Habilitation New Vers 3 [610099]
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 my … development
as a mathematician …the evolution of my academic career.
After completing my Ph.D., I benefited, 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
applied mathematics: Professor Enrique Zuazua, Professor Giuseppe Buttazzo, and Profes-
sor Carlos Conca. I was really impressed by their remarkable ability to connect different
fields of research and I want to express my deep gratitude to all of them, for their support
and guidance and for the willingness to share their 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 field of homogenization theory: Pro-
fessor J. I. D´ ıaz, Professor I. S. Pop, Professor M. Radu-Neuss, Dr. Renata Bunoiu, Professor
G. Sandrakov, Professor G. Savar´ e, D. G´ omez-Castro, Professor M. E. P´ erez, Dr. D. G´ omez.
I want to express my gratitude to all of them, for their hospitality, kindness, and for sharing
with me their love for mathematics.
The work presented here represents a collective effort, the fruit of many encounters I had
over the years with many persons and I am fully conscious about their importance at many
steps in my career. It is impossible to me to thank now all the people that I met in this
scientific journey. Therefore, I shall mention here only the co-authors of my papers on which
this thesis is based on: Professor C. Conca, Professor F. Murat, Professor J.I. D´ ıaz, Professor
A. Li˜ n´ an, Professor H.I. Ene, Dr. A. Capatina, Dr. I. T ¸ent ¸ea, Dr. R. Bunoiu. Working
together was important for my development as a mathematician.
During the last years, I have benefited 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
have always created in our faculty.
3
4 Acknowledgments
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 field of homogenization theory, which represent the core of my
scientific work done during the last fifteen years.
The thesis, written in English, starts by a short summary in Romanian and a brief
overview of the field of homogenization and then summarizes some of my research works
in this field, 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-five 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 Scientific Achievements 13
1 Introduction 15
2 Homogenization of reactive
ows in porous media 19
2.1 Upscaling in stationary reactive flows in periodic porous media . . . . . . . . 20
2.1.1 The model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 The case of a smooth function g. The macroscopic model . . . . . . . 22
2.1.3 The case of a non-smooth function g. The macroscopic model . . . . . 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 coefficients . . . . . . . . . . . 43
3 Homogenization results for unilateral problems 49
3.1 Homogenization results for Signorini’s type problems . . . . . . . . . . . . . . 50
3.1.1 The microscopic problem . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.2 The macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Homogenization results for elliptic problems in perforated domains with mixed-
type boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 Setting of the microscopic problem . . . . . . . . . . . . . . . . . . . . 55
3.2.2 The limit problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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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 . . . . . . . . . . . . . . . . . . . . . . . 92
5 Multiscale modeling of composite media with imperfect interfaces 95
5.1 Multiscale analysis in thermal diffusion problems in composite structures . . 96
5.1.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Diffusion problems with dynamical boundary conditions . . . . . . . . . . . . 100
5.3 Homogenization of a thermal problem with flux jump . . . . . . . . . . . . . 102
5.3.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.2 The macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Other homogenization problems in composite media with imperfect interfaces 113
II Career Evolution and Development Plans 117
6 Scienti c and academic background and research perspectives 119
6.1 Scientific and academic background . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Further research directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 Future plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …???..
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …???…
8
Rezumat
Lucrarea de fat ¸˘ a, preg˘ atit˘ a pentru obt ¸inerea atestatului de abilitare, cuprinde o select ¸ie a
rezultatelor ¸ stiint ¸ifice pe care le-am obt ¸inut ˆ ın domeniul teoriei omogeniz˘ arii dup˘ a dobˆ andirea
titlului de doctor ˆ ın Matematic˘ a. Principala motivat ¸ie din spatele acestui demers o constituie
dorint ¸a de a descrie succint stadiul actual al cunoa¸ sterii ˆ ın domeniul teoriei omogeniz˘ arii, de
a oferi o imagine de ansamblu asupra contribut ¸iilor mele ˆ ın acest vast domeniu ¸ si de a discuta
cˆ ateva probleme deschise ¸ si cˆ ateva posibile perspective de dezvoltare a viitoarei mele cariere
¸ stiint ¸ifice ¸ si academice.
Teza se bazeaz˘ a pe cˆ ateva dintre contribut ¸iile mele originale la aplicarea teoriei omoge-
niz˘ arii, contribut ¸ii cont ¸inuteˆ ın dou˘ azeci ¸ si cinci de articole publicate sau trimse spre publicare
ˆ ın reviste internat ¸ionale cu a larg˘ a audient ¸˘ a, incluzˆ and nu doar matematicieni, ci ¸ si fizicieni,
ingineri ¸ si cercet˘ atori din diferite domenii aplicative. Multe dintre rezultatele cont ¸inute ˆ ın
publicat ¸iile pe care le-am selectat pentru a sprijini aceast˘ a aplicat ¸ie sunt strˆ ans legate sau
motivate de probleme de interes practic. Voi ˆ ıncerca s˘ a ofer o imagine de ansamblu, de sine-
st˘ at˘ atoare, asupra contribut ¸iilor mele ¸ si, acolo unde va fi necesar, s˘ a dau mai multe detalii
care nu sunt prezente ˆ ın lucr˘ arile corespunz˘ atoare publicate, facˆ and astfel ca principalele mele
rezultatele s˘ a fie accesibile unui public mai larg, cu solide cuno¸ stint ¸e generale de matematic˘ a,
dar nu neap˘ arat expert ¸i ˆ ın domeniul specific al teoriei omogeniz˘ arii. Teza prezint˘ a cˆ ateva
dintre cele mai relevante rezultate pe le-am obt ¸inut pe parcursul ultimilor cincisprezece ani
de cercetare efectuat˘ a, singur˘ a sau ˆ ın colaborare, ˆ ın patru arii majore: analiza de multiscar˘ a
a proceselor de react ¸ie-difuzie ˆ ın 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 ˆ ın biologie ¸ si ˆ ın inginerie.
ˆIn afar˘ a de dou˘ a scurte rezumate ˆ ın limba romˆ an˘ a ¸ si ˆ ın limba englez˘ a, teza cuprinde dou˘ a
p˘ art ¸i ¸ si o bibliografie cuprinz˘ atoare. Prima parte, structurat˘ a ˆ ın cinci capitole, este dedicat˘ a
prezent˘ arii principale mele realiz˘ ari ¸ stiint ¸ifice obt ¸inute dup˘ a finalizarea studiilor doctorale.
Dup˘ a un capitol introductiv ˆ ın care este prezentat stadiul actual al cercet˘ arii ˆ ın domeniul
teoriei omogeniz˘ arii ¸ si care ofer˘ a cadrul general ¸ si o motivat ¸ie pentru activitatea mea de
cercetare post-doctoral˘ a ˆ ın acest domeniu, al doilea capitol cuprinde … rezum˘ a principalele
mele contribut ¸ii referitoare la omogenizarea mi¸ sc˘ arilor reactive ˆ ın medii poroase. Mai precis,
sunt prezentate unele rezultate originale de omogenizare pentru curgerile reactive neliniare
stat ¸ionare ˆ ın medii poroase ¸ si, de asemenea, sunt prezentate rezultate privind fenomenele de
adsorbt ¸ie neliniar˘ a ˆ ın medii poroase. Capitolul se bazeaz˘ a pe lucr˘ arile [75], [74], [213] ¸ si [211].
9
10 Rezumat
Al treilea capitol, bazat pe lucr˘ arile [77], [79], [210], [52] ¸ si [55], este dedicat omogeniz˘ arii
unor probleme unilateraleˆ ın 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 ˆ ın 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 ionicˆ ın medii periodiceˆ ıncarcate electric, pentru studiul carcinogenezeiˆ ın celulele
vii sau pentru analiza dinamicii calciului ˆ ın celulele biologice. Rezultatele prezentate ˆ ın acest
capitol sunt cuprinse ˆ ın 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 ˆ ın
domeniul transferului de c˘ aldur˘ a ˆ ın materiale compozite cu interfet ¸e imperfecte ¸ si se bazeaz˘ a
pe lucr˘ arile [48], [49], [50], [215], [216] ¸ si [218]. Definit ¸iile not ¸iunilor de baz˘ a din teoria
omogeniz˘ arii ¸ si rezultatele generale din analiza funct ¸ional˘ a care vor fi folosite pe parcursul
acestei lucr˘ ari pot fi g˘ asite ˆ ın [30], [38], [46], [57], [63], [137], [149], [156], [162], [163], [207].
A doua parte a acestei teze prezint˘ a cˆ ateva planuri de dezvoltare ¸ si de evolut ¸ie ˆ ın carier˘ a.
Dup˘ a o scurt˘ a trecere ˆ ın revist˘ a a parcursului meu ¸ stiint ¸ific ¸ si academic de pˆ ana acum, sunt
prezentate direct ¸iile viitoare de cercetare ¸ si cˆ ateva planuri de dezvoltare pe termen scurt,
mediu ¸ si lung a carierei mele ¸ stiint ¸ifice ¸ si academice.
Lucrarea se ˆ ıncheie cu o bibliografie cuprinz˘ atoare, menit˘ a s˘ a ilustreze stadiul actual al
cunoa¸ sterii ˆ ın acest domeniu vast al teoriei omogeniz˘ arii ¸ si al aplicat ¸iilor sale.
Rezultatele incluse ˆ ın aceast˘ a tez˘ a au fost obt ¸inute singur˘ a sau ˆ ın strˆ ans˘ a colaborare cu
mai multe institut ¸ii academice ¸ si de cercetare din Romˆ 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´ ıaz, Profesor dr. A. Li˜ n´ an, Profesor dr. H.l. Ene, Dr. A. C˘ ap˘ at ¸ ˆ ın˘ 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 ˆ ın aceast˘ a tez˘ a pot fi sintetizate astfel:
•efectuarea unui studiu riguros al proceselor neliniare de react ¸ie-difuzieˆ ın medii poroase,
care includ difuzie, react ¸ii chimice ¸ si diferite tipuri de rate de adsorbt ¸ie;
•obt ¸inerea unor rezultate de omogenizare pentru probleme unilateraleˆ ın medii perforate;
•elaborarea de noi modele matematice pentru fenomenele de transport ionic ˆ ın medii
periodice ˆ ınc˘ arcate electric;
•derivarea de noi modele matematice neliniare pentru studiul proceselor de carcino
genez˘ a ˆ ın celulele umane ¸ si al dinamicii calciului ˆ ın celulele vii;;
•efectuarea unei analize asimptotice riguroase pentru procese de difuzie termic˘ a ˆ ın
structuri compozite;
•rafinarea studiului problemelor de difuzie cu condit ¸ii dinamice pe frontier˘ a;
•obt ¸inerea de noi modele matematice pentru probleme de difuzie cu salt ˆ ın flux.
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 field of homogenization theory after defending my Ph.D.
thesis. The main motivation behind this endeavour is to briefly describe the state of the art
in the field 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
scientific and academic career.
The thesis relies on some of my original contributions to the applications of the homoge-
nization theory, contained in twenty-five 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 fields. 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 specific field of homogenization theory.
The thesis is based on some of the most relevant results I obtained during the last fifteen
years of research conducted, alone or in collaboration, in four major areas: multiscale analysis
of reaction-diffusion processes in porous media, upscaling in unilateral problems, multiscale
modeling of composite media with imperfect interfaces, and mathematical models in biology
and in engineering. Thus, the homogenization theory and its applications represent the core
of my scientific work done during these last fifteen years.
Apart from two short abstracts in Romanian and in English, the thesis comprises two parts
and a comprehensive bibliography. The first part, structured into five chapters, is devoted to
the presentation of my main scientific achievements since the completion of my Ph.D. thesis.
After a brief introductory chapter presenting the state of the art in the field of homogenization
theory and offering the general framework and a motivation for my post-doctoral research
work in this area, the second chapter is divided in two distinct sections, summarizing my
main contributions related to the homogenization of reactive flows in porous media. More
precisely, some original results for upscaling in stationary nonlinear reactive flows in porous
media and, also, results on nonlinear adsorption phenomena in porous media are presented.
The chapter relies on the papers [75], [74], [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 [77], [79], [210], [52], and [55]. 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 first part summarizes the
most important results I achieved, alone or in collaboration, in the field of heat transfer in
composite materials with imperfect interfaces and is based on the articles [48], [49], [50],
[215], [216], and [218]. For the definitions 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], [38], [46], [57], [63], [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 scientific and academic background, further research directions
and some future plans on my scientific 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 field 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´ ıaz, 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-diffusion processes in porous media,
including diffusion, chemical reactions and different types of adsorption rates;
•obtaining new homogenization results for unilateral problems in perforated media;
•elaborating new mathematical models for ionic transport phenomena in periodic
charged media;
•getting original homogenization results for calcium dynamics in living cells.
•deriving new nonlinear mathematical models for carcinogenesis in human cells.
•performing a rigorous multiscale analysis of some relevant thermal diffusion processes
in composite structures;
•refining the study of diffusion problems with dynamical boundary conditions;
•obtaining new mathematical models for diffusion problems with flux 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 coefficients, are discontinuous and,
moreover, highly oscillating. For example, in a composite material, constituted by the fine
mixing of two ore more components, the physical parameters are obviously discontinuous and
they are highly oscillating between different 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 fictitious 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 difficulties related to the explicit determination of a solution of the problem at
the microscale and offering 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 defined on a
15
16 Introduction
fixed domain and they have, in general, simpler or even constant coefficients (the effective or
homogenized coefficients), while the original equations have rapidly oscillating coefficients,
they are defined on a complicated domain and satisfy nonlinear boundary conditions. The
dependence on the real microstructure is given through the homogenized coefficients.
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 field became interesting from a purely mathematical
point of view. This gave rise to a new mathematical discipline, the homogenization theory.
The first rigorous developments of this theory appeared with the seminal works of Y. Babuka
[28], E. De Giorgi and S. Spagnolo [88], A. Bensoussan, J. L. Lions and G. Papanicolaou
[38], 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 [107]
and by G. Allaire and M. Briane [2]. In 1990, T. Arbogast, J. Douglas and U. Hornung [21]
defined a dilation operator in order to study homogenization problems in a periodic medium
with double porosity. An alternative approach was offered by the Bloch-wave homogenization
method [78], 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 finite 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., [57]). Let us finally 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 [102]). It is important to emphasize that homogenization theory
can be applied to non-periodic media, as well. To this end, one can use G- orH-convergence
techniques. Also, it is possible to deal with general geometrical settings, without assuming
periodicity or randomness.
Homogenization methods have been successfully applied to various problems, such as the
convective-diffusive transport in porous media, nonlinear elasticity, 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., [62], [76], [64], [75], [77],
and the references therein). We also mention here some remarkable monographs dedicated
to the mathematical problems of homogenization: [149], [29], [38], [162], [166], [186], [200],
[63], [71], [104].
Multiscale methods offer 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
definitely one of the most interdisciplinary field of mathematics.
My interest in this broad field 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-diffusion processes in porous media, homogenization results for
unilateral problems, multiscale modeling of composite media with imperfect interfaces, and
mathematical models in biology and in engineering.
My research activity in the field 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 fields, 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 field of homogenization theory. As a matter of fact,
the homogenization theory and its applications represent the core of my scientific work done
during the last fifteen 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 first part, structured into five chapters, is devoted to the presentation of my main
scientific achievements since the completion of my Ph.D. thesis. After a brief introductory
chapter presenting the state of the art in the field of homogenization theory and offering
the general framework and a motivation for my post-doctoral research work in this area,
the second chapter is divided in two distinct sections, summarizing my main contributions
related to the homogenization of reactive flows in porous media. More precisely, some original
results for upscaling in stationary nonlinear reactive flows in porous media and, also, results
on nonlinear adsorption phenomena in porous media are presented. The chapter relies on the
papers [75], [74], [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 [77], [79], [210],
[52], and [55]. 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 first part summarizes the most important results
I achieved, alone or in collaboration, in the field of heat transfer in composite materials with
imperfect interfaces and is mainly based on the articles [48], [49], [50], [215], [216], and [218].
For the definitions 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], [38],
17
[46], [57], [63], [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 scientific and academic background, further research directions
and some future plans on my scientific and academic career are presented.
The thesis end by a comprehensive bibliography, illustrating the state of the art in this
vast field 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-five articles. The results included in this thesis have been
obtained during the last fifteen 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´ ıaz, 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 field 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 field of homogenization theory will be presented in the
following chapters):
•performing a rigorous study of nonlinear reaction-diffusion processes in porous media,
including diffusion, chemical reactions and different types of adsorption rates;
•obtaining new homogenization results for unilateral problems in perforated media;
•elaborating new mathematical models for ionic transport phenomena in periodic
charged media;
•getting original homogenization results for calcium dynamics in living cells.
•deriving new nonlinear mathematical models for carcinogenesis in human cells.
•performing a rigorous multiscale analysis of some relevant thermal diffusion processes
in composite structures;
•refining the study of diffusion problems with dynamical boundary conditions;
•obtaining new mathematical models for thermal problems with flux 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 flows 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 flows in heterogeneous porous media has
received considerable attention in the last decades. However, even with the increases in the
power of computers, the complex multiscale structure of the analyzed media constitutes a
critical problem in the numerical treatment of such models and there is a considerable interest
in the development of upscaled or homogenized models in which the effective properties of
the medium vary on a coarse scale which proves to be suitable for efficient computation,
but enough accurately to capture the influence of the fine-scale structure on the coarse-scale
properties of the solution.
Porous media play an important role in many areas, such as hydrology (groundwater
flow, 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 fiber materials, brick manufacturing). There is an extensive literature on the
determination of the effective properties of heterogeneous porous media (see, e.g., [137], [33],
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 flows 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 flow, diffusion, convection, and chemical reactions
is a difficult task. The homogenization theory proves to be a very efficient tool by provid-
ing suitable techniques allowing us to pass from the microscopic scale to the macroscopic
19
20 HOMOGENIZATION RESULTS FOR HETEROGENEOUS 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 first
works containing rigorous homogenization results for reactive flows 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], [33],
[169], [174], [158], [141], [168], [159], [100] and the references therein). For instance, rigorous
homogenization results for reactive flows 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 flows combined with the mechanics of cells, we refer to [148].
Rigorous homogenization techniques for obtaining the effective 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 [99]). Related problems, such that single or two-phase flow 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
flow, diffusion, convection, and reactions in porous media, we refer to [137].
In this chapter, some applications of the homogenization method to the study of reactive
flows 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 [75], [74],
[213], and [211].
2.1 Upscaling in stationary reactive flows in periodic porous
media
We shall discuss now, following [74] and [213], some homogenization results for chemical
reactive flows 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 [33], J. I. D´ ıaz [90], [93], [92], and U. Hornung [137]
and the references therein. We shall be concerned with a problem modeling the stationary
reactive flow of a fluid confined in the exterior of some periodically distributed obstacles,
reacting on the boundaries of the obstacles. More precisely, the challenge in our first paper
dedicated to this subject, namely [74], consists in dealing with Lipschitz or even non-Lipschitz
continuous reaction rates such as Langmuir or Freundlich kinetics, which, at that time, were
open cases in the literature. Our results represent a generalization of some of the results
in [137]. Using rigorous multiscale techniques, we derive a macroscopic model system for
such elliptic problems modeling chemical reactions on the grains of a porous medium. The
effective model preserves all the relevant information from the microscopic level. The case in
which chemical reactions arise inside the grains of a porous medium will be also discussed.
HOMOGENIZATION OF REACTIVE FLOWS 21
Also, we shall present some results obtained in [213], where we have analyzed the effective
behavior of the solution of a nonlinear problem arising in the modeling of enzyme catalyzed
reactions through the exterior of a domain containing periodically distributed reactive solid
obstacles.
2.1.1 The model problem
Let Ω be an open smooth connected bounded set in Rn(n≥3) and let us insert in it a set
of periodically distributed reactive obstacles. As a result, we obtain an open set Ωε, called
theexterior domain ;εrepresents 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 thatF⊂Y. We shall refer to Fas being
the elementary obstacle . We setY∗=Y\F. Ifεis a real parameter taking values in a
sequence of positive numbers converging to zero, for each εand for any integer vector k∈Zn,
setFε
kthe translated image of εFby the vector k,Fε
k=ε(k+F). The setFε
krepresents the
obstacles in Rn. Also, let us denote by Fεthe set of all the obstacles contained in Ω, i.e.
Fε=∪{
Fε
k|Fε
k⊂Ω, k∈Zn}
.
Set Ωε= Ω\Fε. Hence, Ωεis a periodic domain with periodically distributed obstacles
of size of the same order as the period. We remark that the obstacles do not intersect the
boundary∂Ω. LetSε=∪{∂Fε
k|Fε
k⊂Ω, k∈Zn}. So,∂Ωε=∂Ω∪Sε.
We denote by |ω|the Lebesgue measure of any measurable subset ω⊂Rnand, for an
arbitrary function ψ∈L2(Ωε), we denote by eψits extension by zero to the whole of Ω. Also,
throughout this thesis, by Cwe denote a generic fixed strictly positive constant, whose value
can change from line to line.
The first problem we present in this section concerns the stationary reactive flow of a fluid
confined in Ωε, of concentration uε, reacting on the boundary of the obstacles. A simplified
version of this kind of problem can be written as follows:
−Df∆uε=fin Ωε,
−Df∂uε
∂ν=εg(uε) onSε,
uε= 0 on∂Ω.(2.1)
Here,νis the exterior unit normal to Ωε,f∈L2(Ω) andSεis the boundary of our exterior
medium Ω \Ωε. Moreover, for simplicity, we assume that the fluid is is homogeneous and
isotropic, with a constant diffusion coefficient Df>0. We can treat in a similar manner the
more general case in which, instead of considering constant diffusion coefficients, we work
with an heterogeneous medium represented by periodic symmetric bounded matrices which
are assumed to be uniformly coercive.
The semilinear boundary condition imposed on Sεin problem (2.1) describes the chemical
reactions which take place locally at the interface between the reactive fluid and the grains.
