is represented by the existence of a critical size of the
perforations that separates dierent emerging phenomena as the small parameter "tends to
zero. In this critical case, 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 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, F. Murat [68], putting into evidence, in the case of critical holes,
54

Chapter 3. Homogenization results for unilateral problems 55
the appearance of a strange term. Their results were extended, using dierent techniques,
to heterogeneous media by N. Ansini, A. Braides [19], G. Dal Maso, F. Murat [83] and D.
Cioranescu, A. Damlamian, G. Griso, D. Onofrei in [59]. Recently, A. Damlamian, N. Meunier
[85] studied the periodic homogenization for multivalued Leray-Lions operators in perforated
domains. The case of non homogeneous Neumann boundary conditions was considered,
among others, by C. Conca, P. Donato [75] and D. Onofrei [187]. For problems with Robin
or nonlinear boundary conditions we refer, for instance, to D. Cioranescu, P. Donato [61], D.
Cioranescu, P. Donato, H. Ene [63], D. Cioranescu, P. Donato, R. Zaki [64] and A. Capatina,
H. Ene [52]. Also, for Signorini's type problems we mention Yu. A. Kazmerchuk, T. A.
Mel'nyk [153]. The homogenization of problems involving perforated domains with two kinds
of holes of various sizes, was recently considered by D. Cioranescu, Hammouda [67].
The non-standard feature of the problem we present here is given by the presence, in
each period, of two holes of dierent sizes and with dierent conditions (3.10) 2;3imposed on
their boundaries. More precisely, we consider the case of Signorini and, respectively, criti-
cal Neumann holes. The Signorini condition (3.10) 2(see [204]) implies that the variational
formulation (3.11) of our problem is expressed as an inequality, which creates further dicul-
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, J.L. Lions [100] 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
Rn,n3, be a
bounded open set such that j@
j= 0 and let Y=
1
2;1
2n
be the reference cell. We
consider an "Y-periodic perforated structure with two kind of holes: some of size "1and the
other ones of size "2, with1and2depending on "and going to zero as "goes to zero. More
precisely, we consider two open sets BandFwith smooth boundaries such that BY,
FYandB\F=  and we denote the above mentioned holes by
B"1=[
2Zn"(+1B);
F"2=[
2Zn"(+2F):
LetY12=Yn(1B[2F) be the part occupied by the material in the cell and suppose that
it is connected. The perforated domain
";12with holes of size of order "1and of size of
order"2at the same time, is dened by

";12=
n(B"1[F"2) =n
x2
jnx
"o
Y2Y12o
:
LetA2L1(
)nnbe aY-periodic symmetric matrix. We suppose that there exist two
positive constants and, with 0<< , such that
jj2A(y)
jj282Rn;8y2Y :
55

56 3.1. Elliptic problems in perforated domains with mixed-type boundary conditions
Moreover, we assume that Ais continuous at the point 0.
Given aY-periodic function g2L2(@F) and a function f2L2(
), we consider the
following microscopic problem:
8
>>>>>>><
>>>>>>>:div (A"ru";12) =fin
";12;
u";120; A"ru";12
B0; u";12A"ru";12
B= 0 on@B"1;
A"ru";12
F=g"2on@F"2;
u";12= 0 on@ext
";12;(3.10)
where
A"(x) =Ax
"
and
g"2(x) =g1
2nx
"o
Y
a.e.x2@F"2:
In (3.10),BandFare the unit exterior normals to the sets B"1and, respectively, F"2.
In order to obtain a variational formulation of problem (3.10), we introduce the space
V"
12=fv2H1(
";12)jv= 0 on@ext
";12g
and the convex set
K"
12=fv2V"
12jv0 on@B"1g:
Then, the variational formulation of (3.10) is the following variational inequality:
(P";12)8
>>>>>>>><
>>>>>>>>:Findu";122K"
12such that
Z

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

";12f"(vu";12) dx
+Z
@F"2g"2(vu";12) ds8v2K"
12:(3.11)
Classical results for variational inequalities (see, for example, [206], [120], [44]) 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 that "!0 instead of writing ( ";1;2)!
(0;0;0).
56

Chapter 3. Homogenization results for unilateral problems 57
3.2.2 The limit problem
For stating the main convergence result for this problem, we introduce the functional space
KB=fv2L2(Rn) ;rv2L2(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);
Z
YAr(iyi)
rdy= 082H1
per(Y)(3.13)
andbe the solution of the problem
8
>><
>>:2KB; (B) = 1;
Z
RnnBA(0)r
rvdz= 08v2KBwithv(B) = 0:(3.14)
For the special geometry of this problem, we need to introduce, following [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,
we have to introduce some notation. For x2Rn, we denote by [ x]Yits integer part k2Zn,
such thatx[x]Y2Yand we setfxgY=x[x]Yforx2Rn. So, forx2Rn, we have
x="hx
"i
+nx
"o
. LetYk=Y+k, fork2Zn. We consider the following sets:
bZ"=n
k2Znj"Yk
o
;b
"= int[
k2bZ"
"Yk
;"=
nb
":
Denition 3.4 For any function '2Lp(
), withp2[1;1), we dene the periodic unfolding
operatorT":Lp(
)!Lp(
Y)by the formula
T"(')(x;y) =8
<
:'
"hx
"i
Y+"y
for a.e. (x;y)2b
"Y;
0 for a.e. (x;y)2"Y:
The operatorT"is linear and continuous from Lp(
) toLp(
Y). We recall here some
useful properties of this operator (see, for instance, [56]):
(i) if'and are two Lebesgue measurable functions on
, one has
T"(' ) =T"(')T"( );
(ii) for every '2L1(
), one has
1
jYjZ

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

'(x) dxZ
"'(x) dx;
57

58 3.1. Elliptic problems in perforated domains with mixed-type boundary conditions
(iii) iff'"gL2(
) is a sequence such that '"!'strongly in L2(
), then
T"('")!'strongly in Lp(
Y);
(iv) if'2L2(Y) isY-periodic and '"(x) ='(x="), then
T"('")!'strongly in L2(
Y);
(v) if'"*' weakly inH1(
), then there exists a subsequence and b'2L2(
;H1
per(Y))
such that
T"(r'")*rx'+ryb'weakly inL2(
Y):
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

"is dened by


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

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

") and their extensions in H1
0(
).
Denition 3.5 For any'2Lp(
), withp2[1;1), we dene the periodic unfolding operator
T"by the formula
T"(')(x;z) =8
<
:T"(')(z;z)if(x;z)2b
"1
Y;
0 otherwise:
By using the change of variable z= (1=)y, one can obtain similar properties for the operator
T"to those stated for T"(see [59]). Further, following [67] and [187], we brie
y recall here
the denition of the boundary unfolding operator Tb
".
Denition 3.6 For any'2Lp(@B"), withp2[1;1), we dene the boundary unfolding
operatorTb
"by the formula
Tb
"(')(x;z) ='
"hx
"i
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(
)such that
T"(u";12)*u weakly inL2(
;H1(Y)); (3.15)
58

Chapter 3. Homogenization results for unilateral problems 59
whereu2H1
0(
)is the unique solution of the homogenized problem
8
>>>>>><
>>>>>>:u2H1
0(
);
Z

Ahomru
r'dxk2
1Z

u'dx =Z

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

'dx8'2H1
0(
):(3.16)
In (3.16),Ahomis the homogenized matrix, dened, in terms of the solution iof (3.13), by
Ahom
ij=Z
Y
aij(y)nX
k=1aik(y)@j
@yk(y)!
dy
andis the capacity of the set B, given by
=Z
RnnBA(0)rz
rzdz;
whereveries (3.14).
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 eect of the unilateral condition
imposed on the boundary of the Signorini holes: the strange term, depending on the matrix
A, charges only the negative part uof the solution.
Remark 3.9 In the case k1= 0, the extra term generated by the Signorini holes vanishes in
the limit, while for k2= 0the contribution of the Neumann holes disappears.
Proof of Theorem 3.7 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
2Z

";12A"rv
rvdxZ

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

Ahomr'
r'dx+1
2k2
1Z

(')2dx
Z

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

'dx:(3.20)
59

60 3.1. Elliptic problems in perforated domains with mixed-type boundary conditions
For'2D(
), we set
h"(x) ='(x)"nX
i=1@'
@xi(x)ix
"
;
whereiis the solution of the problem (3.13).
If we takev"1=h+
"w"1h
", where
w"1(x) = 11
1nx
"o
Y
8x2Rn;
withgiven by (3.14), we obtain
J"
12(v"1) =I1
"I2
";
where
I1
"=1
2Z

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

";12f"(h+
"w"1h
")dx+Z
@F"2g"(h+
"w"1h
") ds:
Using the periodic unfolding operators T"andT"1(see [58] and [59]), we get
T"(A")(x;y) =A(y) in
Y;
T"1(w"1)(x;z) =T"(w"1)(x;1z) = 1(z) in
Rn;
T"1(rw"1)(x;z) =1
"1rz(z) in
Rn:
We also have (
w"1*1 weakly in H1(
);
h"!'strongly in H1(
):(3.21)
Taking into account the properties of the unfolding operator T"andT"1, we get the following
convergences8
>><
>>:T"(h")!'strongly in L2(
Y);
T"(rh")!rx'+ry'1strongly in L2(
Y);
T"1(rh")!rx'+ry'1strongly in L2(
Rn);(3.22)
where
'1=nX
i=1@'
@xii:
By unfolding and by using the fact that frh
"g"is bounded in ( L2(
))n, we can pass to the
limit in the unfolded form of I1
"and we get
lim
"!0I1
"=1
2Z

YA(r'+ry'1)
(r'+ry'1) dxdy+
60

Chapter 3. Homogenization results for unilateral problems 61
1
2k2
1Z

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

YAhomr'
r'dxdy +1
2k2
1Z

(RnnB)A(0)(')2rz
rzdxdz:(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
"=Z

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

'dx: (3.24)
Putting together (3.23) and (3.24), we are led to
lim
"!0J"
12(v"1) =J0(')8'2D(
) (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
ku";12kH1(
";12)C: (3.27)
Sinceu";122V"
12, we can suppose that, up to a subsequence, there exists u2H1
0(
) such
that8
<
:T"(u";12)*u weakly inL2(
;H1(Y));
ku
";12ukL2(
";12)!0;
T"(u
";12)!ustrongly in L2(
Y):(3.28)
It is not dicult to check that we have
lim inf
"!0Z

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

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

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

Ahomru
rudx+k2
1Z

(u)2dx:(3.30)
Since Z

";12A"r(u
";12h"w"1)
r(u
";12h"w"1) dx0;
61

62 3.1. Elliptic problems in perforated domains with mixed-type boundary conditions
we have
Z

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

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

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

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

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

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

n
21
1
"!2Z

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

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

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

rz(z) dxdz
+2Z

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

RnT"1(A")T"1(h")(1(z))rz
T"1(rh") dxdz:
From (3.27) and (3.28) and the properties of the unfolding operators T"andT"1(see [59]),
it follows that there exist u12L2(
;H1
per(Y)) andU12L2(
;L2
loc(Rn)) such that
8
<
:T"(ru
";12)*ru+ryu1 weakly inL2(
Y);
n
2
1T"1(ru
";12)*rzU1 weakly inL2(
Rn):(3.31)
Therefore, we get
lim inf
"!0Z

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

'2dx
Z

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

(RnnB)A0'rzU1
rzdxdz
+2Z

YA(ru+ryu1)
(r'+ry'1)dxdy8'2H1
0(
):
(3.32)
SinceT"1(u
";12) = 0 on
B, we haveU1= 0 on
B. Thus,W1=U1k1u2L2(
;KB).
On the other hand, from the cell problem (3.14), we obtain
divz(A(0)rz) = 0 inD0(
(RnnB))
which, by Stokes formula, leads to
Z
RnnBA(0)rz
rz dz= (B)Z
@BA(0)rz
Bds8 2KB: (3.33)
62

Chapter 3. Homogenization results for unilateral problems 63
For almost every x2
,W1(x;
)2KBand, so, (3.33) gives
Z
RnnBA(0)rz
rzW1dz=W1(x;B)Z
@BA(0)rz
Bds:
SincerzW1=rzU1andU1(x;B) = 0, we obtain
Z
RnnBA(0)rz
rzU1dz=k1uZ
@BA(0)rz
Bds=k1u
which implies that
2k1Z

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

u'dx:
Taking'=uin (3.32) and using the fact thatnX
i=1@u
@xii=u1, we obtain (3.30).
Finally, from (3.19), (3.26) and by the density D(
),!H1
0(
) , we deduce
lim
"!0J"
12(u";12) =J0(u)J0(')8'2H1
0(
): (3.34)
Asis non-negative, by Lax-Milgram theorem, it follows that the minimum point of the
functionalJ0is unique and this means that the whole sequence T"(u";12) converges to u.
Using a classical technique (see, for instance, [185] and [68]), one can prove that the
functional
P(v) =1
2Z