In fact, from a strictly chemical perspective, such a situation represents, equivalently, the
22 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
effective reaction on the walls of the chemical reactor between the fluid filling Ωεand 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 subdifferential of a convex lower semicontinuous function G. These two general situations
are well illustrated by the following important practical examples:
a)g(v) =αv
1 +βv, α,β > 0 (Langmuir kinetics) (2.2)
and
b)g(v) =|v|p−1v,0<p< 1 (Freundlich kinetics) . (2.3)
The exponent pis called the order of the reaction . We point out that if we assume f≥0,
one can prove (see, e.g. [92]) that uε≥0 in Ω \Ωεanduε>0 in Ωε, although uεis 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 uεis, in practical
applications, a concentration, we can impose suitable conditions on the data to ensure that
0≤uε≤1 (see, e.g., [91]).
As usual in homogenization, our goal is to obtain a suitable description of the asymptotic
behavior, as the small parameter εtends to zero, of the solution uεof problem (2.1) in such
domains.
2.1.2 The case of a smooth function g. The macroscopic model
Let us deal first with the case of a smooth function g. We consider that gis a continuously
differentiable function, monotonously non-decreasing and such that g(v) = 0 if and only if
v= 0. Moreover, we suppose that there exist a positive constant Cand an exponent q, with
0≤q<n/ (n−2), such that dg
dv ≤C(1 +|v|q). (2.4)
We introduce the functional space Vε={
v∈H1(Ωε)|v= 0 on∂Ω}
, endowed with the
norm ∥v∥Vε=∥∇v∥L2(Ωε). The weak formulation of problem (2.1) is:
Finduε∈Vεsuch that
Df∫
Ωε∇uε· ∇φdx+ε∫
Sεg(uε)φdσ=∫
Ωεfφdx∀φ∈Vε.(2.5)
By classical existence results (see [46]), there exists a unique weak solution uε∈Vε∩H2(Ωε)
of problem (2.1). This solution being defined only on Ωε, we need to extend it to the whole of
Ω to be able to state the convergence result. To this end, let us recall the following well-known
extension result (see [71]):
HOMOGENIZATION OF REACTIVE FLOWS 23
Lemma 2.1 There exists a linear continuous extension operator
Pε∈ L(L2(Ωε);L2(Ω))∩ L(Vε;H1
0(Ω))
and a positive constant C, independent of ε, such that, for any v∈Vε,
∥Pεv∥L2(Ω)≤C∥v∥L2(Ωε)
and
∥∇Pεv∥L2(Ω)≤C∥∇v∥L2(Ωε).
Therefore, we have the following Poincar´ e’s inequality in Vε:
Lemma 2.2 There exists a positive constant C, independent of ε, such that
∥v∥L2(Ωε)≤C∥∇v∥L2(Ωε)for anyv∈Vε.
The main convergence result for this case is stated in the following theorem.
Theorem 2.3 ([74]) There exists an extension Pεuεof the solution uεof the variational
problem (2.5) such that
Pεuε⇀u weakly inH1
0(Ω),
whereuis the unique solution of
−n∑
i,j=1qij∂2u
∂xi∂xj+|∂F|
|Y∗|g(u) =finΩ,
u= 0 on∂Ω.(2.6)
Here,Q= ((qij))is the homogenized matrix, whose entries are defined as follows:
qij=Df(
δij+1
|Y∗|∫
Y∗∂χj
∂yidy)
(2.7)
in terms of the functions χi, i= 1,…,n, solutions of the cell problems
−∆χi= 0 inY∗,
∂(χi+yi)
∂ν= 0 on∂F,
χiY−periodic.(2.8)
The constant matrix Qis symmetric and positive-definite.
Proof. The proof of this theorem is divided into four steps.
First step. Letuε∈Vεbe the solution of the variational problem (2.5) and let Pεuεbe
the extension of uεinside the obstacles given by Lemma 2.1. Taking φ=uεas a test function
in (2.5), using Schwartz and Poincar´ e’s inequalities, we get
∥Pεuε∥H1
0(Ω)≤C
24 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
and, by passing to a subsequence, still denoted by Pεuε, we can suppose that there exists
u∈H1
0(Ω) such that
Pεuε⇀u weakly inH1
0(Ω). (2.9)
It remains to determine the limit equation satisfied by u.
Second step . In order to obtain the limit equation satisfied 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 h∈Ls′(∂F), 1≤s′≤ ∞ , the linear form µε
honW1,s
0(Ω)
defined by
⟨µε
h,φ⟩=ε∫
Sεh(x
ε)
φdσ∀φ∈W1,s
0(Ω),
with 1/s+ 1/s′= 1. Then (see [62]),
µε
h→µhstrongly in ( W1,s
0(Ω))′, (2.10)
where
⟨µh,φ⟩=µh∫
Ωφdx,
with
µh=1
|Y|∫
∂Fh(y) dσ.
Ifh∈L∞(∂F) or ifhis constant, we have µε
h→µhstrongly in W−1,∞(Ω) and we denote
byµεthe above introduced measure in the particular case in which h= 1. Notice that in
this caseµhbecomesµ1=|∂F|/|Y|. We shall prove now that for any φ∈ D(Ω) and for any
vε⇀v weakly inH1
0(Ω), one has
φg(vε)⇀φg (v) weakly in W1,q
0(Ω), (2.11)
where
q=2n
q(n−2) +n.
To this end, let us remark that
sup∥∇g(vε)∥Lq(Ω)<∞. (2.12)
Indeed, using the growth condition (2.4) imposed to g, we have
∫
Ω ∂g
∂xi(vε) q
dx≤C∫
Ω(
1 +|vε|qq) ∂vε
∂xi q
dx≤
≤C(
1 +(∫
Ω|vε|qqγdx)1/γ)(∫
Ω|∇vε|qδdx)1/δ
,
HOMOGENIZATION OF REACTIVE FLOWS 25
where we took γandδsuch thatqδ= 2, 1/γ+ 1/δ= 1 andqqγ= 2n/(n−2). Notice that
it is from here that we get q=2n
q(n−2) +n. Also, due to the fact that 0 ≤q <n/ (n−2),
it follows that q>1. Since
sup∥vε∥
L2n
n−2(Ω)<∞,
we easily get (2.12). Therefore, to obtain (2.11), it remains only to show that
g(vε)→g(v) strongly in Lq(Ω). (2.13)
But this convergence is a direct consequence of the following well-known result (see [82], [156]
and [162]):
Theorem 2.4 LetG: Ω×R→Rbe a Carath´ eodory function, i.e.
a) for every vthe function G(·,v)is measurable with respect to x∈Ω.
b) for every (a.e.) x∈Ω, the function G(x,·)is continuous with respect to v.
Moreover, if we assume that there exists a positive constant Csuch that
|G(x,v)| ≤C(
1 +|v|r/t)
,
withr≥1andt <∞, then the map v∈Lr(Ω)7→G(x,v(x))∈Lt(Ω)is continuous in the
strong topologies.
Indeed, since
|g(v)| ≤C(1 +|v|q+1),
applying the above theorem for G(x,v) =g(v),t=qandr= (2n/(n−2))−r′, withr′>0
such thatq+ 1< r/t and using the compact injection H1(Ω),→Lr(Ω) we obtain (2.13).
Now, from (2.10), written for h= 1, and (2.11) written for vε=Pεuε, we get
⟨µε,φg(Pεuε)⟩ →|∂F|
|Y|∫
Ωφg(u) dx∀φ∈ D(Ω) (2.14)
and this completes the proof of this step.
Third step . Letξεbe the gradient of uεin Ωεand let us denote by eξεits extension by
zero to the whole of Ω. Then, eξεis bounded in ( L2(Ω))nand, as a consequence, there exists
ξ∈(L2(Ω))nsuch that
eξε⇀ξ weakly in (L2(Ω))n. (2.15)
Let us identify now the equation satisfied by ξ. If we take φ∈ D(Ω), from (2.5) we get
∫
Ωeξε· ∇φdx+ε∫
Sεg(uε)φdσ=∫
ΩχΩεfφdx (2.16)
and we can pass to the limit, with ε→0, in all the terms of (2.16). For the first one, we have
lim
ε→0∫
Ωeξε· ∇φdx=∫
Ωξ· ∇φdx. (2.17)
26 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
For the second term, using (2.14), we obtain
lim
ε→0ε∫
Sεg(uε)φdσ=|∂F|
|Y|∫
Ωg(u)φdx. (2.18)
It is not difficult to pass to the limit in the right-hand side of (2.16). Since
χΩεf ⇀|Y∗|
|Y|fweakly inL2(Ω),
we get
lim
ε→0∫
ΩχΩεfφdx=|Y∗|
|Y|∫
Ωfφdx. (2.19)
Putting together (2.17)-(2.19), we have
∫
Ωξ· ∇φdx+|∂F|
|Y|∫
Ωg(u)φdx=|Y∗|
|Y|∫
Ωfφdx∀φ∈ D(Ω).
Thus,ξsatisfies
−divξ+|∂F|
|Y|g(u) =|Y∗|
|Y|fin Ω (2.20)
and it remains now only to identify ξ.
Fourth step. In order to identify ξ, we shall use the solutions of the local problems (2.8).
For any fixed i= 1,…,n, let us define
Φiε(x) =ε(
χi(x
ε)
+yi)
∀x∈Ωε, (2.21)
wherey=x/ε. By periodicity,
PεΦiε⇀x iweakly inH1(Ω). (2.22)
Letηε
ibe the gradient of Φ iεin Ωεandeηε
ibe the extension by zero of ηε
iinside the holes.
From (2.21), for the j-component of eηε
iwe have
(
eηε
i)
j=(g∂Φiε
∂xj)
=(
^∂χi
∂yj(y))
+δijχY∗
and, therefore,
(
eηε
i)
j⇀1
|Y|(∫
Y∗∂χi
∂yjdy+|Y∗|δij)
=|Y∗|
|Y|qijweakly inL2(Ω). (2.23)
On the other hand, it is not difficult to see that ηε
isatisfies
{
−divηε
i= 0 in Ωε,
ηε
i·ν= 0 onSε.(2.24)
Letφ∈ D(Ω). Multiplying the first equation in (2.24) by φuεand integrating by parts over
Ωε, we obtain ∫
Ωεηε
i· ∇φuεdx+∫
Ωεηε
i· ∇uεφdx= 0.
HOMOGENIZATION OF REACTIVE FLOWS 27
Thus, ∫
Ωeηε
i· ∇φPεuεdx+∫
Ωεηε
i· ∇uεφdx= 0. (2.25)
On the other hand, taking φΦiεas a test function in (2.5), we get
∫
Ωε(∇uε· ∇φ)Φiεdx+∫
Ωε(∇uε· ∇Φiε)φdx+ε∫
Sεg(uε)φΦiεdσ=∫
ΩεfφΦiεdx,
which, using the definitions of eξεandeηε
i, leads to
∫
Ωeξε· ∇φPεΦiεdx+∫
Ωε∇uε·ηε
iφdx+ε∫
Sεg(uε)φΦiεdσ=∫
ΩfχΩεφPεΦiεdx.
Now, from (2.25), we obtain
∫
Ωeξε· ∇φPεΦiεdx−∫
Ωeηε
i· ∇φPεΦiεdx+ε∫
Sεg(uε)φΦiεdσ=∫
ΩfχΩεφPεΦiεdx.(2.26)
Let us pass to the limit in (2.26). By using (2.15) and (2.22), we get
lim
ε→0∫
Ωeξε· ∇φPεΦiεdx=∫
Ωξ· ∇φxidx. (2.27)
On the other hand, from (2.9) and (2.23) we obtain
lim
ε→0∫
Ωeηε
i· ∇φPεuεdx=|Y∗|
|Y|∫
Ωqi· ∇φudx, (2.28)
whereqiis the vector having the j-component equal to qij.
Since the boundary of Fis of classC2,PεΦiε∈W1,∞(Ω) andPεΦiε→xistrongly in
L∞(Ω).Then, since g(Pεuε)PεΦiε→g(u)xistrongly in Lq(Ω) andg(Pεuε)PεΦiεis bounded
inW1,q(Ω), we have g(Pεuε)PεΦiε⇀g(u)xiweakly inW1,q(Ω). Thus,
lim
ε→0ε∫
Sεg(uε)φΦiεdσ=|∂F|
|Y|∫
Ωg(u)φxidx. (2.29)
Finally, for the limit of the right-hand side of (2.26), since
χΩεf ⇀|Y∗|
|Y|fweakly inL2(Ω),
using (2.22), we have
lim
ε→0∫
ΩfχΩεφPεΦiεdx=|Y∗|
|Y|∫
Ωfφx idx. (2.30)
Thus, we get
∫
Ωξ· ∇φxidx−|Y∗|
|Y|∫
Ωqi· ∇φudx+|∂F|
|Y|∫
Ωg(u)φxidx=|Y∗|
|Y|∫
Ωfφx idx. (2.31)
Using Green’s formula and equation (2.20), we get
−∫
Ωξ· ∇xiφdx+|Y∗|
|Y|∫
Ωqi· ∇uφdx= 0 in Ω.
28 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
The above equality holds true for any φ∈ D(Ω) and this implies that
−ξ· ∇xi+|Y∗|
|Y|qi· ∇u= 0 in Ω. (2.32)
Writing (2.30) by components, differentiating with respect to xi,summing after iand using
(2.19), we are led to
|Y∗|
|Y|n∑
i,j=1qij∂2u
∂xi∂xj= divξ=−|Y∗|
|Y|f+|∂T|
|Y|g(u),
which means that uverifies
−n∑
i,j=1qij∂2u
∂xi∂xj+|∂F|
|Y∗|g(u) =fin Ω.
Sinceu∈H1
0(Ω) (i.e.u= 0 on∂Ω) anduis uniquely determined, the whole sequence Pεuε
converges to uand Theorem 2.3 is proven.
Remark 2.5 The right scaling εin front of the function gmodeling the contribution of the
nonlinear reactions on the boundary of the grains leads in the limit to the presence of a new
term distributed all over the domain Ω. Also, let us emphasize again that if we assume f≥0,
the function gin example a)is indeed a particular example of our first model situation.
Remark 2.6 The results in [74] are obtained for the case n≥3. All of them are still valid,
under our assumptions, in the case n= 2. Of course, for this case, n/(n−2)has to be
replaced by +∞and, hence, (2.4) holds true for 0≤q<∞. 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 [74], 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.,
[67], [57], [58], and [65]), which, apart from a significant simplification 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 R×R, satisfying the condition g(0) = 0, is also treated in [74]. If we denote by
D(g) the domain of g, i.e.D(g) ={ξ∈R|g(ξ)̸=∅}, then we suppose that D(g) =R.
Moreover, we assume that gis continuous and there exist C≥0 and an exponent q, with
0≤q<n/ (n−2), such that
|g(v)| ≤C(1 +|v|q). (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.
HOMOGENIZATION OF REACTIVE FLOWS 29
In this case, there exists a lower semicontinuous convex function GfromRto ]−∞,+∞],
Gproper, i.e. G̸≡+∞, such that gis the subdifferential of G,g=∂G. Let
G(v) =∫v
0g(s) ds.
We define the convex set
Kε={
v∈Vε|G(v)|Sε∈L1(Sε)}
.
For a given function f∈L2(Ω), the weak solution of the problem (2.1) is also the unique
solution of the following variational inequality:
finduε∈Kεsuch that
Df∫
ΩεDuεD(vε−uε) dx−∫
Ωεf(vε−uε) dx+⟨µε,G(vε)−G(uε)⟩ ≥0
∀vε∈Kε.(2.34)
We start by remarking that there exists a unique weak solution uε∈Vε∩H2(Ωε) of the
above variational inequality (see [46]). Moreover, it is well-known that the solution uεof the
variational inequality (2.34) is also the unique solution of the minimization problem
uε∈Kε,
Jε(uε) = inf
v∈KεJε(v),
where
Jε(v) =1
2Df∫
Ωε|Dv|2dx+⟨µε,G(v)⟩ −∫
Ωεfvdx.
If we introduce the following functional defined on H1
0(Ω):
J0(v) =1
2∫
ΩQDvDv dx+|∂F|
|Y∗|∫
ΩG(v) dx−∫
Ωfvdx,
the main convergence result for problem (2.34) can be formulated as follows:
Theorem 2.8 (Theorem 2.6 in [74]) One can construct an extension Pεuεof the solution
uεof the variational inequality (2.34) such that
Pεuε⇀u weakly inH1
0(Ω),
whereuis the unique solution of the minimization problem
Findu∈H1
0(Ω)such that
J0(u) = inf
v∈H1
0(Ω)J0(v).(2.35)
Moreover,G(u)∈L1(Ω). Here,Q= ((qij))is the classical homogenized matrix, whose entries
were defined in (2.7)-(2.8).
30 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Remark 2.9 We notice that ualso satisfies
−n∑
i,j=1qij∂2u
∂xi∂xj+|∂F|
|Y∗|g(u) =finΩ,
u= 0 on∂Ω.
Proof of Theorem 2.8. Letuεbe the solution of the variational inequality (2.34). Using
the same extension Pεuεas in the previous case, it follows immediately that Pεuεis bounded
inH1
0(Ω) and, thus, by passing to a subsequence, we have
Pεuε⇀u weakly inH1
0(Ω). (2.36)
Letφ∈ D(Ω). By classical regularity results, χi∈L∞. So, from the boundedness of χiand
φ, it follows that there exists M≥0 such that
∂φ
∂xi
L∞
χi
L∞<M.
Let
vε=φ+∑
iε∂φ
∂xi(x)χi(x
ε)
. (2.37)
Obviously, vε∈Kεand this will allow us to take it as a test function in (2.34). Moreover, it
is easy to see that vε→φstrongly in L2(Ω). Further,
Dvε=∑
i∂φ
∂xi(x)(
ei+Dχi(x
ε))
+ε∑
iD∂φ
∂xi(x)χi(x
ε)
,
where ei, 1≤i≤n, are the elements of the canonical basis in Rn. Takingvεas a test
function in (2.34), we get
∫
ΩεDuεDvεdx≥∫
Ωεf(vε−uε) dx+∫
ΩεDuεDuεdx− ⟨µε,G(vε)−G(uε)⟩
and
∫
ΩDPεuε^(Dvε) dx≥∫
Ωεf(vε−uε) dx+∫
ΩεDuεDuεdx− ⟨µε,G(vε)−G(uε)⟩.(2.38)
Let us denote
ρQej=1
|Y∗|∫
Y∗(Dχj+ej) dy, (2.39)
whereρ=|Y∗|/|Y|. It is not difficult to pass to the limit in the left-hand side and in the
first term of the right-hand side of (2.38). We have
∫
ΩDPεuεgDvεdx→∫
ΩρQDuDφ dx (2.40)
and ∫
Ωεf(vε−uε) dx=∫
ΩfχΩε(vε−Pεuε) dx→∫
Ωfρ(φ−u) dx. (2.41)
HOMOGENIZATION OF REACTIVE FLOWS 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(Pεuε)⇀G(u) weakly in W1,q
0(Ω)
and, so
⟨µε,G(Pεuε)⟩ →|∂F|
|Y|∫
ΩG(u) dx.
In a similar way, we get
⟨µε,G(vε)⟩ →|∂F|
|Y|∫
ΩG(φ) dx
and, therefore, we have
⟨µε,G(vε)−G(Pεuε)⟩ →|∂F|
|Y|∫
Ω(G(φ)−G(u)) dx. (2.42)
Now, it remains only to pass to the limit in the second term of the right-hand side of (2.38).
To this end, let us write down, for any wε∈H1
0(Ω), the subdifferential inequality
∫
ΩεDuεDuεdx≥∫
ΩεDwεDwεdx+ 2∫
ΩεDwε(Duε−Dwε) dx, (2.43)
Reasoning as before and choosing
wε=φ+∑
iε∂φ
∂xi(x)χi(x
ε)
,
whereφhas similar properties as the corresponding φ, we can pass to the limit in the right-
hand side of the inequality (2.43) and we get
lim inf
ε→0∫
ΩεDuεDuεdx≥∫
ΩρQDφDφdx+ 2∫
ΩρQDφ(Du−Dφ) dx,
for anyφ∈ D(Ω). Sinceu∈H1
0(Ω), taking φ→ustrongly in H1
0(Ω),we obtain
lim inf
ε→0∫
ΩεDuεDuεdx≥∫
ΩρQDuDu dx. (2.44)
Putting together (2.40)-(2.42) and (2.44), we get
∫
ΩρQDuDφ dx≥∫
Ωfρ(φ−u) dx+∫
ΩρQDuDu dx−|∂F|
|Y|∫
Ω(G(φ)−G(u)) dx,
for anyφ∈ D(Ω) and, hence, by density, for any v∈H1
0(Ω). So, we obtain
∫
ΩQDuD (v−u) dx≥∫
Ωf(v−u) dx−|∂F|
|Y∗|∫
Ω(G(φ)−G(u)) dx,
which leads exactly to the limit problem (2.34). This ends the proof of Theorem 2.8.
32 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Remark 2.10 The choice of the test function (2.37) gives, in fact, a first-corrector term for
the weak convergence of Pεuεtou.
Remark 2.11 All the results of this section can be obtained for a general diffusion matrix
Aε(x) =A(x/ε), whereA=A(y)is a matrix-valued function on Rnwhich isY-periodic. We
shall assume that
A∈L∞(Ω)n×n,
Ais a symmetric matrix,
for some 0<γ <λ, γ |ξ|2≤A(y)ξ·ξ≤λ|ξ|2∀ξ, y∈Rn.
Problems similar to the one presented here may arise in various other contexts (see, e.g.
[211] and [213]). In [213], we analyzed the effective behavior of the solution of a nonlinear
problem arising in the modeling of enzyme catalyzed reactions through the exterior of a
domain containing periodically distributed reactive solid obstacles, with period ε. Enzymes
are proteins that speed up the rate of a chemical reaction without being used up. They
are specific 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 affect chemical reactions. Michaelis-Menten equation remains
the most generally applicable equation for describing enzymatic reactions. In this case, we
consider the following elliptic problem:
−Df∆uε+β(uε) =fin Ωε,
−Df∂uε
∂ν=εg(uε) onSε,
uε= 0 on∂Ω.(2.45)
Here, the function βis continuously differentiable, monotonously non-decreasing and such
thatβ(0) = 0. For example, we can take βto be a linear function, i.e. β(v) =λv, or we
can consider the nonlinear case in which βis given by (2.2) (Langmuir kinetics). For the
given function g, we deal here with the case of a single-valued maximal monotone graph with
g(0) = 0, i.e. the case in which gis the subdifferential of a convex lower semicontinuous
functionG. More precisely, we shall consider an important practical example, arising in the
diffusion of enzymes (the Michaelis-Menten model):
g(v) =
δv
v+γ, v ≥0,
0, v< 0,
forδ,γ > 0.
HOMOGENIZATION OF REACTIVE FLOWS 33
The existence and uniqueness of a weak solution of (2.45) is ensured by the classical theory
of monotone problems (see [46] and [101]). Therefore, we know that there exists a unique
weak solution uε∈Vε∩H2(Ωε). Moreover, uεis also the unique solution of the following
variational problem:
Finduε∈Kεsuch that
Df∫
ΩεDuεD(vε−uε) dx+∫
Ωεβ(uε)(vε−uε) dx
−∫
Ωεf(vε−uε) dx+⟨µε,G(vε)−G(uε)⟩ ≥0,∀vε∈Kε,(2.46)
whereµεis the linear form on W1,1
0(Ω) defined by
⟨µε,φ⟩=ε∫
Sεφdσ,∀φ∈W1,1
0(Ω).
The main convergence result in this case, proven in [213], is stated in the following theorem.
Theorem 2.12 The solution uε, properly extended to the whole of Ω, converges to the unique
solution of the following variational inequality:
u∈H1
0(Ω),
∫
ΩQDuD (v−u) dx+∫
Ωβ(u)(v−u) dx≥∫
Ωf(v−u) dx
−|∂F|
|Y∗|∫
Ω(G(v)−G(u)) dx,∀v∈H1
0(Ω).(2.47)
Here,Q= ((qij))is the homogenized matrix, defined in (2.7).
Remark 2.13 Notice that ualso satisfies
−n∑
i,j=1qij∂2u
∂xi∂xj+β(u) +|∂F|
|Y∗|g(u) =finΩ,
u= 0 on∂Ω.
Thus, the asymptotic behavior of the solution of the microscopic problem (2.45) is governed
by a new elliptic boundary-value problem, with an extra zero-order term that captures the
effect of the enzymatic reactions. The effect of the enzymatic reactions initially situated
on the boundaries of the grains spread out in the limit all over the domain, giving the extra
zero-order term which captures this boundary effect. In fact, one could obtain a similar result
by considering interior enzymatic nonlinear chemical reactions given by the same well-known
nonlinear function g. The only difference in the limit equation will be the coefficient appearing
in front of this extra zero-order term. So, one can control the effective behavior of such
reactive flows by choosing different locations for the involved chemical reactions. Moreover,
as we shall see in the next section, we can obtain similar effects by considering transmission
34 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
problems, with an unknown flux on the boundary of each grain, i.e. we can consider the
case in which we have chemical reactions in Ωε, but also inside the grains, instead on their
boundaries. The difference in the limit equation will be the coefficient appearing in front of
this extra zero-order term. Hence, we can control the effective behavior of such reactive flows
by choosing different locations for the involved chemical reactions.
2.1.4 Chemical reactions inside the grains of a porous medium
We shall briefly present now some results obtained in [74] for the case in which we assume that
we have a granular material filling the obstacles and we consider some chemical reactive flows
through the grains. In fact, we consider a perfect transmission problem (with an unknown
flux on the boundary of each grain) between the solutions of two separated equations (for the
case of imperfect transmission problems, see Chapter 5). A simplified version of this kind of
problems can be formulated as follows:
−Df∆uε=fin Ωε,
−Dp∆vε+g(vε) = 0 in Πε
−Df∂uε
∂ν=Dp∂vε
∂νonSε,
uε=vεonSε,
uε= 0 on∂Ω.(2.48)
Here, Πε= Ω\Ωε,νis the exterior unit normal to Ωε,uεandvεare the concentrations in
Ωεand, respectively, inside the grains Πε,Df>0,Dp>0,f∈L2(Ω) andgis a continuous
function, monotonously non-decreasing and such that g(v) = 0 if and only if v= 0. Moreover,
we suppose that there exist a positive constant Cand an exponent q, with 0 ≤q<n/ (n−2),
such that |g(v)| ≤C(1 +|v|q+1). We remark that the above mentioned examples a) andb)
are both covered by this class of functions.
Let us introduce the space
Hε={
wε= (uε,vε) uε∈Vε,vε∈H1(Πε), uε=vεonSε}
,
with the norm
∥wε∥2
Hε=∥∇uε∥2
L2(Ωε)+∥∇vε∥2
L2(Πε).
The variational formulation of problem (2.48) is the following one:
findwε∈Hεsuch that
Df∫
Ωε∇uε· ∇φdx+Dp∫
Πε∇vε· ∇ψdx+∫
Πεg(vε)ψdx=∫
Ωεfφdx
∀(φ,ψ)∈Kε.(2.49)
Under the above hypotheses and the conditions satisfied by Hε, it is well-known (see [46] and
[162]) that (2.49) is a well-posed problem.
HOMOGENIZATION OF REACTIVE FLOWS 35
If we introduce the matrix
A={
DfId inY\F
DpId inF,
then the main result in this situation is stated in the following theorem (for a detailed proof,
see [74]):
Theorem 2.14 One can construct an extension Pεuεof the solution uεof the variational
problem (2.49) such that
Pεuε⇀u weakly inH1
0(Ω),
whereuis the unique solution of
−n∑
i,j=1a0
ij∂2u
∂xi∂xj+|F|
|Y∗|g(u) =finΩ,
u= 0 on∂Ω.(2.50)
Here,A0= ((a0
ij))is the homogenized matrix, defined by
a0
ij=1
|Y|∫
Y(
aij+aik∂χj
∂yk)
dy, (2.51)
whereχi, i= 1,…,n, are the solutions of the cell problems
−div(AD(yj+χj)) = 0 inY,
χj−Yperiodic.(2.52)
The constant matrix A0is symmetric and positive-definite.
Corollary 2.15 Ifuεandvεare the solutions of the problem (2.48), then, passing to a
subsequence, still denoted by ε, there exist u∈H1
0(Ω)andv∈L2(Ω)such that
Pεuε⇀u weakly inH1
0(Ω),
evε⇀v weakly inL2(Ω)
and
v=|F|
|Y|u.
Corollary 2.16 Letθεbe defined by
θε(x) ={uε(x)x∈Ωε,
vε(x)x∈Πε.
Then, there exists θ∈H1
0(Ω)such thatθε⇀ θ weakly inH1
0(Ω), whereθis the unique
solution of
−n∑
i,j=1a0
ij∂2θ
∂xi∂xj+a|F|
|Y∗|g(θ) =finΩ,
θ= 0 on∂Ω,
andA0is given by (2.51)-(2.52), i.e. θ=u, due to the well-posedness of problem (2.49).
36 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Remark 2.17 As already mentioned, the approach used in [74] 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 [43]. Also, one can use the periodic unfolding method (see, e.g., [58], and [65]).
Remark 2.18 The two reactive flows studied above, namely (2.1) and (2.48), lead to com-
pletely different effective behaviors. The macroscopic problem (2.1) arises from the homoge-
nization of a boundary-value problem in the exterior of some periodically distributed obstacles
and the zero-order term occurring in (2.6) reflects the influence of the chemical reactions tak-
ing place on the boundaries of the reactive obstacles. On the other hand, the second model is
a boundary-value problem in the whole domain Ω, with discontinuous coefficients. Its macro-
scopic behavior also involves a zero-order term, but of a completely different nature, emerging
from the chemical reactions occurring inside the grains.
Remark 2.19 In (2.48) we considered that the ratio of the diffusion coefficients in the two
media is of order one in order to compare the case in which the chemical reactions take place
on the boundary of the grains with the case in which the chemical reactions occur inside them.
However, a more interesting problem arises if we consider different orders for the diffusion
in the obstacles and in the pores. More precisely, if one takes the ratio of the diffusion
coefficients to be of order ε2, then the limit model will be the so-called double-porosity model .
This scaling preserves the physics of the flow inside the grains, as ε→0. The less permeable
part of our medium (the grains) contributes in the limit as a nonlinear memory term. In
fact, the effective limit model includes two equations, one in Fand another one in Ω, the
last one containing an extra-term which reflects the remaining influence of the grains (see,
for instance, [21], [42], [43], [75], [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 [62], [75], [77], and
[211]). Also, for the case of a different geometry of the perforated domain and different
transmission conditions, see Chapter 5.
2.2 Nonlinear adsorption of chemicals in porous media
In this section, we shall present some homogenization results, obtained in [75], concerning
the effective behavior of some chemical reactive flows involving diffusion, different types of
adsorption rates and chemical reactions which take place at the boundary of the grains of a
porous material. Such problems arise in many domains, such as chemical engineering or soil
sciences (see, for instance, [137], where the asymptotic behavior of such chemical processes
was analyzed and rigorous convergence results were given for the case of linear adsorption
HOMOGENIZATION OF REACTIVE FLOWS 37
rates and linear chemical reactions). The case of nonlinear adsorption rates, left as open
in [137], was treated in [75]. Two well-known examples of such nonlinear models, namely
the so-called Freundlich and Langmuir kinetics, were studied. We briefly describe here these
results. In a first step, we consider that the surface of the grains is physically and chemically
homogeneous. Then, we assume that the surface of the solid part is physically and chemically
heterogeneous and we allow also a surface diffusion modeled by a Laplace-Beltrami operator
to take place on this surface. In this last case, we show that the effective behavior of our
system is governed by a new boundary-value problem, with an additional microvariable and
a zero-order extra term proving that memory effects are present in this limit model.
2.2.1 The microscopic model and its weak solvability
Our main goal in [75] was to obtain the asymptotic behavior, as ε→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 fluid phase Ωεand a solid skeleton
(grains or pores), Ω \Ωε. We assume that chemical substances are dissolved in the fluid
part. They are transported by diffusion and also, by adsorption, they can change from being
dissolved in the fluid to residing on the surface of the pores, where chemical reactions take
place. Thus, the model consists of a diffusion system in the fluid phase, a reaction system on
the pore surface and a boundary condition coupling them (see (2.54)):
(Vε)
∂vε
∂t(t,x)−Df∆vε(t,x) =h(t,x), x∈Ωε, t> 0,
vε(t,x) = 0, x∈∂Ω, t> 0,
vε(t,x) =v1(x), x∈Ωε, t= 0,(2.53)
−Df∂vε
∂ν(t,x) =εfε(t,x), x∈Sε, t> 0, (2.54)
and
(Wε)
∂wε
∂t(t,x) +awε(t,x) =fε(t,x), x∈Sε, t> 0,
wε(t,x) =w1(x), x∈Sε, t= 0,(2.55)
where
fε(t,x) =γ(g(vε(t,x))−wε(t,x)). (2.56)
Here,vεrepresents the concentration of the solute in the fluid region, wεis the concentration
of the solute on the surface of the skeleton Ω \Ωε,v1∈H1
0(Ω) is the initial concentration of
the solute and w1∈H1
0(Ω) is the initial concentration of the reactants on the surface Sεof
the skeleton. Also, the fluid is assumed to be homogeneous and isotropic, with a constant
diffusion coefficient Df>0,a,γ > 0 are the reaction and, respectively, the adsorption factor
andhis an external source of energy.
The semilinear boundary condition on Sεgives the exchanges of chemical flows across the
boundary of the grains, governed by a non-linear balance law involving the adsorption factor
γ(which, in a first step, is considered to be constant) and the adsorption rate represented
38 HOMOGENIZATION RESULTS FOR HETEROGENEOUS 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 subdifferential of a convex lower semicontinuous
functionG. These two general situations are well illustrated by the two important practical
examplesa) andb) (see (2.2) and (2.3)) mentioned in Section 2.1.1, namely the Langmuir
and, respectively, the Freundlich kinetics.
Let us notice that if vε≥0 in Ωεandvε>0 in Ωε, then the function gin example a)
is indeed a particular example of our first 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
fε(t,x) =γ1g(vε(t,x))−γ2wε(t,x),
withγ1>0 andγ2>0 being the adsorption factor and the desorption factor , respectively
(see [140]).
The existence and uniqueness of a weak solution uε= (vε,wε) of the system (2.53)-(2.56)
can be established by using the classical theory of semilinear monotone problems (see, for
instance, [46] and [162]).
In order to write down the variational formulation of problem (2.53)-(2.56), let us define
some suitable function spaces. Let H=L2(Ω), with the classical scalar product
(u,v)Ω=∫
Ωu(x)v(x) dx,
and let H=L2(0,T;H), with the scalar product
(u,v)Ω,T=∫T
0(u(t),v(t))Ωdt,whereu(t) =u(t,·),v(t) =v(t,·).
Also, letV=H1(Ω), with (u,v)V= (u,v)Ω+ (∇u,∇v)ΩandV=L2(0,T;V), with
(u,v)V=∫T
0(u(t),v(t))Vdt.
We set
W={
v∈ V dv
dt∈ V′}
where V′is the dual space of V,
V0={
v∈ V v= 0 on∂Ω a.e. on (0 ,T)}
,W0=V0∩
W.
In a similar manner, we define the spaces V(Ωε),V(Ωε),V(Sε) and V(Sε). For the latter we
write
⟨u,v⟩Sε=∫
Sεgεuvdσ,
wheregεis the metric tensor on Sε; the rule of partial integration on Sεapplies and, if we
denote the gradient on Sεby∇εand the Laplace-Beltrami operator on Sεby ∆ε, we have
−(∆εu,v)Sε=⟨∇εu,∇εv⟩Sε.
HOMOGENIZATION OF REACTIVE FLOWS 39
Also, for the space of test functions we use the notation D=C∞
0((0,T)×Ω)).
We shall start our analysis with the case in which gis a continuously differentiable func-
tion, monotonously non-decreasing and such that g(v) = 0 if and only if v= 0 (for the
non+mooth case, see Section 2.2.3). Moreover, we shall suppose that there exist a positive
constantCand an exponent q, with 0 ≤q<n/ (n−2), such that
dg
dv ≤C(1 +|v|q). (2.57)
The weak formulation of problem (2.53)-(2.56) is as follows:
findvε∈ W 0(Ωε),vε(0) =v1|Ωεsuch that
−(
vε,dφ
dt)
Ωε,T+ε(fε,φ)Ωε,T=−Df(∇vε,∇φ)Ωε,T+ (h,φ)Ωε,T,
∀φ∈ W 0(Ωε),(2.58)
and
findwε∈ W(Sε),wε(0) =w1|Sεsuch that
−(
wε,dφ
dt)
Sε,T+a(wε,φ)Sε,T= (fε,φ)Sε,T,∀φ∈ W(Sε).(2.59)
Proposition 2.20 There exists a unique weak solution uε= (vε, wε)of the system (2.58)-
(2.59).
Remark 2.21 The solution of (2.59) can be written as
wε(t,x) =w1(x)e−(a+γ)t+γ∫t
0e−(a+γ)(t−s)g(vε(s,x)) ds
or, if we denote by ⋆the convolution with respect to time, as
wε(·,x) =w1(x)e−(a+γ)t+γr(·)⋆g(vε(·,x)),
where
r(ρ) =e−(a+γ)ρ.
The solution vεof problem ( Vε) being defined only on Ωε, we need to extend it to the
whole of Ω to be able to state the convergence result. In order to do that, we use Lemma
2.1. We also recall the following well-known result (see [73]):
Lemma 2.22 There exists a positive constant C, independent of ε, such that, for any v∈Vε,
∥v∥2
L2(Sε)≤C(ε−1∥v∥2
L2(Ωε)+ε∥∇v∥2
L2(Ωε)).
40 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
2.2.2 The main result
Theorem 2.23 (Theorem 2.5 in [75]) One can construct an extension Pεvεof the solution
vεof the problem (Vε)such that
Pεvε⇀v weakly in V,
wherevis the unique solution of the following limit problem:
∂v
∂t(t,x) +F0(t,x)−n∑
i,j=1qij∂2v
∂xi∂xj(t,x) =h(t,x), t> 0, x∈Ω,
v(t,x) = 0, t> 0, x∈∂Ω,
v(t,x) =v1(x), t = 0, x∈Ω,(2.60)
with
F0(t,x) =|∂F|
|Y⋆|γ[
g(v(t,x))−w1(x)e−(a+γ)t−γr(·)⋆g(v(·,x))(t)]
.
Here,Q= ((qij))is the homogenized matrix, whose entries are defined in (2.7). Moreover,
the limit problem for the surface concentration is:
∂w
∂t(t,x) + (a+γ)w(t,x) =γg(v(t,x)), t> 0, x∈Ω,
w(t,x) =w1(x), t = 0, x∈Ω(2.61)
andwcan be written as
w(t,x) =w1(x)e−(a+γ)t+γr(t)⋆g(v(t,x)).
Remark 2.24 The weak formulation of problem (2.60) is:
findv∈ W 0(Ω),v(0) =v1such that
−(
v,dφ
dt)
Ω,T+ (F0,φ)Ω,T=−(Q∇v,∇φ)Ω,T+ (h,φ)Ω,T
∀φ∈ W 0(Ω).(2.62)
Proof of Theorem 2.23. The proof is divided into several steps (see [75]). The first step
consists in proving the uniqueness of the limit problem (2.62). This is stated in the following
proposition, proven in [75]:
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 uε= (vε,wε), asε→0. To achieve this goal, some a priori estimates
on this solution are required (for a detailed proof, see [75]).
HOMOGENIZATION OF REACTIVE FLOWS 41
Proposition 2.26 Letvεandwεbe the solutions of the problem (2.53)-(2.56). There exists
a positive constant C, independent of ε, such that
∥wε(t)∥2
Sε≤(∥wε(0)∥2
Sε+γ
δ∥g(vε)∥2
Sε, t)eγδt,∀t≥0,∀δ>0,
∂wε
∂t
2
Sε, t≤C(∥wε(0)∥2
Sε+∥g(vε)∥2
Sε, t),∀t≥0,
∥vε(t)∥2
Ωε≤C,∥∇vε(t)∥2
Ωε, t≤C
and
∂vε
∂t(t)
2
Ωε≤C.
The last step is the limit passage and the identification of the homogenized problem. Let
vε∈ W 0(Ωε) be the solution of the variational problem (2.58) and let Pεvεbe the extension
ofvεinside 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 ε, such that
∥Pεvε(t)∥Ω+∥∇Pεvε∥Ω,t+∥∂t(Pεvε)(t)∥Ω≤C,
for allt≤T. Therefore, by passing to a subsequence, still denoted by Pεvε, we can assume
that there exists v∈ Vsuch that the following convergences hold:
Pεvε⇀v weakly in V,
∂t(Pεvε)⇀∂ tvweakly in H,
Pεvε→vstrongly in H.
It remains now to identify the limit equation satisfied by v. To this end, we have to pass to
the limit, with ε→0, in all the terms of (2.58). The most difficult part consists in passing
to the limit in the term containing the nonlinear function g. For this one, using the same
techniques as those used in Section 2.1, we can prove that
⟨µε,φg(Pεvε(t))⟩ →|∂F|
|Y|∫
Ωφg(v(t))dx∀φ∈ D.
We are now in a position to use Lebesgue’s convergence theorem. Using the above pointwise
convergence, the a priori estimates stated in Proposition 2.26 and the growth condition (2.57),
we get
lim
ε→0εγ(g(vε),φ)Sε,T=|∂F|
|Y|γ(g(v),φ)Ω, T.
For the rest of the terms, the proof is standard and we obtain immediately (2.60). Since
v∈ W 0(Ω) (i.e.v= 0 on∂Ω) andvis uniquely determined, the whole sequence Pεvε
converges to vand Theorem 2.23 is proven.
42 HOMOGENIZATION RESULTS FOR HETEROGENEOUS 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) =|v|p−1v,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 R×R, satisfying the condition g(0) = 0 and with D(g) =R. Moreover, gis
continuous and satisfies |g(v)| ≤C(1 +|v|). 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.
Letλ>0 be given. We consider the approximating problems:
findvε
λ∈ W 0(Ωε),vε
λ(0) =v1|Ωεsuch that
−(
vε
λ,dφ
dt)Ωε,T+ε(fε
λ,φ)
Ωε,T=−Df(∇vε
λ,∇φ)Ωε,T+ (h,φ)Ωε,T,
∀φ∈ W 0(Ωε)(2.63)
and
findwε
λ∈ W(Sε),wε
λ(0) =w1|Sεsuch that
−(
wε
λ,dφ
dt)
Sε,T+a(wε
λ,φ)Sε,T= (fε
λ,φ)Sε,T,∀φ∈ W(Sε),(2.64)
where
fε
λ=γ(gλ(vε
λ)−wε
λ)
and
gλ=I−Jλ
λ,
with
Jλ= (I+λ∂G)−1.
We remark that gλis a non-decreasing Lipschitz function, with gλ(0) = 0.
Problem (2.63)-(2.64) has a unique solution ( vε
λ,wε
λ), for every λ>0 (see [46] and [162]).
As we saw in Section 2.2.1, we can express wε
λin terms of vε
λ; therefore, it is enough to get
a problem only for vε
λand in what follows we shall focus our attention only on getting the
limit problem for vε
λ.
Mollifyinggλto make it a smooth function (see [34]) and using the results of the previous
subsection, for any λ>0, we get
Pεvε
λ→vλstrongly in H(Ω).