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

uvdx8u;v2H1
0(
):
Therefore, the functional J0is G^ ateaux dierentiable on H1
0(
), which ensures the equivalence
of the minimization problem
J0(u) = min
'2H1
0(
)J0(') (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
, positive on the Signorini holes (see [76]), such that
bu";12*u weakly inH1
0(
): (3.36)
For instance, in a rst step we extend our solution inside the Signorini holes in such a way
that 
bu";12= 0 inB"1;
bu";12=u";12on@B"1;
and, then, we further extend it in a standard way (see, e.g., [75]) inside the Neumann holes.
As a matter of fact, the use of unfolding operators allows us to work without extending our
solution to the whole of
.
63

64 3.1. Elliptic problems in perforated domains with mixed-type boundary conditions
Remark 3.11 We can treat in a similar manner the problem (3.10) for a general matrix
Asatisfying the usual conditions of boundedness and coercivity. In this case, we have to
suppose, like in [59] or [85], that there exist two matrix elds AandA0such that
T"(A")(x;y)!A(x;y)a.e. in
Y
and
T"1(A")(x;z)!A0(x;z)a.e. in
(RnnB):
The only dierence is the fact that in this case the corresponding homogenized matrix, the
cell problems and the strange term depend also on x.
We end this section by pointing out that in Section 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.
64

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 systems having a very complicated
microscopic structure. A lot of eorts have been made in the last years to obtain suitable
mathematical models in biology. Still, despite the all these eorts, many rigorous mathe-
matical models can be viewed as toy models , being far from capturing the complexity of the
phenomena involved in the biological processes.
In this chapter, we shall present some homogenization results for a series of problems
arising in the mathematical modeling of various reaction-diusion processes in biological
tissues. This chapter is based on the papers [212], [213], [214], [217], [221], [223], [220].
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 eective behavior of the solution
of a system of coupled partial dierential equations arising in the modeling of ionic transfer
phenomena, coupled with electrocapillary eects, in periodic charged porous media. The so-
called Nernst-Planck-Poisson system was proposed by W. Nernst and M. Planck (see [199])
for describing the potential dierence 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
the physical aspects behind these models and for a review of the recent relevant literature,
we refer to [115], [145], [199], [201], and [202].
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-
65

66 4.1. Ionic transport phenomena in periodic charged media
Planck system, with suitable boundary and initial conditions. The increased complexity of
the geometry and of the governing equations implies that an asymptotic procedure must be
applied for describing the solution of such a problem. The complicated microstructure will
be homogenized in order to obtain a model that captures its averaged properties.
Via the periodic unfolding method, we can show that the eective behavior of the solution
of our problem is described by a new coupled system of equations (see (4.11)-(4.14)). In
particular, the evolution of the macroscopic electrostatic potential is governed by a new law,
similar to Grahame's law (see [115] and [131]). Apart from a signicant simplication in the
proofs, an advantage of using such an approach based on unfolding operators, which transform
functions dened on oscillating domains into functions acting on xed domains, is that we
can avoid using extension operators and, thus, we can deal in a rigorous manner with media
possessing less regularity than those appearing usually in the literature (it is well-known that
composite materials, biological tissues or semiconductor devices are highly heterogeneous and
their interfaces are not, generally, enough smooth). Moreover, the homogenized equations
being dened on a xed domain
and having simpler coecients will be easier to be handled
numerically than the original equations. The dependence on the initial microstructure can
be seen at the limit in the homogenized coecients.
Related problems have been addressed, using dierent techniques, in [115], [145] or [201].
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 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
in Rn, withj@
j= 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
"the

uid part, by F"the solid part and by S"the inner boundary of the porous medium, i.e. the
interface between the
uid and the solid phases. Since the solid part is not allowed to reach
the outer boundary @
, it follows that S"\@
=;.
In such a periodic microstructure, we shall consider the Poisson-Nernst-Planck system,
with suitable boundary and initial conditions. The diusion in the
uid phase is governed by
the Nernst-Planck equations, while the electric potential which in
uence the ionic transfer is
described by the Poisson equation. Also, we include electrocapillary eects in our analysis.
More precisely, if we denote by [0 ;T], withT > 0, the time interval we are interested in,
we shall analyze the eective behavior, as the small parameter "!0, of the solution of the
66

Chapter 4. Mathematical models in biology 67
following system:
8
>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>:
"=c+
"c
"+D in (0;T)
";
r"
=""G(x;") on (0;T)S";
r"
= 0 in (0 ;T)@
;
@c
"
@tr
(rc
"c
"r") =F(c+
";c
") in (0;T)
";
(rc
"c
"r")
= 0 on (0 ;T)S";
(rc
"c
"r")
= 0 on (0 ;T)@
;
c
"=c
0inft= 0g
":(4.1)
In (4.1),is the unit outward normal to
", "is the electrostatic potential, c
"are the
concentrations of the ions (or the density of electrons and holes in the particular case of van
Roosbroeck model), D2L1(
) is the given doping prole, Gis a nonlinear function which
captures the eect of the electrical double layer phenomenon arising at the interface S"and
Fis a reaction term.
Let us notice that the scaling in the right hand side of the boundary condition on S"for
the electric potential ensures that we keep the in
uence of the double layer at the macroscale.
This scaling is, in fact, physically justied by experiments. For the case in which one considers
dierent scalings in (4.1), see [145] and [222].
We suppose that
"(x) =x
"
;
where=(y) is aY-periodic, bounded, smooth real function such that (y) > 0.
Also, we assume that the electrocapillary adsorption phenomenon at the substrate interface
is modeled by a given nonlinear function G. We address the case in which G=G(x;s) is
continuously dierentiable, monotonously increasing with respect to sfor anyxand with
G(x;0) = 0. Also, we assume that, for n3, there exist C0 and two exponents pandm
such that 8
>>>><
>>>>:@G
@sC(1 +jsjp);
@G
@xiC(1 +jsjm) 1in;(4.2)
with 0pn=(n2) and 0m < n= (n2) +p. Moreover, by using a regulariza-
tion procedure, for example Yosida approximation, as in [74], the hypothesis concerning the
smoothness of the nonlinearity Gcan be relaxed. For instance, we can treat the case of single
or multivalued maximal monotone graphs, as in [74]. Also, our results can be obtained, under
our assumptions, without imposing any growth condition (see [211]).
In practical applications, based on the Gouy-Chapman theory, one can use the Grahame
equation (see [115] and [131]) in which
G(s) =K1sinh(K2s); K 1;K2>0:
67

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 [219] 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) =ax
"
; b"(x) =bx
"
;
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 [109], 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 [74], [110] or [222], 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 [135]). More precisely, we can deal with the case in which
F=F(u;v) is a continuously dierentiable function on R2, which is sublinear and globally
Lipschitz continuous in both variables and such that F(u;v) = 0 foru <0 orv <0. For
other nonlinear reaction rates Fand more general functions G, see [137], [138] and [222].
We suppose that the initial data are non-negative and bounded independently with respect
to"and Z

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

"(c+
"+c
") dx
is conserved and suitable physical equilibrium conditions are veried, both at the microscale
and at the macroscale (see, for details, [115] and [222]). Let us mention that, for simplifying
the notation, we have eliminated in system (4.1) some constant physical relevant parameters.
68

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
diusivity in the
uid phase, but we can extend our analysis to the case of heterogeneous
media given by matrices D
"or to the case of the Stokes-Poisson-Nernst-Planck system, with
Neumann, Dirichlet or even Robin boundary condition. Also, let us remark that we can
address the case in which the electrostatic potential is dened all over the domain
, with
suitable transmission conditions at the interface S", as in [115] or [202].
The weak formulation of problem (4.1) is as follows: nd ( "; c+
"; c
"), with
8
>>>><
>>>>:"2L1(0;T;H1(
"));
c
"2L1(0;T;L2(
"))\L2(0;T;H1(
"));
@c
"
@t2L2(0;T; (H1(
"))0)(4.3)
such that, for any t>0 and for any '1; '22H1(
"), the triple ( "; c+
"; c
") satises:
Z

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

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

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

"F(c+
";c
")'2dx (4.5)
and
c
"(0;x) =c
0(x) in
": (4.6)
The variational problem (4.3)-(4.6) has a unique weak solution ( "; c+
"; c
") (see [115], [145]
or [222]). Moreover, like in [145], 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:
k"kL2((0;T)
")+kr"kL2((0;T)
")C
max
0tTkc
"kL2(
")+ max
0tTkc+
"kL2(
")+krc
"kL2((0;T)
")+krc+
"kL2((0;T)
")+

@c
"
@t

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

@c
"
@t

L2(0;T;(H1(
"))0)C:
Our goal is to obtain, via the periodic unfolding method, the eective 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].
69

70 4.1. Ionic transport phenomena in periodic charged media
For any Lebesgue measurable function 'on
", the periodic unfolding operator T"is the
linear operator dened by
T"(')(x;y) =8
<
:'
"hx
"i
Y+"y
for a.e. (x;y)2b
"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 dicult to see that there exist  2L2(0;T;H1(
)),b2L2((0;T)
;H1
per(Y)),
c2L2(0;T;H1(
)),bc2L2((0;T)
;H1
per(Y)), such that, up to a subsequence,
T"(")* weakly in L2((0;T)
;H1(Y)); (4.7)
T"(r")*r +ryb weakly in L2((0;T)
Y); (4.8)
T"(c
")!cstrongly in L2((0;T)
;H1(Y)); (4.9)
T"(rc
")*rc+rybcweakly inL2((0;T)
Y): (4.10)
4.1.2 The homogenized problem
In order to obtain the needed asymptotic behavior of the solution of our microscopic model,
we shall pass to the limit, with "!0, in the variational formulation of problem (4.1). We
get the following convergence result:
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)
:
8
>><
>>:div(D0r) +1
jYj0G=c+c+D;
@c
@tdiv(D0rcD0cr) =F
0;(4.11)
with the boundary conditions on (0;T)@
:
(
D0r
= 0;
(D0rcD0cr)
= 0(4.12)
and the initial conditions
c(0;x) =c
0(x);8×2
: (4.13)
Here,
0=Z
@F(y) ds;
F
0=F(c+;c) =g(c+c)
70

Chapter 4. Mathematical models in biology 71
andD0=
d0
ij
is the homogenized matrix, dened as follows:
d0
ij=1
jYjZ
Y
ij+@j
@yi(y)
dy;
withj; j= 1;:::;n; solutions of the cell problems
8
>>>><
>>>>:j2H1
per(Y);Z
Yj= 0;

j= 0 inY;
(rj+ej)
= 0 on@F(4.14)
andei,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
'1(t;x) = 0(t;x) +" 1
t;x;x
"
;
with 02D((0;T);C1(
)) and 12D((0;T)
;H1
per(Y)). By unfolding, we obtain
ZT
0Z

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

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

YT"(c+
"c
"+D)T"( 0+" 1)) dxdydt: (4.15)
Using the above convergence results, we can easily compute the limit of the linear terms in
(4.15) dened on
Y(see, for instance, [58], [74] and [222]). For the term containing
the nonlinear function G, let us notice that, exactly like in [74], one can show that if Ris a
continuously dierentiable 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(
), we have
R(x;w")*R(x;w);
weakly inW1;p
0(
), where
p=2n
q(n2) +n:
Using the properties of the unfolding operator T"and Lebesgue's convergence theorem, we
get
ZT
0Z

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

G(x;) 0dxdt:
Therefore, for "!0, we obtain:
ZT
0Z

Y(r(t;x) +ryb(t;x;y )) (r 0(t;x) +ry 1(t;x;y )) dxdydt+
71

72 4.1. Ionic transport phenomena in periodic charged media
0ZT
0Z

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

Y(c+(t;x)c(t;x) +D(x)) 0(t;x) dxdydt: (4.16)
By density, (4.16) is valid for any 02L2(0;T;H1(
)) and 12L2((0;T)
;H1
per(Y)).
Taking 0(t;x) = 0, we get
8
><
>:
yb(t;x;y ) = 0 in (0 ;T)
Y;
ryb
=rx(t;x)
on (0;T)
@F;
b(t;x;y ) periodic in y:
By linearity, we obtain
b(t; x; y ) =nX
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
'2(t;x) = 0(t;x) +" 1
t;x;x
"
;
with 02D((0;T);C1(
)) and 12D((0;T)
;H1
per(Y)), we have:
ZT
0Z

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

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

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

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

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

YF
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(
))
and 12L2((0;T)
;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 [145] and [222]),
the whole sequences of microscopic solutions converge to the solution of the homogenized
problem and this completes the proof of Theorem 4.1.
72

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 [223], where our goal was to analyze,
using homogenization techniques, the eective behavior of a coupled system of reaction-
diusion equations, arising in the modeling of some biochemical processes contributing to
carcinogenesis in living cells. We shall be concerned with the carcinogenic eects produced
in the human cells by Benzo-[a]-pyrene (BP) molecules, found in coal tar, cigarette smoke,
charbroiled food, etc. To understand the complex behavior of these molecules, mathemati-
cal models including reaction-diusion processes and binding and cleaning mechanisms have
been developed. Following [127], we consider here a simplied setting in which BP molecules
invade the cytosol inside of a human cell. There, they diuse freely, but they cannot enter
in the nucleus. Also, they bind to the surface of the endoplasmic reticulum (ER), where
chemical reactions, produced by the enzyme system called MFO (microsomal mixed-function
oxidases), take place, BP being chemically activated to a diol epoxide molecule, Benzo-[a]-
pyrene-7,8-diol-9,10-epoxide (DE). The DE molecules can unbind from the surface of the
endoplasmic reticulum and they can diuse again in the cytosol, where they may enter in
the nucleus. These new molecules can bind to DNA, DNA damage being known as a pri-
mary cause of cancer. Natural cleaning mechanisms occurring in the cytosol that make the
carcinogenic molecules harmless are taken into account in our model, as well. The slow dif-
fusion process taking place at the surface of the endoplasmic reticulum is modeled with the
aid of the Laplace-Beltrami operator, properly scaled. For describing the binding-unbinding
process at the surface of the endoplasmic reticulum, we consider various functions, leading
to dierent homogenized models. Another carcinogenesis model, introduced in [128], will be
brie
y discussed in Section 4.2.3. In this model, BP molecules can bind to the surface of the
endoplasmic reticulum by linking to receptors, the binding process being modeled, based on
the law of mass action, by the product of the concentration of molecules and that of recep-
tors. Also, for a receptor-based model obtained using homogenization techniques, see [79].
For more details about the mechanisms governing carcinogenesis in human cells we refer to
[117] and [191].
Problems closed to the one we treat here were addressed in [6], [79], [127], [128] and
[130]. For papers devoted to the upscaling of reactive transport in porous media, we refer
to [8], [110], [137], [140], [141], [179], [195] and the references therein. For reaction-diusion
problems involving adsorption and desorption, we refer to [6], [73], [74], [98], [137], [159].
For proving our main convergence results, we use the periodic unfolding method (see [56],
[58], [64] and [95]), extended in [128] and [129] for dealing with gradients of functions dened
on smooth periodic manifolds. Our analysis in [223] extended some of the results obtained
in [127] and [128] and were announced in [ ?]. More precisely, we addressed the case in which
the surface of the endoplasmic reticulum is supposed to be heterogeneous and, also, with the
case in which the adsorption is modeled with the aid of a nonlinear isotherm of Langmuir
type. The non-linearity of the model requires strong compactness results for the sequence
of solutions in order to be able to pass to the limit. Also, for passing to the limit in the
terms containing gradients of functions dened on the surface of the endoplasmic reticulum,
73

74 4.2. Multiscale analysis of a carcinogenesis model
we follow the ideas in [128].
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
in Rn, with a
Lipschitz boundary @
and with n2. The domain
, which, as in [128], is assumed to be
representable by a nite union of axis-parallel cuboids with corner coordinates belonging to
Qn, represents a human cell with the domain occupied by the nucleus removed. Following
[223], we denote by Cthe cell membrane and by Nthe boundary of the nucleus. Thus,
@
= C[N. LetY= (0;1)nbe the reference cell and let FYbe an open set with a
Lipschitz continuous boundary @Fthat does not touch the boundary of Y.@Frepresents
the surface of the endoplasmic reticulum. The volume occupied by the cytosol is Y=YnF.
RepeatingYby periodicity, 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, taking values in
a positive real sequence tending to zero and such that
is a nite union of cuboids which
are homothetic to the unit cell with the same ratio ". We set
"=S
k2Zn"(k+Y)\
and
S"=S
k2Zn"(k+@F)\
and we suppose that S"\@
=;. We remark that
is a nite union
of cuboids which are homothetic to the unit cell and that the inclusions do not intersect the
exterior boundary @
.
If we denote by [0 ;T], with 0< T <1, the time interval of interest, we shall analyze
the eective behavior, as the small parameter "!0, of the solution of the following coupled
system of equations:
8
>>>>>>>>>><
>>>>>>>>>>:@u"
@tDu
u"=f(u) in (0;T)
";
u"=ubon (0;T)C;
Duru"
= 0 on (0;T)N;
Duru"
="G1(u";s") on (0;T)S";
u"(0;x) =u0(x) in
";(4.19)
8
>>>>>>>>><
>>>>>>>>>:@v"
@tDv
v"=g(v") in (0;T)
";
v"= 0 on (0;T)N;
Dvrv"
= 0 on (0;T)C;
Duru"
="G2(v";w") on (0;T)S";
v"(0;x) =v0(x) in
";(4.20)
8
<
:@s"
@t"2Ds
"s"=h(s") +G1(u";s") on (0;T)S";
s"(0;x) =s0(x) onS":(4.21)
74

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
",
"is the Laplace-Beltrami operator
onS",u": [0;T]
"!Randv": [0;T]
"!Rrepresent the concentrations of
BP molecules and, respectively, of DE molecules in the cytosol and s": [0;T]S"!R
andw": [0;T]S"!Rare the concentrations of BP molecules and, respectively, of DE
molecules bound to the surface of the ER.
We notice that the diusion on the surface of the endoplasmic reticulum is scaled with
"2, in order to keep the in
uence of the slow surface diusion term at the macroscale. Also,
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 treat in a
similar manner the case in which the binding-unbinding term on S"corresponding to the BP
molecules is scaled with "
and the binding-unbinding term for DE molecules is multiplied
by"m, with
;m2[0;1) (for the linear case, see [127]).
We make the following assumptions on the data:
1.The diusion coecients Du;Dv;Ds;Dw>0 are supposed to be, for simplicity, con-
stant.
2.f,gandhare nonlinear functions modeling the cleaning mechanisms in
"and, respec-
tively, the transformation of the BP molecules to DE molecules bound to the surface of the
endoplasmic reticulum. As in [128], we suppose that the cleaning mechanism is described by
the following nonlinear, nonnegative, increasing, bounded and Lipschitz continuous function:
f(x) =8
<
:ax
x+b; x0;
0; x< 0;
fora;b > 0. The functions gandhare assumed to be of the same form as f, but with
dierent parameters. 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], [195] and [217]).
3.The binding-unbinding phenomena 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, additionally, we impose suitable structural conditions for
ensuring the positivity and L1-estimates of the solution ( u";v";s";w"). A standard choice
is given in [127]. More precisely, the authors consider the linear case in which
G1(u";s") =ls(u"s"); G 2(v";w") =lw(v"w");
wherels;lw>0 represent the binding and unbinding rates to the endoplasmic reticulum,
supposed to be constant. We deal here with two cases, namely the linear Henry isotherm
75

76 4.2. Multiscale analysis of a carcinogenesis model
with highly oscillating coecients and the case of a Langmuir isotherm. More precisely, we
consider, in a rst situation, that
G1(u";s") =l"
uu"l"
ss"; G 2(v";w") =l"
vv"l"
ww"; (4.23)
with
l"
u(x) =lux
"
; l"
s(x) =lsx
"
; l"
v(x) =lvx
"
; l"
w(x) =lwx
"
;
wherelu(y),ls(y),lv(y) andlw(y) areY-periodic, real, smooth, bounded functions with
lu(y)l0
u>0,ls(y)l0
s>0,lv(y)l0
v>0,lw(y)l0
w>0. The fact that we consider
that the model coecients are not constant but vary with respect to the surface variable is
physically justied (for examples where the processes on the membrane are inhomogeneous,
see [80]). As a consequence, in the homogenized limit, additional integral terms are present,
capturing the eect of the cell heterogeneity on the macroscopic behavior of the solution
of system (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 situation we shall analyze here, we consider the case in which G1andG2
are nonlinear functions dened in terms of isotherms of Langmuir type:
G1(u";s") =l"
s1u"
1 +1u"s"
; G 2(v";w") =l"
w2v"
1 +2v"w"
; (4.24)
withi;i>0, fori= 1;2. We denote
g1(u") =1u"
1 +1u"; g 2(v") =2v"
1 +2v": (4.25)
4.The concentration ubof the BP molecules on the cell membrane Cis supposed to be
an element of H1=2(C) (see (4.26)) and the initial values u0(x);v0(x)2L2(
),s0(x);w0(x)2
C1(
) are assumed to be nonnegative and bounded independently with respect to ".
Remark 4.2 Let us notice that, as in [130], we can treat in a similar way the case in which
the Lipschitz continuous functions G1andG2are of the form Gi(p;q) =Gi(p;q)(pq), with
0<Gi;minGi(p;q)Gi;max<1or the case in which Gi(p;q) =Ai(p)Bi(q), withAi
andBiLipschitz continuous and increasing functions, for i= 1;2.
The function g(r) =r=(1+r) is increasing and one to one from R+to [0;= ]. Despite
the fact that gis not dened for r=1=, since we are interested in considering only
non-negative values of the argument r, we can mollify gfor negative values r <0 in such a
way that we get an increasing function on R, growing at most linearly at innity and having
an uniformly bounded derivative (see [6]). Alternatively, since for negative values of the
argument of gsingularities may appear, we can consider, in a rst step, a modied kinetics
g0, obtained by replacing rby its modulusjrjin the denominator of g. This new function is
Lipschitz continuous. Then, proving the existence and uniqueness of a solution of the problem
76

Chapter 4. Mathematical models in biology 77
involving this new kinetics, we show that the solution is non-negative and, therefore, it is
a solution of the initial problem, too. Let us notice that for a small concentration, i.e. for
r1, we are led to a linear function (Henry adsorption isotherm). We point out that,
in fact, from a physical point of view, we can extend the considered rates by zero for all
negative arguments and this would allow a straightforward proof of the fact that the solution
components remain positive if the initial and boundary data are positive.
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 even other isotherms encountered in the literature. The presence of the variable
xin these rates means that we assume that the processes at the surface of the endoplasmic
reticulum are heterogeneous (see [99], [137] and [222]).
In order to write the weak formulation of problem (4.19)-(4.22), we introduce some func-
tion spaces. In the sequel, the space L2(
") is equipped with the classical scalar product and
norm
(u;v)
"=Z

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

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

";T= (u;u)
";T;
whereu(t) =u(t;
);v(t) =v(t;
). Further, following [127] and [128], we set
V(
") =L2((0;T);H1(
"))\H1((0;T);(H1(
"))0);
VN(
") =fv2V(
")jv= 0 on Ng;
VC(
") =fv2V(
")jv=ubon Cg;
V0;C(
") =fv2V(
")jv= 0 on Cg;
where, for an arbitrary Banach space V, we denote by V0its dual. Similar spaces can be
dened for
and S". We use the notation
hu;vi"=Z
S"g"uvdx;
whereg"is the Riemannian tensor on S". Also, let us dene
VN(
") =fv2H1(
")jv= 0 on Ng;
V0;C(
") =fv2H1(
")jv= 0 on Cg; V (S") =H1(S")
and
V(
;Y) =L2((0;T)
;H1
per(Y));V(
;@F) =L2((0;T)
;H1(@F));
77

78 4.2. Multiscale analysis of a carcinogenesis model
whereH1
per(Y) =f'2H1
loc(Rn
1) :'isYperiodicg. Finally, we assume that ub2
H1=2(C), where, for an arbitrary smooth hypersurface 0Rnand for any 0 < r < 1,
we consider the Sobolev-Slobodeckij space
Hr(0) =fu2L2(0) :juj0;r<1g; (4.26)
where
juj2
0;r=Z
00ju(x)u(y)j2
jxyjn1+2rdxdy:
The spaceHr(0) is endowed with the norm kuk2
Hr(0)=kuk2
L2(0)+juj2
0;r(see [116] and
[126]).
Let us give now the variational formulation of problem (4.19)-(4.22).
Problem 1 : nd (u";v";s";w")2VC(
")VN(
")V(S")V(S"), satisfying the initial
condition
(u"(0);v"(0);s"(0);w"(0)) = (u0;v0;s0;w0);
such that, for a.e. t2(0;T) and for any ( '1;'2;)2VC;0(
")VN(
")V(S"), we have
8
>>>>>>>>>>><
>>>>>>>>>>>:@u"
@t;'1

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

"+Dv(rv";r'2) +"hG2(v";w");'2iS"=(g(v");'2)
";
D@s"
@t;E
S"+Dsh"r@Fs";"r@FiS"=hh(s");iS"+hG1(u";s");iS";
D@w"
@t;iS"+Dwh"r@Fw";"r@FiS"=hh(s");iS"+hG2(v";w");iS":(4.27)
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 [128, 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], [95], [128], and [129].
The main feature of these operators is that they map functions dened on the oscillating
domains (0;T)
"and, respectively, (0 ;T)", into functions dened on the xed domains
(0;T)
Yand (0;T)
, respectively. We brie
y recall here the denitions of
78