Then, it is not difficult to see that, proving suitable a priori estimates (classical energy
estimates) on the solutions vλ, we can ensure, via compactness arguments (see [30]), the
strong convergence of vλ, asλ→0, tov, the unique solution of problem (2.60). Hence
vλ→vstrongly in H(Ω).
HOMOGENIZATION OF REACTIVE FLOWS 43
Finally, since
∥Pεvε−v∥Ω,T≤ ∥Pεvε−Pεvε
λ∥Ω,T+∥Pεvε
λ−vλ∥Ω,T+∥vλ−v∥Ω,T,
we get the strong convergence of PεvεtovinH(Ω).
Remark 2.27 The conclusion of the above theorem remains true for more general situations.
It is the case of the so-called zeroth-order reactions, in which, formally, gis given by the
discontinuous function g(v) = 0 , ifv≤0andg(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 β(v) ={0}ifv <0,β(0) = [0,1]andβ(v) ={1}
ifv > 0. The existence and uniqueness of a solution can be found, for instance, in [46].
The solution is obtained by passing to the limit in a sequence of problems associated to a
monotone sequence of Lipschitz functions approximating βand so the results of this section
remain true.
2.2.4 Laplace-Beltrami model with oscillating coefficients
In problem (2.53)-(2.56), the rate aof chemical reactions on Sεand the adsorption coefficient
γwere assumed to be constant. A more realistic model implies to assume that the surface ∂F
is chemically and physically heterogeneous, which means that aandγare rapidly oscillating
functions, i.e.
aε(x) =a(x
ε)
, γε(x) =γ(x
ε)
,
withaandγpositive functions in W1,∞(Ω) which are Y-periodic (for linear adsorption rates,
see [140]). In this case, vεandwεsatisfy the following system of equations:
(Vε)
∂vε
∂t(t,x)−Df∆vε(t,x) =h(t,x), x∈Ωε, t> 0,
vε(t,x) = 0, x∈∂Ω, t> 0,
vε(t,x) =v1(x), x∈Ωε, t= 0,(2.65)
−Df∂vε
∂ν(t,x) =εfε(t,x), x∈Sε, t> 0, (2.66)
and
(Wε){∂wε
∂t(t,x) +aε(x)wε(t,x) =fε(t,x), x∈Sε, t> 0,
wε(t,x) =w1(x), x∈Sε, t= 0,(2.67)
where
fε(t,x) =γε(x)(g(vε(t,x))−wε(t,x)). (2.68)
If we denote y=x/ε, then the main result in this case is the following one:
Theorem 2.28 (Theorem 4.1 in [75]) The effective behavior of vandwis governed by the
following system:
∂v
∂t(t,x) +G0(t,x)−n∑
i,j=1qij∂2v
∂xi∂xj=h(t,x), t> 0, x∈Ω,
v(t,x) = 0t>0, x∈∂Ω,
v(t,x) =v1(x)t= 0, x∈Ω,(2.69)
44 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
and
{∂w
∂t(t,x,y ) + (a(y) +γ(y))w(t,x,y ) =γ(y)g(v(t,x)), t> 0, x∈Ω, y∈∂F
w(t,x,y ) =w1(x)t= 0, x∈Ω, y∈∂F,(2.70)
where
G0(t,x) =1
|Y⋆|∫
∂Ff0(t,x,y ) dσ
and
f0=γ(y)(g(v(t,x))−w(t,x,y )).
Here,Q= ((qij))is the classical homogenized matrix, defined 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)+γ(y))t+γ(y)∫t
0e−(a(y)+γ(y))(t−s)g(v(s,x)) ds,
or, using the convolution notation
w(t,x,y ) =w1(x)e−(a(y)+γ(y))t+γ(y)r(·,y)⋆g(v(·,x))(t),
with
r(τ,y) =e−(a(y)+γ(y))τ.
Moreover, let us notice that (2.69)-(2.70) imply that v(t,x) satisfies the following equation
∂v
∂t(t,x)−n∑
i,j=1qij∂2v
∂xi∂xj(t,x) +F0(t,x) =h(t,x), t> 0, x∈Ω, (2.71)
with
F0(t,x) =1
|Y⋆|∫
∂F{
γ(y)[g(v(t,x))−w1(x)e−(a(y)+γ(y))t−γ(y)r(·,y)⋆g(v(·,x))(t)]}
dσ.
Remark 2.29 The above adsorption model can be slightly generalized by allowing surface
diffusion on Sε. In fact, the chemical substances can creep on the surface and this effect is
similar to a surface-like diffusion. From a mathematical point of view, we can model this
phenomenon by introducing a diffusion term in the law governing the evolution of the surface
concentration wε. This new term is the properly rescaled Laplace-Beltrami operator. This
implies that the first equation in (2.67) has to be replaced by
∂wε
∂t(t,x)−ε2E∆εwε(t,x) +aε(x)wε(t,x) =fε(t,x)x∈Sε, t> 0,
whereE > 0is the diffusion constant on the surface Sεand∆εis the Laplace-Beltrami
operator on Sε.
HOMOGENIZATION OF REACTIVE FLOWS 45
In this case, the homogenized limit is the following one:
∂w
∂t(t,x,y )−E∆∂F
yw(t,x,y ) + (a(y) +γ(y))w(t,x,y ) =γ(y)g(v(t,x)),
fort>0, x∈Ω, y∈∂F. Here, ∆∂Fdenotes the Laplace-Beltrami operator on ∂Fand the
subscriptyindicates the fact that the derivatives are taken with respect to the local variable
y. The limit problem involves the solution of a reaction-diffusion system with respect to an
additional microvariable. Also, notice that the local behavior is no longer governed by an
ordinary differential equation, but by a partial differential one.
Remark 2.30 We notice that the bulk behavior of system (Vε)-(Wε)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 γr⋆g , which shows that we clearly have a memory term in the principal part of
our diffusion-reaction equation (2.71).
The above results can be extended to include the case in which we add a space-dependent
nonlinear reaction rate β=β(x,v) in the interior of the domain and we consider a space-
dependent nonlinear adsorption rate g=g(x,v) and a non-constant diffusion matrix Dε(x).
More precisely, we analyze the asymptotic behavior, as ε→0, of the following coupled system
of equations:
∂vε
∂t(t,x)−div(Dε(x)∇vε(t,x)) +β(x,vε) =h(t,x), x∈Ωε, t> 0,
vε(t,x) = 0, x∈∂Ω, t> 0,
vε(t,x) =v1(x), x∈Ωε, t= 0,(2.72)
−Dε(x)∇vε(t,x)·ν=εfε(t,x), x∈Sε, t> 0, (2.73)
and
∂wε
∂t(t,x) +aε(x)wε(t,x) =fε(t,x), x∈Sε, t> 0,
wε(t,x) =w1(x), x∈Sε, t= 0,(2.74)
where
fε(t,x) =γε(x)(g(x,vε(t,x))−wε(t,x)). (2.75)
Assumptions.
1) The diffusion matrix is defined as being Dε(x) =D(x/ε), whereD=D(y) is a
matrix-valued function on Rnwhich isY-periodic. We shall assume that
D∈L∞(Ω)n×n,
Dis a symmetric matrix,
for some 0<γ <λ, γ |ξ|2≤D(y)ξ·ξ≤λ|ξ|2∀ξ, y∈Rn.
46 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
2) The function β=β(x,v) is continuous, monotonously non-decreasing with respect to
vfor anyxand such that β(x,0) = 0.
3) The function g=g(x,v) is continuously differentiable, monotonously non-decreasing
with respect to vfor anyxand withg(x,0) = 0. We suppose that there exist C≥0 and two
exponentsqandrsuch that
|β(x,v)| ≤C(1 +|v|q) (2.76)
and
∂g
∂v ≤C(1 +|v|q),
∂g
∂xi ≤C(1 +|v|r) 1≤i≤n,(2.77)
with 0 ≤q<n/ (n−2) and with 0 ≤r<n/ (n−2) +q.
Using the theory of semilinear monotone problems (see [46] and [162]), we know that
there exists a unique weak solution uε= (vε,wε) 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 Pεvεof the solution vεof the problem (2.72)-
(2.75) such that
Pεvε⇀v weakly inL2(0,T;H1(Ω)),
wherevis the unique solution of the following limit problem:
∂v
∂t(t,x)−div(D0∇v) +β(x,v) +F0(t,x) =h(t,x), t> 0, x∈Ω,
v(t,x) = 0, t> 0, x∈∂Ω,
v(t,x) =v1(x), t = 0, x∈Ω,(2.78)
with
F0(t,x) =
1
|Y⋆|∫
∂F{
γ(y)[g(x,v(t,x))−w1(x)e−(a(y)+γ(y))t−γ(y)r(·,y)⋆g(x,v(·,x))(t)]}
dσ.
The limit problem for the surface concentration is:
{∂w
∂t(t,x,y ) + (a(y) +γ(y))w(t,x,y ) =γ(y)g(x,v(t,x)), t> 0, x∈Ω, y∈∂F
w(t,x,y ) =w1(x)t= 0, x∈Ω, y∈∂F.(2.79)
Here,D0= ((d0
ij))is the homogenized matrix, defined by:
d0
ij=1
|Y∗|∫
Y∗(
dij(y) +dik(y)∂χj
∂yk)
dy,
in terms of the functions χj, j= 1,…,n, solutions of the cell problems
−divyD(y)(∇yχj+ej) = 0 inY∗,
D(y)(∇χj+ej)·ν= 0 on∂F,
χj∈H1
#Y(Y⋆),∫
Y⋆χj= 0,
where ei,1≤i≤n, are the elements of the canonical basis in Rn.
The constant matrix D0is symmetric and positive-definite.
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)+γ(y))t+γ(y)∫t
0e−(a(y)+γ(y))(t−s)g(x,v(s,x))ds,
or, using the convolution notation
w(t,x,y ) =w1(x)e−(a(y)+γ(y))t+γ(y)r(·,y)⋆g(x,v(·,x))(t),
with
r(τ,y) =e−(a(y)+γ(y))τ.
Remark 2.33 If we consider the case in which we have diffusion of the chemical species on
the surface Sε, i.e.
∂wε
∂t(t,x)−ε2E∆εwε(t,x) +aε(x)wε(t,x) =fε(t,x)x∈Sε, t> 0,
whereE > 0is the diffusion constant on the surface Sεand∆εis the Laplace-Beltrami
operator on Sε, then instead of (2.79) we get the following local partial differential equation:
∂w
∂t(t,x,y )−E∆∂F
yw(t,x,y ) + (a(y) +γ(y))w(t,x,y ) =γ(y)g(x,v(t,x)),(2.80)
fort>0, x∈Ω, y∈∂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 [66], C. Conca, J.I. D´ ıaz and C. Timofte [75], or G. Allaire and H.
Hutridurga [6]). The results presented in this section constitute a generalization of some of
the results obtained in [75] and [137], by considering heterogeneous fluids, 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-diffusion equations involving nonlinear
reaction-terms and nonlinear boundary conditions. Also, it would be of interest to deal with
the case of other geometries of the porous media under consideration or with the case of more
general nonlinear, even discontinuous, kinetics.
47
48 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
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 flow 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 difficult free boundary problems arising in
the study of flow through porous media. We can mention here the representative papers
of G. Fichera [114], G. Signorini [204], G. Stampacchia [206], G. Stampacchia and J. L.
Lions [164], L Boccardo and P. Marcellini [40], L. Boccardo and F. Murat [41], H. Br´ ezis,
U. Mosco [178], R. Glowinski, J.L. Lions and R. Tr´ emoli` eres [121], Duvaut and Lions [101],
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 [46], [51], [101], [155], [121], [185] and [133]. For homogenization
results for variational inequalities, we refer, e.g., to D. Cioranescu and F. Murat [69], Yu. A.
Kazmerchuk and T. A. Mel’nyk [153], or G.A. Yosifian [225].
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 [52], [55], [77], [79], 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 [114]) 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 [46] and [101].
The chapter is based on the papers [52], [55], [77], [79], and [210].
49
50 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
3.1 Homogenization results for Signorini’s type problems
In this section, we shall present some homogenization results, obtained in [77] 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 ε. The main feature of these kind of problems
is the existence of a critical size of the perforations that separates different emerging phe-
nomena as ε→0. In the critical case, it is shown in [77] 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 [77] constitute a generalization of those obtained in the the well-known
pioneering work of D. Cioranescu and F. Murat [69]. 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 uε≥0 acts
on the holes (which includes the classical Dirichlet condition uε= 0 onSε). In [77], 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 Sε.
3.1.1 The microscopic problem
Let Ω be a smooth bounded connected open subset of Rnand letFbe another open bounded
subset of Rn, with a smooth boundary ∂F(of classC2). We assume that 0 belongs to Fand
thatFis star-shaped with respect to 0. Since Fis bounded, to simplify matters, without
loss of generality, we shall assume that F⊂Y, whereY= (−1
2,1
2)nis the representative cell
inRn. We setY∗=Y\F. Letr:R+→R+be a continuous map, characterizing the size of
the holes. We assume that
lim
ε→0r(ε)
ε= 0 andr(ε)<ε/2 (3.1)
or
r(ε)∼ε. (3.2)
The first situation corresponds to the case of small holes , while the last one covers the case
ofbig holes .
For eachεand for any integer vector k∈Zn, letFε
k=εk+r(ε)F. Also, let us denote by
Fεthe set of all the holes contained in Ω, i.e.
Fε=∪{
Fε
k|Fε
k⊂Ω, k∈Zn}
.
HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 51
Set Ωε= Ω\FεandSε=∪{∂Fε
k|Fε
k⊂Ω,k∈Zn}. So,∂Ωε=∂Ω∪Sε.
Let us consider a family of inhomogeneous media occupying the region Ω, parameter-
ized byεand represented by n×nmatricesAε(x) of real-valued coefficients defined on Ω.
Therefore, the positive parameter εwill also define a length scale measuring how densely
the inhomogeneities are distributed in Ω. 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 ε, 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
Aε(x) involves rapidly oscillating coefficients).
We define the following nonempty closed convex subset of H1(Ωε):
Kε={
v∈H1(Ωε)|v= 0 on∂Ω, v≥0 onSε}
. (3.3)
Our main motivation is to study the asymptotic behavior of the solution of the following
variational problem in Ωε:
Finduε∈Kεsuch that∫
ΩεAεDuεD(vε−uε) dx≥∫
Ωεf(vε−uε) dx∀vε∈Kε,(3.4)
wherefis a given function in L2(Ω).
The solution uεof (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 uεand two subsets
Sε
0andSε
+such thatSε
0∪Sε
+=Sε, Sε
0∩Sε
+=∅, and
−div(AεDuε) =fin Ωε,
uε= 0 onSε
0, AεDuε·ν≥0 onSε
0,
uε>0 onSε
+, AεDuε·ν= 0 onSε
+,(3.5)
whereνis the exterior unit normal to the surface Sε. Thus, on Sε, there are two a priori
unknown subsets Sε
0andSε
+whereuεsatisfies complementary boundary conditions coming
from the following global constraints:
uε, AεDuε·ν≥0 anduεAεDuε·ν= 0 onSε. (3.6)
We shall consider periodic structures defined by
Aε(x) =A(x
ε)
,
whereA=A(y) is a matrix-valued function on Rnwhich isY-periodic and satisfies the
following conditions:
A∈L∞(Ω)n×n,
Ais a symmetric matrix,
for some 0<α<β, α |ξ|2≤A(y)ξ·ξ≤β|ξ|2∀ξ, y∈I Rn.
For simplicity, we further assume that Ais continuous with respect to y. Under the above
structural hypotheses and the conditions fulfilled by Kε, 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.
52 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
3.1.2 The macroscopic models
Several situations can occur depending on the asymptotic behavior of the size of the holes
and there exists a critical size that separates different behaviors of the solution uεasε→0.
This size is of order εn/(n−2)ifn≥3 and of order exp( −1/ε2) ifn= 2. For simplicity, in what
follows, we shall discuss only the case n≥3 (the case n= 2 can be treated in an analogous
manner).
In the critical case, it was proven in [77] that the solution uεof problem (3.4), properly
extended to the whole of Ω, converges to the unique solution of a Dirichlet problem in Ω,
associated with a new operator which is the sum of the standard homogenized one and an
extra term strange term that comes in from the special geometry (the size of the holes). More
precisely, we have the following result:
Theorem 3.1 (Theorem 4.1 in [77]) There exists an extension Pεuεof the solution uεof
the variational inequality (3.4), positive inside the holes, such that
Pεuε⇀u weakly inH1
0(Ω),
whereuis the unique solution of
u∈H1
0(Ω),∫
ΩA0DuDv dx−⟨
µ0u−,v⟩
H−1(Ω),H1
0(Ω)=∫
Ωfvdx∀v∈H1
0(Ω).(3.7)
Here,A0is the homogenized matrix, whose entries are defined as follows:
a0
ij=1
|Y|∫
Y(
aij(y) +aik(y)∂χj
∂yk)
dy,
in terms of the functions χj, j= 1,…,n, solutions of the cell problems
−divy(A(y)Dy(yj+χj)) = 0 inY,
χj−Yperiodic
andµ0is given by
µ0= inf
w∈H1(Rn){∫
RnA(0)DwDw dx|w≥1q.e. onF}
. (3.8)
The constant matrix A0is symmetric and positive-definite.
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 reflected by the presence
of the homogenized matrix A0, and those due to the critical size of the holes are reflected by
the appearance of a strange term µ0. The other ingredient contained in (3.7) is the spreading
effect of the unilateral condition uε≥0imposed on Sε, 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 u−that is penalized at the limit.
HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 53
The proof of Theorem 3.1, given in [77], is based on the use of a technical result of E. De
Giorgi [87] for matching boundary conditions for minimizing sequences. This result allowed
us to modify sequences of functions near the holes Br(ε)(εi) 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 [77]. The first
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 uεconverges to the
solution of a classical homogenized Dirichlet problem in Ω associated to the matrix A. The
second case is that when the size of the holes is bigger than the critical one, but still smaller
than the period ε. The holes being big enough, the positivity constraint of the solution uε
imposed only on Sεwill become a positivity condition, u≥0, 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 ε. The solution uε, properly extended to the whole of Ω, converges in this
case to an obstacle problem, associated to the homogenization of a periodic heterogeneous
and perforated medium. The influence of the holes comes twofold: on one hand, they spread
the positivity condition on Sεto the whole of Ω and, on the other one, their particular size
affects the homogenized medium. More precisely, for the case of holes of the same size as the
period, we have the following result (see [77]):
Theorem 3.3 (Theorem 4.6 in [77]) There exists an extension Pεuεof the solution uεof
the variational inequality (3.4), positive inside the holes, such that
Pεuε⇀u weakly inH1
0(Ω),
whereuis the unique solution of
u∈H1
0(Ω), u≥0inΩ,
∫
ΩA0DuDu dx−2∫
Ωfudx≤∫
ΩA0DvDv dx−2∫
Ωfvdx,
∀v∈H1
0(Ω), v≥0inΩ.(3.9)
Here,A0= (a0
ij)is the homogenized matrix, defined by
a0
ij=1
|Y∗|∫
Y∗(
aij(y) +aik(y)∂χj
∂yk)
dy,
in terms of the functions χj, j= 1,…,n, solutions of the cell problems
−divyA(y)(Dyχj+ej) = 0 inY∗,
A(y)(Dχj+ej)·ν= 0 on∂F,
χj∈H1
#Y(Y⋆),∫
Y⋆χjdy= 0,
54 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
where ei,1≤i≤n, are the elements of the canonical basis in Rn. The constant matrix A0
is symmetric and positive-definite.
Let us notice that the variational inequality in (3.9) can be written as
∫
ΩA0DuD (v−u)dx≥∫
Ωf(v−u)dx.
As mentioned before, the method we followed in [77] is the energy method of L. Tar-
tar [207]. However, it is worth mentioning that the Γ-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 [82] and
of A. Braides and A. Defranceschi [44]. 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 fixed domains, by D. Cioranescu, A.
Damlamian, G. Griso in [59], [58] and by A. Damlamian in [85]. Their results were extended
to perforated domains by D. Cioranescu, P. Donato, R. Zaki [65], [67] and, further, by D.
Cioranescu, A. Damlamian, G. Griso, D. Onofrei in [60], by D. Onofrei in [187] and by A.
Damlamian, N. Meunier in [86] for the case of small holes.
The periodic unfolding method brings significant simplifications 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 [52]
and generalizing the corresponding results in [77]. More precisely, the asymptotic behavior
of a class of elliptic equations with highly oscillating coefficients, in an ε-periodic perforated
structure, with two holes of different sizes in each period, will be analyzed. Two distinct
conditions, one of Signorini’s type and another one of Neumann type, are imposed on the
corresponding boundaries of the holes, while on the exterior fixed boundary of the perfo-
rated domain, an homogeneous Dirichlet condition is prescribed. As mentioned in [52], the
main feature of this type of problems is represented by the existence of a critical size of the
perforations that separates different emerging phenomena as the small parameter εtends to
zero. In this critical case, it was proven in [52] 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 [69], putting into evidence, in the case of critical holes,
HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 55
the appearance of a strange term. Their results were extended, using different techniques,
to heterogeneous media by N. Ansini, A. Braides [19], G. Dal Maso, F. Murat [84] and D.
Cioranescu, A. Damlamian, G. Griso, D. Onofrei in [60]. Recently, A. Damlamian, N. Meunier
[86] 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 [76] and D. Onofrei [187]. For problems with Robin
or nonlinear boundary conditions we refer, for instance, to D. Cioranescu, P. Donato [62], D.
Cioranescu, P. Donato, H. Ene [64], D. Cioranescu, P. Donato, R. Zaki [65] and A. Capatina,
H. Ene [53]. 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 [68].
The non-standard feature of the problem we present here is given by the presence, in
each period, of two holes of different sizes and with different conditions (3.10) 2,3imposed on
their boundaries. More precisely, we consider the case of Signorini and, respectively, criti-
cal Neumann holes. The Signorini condition (3.10) 2(see [204]) implies that the variational
formulation (3.11) of our problem is expressed as an inequality, which creates further difficul-
ties. Problems involving such boundary conditions arise in groundwater hydrology, chemical
flows 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 [101] and U. Hornung [137].