Chapter 4. Mathematical models in biology 79
these two operators for our particular geometry. For any '2Lp((0;T)
") and any
p2[1;1], we dene the periodic unfolding operator T":Lp((0;T)
")!Lp((0;T)
Y)
by the formulaT"(')(t;x;y ) ='
t;hx
"i
+"y
. In a similar manner, for any function
2Lp((0;T)"), the periodic boundary unfolding operator T"
b:Lp((0;T)")!
Lp((0;T)
) is dened byT"
b()(t;x;y ) =
t;"hx
i
+"y
.
Using these unfolding operators, we deduce the homogenized limit system.
Theorem 4.4 (Theorem 3.1 in [223]) 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 )2VC(
)
VN(
)V(
;@F)V(
;@F), with (u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0), of the following
macroscopic problem:
8
>>>>>>>>>>><
>>>>>>>>>>>:jYj@u
@t;'1

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

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

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

@F+ (Dwr@F
yw;r@F)
@F(G2(v;w);)
@F= (h(s);)
@F;(4.28)
for('1;'2;)2V0;C(
)VN(
)V(
;@F). Here,AuandAvare the homogenized matrices,
dened by: 8
>><
>>:Au
ij=DuZ
Y
ij+@j
@yi
dy;
Av
ij=DvZ
Y
ij+@j
@yi
dy;(4.29)
in terms of the functions j2H1
per(Y)=R, ,j= 1;:::;n; weak solutions of the cell problems
ry
(ryj+ej) = 0; y2Y;
(ryj+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 [223]) The limit function (u;v;s;w )2VC(
)VN(
)
V(
;@F)V(
;@F), dened in Theorem 4.1 and satisfying
(u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0);
is the unique solution of the following problem:
8
>>>><
>>>>:jYj@u
@tr
(Auru) +Z
@FG1(u;s) dy=jYjf(u)in(0;T)
;
u=ubon(0;T)C;
Auru
= 0 on(0;T)N;(4.31)
79

80 4.2. Multiscale analysis of a carcinogenesis model
8
>>>><
>>>>:jYj@v
@tr
(Avrv) +Z
@FG2(v;w) dy=jYjg(v)in(0;T)
;
v= 0 on(0;T)N;
Avrv
= 0 on(0;T)C;(4.32)
8
>><
>>:@s
@tDs
@F
ysG1(u;s) =h(s)on(0;T)
@F;
@w
@tDw
@F
ywG2(v;w) =h(s)on(0;T)
@F::(4.33)
As in [127, Theorem 14] and [128, Theorem 6.1], it follows that the solution of the macroscopic
problem (4.28) in unique.
Remark 4.6 We remark that the in
uence of the properly scaled binding-unbinding processes
taking 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, we point out that the limit problem
involves an additional microvariable y. This local phenomenon yields a more complicated
microstructure of the eective medium; in (4.31)-(4.32), x2
plays the role of a macroscopic
variable and y2@Fis a microscopic one. The limit model consists of two partial dierential
equations, with global diusion (with respect to the macroscopic variable x), for the limit
of the BP and DE molecules in the cytosol (see (4.31)-(4.32)) and two partial dierential
equations, governing the local behavior of the system, with local diusion (with respect to the
microscopic variable y) on@F(see (4.33)). .
Remark 4.7 We can deal, in a similar manner, with the more general case in which, in-
stead of considering constant diusion coecients, 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 diusion coecients DuandDv, we have two matrices A"
uand, respectively, A"
v.
We suppose that A"
uandA"
vare sequences of matrices in M(;;
)such that
T"(A"
u)!Au;T"(A"
v)!Avstrongly in L1(
Y)nn; (4.34)
for some matrices Au=Au(x;y)andAv=Au(x;y)inM(;;
Y)(see [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
byM(;;
)the set of all the matrices A2(L1(
))nnwith the property that, for any
2Rn,(A(y); )jj2;jA(y)jjj, almost everywhere in
.
Also, for the diusion coecients on the surface S"we can suppose that they are not con-
stant, but they depend on ". For instance, we can work with the diusion tensors D"
s(x) =
Ds(x=")andD"
w(x) =Dw(x="), whereDsandDware two uniformly coercive periodic 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(
@F)such thatT"
b(D"
s)!Dsand
T"
b(D"
w)!Dwstrongly in L1(
@F). .
80

Chapter 4. Mathematical models in biology 81
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).
The following proposition, proven in [223], states that the functions u";v";s"andw"are
nonnegative and bounded from above if the initial data are assumed to be bounded and
nonnegative. The positivity of u";v";s"andw"is a natural requirement, since they represent
concentrations of BP and DE molecules in the cytosol and on the surface of the endoplasmic
reticulum. Also, this property is essential for proving the well-posedness of our problem. On
the other hand, essential boundedness of the solution is necessary from the point of view of
practical applications.
Proposition 4.8 (Proposition 4.1 in [223]) The functions u"andv"are nonnegative for
almost every x2
"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]
"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. Besides,
exactly like in [127, Lemma 2] and [128, Lemma A.2], one can prove the L2-boundedness of
the solution ( u";v";s";w").
Proposition 4.9 (Proposition 4.2 in [223]) There exists a constant C > 0, independent of
", such that
ku"k2

"+kru"k2

";t+kv"k2

"+krv"k2

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

@u"
@t

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

@v"
@t

L2((0;T);(H1
0(
"))0)C:: (4.35)
The above a priori estimates will allow us to apply the periodic unfolding method and to
get the needed convergence results for the solution of problem (4.27). Still, the nonlinear-
ity of the model requires strong compactness results for the sequence of solutions in order
to be able to pass to the limit. We know (see [128]) that u";v"2L2((0;T);H1(
"))\
H1((0;T);(H1
0(
"))0)\L1((0;T)
").
Using suitable extension results (see, for instance, [139], [116] and [171]) and Lemma 5.6
from [130], we know that we can construct two extensions u"andv"that converge strongly
tou;v2L2((0;T);L2(
)). We point out that one can obtain (see e. g. [116] and [171]) the
existence of a linear and bounded extension operator to the whole of
, which preserves the
non-negativity, the essential boundedness and the above priori estimates.
Since the functions g1andg2are Lipschitz, exactly like in Lemma 4.3 in [128], we can
prove thatT"
b(s") andT"
b(w") are Cauchy sequences in L2((0;T)
@F).
81

82 4.2. Multiscale analysis of a carcinogenesis model
Proposition 4.10 (Proposition 4.3 in [223]) For any  >0, there exists "0>0such that
for any 0<"1;"2<"0one has
kT"1
b(s"1)T"2
b(s"2)k(0;T)
@F+kT"1
b(w"1)T"2
b(w"2)k(0;T)
@F<::
This implies that the sequences T"
b(s") andT"
b(w") are strongly convergent in L2((0;T)

@F).
As already mentioned, (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], [95], [128], and [129]), we
get immediately the following compactness result.
Proposition 4.11 (Proposition 4.4 in [223]) Let (u";v";s";w")be the solution of problem
(4.27). Then, there exist u;v2L2((0;T);H1(
));bu;bv2L2((0;T)
;H1
per(Y)),s;w2
L2((0;T)
;H1
per(@F))such that, up to a subsequence, when "!0, we have
8
>>>>>>>>>><
>>>>>>>>>>:T"(u")*u weakly inL2((0;T)
;H1(Y));
T"(v")*v weakly inL2((0;T)
;H1(Y));
u"!u;v"!vstrongly in L2((0;T)
);
T"(ru")*ru+rybuweakly inL2((0;T)
Y);
T"(rv")*rv+rybvweakly inL2((0;T)
Y);
T"
b(s")*s weakly inL2((0;T)
;H1(@F));
T"
b(w")*w weakly inL2((0;T)
;H1(@F));
T"
b(s")!s;T"
b(w")!wstrongly in L2((0;T)
@F)::(4.36)
For passing to the limit in the nonlinear terms containing the functions G1andG2, we have
to show thatT"
b(u")!uandT"
bv")!v, strongly in L2((0;T)
@F). These strong
convergence results follow from the strong convergence of u"andv", respectively, the trace
lemma (see Lemma 3.1 in [116]) and the properties of the unfolding operator T"
b. More
precisely, we have the following result (see [223] and [116]).
Proposition 4.12 (Proposition 4.5 in [223]) Up to a subsequence, one has
T"
b(u")!ustrongly in L2((0;T)
@F)
and
T"
b(v")!vstrongly in L2((0;T)
@F):
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)
Y):
g(T"(v"))!g(v) strongly in L2((0;T)
Y):
In a similar way, we obtain
h(T"
b(s"))!h(s) strongly in L2((0;T)
@F):
82

Chapter 4. Mathematical models in biology 83
For getting the limit behavior of the terms involving G1andG2, in the rst situation, i.e. for
Henry isotherm with rapidly oscillating coecients given by (2.5), we can easily pass to the
limit since these coecients are uniformly bounded in L1(
) and converge strongly therein,
while for the second situation, i.e. isotherm of the form (3.24), we need to use the strong
convergence ofT"
b(u"),T"
b(v"),T"
b(s") andT"
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)
@F)
and
G2(T"
b(v";w"))!G2(v;w) strongly in L2((0;T)
@F):
By classical results (see, for instance, Theorem 2.12 in [56] and Theorem 2.17 in [95] ), 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
'(t;x) ='1(t;x) +"'2
t;x;x
"
; (4.37)
with'12C1
0((0;T);C1(
))'22C1
0((0;T);C1(
;C1
per(Y))).
Integrating with respect to time, 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], [74], [128] and [217]), we get:
ZT
0Z

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

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

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

Yf(u)'1dxdydt: (4.38)
By standard density arguments, it follows that (4.38) is valid for any '12L2(0;T;H1(
)),
'22L2((0;T)
;H1
per(Y)). In a similar manner, for the limit equation for v", we obtain
ZT
0Z

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

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

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

Yg(v)'1dxdydt; (4.39)
for any'12L2((0;T);H1(
)),'22L2((0;T)
;H1
per(Y)).
In order to obtain the limit equations for s"andw", we apply the convergence results
obtained in [128] (see Lemma 2.6 and Theorem 2.9). Indeed, using the boundary unfolding
operatorT"
bin (4:27)3, by passing to the limit we get
ZT
0Z

@F@s
@tdxdydt+DsZT
0Z

@Fr@F
ys
r@F
ydxdydt=
83

84 4.2. Multiscale analysis of a carcinogenesis model
ZT
0Z

@FG1(u;s)dxdydtZT
0Z

@Fh(s)dxdydt; (4.40)
for any2C1
0((0;T);C1(
;C1
per(@F))).
In a similar manner, we get
ZT
0Z

@F@w
@tdxdydt+DwZT
0Z

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

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

@Fh(s)dxdydt; (4.41)
for any2C1
0((0;T);C1(
;C1
per(@F))).
Thus, we get exactly the weak formulation of the limit problem (4.28). Indeed, if we take
'1= 0, we easily get the cell problems (4.30) and
bu=nX
k=1@u
@xkk;bv=nX
k=1@v
@xkk: (4.42)
Then, taking '2= 0 and using (4.42), we obtain (4.28). Moreover, by standard techniques,
we can derive the initial conditions ( u(0);v(0);s(0);w(0)) = (u0;v0;s0;w0). Since the so-
lution (u;v;s;w ) of problem (4.28) is uniquely determined, the above convergences for the
microscopic solution ( u";v";s";w") hold for the whole sequence and this ends the proof of
Theorem 4.7.
4.2.3 A nonlinear carcinogenesis model involving free receptors
We end this section by brie
y discussing a generalization of a recent nonlinear model proposed
in [128] (see, also, [223]) for the carcinogenesis in human cells, involving a new variable
modeling the free receptors present at the surface of the ER. In this new model, the BP
molecules present in the cytosol are transformed into BP molecules bound to the surface of
the ERs"only if they nd a free receptor R". Following [128], let R": [0;T]S"![0;1] be
the relative concentration of free receptors on the surface of the ER. The maximal relative
quantity of free receptors is R= 1. At the surface of the ER, BP molecules bind to receptors
and, using the law of mass action, the binding is described by the product kuu"R", with
a constant rate ku>0. DE molecules are assumed to have a similar behavior. When BP
moleculesu"or DE molecules v"bind to the surface of the endoplasmic reticulum, the quantity
of free receptors decreases and when the molecules s"andw"unbind from the surface of the
endoplasmic reticulum the amount of free receptors increases. If receptors are supposed to
be xed on the surface of the endoplasmic reticulum, then their evolution is governed by (see
[128] 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 nonlinear problem is stated below.
84

Chapter 4. Mathematical models in biology 85
Problem 2: nd (u";v";s";w";R")2 VC(
")VN(
")V(S")V(S")VR(S"),
satisfying the initial condition
(u"(0);v"(0);s"(0);w"(0);R"(0)) = (u0;v0;s0;w0;R);
such that, for a.e. t2(0;T) and for all ( '1;'2;)2VC;0(
")VN(
")V(S"), we have
8
>>>>>>>>>>>>><
>>>>>>>>>>>>>:@u"
@t;'1

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

"+Dv(rv";r'2) +"hkvv"R"lww";'2i"=(g(v");'2)
";
D@s"
@t;E
S"+Dsh"r@Fs";"r@FiS"=hh(s");iS"+hkuu"R"lss";iS";
D@w"
@t;E
S"+Dwh"r@Fw";"r@FiS"=hh(s");iS"+hkvv"R"lww";iS";
h@tR";iS"+hR"jkuu"+kvv"j;iS"=h(RR")jkss"+kww"j;iS":(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")2VC(
")VN(
")V(S")V(S")VR(S")
of the above variational problem is proven in [128, Theorem 4.4]. Also, it is proven in
[128, Lemma 4.1 and Theorem 4.5] that R"is nonnegative and bounded by R > 0 almost
everywhere in [0 ;T]S"andT"
b(R") converges strongly to R2L2((0;T)
@F).
In this case, the homogenized result is stated in the following theorem (see Theorem 5.1
in [128]).
Theorem 4.13 The homogenized limit problem is as follows: nd (u;v;s;w;R )2VC(
)
VN(
)V(
;)V(
;@F)VR(
;@F), with (u(0);v(0);s(0);w(0);R(0)) = (u0;v0;s0;w0;R),
such that
8
>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:jYj@u
@t;'1

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

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

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

@F+ (Dwrw;r)
@F(kvvRlww;)
@F= (h(s);)
@F;
(@tR;)
@F+ (R(kuu+kvv);)
@F= ((RR)(kss+kww);)
@F;
for('1;'2;)2V0;C(
)VN(
)V(
;@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 dierential equation.
All the above results are still valid for the case of highly oscillating coecients k"
u,k"
vand,
respectively, l"
sandl"
w. Moreover, based on the law of mass action, various other functions
G1(R";u") andG2(R";v") can be used to describe the adsorption phenomena at the surface
85

86 4.3. Homogenization results for the calcium dynamics in living cells
of the endoplasmic reticulum. As particular situations, we can mention the case in which
G1=R"g1(u") andG2=R"g2(v"), withg1andg2suitable Lipschitz continuous functions
(e.g. the Langmuir kinetics considered above). In such a case, the equation governing the
evolution of the receptors is
@tR"=R"jg1(u") +g2(v")j+ (RR")jkss"+kww"jon (0;T)S":
The case in which the binding processes at the surface of the endoplasmic reticulum is given
by suitable nonlinear functions G1(x=";R";u") and, respectively, G2(x=";R";v"), can be ad-
dressed, too. Such isotherms were proposed in [137].
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 [221]. More precisely, our goal was to analyze the eective behavior
of a nonlinear system of coupled reaction-diusion equations arising in the modeling of the
dynamics of calcium ions in living cells is analyzed. Calcium is a very important second
messenger in a living cell, contributing to many cellular processes, such as protein synthe-
sis, muscle contraction, cell cycle, or metabolism (see, for instance, [71]). Controlling the
intracellular 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 complex
and highly heterogeneous cellular structure spreads throughout the cytoplasm, generating
various zones with diverse morphology and functions. The study of the dynamics of calcium
ions, acting as messengers between the endoplasmic reticulum and the cytosol inside living
cells, represents a topic of huge interest, which still requires special attention. Many biologi-
cal mechanisms involving the functions of the cytosol and of the endoplasmic reticulum are
not yet perfectly understood.
The model considered in [221] 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. Depending on the magnitude of this term,
dierent models arise at the limit. In a particular case, we obtain, at the macroscale, a
bidomain model, which is largely used for studying the dynamics of the calcium ions in
human cells. The calcium bidomain system consists of two reaction-diusion equations, one
for the concentration of calcium ions in the cytosol and one for the concentration of calcium
ions in the endoplasmic reticulum, coupled through a reaction term. For details about the
physiological background of such a model, we refer to [154]. The problem of obtaining the
calcium bidomain equations using homogenization techniques was addressed by a formal
approach in [122] and by a rigorous one, based on the use of the two-scale convergence
method, in [130]. Our results in [221] constitute a generalization of some of the results
contained in [122] and [130]. The proper scaling of the interfacial exchange term has an
86

Chapter 4. Mathematical models in biology 87
important in
uence on the limit problem and, using some techniques from [95], we extend
the analysis from [130] to the case in which the parameter
arising in the exchange term
belongs to R.
Bidomain models arise also in other contexts, such as the modeling of diusion processes
in partially ssured media (see [31], [27] and [102]) or the modeling of the electrical activity
of the heart (see [15], [13] and [189]).
4.3.1 Setting of the problem
Let us start by describing the geometry of the problem. Let
be a bounded domain in
Rn, withn3, having a Lipschitz boundary @
made up of a nite number of connected
components. The domain
is assumed to be a periodic structure formed by two connected
parts,
"
1and
"
2, separated by an interface ". We suppose that only the phase
"
1reaches the
exterior xed boundary @
. 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
"
1models the cytosol, while the phase
"
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 [102]).
For any"2(0;1), let
Z"=fk2Znj"k+"Y
g;
K"=fk2Z"j"k"ei+"Y
;8i= 1;:::;ng;
whereeiare the vectors of the canonical basis of Rn. We set

"
2= int([
k2K"("k+"Y2));
"
1=
n
"
2:
For1;12R, with 0< 1< 1, we denote byM(1;1;Y) the collection of all the matrices
A2(L1(Y))nnwith the property that, for any 2Rn, (A(y); )1jj2;jA(y)j1jj,
almost everywhere in Y. We consider the matrices A"(x) =A(x=") dened on
, where
A2M (1;1;Y) is aY-periodic symmetric matrix and we denote the matrix AbyA1inY1
and, respectively, by A2inY2.
If we denote by (0 ;T) the time interval of interest, we shall be concerned with the eective
87

88 4.3. Homogenization results for the calcium dynamics in living cells
behavior of the solutions of the following microscopic system:
8
>>>>>>>>>>>>><
>>>>>>>>>>>>>:@u"
1
@tdiv (A1"ru"
1) =f(u"
1) in (0;T)
"
1;
@u"
2
@tdiv (A2"ru"
2) =g(u"
2) in (0;T)
"
2;
A1"ru"
1
=A2"ru"
2
on (0;T)";
A1"ru"
1
="
h(u"
1;u"
2) on (0;T)";
u"
1= 0 on (0 ;T)@
;
u"
1(0;x) =u0
1(x) in
"
1; u"
2(0;x) =u0
2(x) in
"
2;(4.44)
whereis the unit outward normal to
"
1and the parameter
is 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 assume that the initial conditions are
non-negative and that the functions fandgare 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)
whereh"
0(x) =h0(x=") andh0=h0(y) is a realY-periodic function in L1(), withh0(y)
>0. Let
H=Z
h0(y) dy6= 0:
Exactly like in [130], 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<hminh(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
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 [154] and [136]).
In order to write the weak form of our microscopic problem, let us introduce the needed
function spaces and norms. Let
H1
@
(
"
1) =fv2H1(
"
1)jv= 0 on@
\@
"
1g;
V(
"
1) =L2(0;T;H1
@
(
"
1));V(
"
1) =n
v2V(
"
1)j@v
@t2L2((0;T)
"
1)o
;
V(
"
2) =L2(0;T;H1(
"
2));V(
"
2) =n
v2V(
"
2)j@v
@t2L2((0;T)
"
2)o
;
88

Chapter 4. Mathematical models in biology 89
with
(u(t);v(t))
"=Z

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

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

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

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

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

;t= (u;u)
;t
and
V0(
) =fv2V(
)jv= 0 on@
a.e. on (0 ;T)g;V0(
) =V0(
)\V(
):
The variational formulation of problem (4.44) is the following one: nd ( u"
1;u"
2)2V(
"
1)
V(
"
2), with (u"
1(0;x);u"
2(0;x)) = (u0
1(x);u0
2(x))2(L2(
))2and
@u"
1
@t(t);'(t)

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

"
2+
(A"
1(t)ru"
1;r'(t))
"
1+ (A"
2(t)ru"
2;r (t))
"
2
"
(h(u"
1;u"
2);'(t) (t))"= (f(u"
1(t));'(t))
"
1+ (g(u"
2(t)); (t))
"
2; (4.47)
for a.e.t2(0;T) and any ( '; )2V(
"
1)V(
"
2).
Following the same techniques used in [130], it is not dicult to prove that (4.47) is a
well-posed problem and that u"andv"are non-negative and bounded almost everywhere.
Taking (u"
1;u"
2) as test function in (4.47), integrating with respect to time and taking into
account that u"
1andu"
2are bounded and non-negative, it follows that there exists a constant
C0, independent of ", such that
ku"
1(t)k2

"
1+ku"
2(t)k2

"
2+kru"
1k2

"
1;t+kru"
2k2

"
2;t+"
(h(u"
1;u"
2);u"
1u"
2)";tC;
for a.e.t2(0;T). Also, as in [130] or [215], we can see that there exists a positive constant
C0, independent of ", such that

@u"
1
@t(t)

2

"
1+

@u"
2
@t(t)

2

"
2C;
for
1 and

@u"
1
@t

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

@u"
2
@t

L2(0;T;H1(
"
2))C;
for
<1.
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
89

90 4.3. Homogenization results for the calcium dynamics in living cells
functions dened on xed domains (see [56], [58] and [95]). 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 [95]):
bZ"=n
k2Znj"Yk
o
;b
"= int[
k2bZ"
"Yk
;"=
nb
";
b
"
=[
k2bZ"
"Yk

;"
=
"
nb
"
;b"=@b
"
2:
Denition 4.14 For any Lebesgue measurable function 'on
"
,2f1;2g, we dene the
periodic unfolding operators by the formula
T"
(')(x;y) =8
<
:'
"x
"
Y+"y

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

for a.e. (x;y)2b
"
0 for a.e. (x;y)2"
Remark 4.15 We notice that if '2H1(
"
), thenT"
b(') =T"
(')jb
".
We recall here some useful properties of these operators (see, for instance, [56], [94], and
[95]).
Proposition 4.16 Forp2[1;1)and= 1;2, the operatorsT"
are linear and continuous
fromLp(
"
)toLp(
Y)and
(i) if'and are two Lebesgue measurable functions on
"
, one has
T"
(' ) =T"
(')T"
( );
(ii) for every '2L1(
"
), one has
1
jYjZ

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

"'(x) dxZ
"'(x) dx;
(iii) iff'"g"Lp(
)is a sequence such that '"!'strongly in Lp(
), then
T"
('")!'strongly in Lp(
Y);
(iv) if'2Lp(Y)isY-periodic and '"(x) ='(x="), then
T"
('")!'strongly in Lp(
Y);
90

Chapter 4. Mathematical models in biology 91
(v) if'2W1;p(
"
), thenry(T"
(')) ="T"
(r')andT"
(')belongs toL2

;W1;p(Y)

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

T"
b(')(x;y) dxdy:
For
= 1, using the obtained a priori estimates and the properties of the operators
T"
1andT"
2, it follows that there exist u12L2(0;T;H1
0(
)),u22L2(0;T;H1(
)),bu12
L2((0;T)
;H1
per(Y1)),bu22L2((0;T)
;H1
per(Y2)) such that, passing to a subsequence,
for"!0, we have:
8
>><
>>:T"
1(u"
1)!u1strongly in L2((0;T)
;H1(Y1));
T"
1(ru"
1)*ru1+rybu1weakly inL2((0;T)
Y1);
T"
2(u"
2)*u 2weakly inL2((0;T)
;H1(Y2));
T"
2(ru"
2)*ru2+rybu2weakly inL2((0;T)
Y2):(4.48)
Moreover, as in [130] and [215],@u1
@t2L2(0;T;L2(
));@u2
@t2L2(0;T;L2(
)) andu12
C0([0;T];H1
0(
)); u22C0([0;T];H1(
)). So,u12V0(
) andu22V(
).
Let us mention that, in fact, under our hypotheses, passing to a subsequence, T"
1(u"
1)
converges strongly to u1inLp((0;T)
Y1), for 1p<1. As a consequence, since the
Nemytskii operator corresponding to the nonlinear function fis continuous, it follows that
f(T"
1(u"
1)) converges to f(u1). A similar result holds true for u"
2.
Since
kT"
1(u"
1)T"
2(u"
2)kL2((0;T)
)C"1

2
it follows that for the case
= 0 and
=1 we have, at the macroscale, u1=u2=u02
V0(
). Moreover, for
=1, following the techniques from [95], one can prove that
T"
1(u"
1)T"
2(u"
2)
"*bu1bu2weakly inL2((0;T)
):
4.3.2 The main convergence results
We present now, without proofs, the main convergence results obtained in [221].
Theorem 4.17 (Theorem 1 in [221]) 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(A1ru1)H(u2u1) =f(u1)in(0;T)
;
(1)@u2
@tdiv(A2ru2) +H(u2u1) = (1)g(u2)in(0;T)
;
u1(0;x) =u0
1(x); u 2(0;x) =u0
2(x)in
:(4.49)
Here,
H=Z
h0(y) dy
91