3.2.1 Setting of the microscopic problem
Let us briefly describe now the new geometry of the problem. Let Ω ⊂Rn,n≥3, be a
bounded open set such that |∂Ω|= 0 and let Y=(
−1
2,1
2)n
be the reference cell. We
consider an εY-periodic perforated structure with two kind of holes: some of size εδ1and the
other ones of size εδ2, withδ1andδ2depending on εand going to zero as εgoes to zero. More
precisely, we consider two open sets BandFwith smooth boundaries such that B⊂⊂Y,
F⊂⊂YandB∩F= Ø and we denote the above mentioned holes by
Bεδ1=∪
ξ∈Znε(ξ+δ1B),
Fεδ2=∪
ξ∈Znε(ξ+δ2F).
LetYδ1δ2=Y\(δ1B∪δ2F) be the part occupied by the material in the cell and suppose that
it is connected. The perforated domain Ω ε,δ1δ2with holes of size of order εδ1and of size of
orderεδ2at the same time, is defined by
Ωε,δ1δ2= Ω\(Bεδ1∪Fεδ2) ={
x∈Ω|{x
ε}
Y∈Yδ1δ2}
.
LetA∈L∞(Ω)n×nbe aY-periodic symmetric matrix. We suppose that there exist two
positive constants αandβ, with 0<α<β , such that
α|ξ|2≤A(y)ξ·ξ≤β|ξ|2∀ξ∈Rn,∀y∈Y .
56 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Moreover, we assume that Ais continuous at the point 0.
Given aY-periodic function g∈L2(∂F) and a function f∈L2(Ω), we consider the
following microscopic problem:
−div (Aε∇uε,δ1δ2) =fin Ω ε,δ1δ2,
uε,δ1δ2≥0, Aε∇uε,δ1δ2·νB≥0, u ε,δ1δ2Aε∇uε,δ1δ2·νB= 0 on∂Bεδ1,
Aε∇uε,δ1δ2·νF=gεδ2on∂Fεδ2,
uε,δ1δ2= 0 on∂extΩε,δ1δ2,(3.10)
where
Aε(x) =A(x
ε)
and
gεδ2(x) =g(1
δ2{x
ε}
Y)
a.e.x∈∂Fεδ2.
In (3.10),νBandνFare the unit exterior normals to the sets Bεδ1and, respectively, Fεδ2.
In order to obtain a variational formulation of problem (3.10), we introduce the space
Vε
δ1δ2={v∈H1(Ωε,δ1δ2)|v= 0 on∂extΩε,δ1δ2}
and the convex set
Kε
δ1δ2={v∈Vε
δ1δ2|v≥0 on∂Bεδ1}.
Then, the variational formulation of (3.10) is the following variational inequality:
(Pε,δ1δ2)
Finduε,δ1δ2∈Kε
δ1δ2such that
∫
Ωε,δ1δ2Aε∇uε,δ1δ2·(∇v− ∇uε,δ1δ2) dx≥∫
Ωε,δ1δ2fε(v−uε,δ1δ2) dx
+∫
∂Fεδ2gεδ2(v−uε,δ1δ2) ds∀v∈Kε
δ1δ2.(3.11)
Classical results for variational inequalities (see, for example, [206], [121], [45]) 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
ε, δ1, δ2→0. Following [52], we consider the case in which
k1= lim
ε→0δn
2−1
1
ε,0<k1<∞,
k2= lim
ε→0δn−1
2
ε,0<k2<∞,(3.12)
which signifies 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 ε→0 instead of writing ( ε,δ1,δ2)→
(0,0,0).
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={v∈L2∗(Rn) ;∇v∈L2(Rn), v= constant on B}
where 2∗is the Sobolev exponent2n
n−2associated to 2. Also, for i= 1,…,n , letχibe the
solution of the cell problem
χi∈H1
per(Y),
∫
YA∇(χi−yi)· ∇ϕdy= 0∀ϕ∈H1
per(Y)(3.13)
andθbe the solution of the problem
θ∈KB, θ(B) = 1,
∫
Rn\BA(0)∇θ· ∇vdz= 0∀v∈KBwithv(B) = 0.(3.14)
For the special geometry of this problem, we need to introduce, following [57] and [60], two
unfolding operators TεandTεδ, the first one corresponding to the case of fixed domains and
the second one to the case of domains with small inclusions. For defining the first operator,
we have to introduce some notation. For x∈Rn, we denote by [ x]Yits integer part k∈Zn,
such thatx−[x]Y∈Yand we set {x}Y=x−[x]Yforx∈Rn. So, forx∈Rn, we have
x=ε([x
ε]
+{x
ε})
. LetYk=Y+k, fork∈Zn. We consider the following sets:
bZε={
k∈Zn|εYk⊂Ω}
,bΩε= int∪
k∈bZε(
εYk)
,Λε= Ω\bΩε.
De nition 3.4 For any function φ∈Lp(Ω), withp∈[1,∞), we define the periodic unfolding
operator Tε:Lp(Ω)→Lp(Ω×Y)by the formula
Tε(φ)(x,y) =
φ(
ε[x
ε]
Y+εy)
for a.e. (x,y)∈bΩε×Y,
0 for a.e. (x,y)∈Λε×Y.
The operator Tεis linear and continuous from Lp(Ω) toLp(Ω×Y). We recall here some
useful properties of this operator (see, for instance, [57]):
(i) ifφandψare two Lebesgue measurable functions on Ω, one has
Tε(φψ) =Tε(φ)Tε(ψ);
(ii) for every φ∈L1(Ω), one has
1
|Y|∫
Ω×YTε(φ)(x,y) dxdy=∫
bΩεφ(x) dx=∫
Ωφ(x) dx−∫
Λεφ(x) dx;
58 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
(iii) if {φε} ⊂L2(Ω) is a sequence such that φε→φstrongly in L2(Ω), then
Tε(φε)−→φstrongly in Lp(Ω×Y);
(iv) ifφ∈L2(Y) isY-periodic and φε(x) =φ(x/ε), then
Tε(φε)−→φstrongly in L2(Ω×Y);
(v) ifφε⇀φ weakly inH1(Ω), then there exists a subsequence and bφ∈L2(Ω;H1
per(Y))
such that
Tε(∇φε)⇀∇xφ+∇ybφweakly inL2(Ω×Y).
For domains with small holes, we need to introduce an unfolding operator depending on
two parameters εandδ. We recall now its definition (for details, see [60]). To this end, let
us consider domains with εY-periodically distributed holes of size εδ, forδ>0 going to zero
withεtending to zero. More precisely, for a given open set B⊂⊂Y, we denote Y∗
δ=Y\δB.
The perforated domain Ω∗
εδis defined by
Ω∗
εδ={
x∈Ω {x
ε}
∈Y∗
δ}
.
If we consider functions which vanish on the whole boundary of the perforated domain, i.e.
functions belonging to H1
0(Ω∗
εδ), then we can extend them by zero to the whole of Ω. In this
case, we shall not distinguish between functions in H1
0(Ω∗
εδ) and their extensions in H1
0(Ω).
De nition 3.5 For anyφ∈Lp(Ω), withp∈[1,∞), we define the periodic unfolding operator
Tεδby the formula
Tεδ(φ)(x,z) =
Tε(φ)(z,δz)if(x,z)∈bΩε×1
δY,
0 otherwise.
By using the change of variable z= (1/δ)y, one can obtain similar properties for the operator
Tεδto those stated for Tε(see [60]). Further, following [68] and [187], we briefly recall here
the definition of the boundary unfolding operator Tb
εδ.
De nition 3.6 For anyφ∈Lp(∂Bεδ), withp∈[1,∞), we define the boundary unfolding
operator Tb
εδby the formula
Tb
εδ(φ)(x,z) =φ(
ε[x
ε]
Y+εδz)
a.e. forx∈Rn,z∈∂B.
The main convergence result obtained in [52] is stated in the following theorem.
Theorem 3.7 (Theorem 3.1 in [52]) Let uε,δ1δ2be the solution of the variational inequality
(3.11). Under the above hypotheses, there exists u∈H1
0(Ω)such that
Tε(uε,δ1δ2)⇀u weakly inL2(Ω;H1(Y)), (3.15)
HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 59
whereu∈H1
0(Ω)is the unique solution of the homogenized problem
u∈H1
0(Ω),
∫
ΩAhom∇u· ∇φdx−k2
1∫
Ωµu−φdx =∫
Ωfφdx
+k2|∂F|M∂F(g)∫
Ωφdx∀φ∈H1
0(Ω).(3.16)
In (3.16),Ahomis the homogenized matrix, defined, in terms of the solution χiof (3.13), by
Ahom
ij=∫
Y(
aij(y)−n∑
k=1aik(y)∂χj
∂yk(y))
dy
andµis the capacity of the set B, given by
µ=∫
Rn\BA(0)∇zθ· ∇zθdz,
whereθverifies (3.14).
Remark 3.8 The limit problem (3.16) contains two extra terms, generated by the suitable
sizes of our holes. Also, let us remark in (3.16) the spreading effect of the unilateral condition
imposed on the boundary of the Signorini holes: the strange term, depending on the matrix
A, charges only the negative part u−of 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:
{finduε,δ1δ2∈Kε
δ1δ2such that
Jε
δ1δ2(uε,δ1δ2)≤Jε
δ1δ2(v)∀v∈Kε
δ1δ2,(3.17)
where
Jε
δ1δ2(v) =1
2∫
Ωε,δ1δ2Aε∇v· ∇vdx−∫
Ωε,δ1δ2fvdx−∫
∂Fεδ2gεδ2vds. (3.18)
Let us prove that
lim sup
ε→0Jε
δ1δ2(uε,δ1δ2)≤J0(φ)∀φ∈ D(Ω), (3.19)
where
J0(φ) =1
2∫
ΩAhom∇φ· ∇φdx+1
2k2
1∫
Ωµ(φ−)2dx
−∫
Ωfφdx+k2|∂F|M∂F(g)∫
Ωφdx.(3.20)
60 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Forφ∈ D(Ω), we set
hε(x) =φ(x)−εn∑
i=1∂φ
∂xi(x)χi(x
ε)
,
whereχiis the solution of the problem (3.13).
If we takevεδ1=h+
ε−wεδ1h−
ε, where
wεδ1(x) = 1−θ(1
δ1{x
ε}
Y)
∀x∈Rn,
withθgiven by (3.14), we obtain
Jε
δ1δ2(vεδ1) =I1
ε−I2
ε,
where
I1
ε=1
2∫
Ωε,δ1δ2Aε(∇h+
ε−wεδ1∇h−
ε−h−
ε∇wεδ1)·(∇h+
ε−wεδ1∇h−
ε−h−
ε∇wεδ1) dx,
I2
ε=∫
Ωε,δ1δ2fε(h+
ε−wεδ1h−
ε)dx+∫
∂Fεδ2gε(h+
ε−wεδ1h−
ε) ds.
Using the periodic unfolding operators TεandTεδ1(see [59] and [60]), we get
Tε(Aε)(x,y) =A(y) in Ω ×Y,
Tεδ1(wεδ1)(x,z) =Tε(wεδ1)(x,δ1z) = 1−θ(z) in Ω ×Rn,
Tεδ1(∇wεδ1)(x,z) =−1
εδ1∇zθ(z) in Ω ×Rn.
We also have {
wεδ1⇀1 weakly in H1(Ω),
hε→φstrongly in H1(Ω).(3.21)
Taking into account the properties of the unfolding operator TεandTεδ1, we get the following
convergences
Tε(hε)→φstrongly in L2(Ω×Y),
Tε(∇hε)→ ∇ xφ+∇yφ1strongly in L2(Ω×Y),
Tεδ1(∇hε)→ ∇ xφ+∇yφ1strongly in L2(Ω×Rn),(3.22)
where
φ1=n∑
i=1∂φ
∂xiχi.
By unfolding and by using the fact that {∇h−
ε}εis bounded in ( L2(Ω))n, we can pass to the
limit in the unfolded form of I1
εand we get
lim
ε→0I1
ε=1
2∫
Ω×YA(∇φ+∇yφ1)·(∇φ+∇yφ1) dxdy+
HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 61
1
2k2
1∫
Ω×(Rn\B)A(0)(φ−)2∇zθ· ∇zθdxdz,
which, together with (3.13), yields
lim
ε→0I1
ε=1
2∫
Ω×YAhom∇φ· ∇φdxdy +1
2k2
1∫
Ω×(Rn\B)A(0)(φ−)2∇zθ· ∇zθdxdz.(3.23)
Exactly like in [68], i.e. using the boundary unfolding operator Tb
εδ2, we can pass to the limit
inI2
εand we obtain
lim
ε→0I2
ε=∫
Ωfφdx+k2|∂F|M∂F(g)∫
Ωφdx. (3.24)
Putting together (3.23) and (3.24), we are led to
lim
ε→0Jε
δ1δ2(vεδ1) =J0(φ)∀φ∈ D(Ω) (3.25)
and, thus, we get (3.19).
Let us show now that
lim inf
ε→0Jε,δ1δ2(uε,δ1δ2)≥J0(u). (3.26)
To this end, we decompose our solution into its positive and, respectively, its negative part,
i.e.
uε,δ1δ2=u+
ε,δ1δ2−u−
ε,δ1δ2.
From the problem ( Pε,δ1δ2), it follows that there exists a constant Csuch that
∥uε,δ1δ2∥H1(Ωε,δ1δ2)≤C. (3.27)
Sinceuε,δ1δ2∈Vε
δ1δ2, we can suppose that, up to a subsequence, there exists u∈H1
0(Ω) such
that
Tε(uε,δ1δ2)⇀u weakly inL2(Ω;H1(Y)),
∥u−
ε,δ1δ2−u−∥L2(Ωε,δ1δ2)→0,
Tε(u−
ε,δ1δ2)→u−strongly in L2(Ω×Y).(3.28)
It is not difficult to check that we have
lim inf
ε→0∫
Ωε,δ1δ2Aε∇u+
ε,δ1δ2· ∇u+
ε,δ1δ2dx≥∫
ΩAhom∇u+· ∇u+dx. (3.29)
In order to get (3.26), taking into account that the linear terms pass immediately to the
limit, it remains only to prove that
lim inf
ε→0∫
Ωε,δ1δ2Aε∇u−
ε,δ1δ2· ∇u−
ε,δ1δ2dx≥∫
ΩAhom∇u−· ∇u−dx+k2
1∫
Ωµ(u−)2dx.(3.30)
Since ∫
Ωε,δ1δ2Aε∇(u−
ε,δ1δ2−hεwεδ1)· ∇(u−
ε,δ1δ2−hεwεδ1) dx≥0,
62 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
we have
∫
Ωε,δ1δ2Aε∇u−
ε,δ1δ2· ∇u−
ε,δ1δ2dx≥ −∫
Ωε,δ1δ2Aε(hε)2∇wεδ1· ∇wεδ1dx
−∫
Ωε,δ1δ2Aε(wεδ1)2∇hε· ∇hεdx+ 2∫
Ωε,δ1δ2Aεhε∇u−
ε,δ1δ2· ∇wεδ1dx
+2∫
Ωε,δ1δ2Aεwεδ1∇u−
ε,δ1δ2· ∇hεdx−2∫
Ωε,δ1δ2Aεhεwεδ1∇wεδ1· ∇hεdx
−(
δn
2−1
1
ε)2∫
Ω×RnTεδ1(Aε)(Tεδ1(hε))2∇zθ· ∇zθdxdz
−∫
Ω×YTε(Aε)Tε(∇hε)· Tε(∇hε)(Tε(wεδ1))2dxdy
−2δn
2−1
1
ε∫
Ω×RnTεδ1(Aε)Tεδ1(hε)[
δn/2
1Tεδ1(∇u−
ε,δ1δ2)]
· ∇zθ(z) dxdz
+2∫
Ω×YTε(Aε)Tε(wεδ1)Tε(∇u−
ε,δ1δ2)· Tε(∇hε) dx
+2δn
2−1
1
εδn/2
1∫
Ω×RnTεδ1(Aε)Tεδ1(hε)(1−θ(z))∇zθ· Tεδ1(∇hε) dxdz.
From (3.27) and (3.28) and the properties of the unfolding operators TεandTεδ1(see [60]),
it follows that there exist u1∈L2(Ω;H1
per(Y)) and U1∈L2(Ω;L2
loc(Rn)) such that
Tε(∇u−
ε,δ1δ2)⇀∇u−+∇yu1 weakly inL2(Ω×Y),
δn
2
1Tεδ1(∇u−
ε,δ1δ2)⇀∇zU1 weakly inL2(Ω×Rn).(3.31)
Therefore, we get
lim inf
ε→0∫
Ωε,δ1δ2Aε∇u−
ε,δ1δ2· ∇u−
ε,δ1δ2dx≥ −k2
1∫
Ωµφ2dx
−∫
Ω×YA(∇φ+∇yφ1)·(∇φ+∇yφ1) dxdy−2k1∫
Ω×(Rn\B)A0φ∇zU1· ∇zθdxdz
+2∫
Ω×YA(∇u−+∇yu1)·(∇φ+∇yφ1)dxdy ∀φ∈H1
0(Ω).
(3.32)
SinceTεδ1(u−
ε,δ1δ2) = 0 on Ω ×B, we have U1= 0 on Ω ×B. Thus,W1=U1−k1u−∈L2(Ω;KB).
On the other hand, from the cell problem (3.14), we obtain
−divz(A(0)∇zθ) = 0 in D′(Ω×(Rn\¯B))
which, by Stokes formula, leads to
∫
Rn\BA(0)∇zθ· ∇zψdz=ψ(B)∫
∂BA(0)∇zθ·νBds∀ψ∈KB. (3.33)
HOMOGENIZATION RESULTS FOR UNILATERAL PROBLEMS 63
For almost every x∈Ω,W1(x,·)∈KBand, so, (3.33) gives
∫
Rn\BA(0)∇zθ· ∇zW1dz=W1(x,B)∫
∂BA(0)∇zθ·νBds.
Since ∇zW1=∇zU1andU1(x,B) = 0, we obtain
∫
Rn\BA(0)∇zθ· ∇zU1dz=−k1u−∫
∂BA(0)∇zθ·νBds=−k1u−µ
which implies that
−2k1∫
Ω×(Rn\B)A(0)φ∇zU1· ∇zθdx= 2k2
1∫
Ωµu−φdx.
Takingφ=u−in (3.32) and using the fact thatn∑
i=1∂u−
∂xiχi=u1, we obtain (3.30).
Finally, from (3.19), (3.26) and by the density D(Ω),→H1
0(Ω) , we deduce
lim
ε→0Jε
δ1δ2(uε,δ1δ2) =J0(u)≤J0(φ)∀φ∈H1
0(Ω). (3.34)
Asµis non-negative, by Lax-Milgram theorem, it follows that the minimum point of the
functionalJ0is unique and this means that the whole sequence Tε(uε,δ1δ2) converges to u.
Using a classical technique (see, for instance, [185] and [69]), one can prove that the
functional
P(v) =1
2∫
Ω(v−)2dx∀v∈H1
0(Ω)
is Fr´ echet (and thus Gˆ ateaux) differentiable and its gradient is given by
P′(u)·v=−∫
Ωu−vdx∀u,v∈H1
0(Ω).
Therefore, the functional J0is Gˆ ateaux differentiable on H1
0(Ω), which ensures the equivalence
of the minimization problem
J0(u) = min
φ∈H1
0(Ω)J0(φ) (3.35)
with the problem (3.16). This completes the proof of Theorem 3.7.
Remark 3.10 From (3.27), it follows that there exists an extension buε,δ1δ2of our solution
to the whole of Ω, positive on the Signorini holes (see [77]), such that
buε,δ1δ2⇀u weakly inH1
0(Ω). (3.36)
For instance, in a first step we extend our solution inside the Signorini holes in such a way
that {∆buε,δ1δ2= 0 inBεδ1,
buε,δ1δ2=uε,δ1δ2on∂Bεδ1,
and, then, we further extend it in a standard way (see, e.g., [76]) 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 Ω.
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 [60] or [86], that there exist two matrix fields AandA0such that
Tε(Aε)(x,y)→A(x,y)a.e. in Ω×Y
and
Tεδ1(Aε)(x,z)→A0(x,z)a.e. in Ω×(Rn\B).
The only difference is the fact that in this case the corresponding homogenized matrix, the
cell problems and the strange term µdepend also on x.
We end this section by pointing out that in Section 5.4 we shall briefly mention some
related models, obtained via the periodic unfolding method in [55]. 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 efforts have been made in the last years to obtain suitable
mathematical models in biology. Still, despite the all these efforts, 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-diffusion 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 effective behavior of the solution
of a system of coupled partial differential equations arising in the modeling of ionic transfer
phenomena, coupled with electrocapillary effects, in periodic charged porous media. The so-
called Nernst-Planck-Poisson system was proposed by W. Nernst and M. Planck (see [199])
for describing the potential difference in galvanic cells. Such a model has nowadays broad
applicability in electrochemistry, in biology, in plasma physics or in the semiconductor device
modeling, where this system is also known as van Roosbroeck system . For more details about
the physical aspects behind these models and for a review of the recent relevant literature,
we refer to [117], [197], [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 HOMOGENIZATION RESULTS FOR HETEROGENEOUS 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 effective 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 [117] and [131]). Apart from a significant simplification in the
proofs, an advantage of using such an approach based on unfolding operators, which transform
functions defined on oscillating domains into functions acting on fixed 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 defined on a fixed domain Ω and having simpler coefficients 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 coefficients.
Related problems have been addressed, using different techniques, in [117], [197] 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 specific terminology for this case. Thus,
we consider a bounded connected smooth open set Ω in Rn, with |∂Ω|= 0 and with n≥2.