92 4.3. Homogenization results for the calcium dynamics in living cells
andA1andA2are the homogenized matrices, given by:
A1
ij=Z
Y1
a1
ij+nX
k=1a1
ik@1j
@yk!
dy;
A2
ij=Z
Y2
a2
ij+nX
k=1a2
ik@2j
@yk!
dy;
wherea1
ij=A1
ij,a2
ij=A2
ijand1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n , are the
weak solutions of the cell problems
8
<
:divy(A1(y)(ry1k+ek)) = 0; y2Y1;
A1(y)(ry1k+ek)
= 0; y2;(4.50)
8
<
:divy(A2(y)(ry2k+ek)) = 0; y2Y2;
A2(y)(ry2k+ek)
= 0; y2:(4.51)
At a macroscopic scale, we obtain a continuous model, a so-called bidomain model , similar to
those arising in the context of the modeling of diusion processes in partially ssured media
(see [31] and [102]) or in the case of the modeling of the electrical activity of the heart (see
[15], [13] and [189]). 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(A0ru0) =f(u0) + (1)g(u0)in(0;T)
;
u0(0;x) =u0
1(x) +u0
2(x)in
:(4.52)
Here, the eective matrix A0is given by:
A0
ij=Z
Y1
a1
ij+nX
k=1a1
ik@1j
@yk!
dy+Z
Y2
a2
ij+nX
k=1a2
ik@2j
@yk!
dy;
in terms of the functions 1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n; weak solutions
of the local problems
8
<
:divy(A1(y)(ry1k+ek)) = 0; y2Y1;
A1(y)(ry1k+ek)
= 0; y2;(4.53)
92

Chapter 4. Mathematical models in biology 93
8
<
:divy(A2(y)(ry2k+ek)) = 0; y2Y2;
A2(y)(ry2k+ek)
= 0; y2:(4.54)
In this case, the exchange at the interface leads to the modication of the limiting diusion
matrix, but the insulation is not enough strong to impose the existence of two dierent limit
concentrations.
Theorem 4.19 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 eective concentration eld u0
satises:8
<
:@u0
@tdiv(A0ru0) =f(u0) + (1)g(u0)in(0;T)
;
u0(0;x) =u0
1(x) +u0
2(x)in
:(4.55)
The eective coecients are given by:
A0;ij=Z
Y1
a1
ij+nX
k=1a1
ik@w1j
@yk!
dy+Z
Y2
a2
ij+nX
k=1a2
ik@w2j
@yk!
dy;
wherew1k2H1
per(Y1)=R; w2k2H1
per(Y2)=R,k= 1;:::;n; are the weak solutions of the cell
problems
8
>><
>>:divy(A1(y)(ryw1k+ek)) = 0; y2Y1;
divy(A2(y)(ryw2k+ek)) = 0; y2Y2;
(A1(y)ryw1k)
= (A2(y)ryw2k)
; y2;
(A1(y)ryw1k)
+h0(y)(w1kw2k) =A1(y)ek
; y2:(4.56)
It is important to notice that the diusion coecients depend now on h0. A similar result
holds true for the case in which his given by (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 eect
was noticed in [6].
Remark 4.20 To simplify the presentation, we address here only the relevant cases
=
1;0;1. For the case
2(1;1), we get, at the limit, the macroscopic problem (4.50), while
for
>1, we obtain a problem similar to (4.49), but without the exchange term H(u2u1)
orjjh(u1;u2), respectively. Finally, for the case
<1, we obtain, at the limit, a standard
composite medium without any barrier resistance. It is worth mentioning that in this case we
getw1k=w2kon, fork= 1;:::;n .
Remark 4.21 The conditions imposed on the nonlinear functions f;g, andhcan be relaxed.
For example, we can consider that fandgare maximal monotone graphs, satisfying suitable
growth conditions (see, e.g., [74]). Also, as in [179], [195] and [223], we can work with more
general functions h.
93

94 4.3. Homogenization results for the calcium dynamics in living cells
94

Chapter 5
Multiscale modeling of composite
media with imperfect interfaces
In the last decades, the study of the eective properties of heterogeneous composite mate-
rials 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 [152]). 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 [165] and [176]). The imperfect con-
tact between the constituents of a composite material can be generated by various causes:
the presence of impurities at the boundaries, the presence of a thin interphase, the inter-
face damage, 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 ap-
propriate macroscopic continuum models, obtained by averaging the rapid oscillations of the
material properties. Besides, such eective models have the advantage of avoiding extensive
numerical computations arising when dealing with the small scale behavior of the system.
The homogenization of a thermal problem in a two-component composite with interfacial
barrier, with jump of the temperature and continuity of the
ux, 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 [96], [177], [104], [95] and [218], to cite just a few of them. 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 completely dierent
macroscopic problems.
Problems like 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 [161], [194], [193], [192], [215] and [216]. The case in which only one
95

96 5.1. Multiscale analysis in thermal diusion problems in composite structures
phase is connected, while the other one is disconnected was considered, for instance, in [96],
[177], and [95]. For similar homogenization problems of parabolic or hyperbolic type, we refer
to [93] and [150]. 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],
[97], [106], [108], [107], [142], [189], and [223].
In this chapter, we shall present some recent homogenization results for diusion problems
in composite media with imperfect imperfect interfaces. We start by brie
y describing the
results obtained in [218] for a thermal diusion problem in a so-called bi-connected structure .
In the same geometry, we shall present some homogenization results contained in [215] and
[216] for diusion 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 dierential 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 [106]) or of a dynamic coupled thermoelasticity problem in
composite media with imperfect interfaces and various geometries (see [108] and [107]). We
end this chapter by presenting some recent results obtained in [47], [48], and [49] for diusion
problems involving jumps both in the solution and in the
ux.
This chapter is based on the papers [47], [48], [49], [215], [216], and [218].
5.1 Multiscale analysis in thermal diusion problems in com-
posite structures
In [218], we have analyzed, using the periodic unfolding method, the eective thermal transfer
in a periodic composite material formed by two constituents, separated by an imperfect
interface. Our results were set in the framework of thermal transfer, but they are true for
more general reaction-diusion processes. We assumed that we have nonlinear sources acting
in each phase and that at the interface between the two constituents the
ux is continuous,
while the temperature eld has a jump. We were interested in describing 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. 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 considered and three dierent limit problems are obtained starting
from the same type of micromodel. The results in [218] constitute a generalization of those
obtained in [102], [190], [215] and [216]. For heat conduction problems in periodic materials
with a dierent geometry, we refer to [95] and [177] and the references therein.
In [218], for simplicity, we dealt only with the stationary case, but the dynamic one can
be treated in a similar manner (see [74] and [190]). Similar problems have been addressed,
using dierent techniques, formal or not, in [26], [27], [165] and [102].
96

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
in Rn, withn3, with a Lipschitz-continuous boundary @
. We assume
that
is formed by two constituents,
"
1and
"
2, representing two materials with dierent
thermal characteristics, separated by an imperfect interface ". We assume that both phases

"
1and
"
2=
n
"
1are connected, but only
"
1reaches the external xed boundary @
.
Here,"represents a small parameter related to the characteristic size of the two constituents.
Our goal in [218] was to describe the eective behavior of the solution ( u"
1;u"
2) of the
following coupled system of equations:
8
>>>><
>>>>:div (A"
1ru"
1) +(u"
1) =fin
"
1;
div (A"
2ru"
2) +(u"
2) =fin
"
2;
A"
1ru"
1
=A"
2ru"
2
on ";
A"
1ru"
1
="
h(u"
1;u"
2) on ";
u"
1= 0 on@
:(5.1)
In (5.1),is the unit outward normal to
"
1,A"=A(x="), whereA2M (1;1;Y) is a
Y-periodic symmetric matrix (as in Section 4.3.1), and f2L2(
). We remark that the
ux is
continuous across the boundary ", but the continuity of temperatures is replaced by a Biot
boundary condition. The functions =(r) and=(r) are supposed to be continuous,
monotonously non-decreasing with respect to rand such that (0) = 0 and (0) = 0. Further,
we suppose that there exist C0 andq, with 0q<n= (n2), such that
j(r)jC(1 +jrjq) (5.2)
and
j(r)jC(1 +jrjq): (5.3)
We also assume, as in Section 4.3, that
h(u"
1;u"
2) =h"
0(x)(u"
2u"
1);
whereh"
0(x) =h0(x=") andh0(y) is aY-periodic, smooth real function with h0(y) >0.
Moreover, let
H=Z
h0(y) d6= 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 [74].
For proving the well posedness of problem (5.1), we refer to [15], [13], [190], [189], [215]
and [216]. 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.
97

98 5.1. Multiscale analysis in thermal diusion problems in composite structures
5.1.2 The main results
We describe now the eective behavior of the solutions of the microscopic model (5.1) for the
above mentioned three 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(
),u22H1(
),bu12L2(
;H1
per(Y1)),bu22L2(
;H1
per(Y2)) such that,
passing to a subsequence, for "!0, we have:
8
>><
>>:T"
1(u"
1)!u1strongly in L2(
;H1(Y1));
T"
1(ru"
1)*ru1+rybu1weakly inL2(
Y1);
T"
2(u"
2)*u 2weakly inL2(
;H1(Y2));
T"
2(ru"
2)*ru2+rybu2weakly inL2(
Y2):(5.4)
The main convergence result in this case is stated in the following theorem, proven in [218].
Theorem 5.1 For
= 1, 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(
), of the following
macroscopic problem:
(
div(A1ru1) +(u1)H(u2u1) =fin
;
div(A2ru2) + (1)(u2) +H(u2u1) = (1)fin
:(5.5)
In (5.5),A1andA2are the homogenized matrices, dened by:
A1
ij=Z
Y1
aij+aik@1j
@yk
dy;
A2
ij=Z
Y2
aij+aik@2j
@yk
dy
and1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n , are the weak solutions of the cell
problems8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
So, at a macroscopic scale, the composite medium, despite of its discrete structure, can be
represented by a continuous model, which is similar to the so-called bidomain model , arising
in the context of diusion in partially ssured media (see [31] and [102]) or in the case of
electrical activity of the heart (see [15], [13] and [189]). The composite medium is conceived
98

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 99
as the 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(
)andu0satises:
div(A0ru0) +(u0) + (1)(u0) =fin
: (5.6)
Here, the eective matrix A0is given by:
A0
ij=Z
Y1
aij+aik@1j
@yk
dy+Z
Y2
aij+aik@2j
@yk
dy;
in terms of the functions 1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n; weak solutions
of the cell problems
8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
In this case, the insulation provided by the interface is sucient to modify the limiting
diusion matrix, but it is not strong enough to force the existence of two dierent limit
phases.
Theorem 5.3 For the case
=1, i.e. for weak contact resistance, we also get, at the
limit,u1=u2=u02H1
0(
)and, in this case, the eective temperature eld u0satises:
div(A0ru0) +(u0) + (1)(u0) =fin
: (5.7)
The macroscopic coecients are given by:
A0;ij=Z
Y1
aij+aik@w1j
@yk
dy+Z
Y2
aij+aik@w2j
@yk
dy;
wherew1k2H1
per(Y1)=R; w2k2H1
per(Y2)=R,k= 1;:::;n; are the weak solutions of the cell
problems8
>><
>>:ry
(A1(y)ryw1k) =ryA1(y)ek; y2Y1;
ry
(A2(y)ryw2k) =ryA2(y)ek; y2Y2;
(A1(y)ryw1k)
= (A2(y)ryw2k)
 y2;
(A1(y)ryw1k)
+h0(y)(w1kw2k) =A1(y)ek
 y2:
In this case, as expected, the eective coecients depend on h0.
99

100 5.2. Diusion problems with dynamical boundary conditions
5.2 Diusion problems with dynamical boundary conditions
Recent computational and theoretical studies investigating the macroscopic behavior of com-
posite 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 do-
mains. In this section, we shall present some homogenization results obtained by using the
oscillating test function method of L. Tartar in [215] and generalized later, via the periodic
unfolding method, in [216]. The aim of these papers was to analyze the asymptotic behav-
ior of the solution of a nonlinear problem arising in the modeling of thermal diusion in a
two-component composite material. We assume that we have nonlinear sources and that at
the interface between the two materials the
ux is continuous and depends in a dynamical
nonlinear way on the jump of the temperature eld. 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"
1ru"
1) +(u"
1) =fin
"
1(0;T);
div (A"
2ru"
2) =f; in
"
2(0;T);
A"
1ru"
1
=A"
2ru"
2
on "(0;T);
A"
1ru"
1
+a"@
@t(u"
1u"
2) ="g(u"
2u"
1) on "(0;T);
u"
1= 0 on@
(0;T);
u"
1(0;x)u"
2(0;x) =c0(x);on ":(5.8)
Here,f2L2(0;T;L2(
)),c02H1
0(
) anda>0. The function is continuous, monotonously
non-decreasing and such that (0) = 0 and the function gis continuously dierentiable,
monotonously non-decreasing and with g(0) = 0. We shall assume that there exist a positive
constantCand an exponent q, with 0q<n= (n2), such that
j(v)jC(1 +jvjq);dg
dvC(1 +jvjq): (5.9)
As particular examples of such functions we can consider, for instance, the following important
practical ones: (v) =v
1 +
v; ;
> 0 (Langmuir kinetics), (v) =jvjp1v;0<p< 1
(Freundlich kinetics), g(v) =avorg(v) =av3, witha>0.
Well posedness of problem (5.8) in suitable function spaces and proper energy estimates
have been obtained in [15], [45] and [189]. Using Tartar's method of oscillating test functions
(see [207]), coupled with monotonicity methods and results from the theory of semilinear
problems (see [45] and [74]), we can prove that the asymptotic behavior of the solution of
problem (5.8) is governed by a new nonlinear system, similar to Barenblatt's model (see [31]
and [102]), with additional terms capturing the eect of the interfacial barrier, of the dynam-
ical boundary condition and of the nonlinear sources. Our results constitute a generalization
of those obtained in [31] and [102], by considering nonlinear sources, nonlinear dynamical
transmission conditions and dierent techniques in the proofs. Similar problems have been
considered, using dierent techniques, in [13] and [189], for studying electrical conduction in
biological tissues.
100

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 101
Using well-known extension results (see, for instance, [70], [61] and [189]) and suitable test
functions, we can take the limit in the variational formulation of problem (5.8) and obtain
the eective behavior of the solution of our microscopic model. The main convergence result
can be formulated as follows:
Theorem 5.4 (Theorem 2.1 in [215]) One can construct two extensions P"u"
1andP"u"
2
of the solutions u"
1andu"
2of problem (4.8) such that P"u"
1* u 1; P"u"
2* v , weakly in
L2(0;T;H1
0(
)), where
8
>>>>>>><
>>>>>>>:ajj@
@t(u1u2)div(A1ru1) +(u1)jjg(u2u1) =
=fin
(0;T);
ajj@
@t(u2u1)div(A2ru2)+jjg(u2u1) =
= (1)fin
(0;T);
u1(0;x)u2(0;x) =c0(x)on
:(5.10)
Here,A1andA2are the homogenized matrices, dened by:
A1
ij=Z
Y1
aij+aik@1j
@yk
dy;
A2
ij=Z
Y2
aij+aik@2j
@yk
dy;
in terms of the functions 1k2H1
per(Y1)=R; 2k2H1
per(Y2)=R,k= 1;:::;n , weak solutions
of the cell problems
8
<
:ry
((A1(y)ry1k) =ryA1(y)ek; y2Y1;
(A1(y)ry1k)
=A1(y)ek
; y2;
8
<
:ry
((A2(y)ry2k) =ryA2(y)ek; y2Y2;
(A2(y)ry2k)
=A2(y)ek
; y2:
Thus, in the limit, we obtain a system similar to the so-called Barenblatt model . Also, such
a system is similar to the bidomain model appearing in the context of electrical activity of
the heart. At a macroscopic level, despite of the discrete cellular structure, the composite
material can be represented by a continuous model, describing the averaged properties of the
complex structured composite material. We point out that the above model is a degenerate
parabolic system, as the time derivatives involve the unknown vu, instead of the unknowns
uandvoccurring in the second-order conduction term.
These results were generalized in [216] where we used the periodic unfolding method and
we considered more general nonlinearities (x;u") andg(x;v"u")).
101

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, through homogenization techniques, the macroscopic thermal
transfer 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 the

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 the
uxes arise in
linear elasticity, in the theory of semiconductors, in the study of photovoltaic systems or
of problems in media with cracks (see, e.g., [26], [38], [46], [134], and [157]). In general,
only formal methods of averaging were used in the literature to deal with such imperfect
transmission problems and obtaining rigorous homogenization results is still a dicult task.
Nevertheless, we can mention here the rigorous results obtained in [145], [116] or [111].
The essential novelty brought by us in [47] resides in assuming, apart from the disconti-
nuity 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 dierent homogenized problems (stated in Theorem 5.10 and
Theorem 5.15).
5.3.1 Setting of the problem
Let us consider a composite material occupying an open bounded set
in Rn, withn2, with
a Lipschitz-continuous boundary @
. We assume that
is made up of two parts denoted
"
1
and
"
2, separated by an imperfect interface ". We suppose that the phase
"
1is connected
and reaches the exterior boundary @
and that the second phase,
"
2, is disconnected, being
the union of domains of size ", periodically distributed in
with periodicity ". More precisely,
we assume that Y1andY2are two non-empty disjoint connected open subsets of the reference
cellY= (0;1)ninRn, such that Y2YandY=Y1[Y2. We also suppose that = @Y2
is Lipschitz continuous and that Y2is connected. For each k2ZN, we denote Yk=k+Y
andYk
=k+Y, for= 1;2. For each ", we dene, Z"=n
k2ZN:"Yk
2
o
and we
set
"
2=S
k2Z"
"Yk
2

and
"
1=
n
"
2. The boundary of
"
2is denoted by "andis the
unit outward normal to
"
1.
We aim at describing the asymptotic behavior, as "!0, of the solution u"= (u"
1;u"
2) of
the following problem:
8
>>>>>>><
>>>>>>>:div (A"ru"
1) =fin
"
1;
div (A"ru"
2) =fin
"
2;
A"ru"
1
=h"
"(u"
1u"
2)G"on ";
A"ru"
2
=h"
"(u"
1u"
2) on ";
u"
1= 0 on@
:(5.11)
Remark 5.5 We point out that
A"ru"
2
A"ru"
1
=G"; (5.12)
102

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(
) is given. If his aY{periodic function in L1() such that there
existsh02Rwith 0<h 0<h(y) a.e. on , then we take
h"(x) =hx
"
a.e. on ":
For aY-periodic symmetric matrix A2M (1;1;Y) (see Section 4.3.1), we set
A"(x) =Ax
"
a.e. in
:
Letgbe aY-periodic function that belongs to L2(). We dene
g"(x) =gx
"
a.e. on ":
As already mentioned, for the given function G"in (5:11) we consider the following relevant
forms (see, also, [55]):
Case 1 :G"="gx
"
, ifM(g)6= 0.
Case 2 :G"=gx
"
, ifM(g) = 0.
Here,M(g) =1
jjZ
g(y) dydenotes the mean value of the function gon .
For writing the variational formulation of problem (5.11), we consider, for every positive
"<1, the Hilbert space
H"=V"H1(
"
2); (5.13)
where the space V"=
v2H1(
"
1); v= 0 on@

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

"
1ru1rv1dx+Z

"
2ru2rv2dx+1
"Z
"(u1u2)(v1v2) dx (5.14)
whereu= (u1;u2) andv= (v1;v2) belong to H". The norm induced by the scalar product
(5.14) is given by
kvk2
H"=krv1k2
L2(
"
1)+krv2k2
L2(
"
2)+1
"kv1v2k2
L2("): (5.15)
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) =Z

"
1A"ru1rv1dx+Z

"
2A"ru2rv2dx+Z
"h"
"(u1u2)(v1v2) dx
103

104 5.3. Homogenization of a thermal problem with
ux jump
and
l(v) =Z

"
1fv1dx+Z

"
2fv2dx+Z
"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
kru"
1kL2(
"
1)C;kru"
2kL2(
"
2)C (5.17)
and
ku"
1u"
2kL2(")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
"
1,
"
2and, respectively, ", into functions dened on the
xed domains
Y1,
Y2and
, respectively.
From [95], we know that if u"= (u"
1;u"
2) is a sequence in H", then
1
"jYjZ

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

h(y) (T"
1(u"
1)T"
2(u"
2))T"
(') dxdy:
Moreover, the following general compactness results were obtained in [95].
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
kT"
1(ru"
1)kL2(
Y1)C;
kT"
2(ru"
2)kL2(
Y2)C;
kT"
2(u"
1)T"
1(u"
2)kL2(
)C":
Theorem 5.8 Letu"= (u"
1;u"
2)be a bounded sequence in H". Then, up to a subsequence,
still denoted by ", there exist u12H1
0(
),u22L2(
),bu12L2

;H1
per(Y1)

andbu22
104

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 105
L2

;H1(Y2)

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

;H1(Y1)

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

;H1(Y1)

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

;H1(Y2)

:
5.3.2 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, we shall denote their common value by
u. We notice that ubelongs toH1
0(
).
From the priori estimates (5.17)-(5.18) and the above mentioned general compactness
results, it follows that there exist u2H1
0(
),bu12L2(
;H1
per(Y1)),bu22L2(
;H1(Y2)) such
thatM(bu1) = 0 and up to a subsequence, for "!0, we have:
T"
1(u"
1)!ustrongly in L2(
;H1(Y1));
T"
1(ru"
1)*ru+rybu1weakly inL2(
Y1);
T"
2(u"
2)*u weakly inL2(
;H1(Y2));
T"
2(ru"
2)*rybu2weakly inL2(
Y2);
eu"
*jYj
jYjuweakly inL2(
);  = 1;2:(5.19)
Besides, one has
T"
1(u"
1)T"
2(u"
2)
"*bu1u2weakly inL2(
); (5.20)
where u22L2(
;H1(Y2)) is dened by
u2=bu2yru;
105

106 5.3. Homogenization of a thermal problem with
ux jump
for some2L2(
).
LetWper(Y1) =fv2H1
per(Y1)jM (v) = 0g. We consider the space
V=H1
0(
)L2(
;Wper(Y1))L2

;H1(Y2)

;
endowed with the norm
kVk2
V=krv+rybv1k2
L2(
Y1)+krv+ryv2k2
L2(
Y2)+kbv1v2k2
L2(
);
for allV= (v;bv1;v2)2V.
In order to pass to the limit, we have to treat separately the above mentioned two cases
for the function G".
Case 1 :G"="gx
"
, ifM(g)6= 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)2V of the
following unfolded limit problem:
1
jYjZ

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

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

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

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

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

F(x)'(x) dx;
with
F(x) =f(x) +jj
jYjM(g):
Corollary 5.10 (Corollary 1 in [47]) The function u2H1
0(
)dened by (5.19) is the unique
solution of the following homogenized equation:
div(Ahomru) =f+jj
jYjM(g)in
; (5.22)
whereAhomis the homogenized matrix whose entries are given, for i;j= 1;:::;n , by
Ahom
ij=1
jYjZ
Y1
aijnX
k=1aik@j
1
@yk!
dy+1
jYjZ
Y2
aijnX
k=1aik@j
2
@yk!
dy; (5.23)
106

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 107
in terms of j
12H1
per(Y1)andj
22H1(Y2),j= 1;:::;n , the weak solutions of the following
cell problems:
8
>>>>><
>>>>>:divy(A(y)(ryj
1+ej)) = 0; y2Y1;
divy(A(y)(ryj
2+ej)) = 0; y2Y2;
(A(y)ryj
1)
= (A(y)ryj
2)
; y2;
(A(y)(ryj
1+ej))
+h(y)(j
1j
2) = 0; y2:
M(j
1) = 0:(5.24)
whereis the exterior unit normal to Y1.
Remark 5.11 We notice that the homogenized matrix Ahomdepends on the function h.
Thus, the eect 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 coecients.
Case 2 :G"(x) =gx
"
, 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)2V of the
following unfolded limit problem:
1
jYjZ

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

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

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

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

g(y)1(x;y) dxdy; (5.25)
for all'2H1
0(
),12L2(
;H1
per(Y1)),22L2(
;H1(Y2)).
Let us note that the term1
jYjZ

g(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 that, apart from the standard solutions j
1andj
2
of 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(
) dened in (5.19) is the
unique solution of the homogenized equation
div(Ahomru) =fin
; (5.26)
107

108 5.3. Homogenization of a thermal problem with
ux jump
whereAhomis the homogenized matrix whose entries are given in (5.26). Besides, we have
bu1(x;y) =NX
j=1@u
@xj(x)j
1(y) +1(y);
u2(x;y) =NX
j=1@u
@xj(x)j
2(y) +2(y);
wherej
1andj
2are dened by (5.27) and the function (1;2)is the unique solution of the
cell problem8
>>>><
>>>>:divy(A(y)r1) = 0 inY1;
divy(A(y)r2) = 0 inY2;
A(y)r1
=h(y)(12)g(y)on;
A(y)r2
=h(y)(12)on;
M(1) = 0:
Proof. By choosing '= 0 in (5.25), we obtain:
1
jYjZ

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

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

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

g(y)1(x;y) dxdy: (5.27)
By choosing now suitable test functions  1and  2in (5.27), we obtain
divy(A(y)rybu1) = divy(A(y)ru) a.e. in
Y1; (5.28)
divy(A(y)ryu2) = divy(A(y)ru) a.e. in
Y2; (5.29)
A(y)(ru+ryu2)
=h(y)(bu1u2) a.e. on
; (5.30)
A(y)(ru+rybu1)
=h(y)(bu1u2)g(y) a.e. on
: (5.31)
We remark that we have a discontinuity type condition:
A(y)(ru+ryu2)
A(y)(ru+rybu1)
=g(y) a.e. on
: (5.32)
As in [47], we search bu1and u2in the following nonstandard form:
bu1(x;y) =nX
j=1@u
@xj(x)j
1(y) +1(y); (5.33)
u2(x;y) =nX
j=1@u
@xj(x)j
2(y) +2(y); (5.34)
108