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 fluid phase Y∗and, 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 fluid region is connected. We denote by Ωεthe
fluid part, by Fεthe solid part and by Sεthe inner boundary of the porous medium, i.e. the
interface between the fluid and the solid phases. Since the solid part is not allowed to reach
the outer boundary ∂Ω, it follows that Sε∩∂Ω =∅.
In such a periodic microstructure, we shall consider the Poisson-Nernst-Planck system,
with suitable boundary and initial conditions. The diffusion in the fluid phase is governed by
the Nernst-Planck equations, while the electric potential which influence the ionic transfer is
described by the Poisson equation. Also, we include electrocapillary effects in our analysis.
More precisely, if we denote by [0 ,T], withT > 0, the time interval we are interested in,
we shall analyze the effective behavior, as the small parameter ε→0, of the solution of the
MATHEMATICAL MODELS IN BIOLOGY 67
following system:
−∆ Φ ε=c+
ε−c−
ε+D in (0,T)×Ωε,
−∇Φε·ν=εσεG(x,Φε) on (0,T)×Sε,
∇Φε·ν= 0 in (0 ,T)×∂Ω,
∂c±
ε
∂t− ∇ · (∇c±
ε±c±
ε∇Φε) =F±(c+
ε,c−
ε) in (0,T)×Ωε,
(∇c±
ε±c±
ε∇Φε)·ν= 0 on (0 ,T)×Sε,
(∇c±
ε±c±
ε∇Φε)·ν= 0 on (0 ,T)×∂Ω,
c±
ε=c±
0in{t= 0} ×Ωε.(4.1)
In (4.1),νis the unit outward normal to Ωε, Φεis the electrostatic potential, c±
εare the
concentrations of the ions (or the density of electrons and holes in the particular case of van
Roosbroeck model), D∈L∞(Ω) is the given doping profile, Gis a nonlinear function which
captures the effect of the electrical double layer phenomenon arising at the interface Sεand
F±is a reaction term.
Let us notice that the scaling in the right hand side of the boundary condition on Sεfor
the electric potential ensures that we keep the influence of the double layer at the macroscale.
This scaling is, in fact, physically justified by experiments. For the case in which one considers
different scalings in (4.1), see [197] and [222].
We suppose that
σε(x) =σ(x
ε)
,
whereσ=σ(y) is aY-periodic, bounded, smooth real function such that σ(y)≥δ > 0.
Also, we assume that the electrocapillary adsorption phenomenon at the substrate interface
is modeled by a given nonlinear function G. We address the case in which G=G(x,s) is
continuously differentiable, monotonously increasing with respect to sfor anyxand with
G(x,0) = 0. Also, we assume that, for n≥3, there exist C≥0 and two exponents pandm
such that
∂G
∂s ≤C(1 +|s|p),
∂G
∂xi ≤C(1 +|s|m) 1≤i≤n,(4.2)
with 0 ≤p≤n/(n−2) and 0 ≤m < n/ (n−2) +p. Moreover, by using a regulariza-
tion procedure, for example Yosida approximation, as in [75], 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 [75]. 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 [117] and [131]) in which
G(s) =K1sinh(K2s), K 1,K2>0.
68 HOMOGENIZATION RESULTS FOR HETEROGENEOUS 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 x≈xor sinhx≈x+x3/3!.
For the reaction terms, in [219] we addressed the case in which
F±(c+
ε,c−
ε) =∓(c+
ε−c−
ε).
Of course, the case in which
F±(c+
ε,c−
ε) =∓(aεc+
ε−bεc−
ε),
with
aε(x) =a(x
ε)
, bε(x) =b(x
ε)
,
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+
ε,c−
ε) =∓g(c+
ε−c−
ε),
withgan increasing locally Lipschitz continuous function on R, withg(0) = 0. In particular,
this setting includes the case treated in [111], i.e. the one in which
F±(c+
ε,c−
ε) =∓(f1(c+
ε)−f2(c−
ε)),
withfi, fori∈1,2, increasing Lipschitz continuous functions satisfying conditions which
guarantee the positivity and the necessary uniform upper bounds for the concentration fields.
Using the techniques from [75], [112] 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 differentiable function on R2, which is sublinear and globally
Lipschitz continuous in both variables and such that F±(u,v) = 0 foru <0 orv <0. For
other nonlinear reaction rates F±and more general functions G, see [137], [138] and [222].
We suppose that the initial data are non-negative and bounded independently with respect
toεand ∫
Ωε(c+
0−c−
0+D) dx=ε∫
SεσεG(x,Φε) ds.
Moreover, we suppose that the mean value in Ωεof the potential Φ εis zero.
From the Nernst-Planck equation, it is not difficult to see that the the total mass
M=∫
Ωε(c+
ε+c−
ε) dx
is conserved and suitable physical equilibrium conditions are verified, both at the microscale
and at the macroscale (see, for details, [117] and [222]). Let us mention that, for simplifying
the notation, we have eliminated in system (4.1) some constant physical relevant parameters.
MATHEMATICAL MODELS IN BIOLOGY 69
We consider here only two oppositely charged species, i.e. positively and negatively
charged particles, with concentrations c±
ε, but all the results can be easily generalized for the
case ofNspecies. Also, let us notice that we deal here only with the case of an isotropic
diffusivity in the fluid phase, but we can extend our analysis to the case of heterogeneous
media given by matrices D±
εor 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 defined all over the domain Ω, with
suitable transmission conditions at the interface Sε, as in [117] or [202].
The weak formulation of problem (4.1) is as follows: find (Φ ε, c+
ε, c−
ε), with
Φε∈L∞(0,T;H1(Ωε)),
c±
ε∈L∞(0,T;L2(Ωε))∩L2(0,T;H1(Ωε)),
∂c±
ε
∂t∈L2(0,T; (H1(Ωε))′)(4.3)
such that, for any t>0 and for any φ1, φ2∈H1(Ωε), the triple (Φ ε, c+
ε, c−
ε) satisfies:
∫
Ωε∇Φε· ∇φ1dx−∫
Sε∇Φε·νφ1dσ=∫
Ωε(c+
ε−c−
ε+D)φ1dx, (4.4)
⟨∂c±
ε
∂t, φ2⟩
(H1)′,H1+∫
Ωε(∇c±
ε±c±
ε∇Φε)· ∇φ2dx=∫
ΩεF±(c+
ε,c−
ε)φ2dx (4.5)
and
c±
ε(0,x) =c±
0(x) in Ωε. (4.6)
The variational problem (4.3)-(4.6) has a unique weak solution (Φ ε, c+
ε, c−
ε) (see [117], [197]
or [222]). Moreover, like in [197], we can prove that the concentrations are non-negative. As
a matter of fact, they are bounded from below and above, uniformly in ε.
Under the above hypotheses, it follows that there exists a constant C∈R+, independent
ofε, such that the following a priori estimates hold true:
∥Φε∥L2((0,T)×Ωε)+∥∇Φε∥L2((0,T)×Ωε)≤C
max
0≤t≤T∥c−
ε∥L2(Ωε)+ max
0≤t≤T∥c+
ε∥L2(Ωε)+∥∇c−
ε∥L2((0,T)×Ωε)+∥∇c+
ε∥L2((0,T)×Ωε)+
∂c±
ε
∂t
L2(0,T;(H1(Ωε))′)+
∂c±
ε
∂t
L2(0,T;(H1(Ωε))′)≤C.
Our goal is to obtain, via the periodic unfolding method, the effective behavior, as ε→0, of
the solution (Φ ε, c+
ε, c−
ε) of problem (4.3)-(4.6).
Let us briefly recall here the definition of the unfolding operator Tεintroduced in [59]
and [65] 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 [58], [57], [65], and
[67].
70 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
For any Lebesgue measurable function φon Ωε, the periodic unfolding operator Tεis the
linear operator defined by
Tε(φ)(x,y) =
φ(
ε[x
ε]
Y+εy)
for a.e. (x,y)∈bΩε×Y∗,
0 for a.e. ( x,y)∈Λε×Y∗.
The periodic unfolding operator Tεhas similar properties as the corresponding operator Tε
defined for fixed domains in Section 3.2.2.
Using the properties of the unfolding operator Tεand the above a priori estimates, it
is not difficult to see that there exist Φ ∈L2(0,T;H1(Ω)),bΦ∈L2((0,T)×Ω;H1
per(Y∗)),
c±∈L2(0,T;H1(Ω)),bc±∈L2((0,T)×Ω;H1
per(Y∗)), such that, up to a subsequence,
Tε(Φε)⇀Φ weakly in L2((0,T)×Ω;H1(Y∗)), (4.7)
Tε(∇Φε)⇀∇Φ +∇ybΦ weakly in L2((0,T)×Ω×Y∗), (4.8)
Tε(c±
ε)→c±strongly in L2((0,T)×Ω;H1(Y∗)), (4.9)
Tε(∇c±
ε)⇀∇c±+∇ybc±weakly inL2((0,T)×Ω×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 ε→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 (Φε, c+
ε,c−
ε)
of system (4.1) converges, in the sense of (4.7)-(4.10), as ε→0, to the unique solution
(Φ, c+,c−)of the following macroscopic problem in (0,T)×Ω:
−div(D0∇Φ) +1
|Y∗|σ0G=c+−c−+D,
∂c±
∂t−div(D0∇c±±D0c±∇Φ) =F±
0,(4.11)
with the boundary conditions on (0,T)×∂Ω:
{
D0∇Φ·ν= 0,
(D0∇c±±D0c±∇Φ)·ν= 0(4.12)
and the initial conditions
c±(0,x) =c±
0(x),∀x∈Ω. (4.13)
Here,
σ0=∫
∂Fσ(y) ds,
F±
0=F±(c+,c−) =∓g(c+−c−)
MATHEMATICAL MODELS IN BIOLOGY 71
andD0=(
d0
ij)
is the homogenized matrix, defined as follows:
d0
ij=1
|Y∗|∫
Y∗(
δij+∂χj
∂yi(y))
dy,
withχj, j= 1,…,n, solutions of the cell problems
χj∈H1
per(Y∗),∫
Y∗χj= 0,
−∆χj= 0 inY∗,
(∇χj+ej)·ν= 0 on∂F(4.14)
andei,1≤i≤n, the vectors in the canonical basis of Rn.
Proof. In order to prove Theorem 4.1, let us first take in the Poisson equation (4.4) the
test function
φ1(t,x) =ψ0(t,x) +εψ1(
t,x,x
ε)
,
withψ0∈ D((0,T);C∞(Ω)) andψ1∈ D((0,T)×Ω;H1
per(Y∗)). By unfolding, we obtain
∫T
0∫
Ω×Y∗Tε(∇Φε)Tε(∇(ψ0+εψ1)) dxdydt+
∫T
0∫
Ω×∂FTε(σ)Tε(G(Φε))Tε(ψ0+εψ1)) dxdsdt=
∫T
0∫
Ω×Y∗Tε(c+
ε−c−
ε+D)Tε(ψ0+εψ1)) dxdydt. (4.15)
Using the above convergence results, we can easily compute the limit of the linear terms in
(4.15) defined on Ω ×Y∗(see, for instance, [59], [75] and [222]). For the term containing
the nonlinear function G, let us notice that, exactly like in [75], one can show that if Ris a
continuously differentiable function, monotonously increasing, with R(x,v) = 0 if and only if
v= 0 and fulfilling the assumption (4.2), then, for any wε⇀w weakly inH1
0(Ω), we have
R(x,wε)⇀R(x,w),
weakly inW1,p
0(Ω), where
p=2n
q(n−2) +n.
Using the properties of the unfolding operator Tεand Lebesgue’s convergence theorem, we
get
∫T
0∫
Ω×∂FTε(σ)Tε(G(x,Φε))Tε(ψ0+εψ1)) dxdsdt→σ0∫T
0∫
ΩG(x,Φ)ψ0dxdt.
Therefore, for ε→0, we obtain:
∫T
0∫
Ω×Y∗(∇Φ(t,x) +∇ybΦ(t,x,y )) (∇ψ0(t,x) +∇yψ1(t,x,y )) dxdydt+
72 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
σ0∫T
0∫
ΩG(x,Φ(t,x))ψ0(t,x) dxdt=
∫T
0∫
Ω×Y∗(c+(t,x)−c−(t,x) +D(x))ψ0(t,x) dxdydt. (4.16)
By density, (4.16) is valid for any ψ0∈L2(0,T;H1(Ω)) andψ1∈L2((0,T)×Ω;H1
per(Y∗)).
Takingψ0(t,x) = 0, we get
−∆ybΦ(t,x,y ) = 0 in (0 ,T)×Ω×Y∗,
∇ybΦ·ν=−∇ xΦ(t,x)·νon (0,T)×Ω×∂F,
bΦ(t,x,y ) periodic in y.
By linearity, we obtain
bΦ(t, x, y ) =n∑
j=1χj(y)∂Φ
∂xj(t, x), (4.17)
whereχj, j=1, n, are the solutions of the local problems (4.14). Taking ψ1(t,x,y ) = 0,
integrating with respect to the variable xand using (4.17), we easily get the homogenized
problem for the electrostatic potential Φ.
In a second step, taking in the Nernst-Planck equation the test function
φ2(t,x) =ψ0(t,x) +εψ1(
t,x,x
ε)
,
withψ0∈ D((0,T);C∞(Ω)) andψ1∈ D((0,T)×Ω;H1
per(Y∗)), we have:
−∫T
0∫
Ω×Y∗Tε(c±
ε)Tε(∂
∂t(ψ0+εψ1))
dxdydt+
∫T
0∫
Ω×Y∗Tε(∇c±
ε±c±
ε∇Φε)Tε(ψ0+εψ1)) dxdydt=
∫T
0∫
Ω×Y∗Tε(F±(c+
ε,c−
ε))Tε(ψ0+εψ1) dxdydt.
Passing to the limit with ε→0, we obtain
−∫T
0∫
Ω×Y∗c±(t,x)∂
∂tψ0(t,x) dxdydt+
∫T
0∫
Ω×Y∗(∇c±(t,x) +∇ybc±(t,x,y ))(∇ψ0(t,x) +∇yψ1(t,x,y )) dxdydt=
∫T
0∫
Ω×Y∗F±
0(c+,c−)ψ0(t,x) dxdydt. (4.18)
Using again standard density arguments, (4.18) can be written for any ψ0∈L2(0,T;H1(Ω))
andψ1∈L2((0,T)×Ω;H1
per(Y∗)). Then, taking ψ0(t,x) = 0 and, then, ψ1(t,x,y ) = 0, we
get exactly the homogenized problem for the concentrations c±.
Due to the uniqueness of the solutions Φ and c±of problem (4.3)-(4.6) (see [197] 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.
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 effective behavior of a coupled system of reaction-
diffusion equations, arising in the modeling of some biochemical processes contributing to
carcinogenesis in living cells. We shall be concerned with the carcinogenic effects produced
in the human cells by Benzo-[a]-pyrene (BP) molecules, found in coal tar, cigarette smoke,
charbroiled food, etc. To understand the complex behavior of these molecules, mathemati-
cal models including reaction-diffusion processes and binding and cleaning mechanisms have
been developed. Following [127], we consider here a simplified setting in which BP molecules
invade the cytosol inside of a human cell. There, they diffuse freely, but they cannot enter
in the nucleus. Also, they bind to the surface of the endoplasmic reticulum (ER), where
chemical reactions, 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 diffuse 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 different homogenized models. Another carcinogenesis model, introduced in [128], will be
briefly 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 [80].
For more details about the mechanisms governing carcinogenesis in human cells we refer to
[119] and [191].
Problems closed to the one we treat here were addressed in [6], [80], [127], [128] and
[130]. For papers devoted to the upscaling of reactive transport in porous media, we refer
to [8], [112], [137], [140], [141], [179], [195] and the references therein. For reaction-diffusion
problems involving adsorption and desorption, we refer to [6], [74], [75], [99], [137], [159].
For proving our main convergence results, we use the periodic unfolding method (see [57],
[59], [65] and [96]), extended in [128] and [129] for dealing with gradients of functions defined
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 defined on the surface of the endoplasmic reticulum,
74 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
we follow the ideas in [128].
4.2.1 The microscopic problem
Let us describe now briefly the geometry of the problem, which is similar to the one considered
in Section 2.1. More precisely, we consider a bounded connected open set Ω in Rn, with a
Lipschitz boundary ∂Ω and with n≥2. The domain Ω, which, as in [128], is assumed to be
representable by a finite 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 Γ Cthe cell membrane and by Γ Nthe boundary of the nucleus. Thus,
∂Ω = Γ C∪ΓN. LetY= (0,1)nbe the reference cell and let F⊂Ybe 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∗=Y\F.
RepeatingYby periodicity, the union of all Y∗is a connected set in Rn, denoted by Rn
1.
Letε∈(0,1) be a small parameter related to the periodicity length, taking values in
a positive real sequence tending to zero and such that Ω is a finite union of cuboids which
are homothetic to the unit cell with the same ratio ε. We set Ωε=∪
k∈Znε(k+Y∗)∩Ω and
Sε=∪
k∈Znε(k+∂F)∩Ω and we suppose that Sε∩∂Ω =∅. We remark that Ω is a finite union
of cuboids which are homothetic to the unit cell and that the inclusions do not intersect the
exterior boundary ∂Ω.
If we denote by [0 ,T], with 0< T < ∞, the time interval of interest, we shall analyze
the effective behavior, as the small parameter ε→0, of the solution of the following coupled
system of equations:
∂uε
∂t−Du∆uε=−f(uϵ) in (0,T)×Ωε,
uε=ubon (0,T)×ΓC,
−Du∇uε·ν= 0 on (0,T)×ΓN,
−Du∇uε·ν=εG1(uε,sε) on (0,T)×Sε,
uε(0,x) =u0(x) in Ωε,(4.19)
∂vε
∂t−Dv∆vε=−g(vε) in (0,T)×Ωε,
vε= 0 on (0,T)×ΓN,
−Dv∇vε·ν= 0 on (0,T)×ΓC,
−Du∇uε·ν=εG2(vε,wε) on (0,T)×Sε,
vε(0,x) =v0(x) in Ωε,(4.20)
∂sε
∂t−ε2Ds∆εsε=−h(sε) +G1(uε,sε) on (0,T)×Sε,
sε(0,x) =s0(x) onSε.(4.21)
MATHEMATICAL MODELS IN BIOLOGY 75
∂wε
∂t−ε2Dw∆εwε=h(sε) +G2(vε,wε) on (0,T)×Sε,
wε(0,x) =w0(x) onSε.(4.22)
In (4.19)-(4.22), νis the outward unit normal to Ωε, ∆εis the Laplace-Beltrami operator
onSε,uε: [0,T]×Ωε→Randvε: [0,T]×Ωε→Rrepresent the concentrations of
BP molecules and, respectively, of DE molecules in the cytosol and sε: [0,T]×Sε→R
andwε: [0,T]×Sε→Rare the concentrations of BP molecules and, respectively, of DE
molecules bound to the surface of the ER.
We notice that the diffusion on the surface of the endoplasmic reticulum is scaled with
ε2, in order to keep the influence of the slow surface diffusion term at the macroscale. Also,
the scaling in the right-hand side of the boundary conditions (4.19) and (4.20) on Sεensures
that we keep the influence of the binding processes at the macroscale. We can treat in a
similar manner the case in which the binding-unbinding term on Sεcorresponding to the BP
molecules is scaled with εγand the binding-unbinding term for DE molecules is multiplied
byεm, withγ,m∈[0,1) (for the linear case, see [127]).
We make the following assumptions on the data:
1.The diffusion coefficients Du,Dv,Ds,Dw>0 are supposed to be, for simplicity, con-
stant.
2.f,gandhare nonlinear functions modeling the cleaning mechanisms in Ωεand, 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) =
ax
x+b, x≥0,
0, x< 0,
fora,b > 0. The functions gandhare assumed to be of the same form as f, but with
different 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 ( uε,vε,sε,wε) 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 L∞-estimates of the solution ( uε,vε,sε,wε). A standard choice
is given in [127]. More precisely, the authors consider the linear case in which
G1(uε,sε) =ls(uε−sε), G 2(vε,wε) =lw(vε−wε),
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
76 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
with highly oscillating coefficients and the case of a Langmuir isotherm. More precisely, we
consider, in a first situation, that
G1(uε,sε) =lε
uuε−lε
ssε, G 2(vε,wε) =lε
vvε−lε
wwε, (4.23)
with
lε
u(x) =lu(x
ε)
, lε
s(x) =ls(x
ε)
, lε
v(x) =lv(x
ε)
, lε
w(x) =lw(x
ε)
,
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 coefficients are not constant but vary with respect to the surface variable is
physically justified (for examples where the processes on the membrane are inhomogeneous,
see [81]). As a consequence, in the homogenized limit, additional integral terms are present,
capturing the effect 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 defined in terms of isotherms of Langmuir type:
G1(uε,sε) =lε
s(α1uε
1 +β1uε−sε)
, G 2(vε,wε) =lε
w(α2vε
1 +β2vε−wε)
, (4.24)
withαi,βi>0, fori= 1,2. We denote
g1(uε) =α1uε
1 +β1uε, g 2(vε) =α2vε
1 +β2vε. (4.25)
4.The concentration ubof the BP molecules on the cell membrane Γ Cis supposed to be
an element of H1/2(ΓC) (see (4.26)) and the initial values u0(x),v0(x)∈L2(Ω),s0(x),w0(x)∈
C1(¯Ω) are assumed to be nonnegative and bounded independently with respect to ε.
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)(p−q), with
0<G i,min≤Gi(p,q)≤Gi,max<∞or 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) =αr/(1+βr) is increasing and one to one from R+to [0,α/β ]. Despite
the fact that gis not defined for r=−1/β, 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 infinity 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 first step, a modified kinetics
g0, obtained by replacing rby its modulus |r|in the denominator of g. This new function is
Lipschitz continuous. Then, proving the existence and uniqueness of a solution of the problem
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
βr≪1, 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 Sεare given by some rates Gε
1(x,uε,sε)andGε
2(x,vε,wε)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 [100], [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(Ωε) is equipped with the classical scalar product and
norm
(u,v)Ωε=∫
Ωεu(x)v(x) dx,∥u∥2
Ωε= (u,u)Ωε,
and the space L2((0,T),L2(Ωε)) is endowed with
(u,v)Ωε,T=∫T
0(u(t),v(t))Ωdt,∥u∥2
Ωε,T= (u,u)Ωε,T,
whereu(t) =u(t,·),v(t) =v(t,·). Further, following [127] and [128], we set
V(Ωε) =L2((0,T),H1(Ωε))∩H1((0,T),(H1(Ωε))′),
VN(Ωε) ={v∈ V(Ωε)|v= 0 on Γ N},
VC(Ωε) ={v∈ V(Ωε)|v=ubon Γ C},
V0,C(Ωε) ={v∈ V(Ωε)|v= 0 on Γ C},
where, for an arbitrary Banach space V, we denote by V′its dual. Similar spaces can be
defined for Ω and Sε. We use the notation
⟨u,v⟩Γε=∫
Sεgεuvdσx,
wheregεis the Riemannian tensor on Sε. Also, let us define
VN(Ωε) ={v∈H1(Ωε)|v= 0 on Γ N},
V0,C(Ωε) ={v∈H1(Ωε)|v= 0 on Γ C}, V (Sε) =H1(Sε)
and
V(Ω,Y) =L2((0,T)×Ω,H1
per(Y∗)),V(Ω,∂F) =L2((0,T)×Ω,H1(∂F)),
78 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
whereH1
per(Y∗) ={φ∈H1
loc(Rn
1) :φisY−periodic }. Finally, we assume that ub∈
H1/2(ΓC), where, for an arbitrary smooth hypersurface Γ 0⊆Rnand for any 0 < r < 1,
we consider the Sobolev-Slobodeckij space
Hr(Γ0) ={u∈L2(Γ0) :|u|Γ0,r<∞}, (4.26)
where
|u|2
Γ0,r=∫
Γ0×Γ0|u(x)−u(y)|2
|x−y|n−1+2rdσxdσy.
The spaceHr(Γ0) is endowed with the norm ∥u∥2
Hr(Γ0)=∥u∥2
L2(Γ0)+|u|2
Γ0,r(see [118] and
[126]).
Let us give now the variational formulation of problem (4.19)-(4.22).
Problem 1 : find (uε,vε,sε,wε)∈ VC(Ωε)×VN(Ωε)×V(Sε)×V(Sε), satisfying the initial
condition
(uε(0),vε(0),sε(0),wε(0)) = (u0,v0,s0,w0),
such that, for a.e. t∈(0,T) and for any ( φ1,φ2,ϕ)∈VC,0(Ωε)×VN(Ωε)×V(Sε), we have
(∂uε
∂t,φ1)
Ωε+Du(∇uε,∇φ1) +ε⟨G1(uε,sε),φ1⟩Sε=−(f(uε),φ1)Ωε,
(∂vε
∂t,φ2)
Ωε+Dv(∇vε,∇φ2) +ε⟨G2(vε,wε),φ2⟩Sε=−(g(vε),φ2)Ωε,
⟨∂sε
∂t,ϕ⟩
Sε+Ds⟨ε∇∂Fsε,ε∇∂Fϕ⟩Sε=−⟨h(sε),ϕ⟩Sε+⟨G1(uε,sε),ϕ⟩Sε,
⟨∂wε
∂t,ϕ⟩Sε+Dw⟨ε∇∂Fwε,ε∇∂Fϕ⟩Sε=⟨h(sε),ϕ⟩Sε+⟨G2(vε,wε),ϕ⟩Sε.(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 ( uε,vε,sε,wε) 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 ( uε,vε,sε,wε) 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 ε→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 TεandTε
bdefined, e.g., in [57], [59], [65], [96], [128], and [129].
The main feature of these operators is that they map functions defined on the oscillating
domains (0,T)×Ωεand, respectively, (0 ,T)×Γε, into functions defined on the fixed domains
(0,T)×Ω×Y∗and (0,T)×Ω×Γ, respectively. We briefly recall here the definitions of
MATHEMATICAL MODELS IN BIOLOGY 79
these two operators for our particular geometry. For any φ∈Lp((0,T)×Ωε) and any
p∈[1,∞], we define the periodic unfolding operator Tε:Lp((0,T)×Ωε)→Lp((0,T)×Ω×Y∗)
by the formula Tε(φ)(t,x,y ) =φ(
t,ϵ[x
ε]
+εy)
. In a similar manner, for any function
ϕ∈Lp((0,T)×Γε), the periodic boundary unfolding operator Tε
b:Lp((0,T)×Γε)→
Lp((0,T)×Ω×Γ) is defined by Tε
b(ϕ)(t,x,y ) =ϕ(
t,ε[x
ϵ]
+εy)
.
Using these unfolding operators, we deduce the homogenized limit system.
Theorem 4.4 (Theorem 3.1 in [223]) The solution (uε,vε,sε,wε)of system (4.19)-(4.22)
converges, as ε→0, in the sense of (3.36), to the unique solution (u,v,s,w )∈ V C(Ω)×
VN(Ω)× V(Ω,∂F)× V(Ω,∂F), with (u(0),v(0),s(0),w(0)) = (u0,v0,s0,w0), of the following
macroscopic problem:
|Y∗|(∂u
∂t,φ1)
Ω+ (Au∇u,∇φ1)Ω+ (G1(u,s),φ1)Ω×∂F=−|Y∗|(f(u),φ1)Ω,
|Y∗|(∂v
∂t,φ2)
Ω+ (Av∇v,∇φ2)Ω+ (G2(v,w),φ2)Ω×Γ=−|Y∗|(g(v),φ2)Ω,
(∂s
∂t,ϕ)
Ω×∂F+ (Ds∇∂F
ys,∇∂Fϕ)Ω×∂F−(G1(u,s),ϕ)Ω×∂F=−(h(s),ϕ)Ω×∂F,
(∂w
∂t,ϕ)
Ω×∂F+ (Dw∇∂F
yw,∇∂Fϕ)Ω×∂F−(G2(v,w),ϕ)Ω×∂F= (h(s),ϕ)Ω×∂F,(4.28)
for(φ1,φ2,ϕ)∈V0,C(Ω)×VN(Ω)×V(Ω,∂F). Here,AuandAvare the homogenized matrices,
defined by:
Au
ij=Du∫
Y∗(
δij+∂χj
∂yi)
dy,
Av
ij=Dv∫
Y∗(
δij+∂χj
∂yi)
dy,(4.29)
in terms of the functions χj∈H1
per(Y∗)/R, ,j= 1,…,n, weak solutions of the cell problems
{−∇ y·(∇yχj+ej) = 0, y∈Y∗,
(∇yχj+ej)·ν= 0, y∈∂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 )∈ V C(Ω)× V N(Ω)×
V(Ω,∂F)× V(Ω,∂F), defined 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:
|Y∗|∂u
∂t− ∇ · (Au∇u) +∫
∂FG1(u,s) dσy=−|Y∗|f(u)in(0,T)×Ω,
u=ubon(0,T)×ΓC,
Au∇u·ν= 0 on(0,T)×ΓN,(4.31)
80 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
|Y∗|∂v
∂t− ∇ · (Av∇v) +∫
∂FG2(v,w) dσy=−|Y∗|g(v)in(0,T)×Ω,
v= 0 on(0,T)×ΓN,
Av∇v·ν= 0 on(0,T)×ΓC,(4.32)
∂s
∂t−Ds∆∂F
ys−G1(u,s) =−h(s)on(0,T)×Ω×∂F,
∂w
∂t−Dw∆∂F
yw−G2(v,w) =h(s)on(0,T)×Ω×∂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 influence of the properly scaled binding-unbinding processes
taking place at the surface of the endoplasmic reticulum is reflected 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 effective medium; in (4.31)-(4.32), x∈Ωplays the role of a macroscopic
variable and y∈∂Fis a microscopic one. The limit model consists of two partial differential
equations, with global diffusion (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 differential
equations, governing the local behavior of the system, with local diffusion (with respect to the
microscopic variable y) on∂F(see (4.33)). .
Remark 4.7 We can deal, in a similar manner, with the more general case in which, in-
stead of considering constant diffusion coefficients, we work with an heterogeneous medium
represented by periodic symmetric bounded matrices which are assumed to be uniformly coer-
cive. Moreover, all the above results can be extended to the situation in which, instead of the
constant diffusion coefficients DuandDv, we have two matrices Aε
uand, respectively, Aε
v.
We suppose that Aε
uandAε
vare sequences of matrices in M(α,β, Ω)such that
Tε(Aε
u)→Au,Tε(Aε
v)→Avstrongly in L1(Ω×Y)n×n, (4.34)
for some matrices Au=Au(x,y)andAv=Au(x,y)inM(α,β, Ω×Y)(see [59]). In this
case, since the correctors χjdepend also on x, the new homogenized matrices Ahom
uandAhom
v
are no longer constant, but depend on x. Here, for α,β∈R, with 0< α≤β, we denote
byM(α,β, Ω)the set of all the matrices A∈(L∞(Ω))n×nwith the property that, for any
ξ∈Rn,(A(y)ξ, ξ)≥α|ξ|2,|A(y)ξ| ≤β|ξ|, almost everywhere in Ω.
Also, for the diffusion coefficients on the surface Sεwe can suppose that they are not con-
stant, but they depend on ε. For instance, we can work with the diffusion tensors Dε
s(x) =
Ds(x/ε)andDε
w(x) =Dw(x/ε), whereDsandDware two uniformly coercive periodic sym-
metric given tensors Ds(y)andDw(y), with entries belonging to L∞(∂F). Moreover, we
can also address the case in which we suppose that Dε
sandDε
ware such that there exist
Ds=Ds(x,y)andDw=Dw(x,y)with entries in L∞(Ω×∂F)such that Tε
b(Dε
s)→Dsand
Tε
b(Dε
w)→Dwstrongly in L1(Ω×∂F). .
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 ( uε,vε,sε,wε) of the
problem (4.27).
The following proposition, proven in [223], states that the functions uε,vε,sεandwεare
nonnegative and bounded from above if the initial data are assumed to be bounded and
nonnegative. The positivity of uε,vε,sεandwεis 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 uεandvεare nonnegative for
almost every x∈Ωεandt∈[0,T]and the functions sεandwεare nonnegative for almost
everyx∈Sεandt∈[0,T]. Also, the functions uεandvεare bounded independently of ε
almost everywhere in [0,T]×Ωεand the functions sεandwεare bounded independently of ε
almost everywhere in [0,T]×Sε.
Hence, the solution ( uε,vε,sε,wε) of problem (4.27) is bounded in the L∞-norm. Besides,
exactly like in [127, Lemma 2] and [128, Lemma A.2], one can prove the L2-boundedness of
the solution ( uε,vε,sε,wε).
Proposition 4.9 (Proposition 4.2 in [223]) There exists a constant C > 0, independent of
ε, such that
∥uε∥2
Ωε+∥∇uε∥2
Ωε,t+∥vε∥2
Ωε+∥∇vε∥2
Ωε,t≤C,
ε∥sε∥2
Sε+ε3∥∇∂Fsε∥2
Sε,t+ε∥wε∥2
Sε+ε3∥∇∂Fwε∥2
Sε,t≤C,
ε∥G1(uε,sε)∥2
Sε,t+ε∥G2(vε,wε)∥2
Sε,t≤C,
for almost every t∈[0,T]. Also, one gets
∂uε
∂t
L2((0,T),(H1
0(Ωε))′)+
∂vε
∂t
L2((0,T),(H1
0(Ωε))′)≤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 uε,vε∈L2((0,T),H1(Ωε))∩
H1((0,T),(H1
0(Ωε))′)∩L∞((0,T)×Ωε).
Using suitable extension results (see, for instance, [139], [118] and [171]) and Lemma 5.6
from [130], we know that we can construct two extensions uεandvεthat converge strongly
tou,v∈L2((0,T),L2(Ω)). We point out that one can obtain (see e. g. [118] and [171]) the
existence of a linear and bounded extension operator to the whole of Ω, 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 that Tε
b(sε) and Tε
b(wε) are Cauchy sequences in L2((0,T)×Ω×∂F).
82 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Proposition 4.10 (Proposition 4.3 in [223]) For any δ >0, there exists ε0>0such that
for any 0<ε1,ε2<ε0one has
∥Tε1
b(sε1)− Tε2
b(sε2)∥(0,T)×Ω×∂F+∥Tε1
b(wε1)− Tε2
b(wε2)∥(0,T)×Ω×∂F<δ..
This implies that the sequences Tε
b(sε) and Tε
b(wε) are strongly convergent in L2((0,T)×
Ω×∂F).
As already mentioned, (4.27) is a well-posed problem. Using the above a priori estimates
and the properties of the operators TεandTε
b(see [57], [59], [65], [96], [128], and [129]), we
get immediately the following compactness result.
Proposition 4.11 (Proposition 4.4 in [223]) Let (uε,vε,sε,wε)be the solution of problem
(4.27). Then, there exist u,v∈L2((0,T),H1(Ω)),bu,bv∈L2((0,T)×Ω,H1
per(Y∗)),s,w∈
L2((0,T)×Ω,H1
per(∂F))such that, up to a subsequence, when ε→0, we have
Tε(uε)⇀u weakly inL2((0,T)×Ω,H1(Y∗)),
Tε(vε)⇀v weakly inL2((0,T)×Ω,H1(Y∗)),
uε→u,vε→vstrongly in L2((0,T)×Ω),
Tε(∇uε)⇀∇u+∇ybuweakly inL2((0,T)×Ω×Y∗),
Tε(∇vε)⇀∇v+∇ybvweakly inL2((0,T)×Ω×Y∗),
Tε
b(sε)⇀s weakly inL2((0,T)×Ω,H1(∂F)),
Tε
b(wε)⇀w weakly inL2((0,T)×Ω,H1(∂F)),
Tε
b(sε)→s,Tε
b(wε)→wstrongly in L2((0,T)×Ω×∂F)..(4.36)
For passing to the limit in the nonlinear terms containing the functions G1andG2, we have
to show that Tε
b(uε)→uandTε
bvε)→v, strongly in L2((0,T)×Ω×∂F). These strong
convergence results follow from the strong convergence of uεandvε, respectively, the trace
lemma (see Lemma 3.1 in [118]) and the properties of the unfolding operator Tε
b. More
precisely, we have the following result (see [223] and [118]).
Proposition 4.12 (Proposition 4.5 in [223]) Up to a subsequence, one has
Tε
b(uε)→ustrongly in L2((0,T)×Ω×∂F)
and
Tε
b(vε)→vstrongly in L2((0,T)×Ω×∂F).
We remark that from the strong convergence of uεandvεand the continuity of the Nemytskii
operator for fand, respectively, g, which are bounded and continuous functions, we have
f(Tε(uε))→f(u) strongly in L2((0,T)×Ω×Y∗).
g(Tε(vε))→g(v) strongly in L2((0,T)×Ω×Y∗).
In a similar way, we obtain
h(Tε
b(sε))→h(s) strongly in L2((0,T)×Ω×∂F).
MATHEMATICAL MODELS IN BIOLOGY 83
For getting the limit behavior of the terms involving G1andG2, in the first situation, i.e. for
Henry isotherm with rapidly oscillating coefficients given by (2.5), we can easily pass to the
limit since these coefficients are uniformly bounded in L∞(Ω) and converge strongly therein,
while for the second situation, i.e. isotherm of the form (3.24), we need to use the strong
convergence of Tε
b(uε),Tε
b(vε),Tε
b(sε) andTε
b(wε) and the properties of the functions g1and
g2. Therefore, we obtain
G1(Tε
b(uε,sε))→G1(u,s) strongly in L2((0,T)×Ω×∂F)
and
G2(Tε
b(vε,wε))→G2(v,w) strongly in L2((0,T)×Ω×∂F).
By classical results (see, for instance, Theorem 2.12 in [57] and Theorem 2.17 in [96] ), uand
vare independent of y.
Proof of Theorem 4.7. For getting the limit problem (4.28), we take in the first equation
in (4.27) the admissible test function
φ(t,x) =φ1(t,x) +εφ2(
t,x,x
ε)
, (4.37)
withφ1∈C∞
0((0,T),C∞(Ω))φ2∈C∞
0((0,T),C∞(Ω,C∞
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, [59], [75], [128] and [217]), we get:
∫T
0∫
Ω×Y∗∂u
∂tφ1dxdydt+Du∫T
0∫
Ω×Y∗(∇u+∇ybu)·(∇φ1+∇yφ2) dxdydt+
∫T
0∫
Ω×∂FG1(u,s)φ1dxdσydt=−∫T
0∫
Ω×Y∗f(u)φ1dxdydt. (4.38)
By standard density arguments, it follows that (4.38) is valid for any φ1∈L2(0,T;H1(Ω)),
φ2∈L2((0,T)×Ω;H1
per(Y∗)). In a similar manner, for the limit equation for vε, we obtain
∫T
0∫
Ω×Y∗∂v
∂tφ1dxdydt+Dv∫T
0∫
Ω×Y∗(∇v+∇ybv)·(∇φ1+∇yφ2) dxdydt+
∫T
0∫
Ω×∂FG2(v,w)φ1dxdσydt=−∫T
0∫
Ω×Y∗g(v)φ1dxdydt, (4.39)
for anyφ1∈L2((0,T),H1(Ω)),φ2∈L2((0,T)×Ω,H1
per(Y∗)).
In order to obtain the limit equations for sεandwε, we apply the convergence results
obtained in [128] (see Lemma 2.6 and Theorem 2.9). Indeed, using the boundary unfolding
operator Tε
bin (4.27)3, by passing to the limit we get
∫T
0∫
Ω×∂F∂s
∂tϕdxdσydt+Ds∫T
0∫
Ω×∂F∇∂F
ys· ∇∂F
yϕdxdσydt=
84 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
∫T
0∫
Ω×∂FG1(u,s)ϕdxdσydt−∫T
0∫
Ω×∂Fh(s)ϕdxdσydt, (4.40)
for anyϕ∈C∞
0((0,T),C∞(Ω,C∞
per(∂F))).
In a similar manner, we get
∫T
0∫
Ω×∂F∂w
∂tϕdxdσydt+Dw∫T
0∫
Ω×∂F∇∂F
yw· ∇∂F
yϕdxdσydt=
∫T
0∫
Ω×∂FG2(v,w)ϕdxdσydt+∫T
0∫
Ω×∂Fh(s)ϕdxdσydt, (4.41)
for anyϕ∈C∞
0((0,T),C∞(Ω,C∞
per(∂F))).
Thus, we get exactly the weak formulation of the limit problem (4.28). Indeed, if we take
φ1= 0, we easily get the cell problems (4.30) and
bu=n∑
k=1∂u
∂xkχk,bv=n∑
k=1∂v
∂xkχk. (4.42)
Then, taking φ2= 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 ( uε,vε,sε,wε) 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 briefly 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 ERsεonly if they find a free receptor Rε. Following [128], let Rε: [0,T]×Sε→[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 kuuεRε, with
a constant rate ku>0. DE molecules are assumed to have a similar behavior. When BP
moleculesuεor DE molecules vεbind to the surface of the endoplasmic reticulum, the quantity
of free receptors decreases and when the molecules sεandwεunbind from the surface of the
endoplasmic reticulum the amount of free receptors increases. If receptors are supposed to
be fixed on the surface of the endoplasmic reticulum, then their evolution is governed by (see
[128] for details):
∂Rε
∂t=−Rε|kuuε+kvvε|+ (R−Rε)|kssε+kwwε|on (0,T)×Sε.
Here,ks,kw>0 are supposed to be multiples of lsand, respectively, of lw.
The variational formulation of this nonlinear problem is stated below.
MATHEMATICAL MODELS IN BIOLOGY 85
Problem 2: find (uε,vε,sε,wε,Rε)∈ V C(Ωε)× V N(Ωε)× V(Sε)× V(Sε)× V R(Sε),
satisfying the initial condition
(uε(0),vε(0),sε(0),wε(0),Rε(0)) = (u0,v0,s0,w0,R),
such that, for a.e. t∈(0,T) and for all ( φ1,φ2,ϕ)∈VC,0(Ωε)×VN(Ωε)×V(Sε), we have
(∂uε
∂t,φ1)
Ωε+Du(∇uε,∇φ1) +ε⟨kuuεRε−lssε,φ1⟩Sε=−(f(uε),φ1)Ωε,
(∂vε
∂t,φ2)
Ωε+Dv(∇vε,∇φ2) +ε⟨kvvεRε−lwwε,φ2⟩Γε=−(g(vε),φ2)Ωε,
⟨∂sε
∂t,ϕ⟩
Sε+Ds⟨ε∇∂Fsε,ε∇∂Fϕ⟩Sε=−⟨h(sε),ϕ⟩Sε+⟨kuuεRε−lssε,ϕ⟩Sε,
⟨∂wε
∂t,ϕ⟩
Sε+Dw⟨ε∇∂Fwε,ε∇∂Fϕ⟩Sε=⟨h(sε),ϕ⟩Sε+⟨kvvεRε−lwwε,ϕ⟩Sε,
⟨∂tRε,ϕ⟩Sε+⟨Rε|kuuε+kvvε|,ϕ⟩Sε=⟨(R−Rε)|kssε+kwwε|,ϕ⟩Sε.(4.43)
In (4.43), VR(Sε) ={u∈L2((0,T),L2(Sε))|∂tu∈L2((0,T),L2(Sε))}.