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 109
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)r1) = 0 inY1;
divy(A(y)r2) = 0 inY2;
A(y)r1
=h(y)(12)g(y) on ;
A(y)r2
=h(y)(12) on ;
M(1) = 0:(5.35)
Since
A(y)r2
A(y)r1
=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.
By the Lax-Milgram theorem, this problem has a unique solution in the space
H=Wper(Y1)H1(Y2);
endowed with the scalar product
(;)H= (r1;r1)L2(Y1)+ (r2;r2)L2(Y2)+ (12;12)L2():
By choosing now  1=  2= 0 in (5.25), we obtain:
1
jYjZ

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

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

f(x)'(x) dx: (5.37)
Integrating it by parts with respect to x, we have:
divx1
jYjZ
Y1A(y)(ru+rybu1) dy+1
jYjZ
Y2A(y)(ru+ryu2) dy
=f(x) in
:
By using here the particular form (5.33) and (5.34) of the functions bu1and u2and the
denition of the matrix Ahom, we get:
divx(Ahomru) =f+ divx1
jYjZ
Y1A(y)r1(y)dy+1
jYjZ
Y2A(y)r2(y)dy
in
;
which leads immediately to 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.
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;
)such that
T"
(A")!Astrongly in L1(
Y);
for some matrix A=A(x;y)inM(1;1;
Y). The heterogeneity of the medium given
by such a matrix generates dierent eects in the limit problems. More precisely, in both
109

110 5.4. Other homogenization problems in media with imperfect interfaces
cases, due to the fact that the correctors j
depend also on x, the homogenized matrix Ahom
x
depends on x. Moreover, an interesting eect 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
xru) =
f+divx1
jYjZ
Y1A(x;y)r1(x;y)dy+1
jYjZ
Y2A(x;y)r2(x;y)dy
in
:
A similar eect 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 diusion problem in a highly heterogeneous medium
formed by two constituents,
"
1and
"
2, separated by an imperfect interface was addressed
in [48], as well. The main characteristics of the medium are the discontinuity of the thermal
conductivity over the domain as we go from one constituent to another and the presence of
an imperfect interface between the two constituents, where both the temperature and the

ux exhibit jumps. The order of magnitude of the thermal conductivity of the material
occupying the domain
"
2is"2, while the conductivity of the material occupying the domain

"
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. The limit problem,
obtained via the periodic unfolding method, captures the in
uence of the jumps in the limit
temperature eld, in an additional source term, and in the correctors, too.
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 mention here brie
y some results we obtained recently for such problems.
For instance, our goal in [106] 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
110

Chapter 5. Multiscale modeling of composite media with imperfect interfaces 111
such a model, called a spring type interface model in the literature, the imperfect interface
conditions are equivalent to the eect produced by a very thin and soft (i.e. very compliant)
elastic interphase between the two media. Another interesting imperfect interface condition
arises in the case of a thin and sti interphase, characterized by a jump of the traction
vector across the interface between the two media (see [38] and [134]). 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
[186]). For more details concerning the corresponding mechanical models, we refer to [38],
[134], [165], [170], and [186].
In [106], 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 dierent important cases. More precisely, we obtain, at the macroscale,
one or two equations, with dierent stiness tensors: (i) if the jump is of order "1, we
obtain only one equation at the macroscale, with the stiness tensor depending on the jump
coecient; (ii) if the jump is of order ", we get a system of two coupled equations with
classical stiness tensors; (iii) 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 that may
arise from the presence of impurities at the boundaries between the phases or from chemical
reactions between the constituents. An example of such a material is represented by a concrete
structure. Also, our techniques can be applied for dealing with other geomaterials, such as
mortar, soils or rocks. Similar problems have been considered, using dierent techniques,
formal or not, in [103], [160], [165], and [175]. 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 [118] and [205]. The dynamic case can be addressed in
a similar manner.
Homogenization results for a dynamic coupled thermoelasticity problem in a periodic
composite material made up of two connected constituents with imperfect contact at their
interface were obtained in [107]. The homogenized problem, derived via the periodic un-
folding method, comprises new coupling terms involving the macroscopic displacement and
temperature elds, generated by the imperfect bonding at the interface between the two
phases of the composite. Related problems have been studied, with various methods, over
the last years. For a nice presentation of the classical theory of thermoelasticity, the reader
is referred to [144]. Also, for some interesting thermoelasticity models, we refer to [10], [102],
[103] and [113]. In [108], a similar model was considered, but in a dierent geometry and with
dierent scalings of the temperature-displacement tensors of the two constituents, leading to
dierent homogenized results. More precisely, the domain
was considered to be the union
of a connected part
"
1and a disconnected one
"
2and the temperature-displacement tensor
111

112 5.4. Other homogenization problems in media with imperfect interfaces
was supposed to be of order of unity in the connected part of the medium and, respectively,
of order"in the disconnected one. As a consequence, the macroscopic elasticity tensor, the
temperature-displacement tensor and the thermic-conductivity tensor corresponding to the
disconnected part canceled at the limit. In our case, we keep these tensors in the macroscopic
system and, in addition, we get a dierent specic heat coecient in the equation for the
macroscopic temperature eld coming from the disconnected part. Moreover, let us mention
the presence of new coupling terms in the macroscopic system and the dierent functional
setting compared to the one used in [107].
In [54], we have been concerned with the derivation of macroscopic models for some elas-
ticity 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. Several nonlin-
ear conditions on the boundary of the rigid inclusions were considered. More precisely, we
treated the case when a nonlinear Robin condition is imposed and, respectively, the case
when unilateral contact with given friction is taken into consideration. For the Robin prob-
lem, we extended, via the periodic unfolding method, some of the results contained in [119]
and [146], 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 diculties of this problem came from the fact that the unilateral condi-
tion 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 dierent
from the one addressed in [60]. In particular, for the frictionless contact case, we regained a
result obtained, under more restrictive assumptions, in [146]. This frictionless problem was
also addressed in [147], by the two-scale convergence method, for more general geometric
structures of the inclusions on which the Signorini conditions act.
112

Part II
Career Evolution and Development
Plans
113

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,
115

116 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 collab-
orations with 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), 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-diusion processes in porous media; 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 also at graduate level: Real Anal-
ysis, Complex Analysis, Ordinary Dierential Equations, Complements of Mathematics, etc.
I published ten books or chapters in books.
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 60 papers in peer-reviewed journals;
I published more than 20 papers in proceedings of national and international
conferences;
I am single author for more than 50 papers;
I published more than 70 papers after completing my PhD thesis;
I gave more than 40 talks at international conferences;
I gave … invited talks at international conferences;
I gave …. invited seminars abroad;
I obtained four post-doctoral fellowships;
I was director of … research projects and member of ….other … 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 30 international journals
(SIAM Journal of Applied Mathematics, Networks and Heterogeneous Media, etc).
116

Chapter 6. Scientic and academic background and research perspectives 117
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
recruitment 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 analyz-
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 [121] 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
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,
117

118 6.2. Further research directions
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 eects produced in the skin by the electrical pulses in order to design useful electrop-
ermeabilization protocols. The problem is complex, involving a very complicated geometry
and the nonlinear coupling of a diusion equation for the drug molecules, of a heat equation,
and of an equation for the electric potential. We shall make simplifying assumptions in order
to capture the essential features of the model, while making it tractable. Modeling the skin
as a a composite medium, our goal is to analyze the eective behavior, as the period of the
microstructure tends to zero, of the solutions of this coupled system of partial dierential
equations. We shall analytically investigate the eect of various parameters on the eective
temperature eld in the tissue exposed to permeabilizing electric pulses. The results can be
used for designing skin electropermeabilization protocols for cancer treatment planning.
I.3. Mathematical models for carcinogenesis in living cells. I shall be concerned
with the carcinogenic eects produced in the human cells by Benzo-[a]-pyrene molecules (BP),
which are reactive toxic molecules found in coal tar, cigarette smoke, charbroiled food, etc.
I plan to generalize the results obtained in [128] and [223]. The microscopic mathematical
model, including reaction-diusion processes and binding and cleaning mechanisms, will be
homogenized in order to reduce its complexity and to make it numerically treatable and not
so computationally expensive. We shall consider that BP molecules enter in the cytosol inside
of a human cell. There, they diuse freely, but they cannot enter in the nucleus. Also, they
bind to the surface of the endoplasmic reticulum (ER), where chemical reactions take place,
BP molecules being chemically activated to Benzo-[a]-pyrene-7,8-diol-9,10-epoxide molecules
(DE). These molecules can unbind from the surface of the ER and they can diuse again in
the cytosol, entering in the nucleus. Natural cleaning mechanisms occurring in the cytosol are
taken into account, too. For describing the binding-unbinding process at the surface of the
ER, we shall consider various nonlinear functions, with various scalings, leading to dierent
homogenized models. We shall deal with the case of general nonlinear (even discontinuous)
isotherms, similar to those used in [79], [159], and [195], and of multiple metabolisms BP !
DE. I shall also generalize a carcinogenesis model, introduced in [128], involving free receptors
on the surface of the ER (see, also, Section 4.2.3).
II.1. Homogenization of reaction-diusion 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 diusion problem in a highly
118

Chapter 6. Scientic and academic background and research perspectives 119
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-diusion 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 diusion in composites, in the theory of semiconductors,
in linear elasticity or in reaction-diusion 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 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, we
shall address the problem of nding the eective parameters for electromagnetic periodic
composite materials in the quasi-static case. The developed strategy will allow us to deal
with quite general microscopic geometries and can be applied to other heterogeneous materials
in which the scale of the period is much smaller than the wavelength of the electromagnetic
eld. We shall generalize some of the results obtained in [188].
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 perfo-
rated domains with critical inclusions (see [151] and [125], [123], and [124]). 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 eective properties.
II.4. Multiscale modeling of thermoelastic microstructured materials. The
prediction of the eective 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
of thermoelastic composites with imperfect interfaces has attracted a lot of interest in the
last years, due to the great importance of such heterogeneous materials in many engineering
applications. For instance, there are important applications of the interphase eects on the
thermoelastic response of polymer nanocomposite materials. We shall try to generalize our
results in [106], [107], and [108] to include more general interface eects. The case in which
the strain-stress law is viscoelastic and the case in which we consider thermal eects in the
history of a composite material will be treated, as well.
119

120 6.2. Further research directions
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 re-
search activity in the eld of homogenization is interdisciplinary in its nature and in the last
years I tried to publish my results in more application-oriented high quality journals, with
a broad audience, including not only mathematicians, but also physicists, engineers, and
scientists from various applied elds, such as biology or geology. Also, 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.
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.
120

List of publications
This habilitation thesis is based on the following publications:
[1]C. Conca, J. I. D az, C. Timofte ,Eective 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 Dierential 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 diusion 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 Analy
sis,80(1-2), 45-56, 2012.
[12] C. Timofte ,Multiscale analysis of diusion processes in composite media , Comp. Math.
Appl., 66(9), 1573-1580, 2013.
121

122 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.
[16] H. I. Ene, C. Timofte ,Microstructure models for composites with imperfect interface
via the periodic unfolding method , Asymptot. Anal., 89(1-2), 111-122, 2014.
[17] H. I. Ene, C. Timofte , I. T  ent ea, Homogenization of a thermoelasticity model for a
composite with imperfect interface , Bull. Math. Soc. Sci. Math. Roumanie, 58(106), 2,
147-160, 2015.
[18] H. I. Ene, C. Timofte ,Homogenization results for a dynamic coupled thermoelasticity
problem , Romanian Reports in Physics, 68, 979-989, 2016.
[19] C. Timofte ,Homogenization results for the calcium dynamics in living cells , Math.
Comput. Simulat., in press, 2016, doi:10.1016/j.matcom.2015.06.01 2015.
[20] C. Timofte ,Multiscale analysis of a carcinogenesis model , Math.Comput. Simulat., in
press, 2016, DOI: 10.1016/j.matcom.2016.06.008.
[21]R. Bunoiu, C. Timofte ,Homogenization of a thermal problem with
ux jump , Networks
and Heterogeneous Media, 11(4), 545{562, 2016.
[22]R. Bunoiu, C. Timofte ,On the homogenization of a two-conductivity problem with
ux
jump , to appear in Communication in Mathematical Sciences, 2016.
[23] R. Bunoiu, C. Timofte ,On the homogenization of a diusion problem with
ux jump ,
in preparation, 2016.
[24]A. Capatina, C. Timofte ,Homogenization results for micro-contact elasticity problems ,
Journal of Mathematical Analysis and Applications, 441(1), 462-474, 2016.
[25] C. Timofte ,Homogenization of the Stokes-Poisson-Nernst-Planck system via the peri-
odic unfolding method , in preparation, 2016.
122

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