The existence of a solution ( uε,vε,sε,wε,Rε)∈ VC(Ωε)×VN(Ωε)×V(Sε)×V(Sε)×VR(Sε)
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 Rεis nonnegative and bounded by R > 0 almost
everywhere in [0 ,T]×SεandTε
b(Rε) converges strongly to R∈L2((0,T)×Ω×∂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: find (u,v,s,w,R )∈ VC(Ω)×
VN(Ω)×V(Ω,Γ)×V(Ω,∂F)×VR(Ω,∂F), with (u(0),v(0),s(0),w(0),R(0)) = (u0,v0,s0,w0,R),
such that
|Y∗|(∂u
∂t,φ1)
Ω+ (Au∇u,∇φ1)Ω+ (kuuR−lss,φ1)Ω×Γ=−|Y∗|(f(u),φ1)Ω,
|Y∗|(∂v
∂t,φ2)
Ω+ (Av∇v,∇φ2)Ω+ (kvvR−lww,φ 2)Ω×∂F=−|Y∗|(g(v),φ2)Ω,
(∂s
∂t,ϕ)
Ω×Γ+ (Ds∇Γs,∇Γϕ)Ω×Γ−(kuuR−lss,ϕ)Ω×∂F=−(h(s),ϕ)Ω×∂F,
(∂w
∂t,ϕ)
Ω×∂F+ (Dw∇Γw,∇Γϕ)Ω×∂F−(kvvR−lww,ϕ)Ω×∂F= (h(s),ϕ)Ω×∂F,
(∂tR,ϕ)Ω×∂F+ (R(kuu+kvv),ϕ)Ω×∂F= ((R−R)(kss+kww),ϕ)Ω×∂F,
for(φ1,φ2,ϕ)∈V0,C(Ω)×VN(Ω)×V(Ω,∂F).
Notice that the homogenized matrices AuandAvare given by (4.29). We point out that the
evolution of the receptors is governed by an ordinary differential equation.
All the above results are still valid for the case of highly oscillating coefficients kε
u,kε
vand,
respectively, lε
sandlε
w. Moreover, based on the law of mass action, various other functions
G1(Rε,uε) andG2(Rε,vε) can be used to describe the adsorption phenomena at the surface
86 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
of the endoplasmic reticulum. As particular situations, we can mention the case in which
G1=Rεg1(uε) andG2=Rεg2(vε), withg1andg2suitable Lipschitz continuous functions
(e.g. the Langmuir kinetics considered above). In such a case, the equation governing the
evolution of the receptors is
∂tRε=−Rε|g1(uε) +g2(vε)|+ (R−Rε)|kssε+kwwε|on (0,T)×Sε.
The case in which the binding processes at the surface of the endoplasmic reticulum is given
by suitable nonlinear functions G1(x/ε,Rε,uε) and, respectively, G2(x/ε,Rε,vε), 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, we shall analyze the effective behavior of a non-
linear system of coupled reaction-diffusion equations arising in the modeling of the dynamics
of calcium ions in living cells is analyzed. We deal, at the microscale, with two reaction-
diffusion equations governing the concentration of calcium ions in the endoplasmic reticulum
and, respectively, in the cytosol, coupled through an interfacial exchange term. Depending on
the magnitude of this term, various models arise at the macroscale. In particular, we obtain,
at the limit, a bidomain model. Such a model is widely used for studying the dynamics of
the calcium ions, which are recognized to be important intracellular messengers between the
endoplasmic reticulum and the cytosol inside the biological cells.
Calcium is a very important second messenger in a living cell, participating in many
cellular processes, such as protein synthesis, muscle contraction, cell cycle, metabolism or
apoptosis (see, for instance, [72]). Intracellular free calcium concentrations must be very well
regulated and many buffer proteins, pumps or carriers of calcium take part at this complicated
process. The finely structured endoplasmic reticulum, which is surrounded by the cytosol,
is an important multifunctional intracellular organelle involved in calcium homeostasis and
many of its functions depend on the calcium dynamics. The endoplasmic reticulum plays
an important role in the metabolism of human cells. It performs diverse functions, such as
protein synthesis, translocation across the membrane, folding, etc. This complex and highly
heterogeneous cellular structure spreads throughout the cytoplasm, generating various zones
with diverse morphology and functions. The study of the dynamics of calcium ions, acting as
messengers between the endoplasmic reticulum and the cytosol inside living cells, represents
a topic of huge interest, which still requires special attention. Many biological mechanisms
involving the functions of the cytosol and of the endoplasmic reticulum are not yet perfectly
understood.
Our goal in [221] was to rigorously analyze, using the periodic unfolding method, the
macroscopic behavior of a nonlinear system of coupled reaction-diffusion equations arising
in the modeling of calcium dynamics in living cells. We consider, at the microscale, two
MATHEMATICAL MODELS IN BIOLOGY 87
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, different 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-
diffusion equations, one for the concentration of calcium ions in the cytosol and one for the
concentration of calcium ions in the endoplasmic reticulum, coupled through a reaction term.
For details about the physiological background of such a model, the reader is referred to [154].
Bidomain models arise also in other contexts, such as the modeling of diffusion processes in
partially fissured media (see [31], [27] and [103]) or the modeling of the electrical activity of
the heart (see [15], [13] and [189]).
Our models can serve as a tool for biophysicists to analyze the complex mechanisms
involved in the calcium dynamics in living cells, justifying in a rigorous manner some biological
points of view concerning such processes.
The problem of obtaining the calcium bidomain equations using homogenization tech-
niques was addressed by a formal approach in [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 important influence on the limit problem and, using some techniques
from [96], we extend the analysis from [130] to the case in which the parameter γarising in
the exchange term belongs to R.
4.3.1 Setting of the problem
Let us start by describing the geometry of the problem. Let Ω be a bounded domain in
Rn, withn≥3, having a Lipschitz boundary ∂Ω formed by a finite number of connected
components. The domain Ω is supposed to be a periodic structure made up of two connected
parts, Ωε
1and Ωε
2, separated by an interface Γε. We assume that only the phase Ωε
1reaches
the outer fixed boundary ∂Ω. Here,εis considered to be a small positive real parameter
related to the characteristic dimension of our two regions. For modeling the dynamics of the
concentration of calcium ions in a biological cell, the phase Ωε
1represents the cytosol, while
the phase Ωε
2is the endoplasmic reticulum. Let Y1be an open connected Lipschitz subset of
the elementary cell Y= (0,1)nandY2=Y\Y1. We consider that the boundary Γ of Y2is
locally Lipschitz and that its intersections with the boundary of Yare reproduced identically
on the opposite faces of the elementary cell. Moreover, 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 [103]).
For anyε∈(0,1), let
Zε={k∈Zn|εk+εY⊆Ω},
Kε={k∈Zε|εk±εei+εY⊆Ω,∀i= 1,…,n },
88 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
whereeiare the vectors of the canonical basis of Rn. We denote (see Figure ….)
Ωε
2= int(∪
k∈Kε(εk+εY2)),Ωε
1= Ω\Ωε
2
and we set θ= Y\Y2 .
Forα1,β1∈R, with 0<α 1<β1, we denote by M(α1,β1,Y) the collection of all the ma-
tricesA∈(L∞(Y))n×nwith the property that, for any ξ∈Rn, (A(y)ξ, ξ)≥α1|ξ|2,|A(y)ξ| ≤
β1|ξ|, almost everywhere in Y. We consider the matrices Aε(x) =A(x/ε) defined on Ω, where
A∈ M (α1,β1,Y) is aY-periodic smooth symmetric matrix and we denote the matrix Aby
A1inY1and, respectively, by A2inY2.
If (0,T) is the time interval under consideration, we shall be concerned with the macro-
scopic behavior of the solutions of the following microscopic system:
∂uε
1
∂t−div (A1ε∇uε
1) =f(uε
1) in (0,T)×Ωε
1,
∂uε
2
∂t−div (A2ε∇uε
2) =g(uε
2) in (0,T)×Ωε
2,
A1ε∇uε
1·ν=A2ε∇uε
2·νon (0,T)×Γε,
A1ε∇uε
1·ν=εγh(uε
1,uε
2) on (0,T)×Γε,
uε
1= 0 on (0 ,T)×∂Ω,
uε
1(0,x) =u0
1(x) in Ωε
1, uε
2(0,x) =u0
2(x) in Ωε
2,(4.44)
whereνis the unit outward normal to Ωε
1and the scaling exponent γis a given real number,
related to the speed of the interfacial exchange. As we shall see, three important cases will
arise at the limit, i.e. γ= 1,γ= 0 andγ=−1 (see, also, Remark 4.24). We assume that the
initial conditions are non-negative and that the functions fandgare Lipschitz-continuous,
withf(0) =g(0) = 0. We also suppose that
h(uε
1,uε
2) =hε
0(x)(uε
2−uε
1), (4.45)
wherehε
0(x) =h0(x/ε) andh0=h0(y) is a realY-periodic function in L∞(Γ), withh0(y)≥
δ>0. Besides, we consider that
H=∫
Γh0(y) dσy̸= 0.
As in [130], we can treat in a similar way the case in which the function his Lipschitz-
continuous in both arguments and is given by:
h(r,s) =h(r,s)(s−r), (4.46)
with 0<h min≤h(r,s)≤hmax<∞.
Since it is not easy to find an explicit solution of the well-posed microscopic problem (4.44),
we need to apply an homogenization procedure for obtaining a suitable model that describes
the averaged properties of the complicated microstructure. Using the periodic unfolding
MATHEMATICAL MODELS IN BIOLOGY 89
method, we can find the asymptotic behavior of the solution of our problem. For the case
γ= 1, this behavior is described by a new nonlinear system (see (4.48)), a bidomain model .
So, in this case, at a macroscopic scale, our medium can be represented by a continuous
model, i.e. the superimposition of two interpenetrating continuous media, the cytosol and
the endoplasmic reticulum, which coexist at any point. For the other two relevant cases,
see (4.49) and (4.50). For the particular geometry considered in this section, we use two
unfolding operators, mapping functions defined on oscillating domains into functions given
on fixed domains (see [96]).
It might seem that these simplified assumptions about the complex calcium dynamics
inside a cell are quite strong. However, the homogenized solution fits well with experimental
data (see [154]). Also, one could argue that the periodicity of the microstructure is not
a realistic assumption and it would be interesting to work with a random microstructure.
Still, such a periodic structure provides a very good description, in agreement with all the
experimental findings (see [136]).
We can deal, in a similar manner, with the more general case of an heterogeneous medium
represented by a matrix Aε
0=A0(x,x/ε ) or by a matrix Dε=D(t,x/ε ), under reasonable
assumptions on the matrices A0andD. For instance, we can suppose that Dis a symmetric
matrix, with D,∂D
∂t∈L∞(0;T;L∞
per(Y))n×nand such that, for any ξ∈Rn, (D(t,x)ξ,ξ)≥
α2|ξ|2and|D(t,x)ξ| ≤β2|ξ|, almost everywhere in (0 ,T)×Y, for 0<α 2<β2.
In order to prove the main convergence results for our problem, obtained via the periodic
unfolding method, let us introduce now the function spaces and norms we shall work with in
the sequel. Let
H1
∂Ω(Ωε
1) ={v∈H1(Ωε
1)|v= 0 on∂Ω∩∂Ωε
1},
V(Ωε
1) =L2(0,T;H1
∂Ω(Ωε
1)),V(Ωε
1) ={
v∈V(Ωε
1)|∂v
∂t∈L2((0,T)×Ωε
1)}
,
V(Ωε
2) =L2(0,T;H1(Ωε
2)),V(Ωε
2) ={
v∈V(Ωε
2)|∂v
∂t∈L2((0,T)×Ωε
2)}
,
with
(u(t),v(t))Ωεα=∫
Ωεαu(t,x)v(t,x) dx,∥u(t)∥2
Ωεα= (u(t),u(t))Ωεα,
(u,v)Ωεα,t=∫t
0(u(t),v(t))Ωεαdt,∥u∥2
Ωεα,t= (u,u)Ωεα,t,
forα= 1,2. Also, let
V(Ω) =L2(0,T;H1(Ω)),V(Ω) ={
v∈V(Ω)|∂v
∂t∈L2((0,T)×Ω)}
,
with
(u(t),v(t))Ω=∫
Ωu(t,x)v(t,x) dx,∥u(t)∥2
Ω= (u(t),u(t))Ω,
(u,v)Ω,t=∫t
0(u(t),v(t))Ωdt,∥u∥2
Ω,t= (u,u)Ω,t
90 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
and
V0(Ω) = {v∈V(Ω)|v= 0 on∂Ω a.e. on (0 ,T)},V0(Ω) =V0(Ω)∩ V(Ω).
The variational formulation of problem (4.44) is as follows: find ( uε
1,uε
2)∈ V(Ωε
1)×V(Ωε
2),
with (uε
1(0,x),uε
2(0,x)) = (u0
1(x),u0
2(x))∈(L2(Ω))2and
(∂uε
1
∂t(t),φ(t))
Ωε
1+(∂uε
2
∂t(t),ψ(t))
Ωε
2+
(Aε
1(t)∇uε
1,∇φ(t))Ωε
1+ (Aε
2(t)∇uε
2,∇ψ(t))Ωε
2−
εγ(h(uε
1,uε
2),φ(t)−ψ(t))Γε= (f(uε
1(t)),φ(t))Ωε
1+ (g(uε
2(t)),ψ(t))Ωε
2, (4.47)
for a.e.t∈(0,T) and any ( φ,ψ)∈V(Ωε
1)×V(Ωε
2).
Following the same techniques used in [130], it is not difficult to prove that (4.47) is a
well-posed problem and that uεandvεare non-negative and bounded almost everywhere.
Taking (uε
1,uε
2) as test function in (4.47), integrating with respect to time and taking into
account that uε
1anduε
2are bounded and non-negative, it follows that there exists a constant
C≥0, independent of ε, such that
∥uε
1(t)∥2
Ωε
1+∥uε
2(t)∥2
Ωε
2+∥∇uε
1∥2
Ωε
1,t+∥∇uε
2∥2
Ωε
2,t+εγ(h(uε
1,uε
2),uε
1−uε
2)Γε,t≤C,
for a.e.t∈(0,T). Also, as in [130] or [215], we can see that there exists a positive constant
C≥0, independent of ε, such that
∂uε
1
∂t(t)
2
Ωε
1+
∂uε
2
∂t(t)
2
Ωε
2≤C,
forγ≥1 and
∂uε
1
∂t
L2(0,T;H−1(Ωε
1))+
∂uε
2
∂t
L2(0,T;H−1(Ωε
2))≤C,
forγ <1. These a priori estimates allow us to use the periodic unfolding method (for a two-
component domain) and to obtain the needed convergence results in all the above mentioned
relevant cases.
More precisely, for retrieving the macroscopic behavior of the solution of problem (4.47),
we use two unfolding operators, Tε
1andTε
2, which transform functions defined on oscillating
domains into functions defined on fixed domains (see [57], [59] and [96]). We briefly recall
here the definitions and the main properties of these unfolding operators.
Forx∈Rn, we denote by [ x]Yits integer part k∈Zn, such that x−[x]Y∈Yand we set
{x}Y=x−[x]Yfor a.e.x∈Rn. So, for almost every x∈Rn, we havex=ε([x
ε]
Y+{x
ε}
Y)
.
For defining the above mentioned periodic unfolding operators, we consider the following sets
(see [96]):
bZε={
k∈Zn|εYk⊂Ω}
,bΩε= int∪
k∈bZε(
ε¯Yk)
,Λε= Ω\bΩε,
bΩε
α=∪
k∈bZε(
εYk
α)
,Λε
α= Ωε
α\bΩε
α,bΓε=∂bΩε
2.
MATHEMATICAL MODELS IN BIOLOGY 91
De nition 4.14 For any Lebesgue measurable function φonΩε
α,α∈ {1,2}, we define the
periodic unfolding operators by the formula
Tε
α(φ)(x,y) =
φ(
ε[x
ε]
Y+εy)
for a.e. (x,y)∈bΩε×Yα
0 for a.e. (x,y)∈Λε×Yα
Ifφis a function defined in Ω, for simplicity, we write Tε
α(φ)instead of Tε
α(φ|Ωεα).
For any function ϕwhich is Lebesgue-measurable on Γε, the periodic boundary unfolding
operator Tε
bis defined by
Tε
b(ϕ)(x,y) =
ϕ(
ε[x
ε]
Y+εy)
for a.e. (x,y)∈bΩε×Γ
0 for a.e. (x,y)∈Λε×Γ
Remark 4.15 We notice that if φ∈H1(Ωε
α), then Tε
b(φ) =Tε
α(φ)|bΩε×Γ.
We recall here some useful properties of these operators (see, for instance, [57], [95], and
[96]).
Proposition 4.16 Forp∈[1,∞)andα= 1,2, the operators Tε
αare linear and continuous
fromLp(Ωε
α)toLp(Ω×Yα)and
(i) ifφandψare two Lebesgue measurable functions on Ωε
α, one has
Tε
α(φψ) =Tε
α(φ)Tε
α(ψ);
(ii) for every φ∈L1(Ωε
α), one has
1
|Y|∫
Ω×YαTε
α(φ)(x,y) dxdy=∫
bΩεαφ(x) dx=∫
Ωεαφ(x) dx−∫
Λεφ(x) dx;
(iii) if {φε}ε⊂Lp(Ω)is a sequence such that φε−→φstrongly in Lp(Ω), then
Tε
α(φε)−→φstrongly in Lp(Ω×Yα);
(iv) ifφ∈Lp(Yα)isY-periodic and φε(x) =φ(x/ε), then
Tε
α(φε)−→φstrongly in Lp(Ω×Yα);
(v) ifφ∈W1,p(Ωε
α), then ∇y(Tε
α(φ)) =εTε
α(∇φ)andTε
α(φ)belongs toL2(
Ω;W1,p(Yα))
.
Moreover, for every φ∈L1(Γε), one has
∫
bΓεφ(x) dσx=1
ε|Y|∫
Ω×ΓTε
b(φ)(x,y) dxdσy.
Forγ= 1, using the obtained a priori estimates and the properties of the operators
Tε
1andTε
2, it follows that there exist u1∈L2(0,T;H1
0(Ω)),u2∈L2(0,T;H1(Ω)),bu1∈
92 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
L2((0,T)×Ω;H1
per(Y1)),bu2∈L2((0,T)×Ω;H1
per(Y2)) such that, passing to a subsequence,
forε→0, we have:
Tε
1(uε
1)→u1strongly in L2((0,T)×Ω,H1(Y1)),
Tε
1(∇uε
1)⇀∇u1+∇ybu1weakly inL2((0,T)×Ω×Y1),
Tε
2(uε
2)⇀u 2weakly inL2((0,T)×Ω,H1(Y2)),
Tε
2(∇uε
2)⇀∇u2+∇ybu2weakly inL2((0,T)×Ω×Y2).(4.48)
Moreover, as in [130] and [215],∂u1
∂t∈L2(0,T;L2(Ω)),∂u2
∂t∈L2(0,T;L2(Ω)) andu1∈
C0([0,T];H1
0(Ω)), u2∈C0([0,T];H1(Ω)). So,u1∈ V0(Ω) andu2∈ V(Ω).
Let us mention that, in fact, under our hypotheses, passing to a subsequence, Tε
1(uε
1)
converges strongly to u1inLp((0,T)×Ω×Y1), for 1 ≤p<∞. As a consequence, since the
Nemytskii operator corresponding to the nonlinear function fis continuous, it follows that
f(Tε
1(uε
1)) converges to f(u1). A similar result holds true for uε
2.
Since
∥Tε
1(uε
1)− Tε
2(uε
2)∥L2((0,T)×Ω×Γ)≤Cε1−γ
2
it follows that for the case γ= 0 andγ=−1 we have, at the macroscale, u1=u2=u0∈
V0(Ω). Moreover, for γ=−1, following the techniques from [96], one can prove that
Tε
1(uε
1)− Tε
2(uε
2)
ε⇀bu1−bu2weakly inL2((0,T)×Ω×Γ).
4.3.2 The main convergence results
We present now the main convergence results obtained in [221].
Theorem 4.17 (Theorem 1 in [221]) If γ= 1, the solution (uε
1, uε
2)of system (4.44) con-
verges in the sense of (4.48), as ε→0, to the unique solution (u1, u2)of the following
macroscopic problem:
θ∂u1
∂t−div(A1∇u1)−H(u2−u1) =θf(u1)in(0,T)×Ω,
(1−θ)∂u2
∂t−div(A2∇u2) +H(u2−u1) = (1 −θ)g(u2)in(0,T)×Ω,
u1(0,x) =u0
1(x), u 2(0,x) =u0
2(x)inΩ.(4.49)
Here,
H=∫
Γh0(y) dσy
andA1andA2are the homogenized matrices, given by:
A1
ij=∫
Y1(
a1
ij+n∑
k=1a1
ik∂χ1j
∂yk)
dy,
A2
ij=∫
Y2(
a2
ij+n∑
k=1a2
ik∂χ2j
∂yk)
dy,
MATHEMATICAL MODELS IN BIOLOGY 93
wherea1
ij=A1
ij,a2
ij=A2
ijandχ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n , are the
weak solutions of the cell problems
−divy(A1(y)(∇yχ1k+ek)) = 0, y∈Y1,
A1(y)(∇yχ1k+ek)·ν= 0, y∈Γ,
−divy(A2(y)(∇yχ2k+ek)) = 0, y∈Y2,
A2(y)(∇yχ2k+ek)·ν= 0, y∈Γ.
At a macroscopic scale, we obtain a continuous model, a so-called bidomain model , similar to
those arising in the context of the modeling of diffusion processes in partially fissured media
(see [31] and [103]) or in the case of the modeling of the electrical activity of the heart (see
[15], [13] and [189]). If we assume that his given by (4.46), then, at the limit, the exchange
term appearing in (4.49) is of the form |Γ|h(u1,u2).
Theorem 4.18 Forγ= 0, i.e. for high contact resistance, we obtain, at the macroscale,
only one concentration field. So, u1=u2=u0andu0is the unique solution of the following
problem:
∂u0
∂t−div(A0∇u0) =θf(u0) + (1 −θ)g(u0)in(0,T)×Ω,
u0(0,x) =u0
1(x) +u0
2(x)inΩ.(4.50)
Here, the effective matrix A0is given by:
A0
ij=∫
Y1(
a1
ij+n∑
k=1a1
ik∂χ1j
∂yk)
dy+∫
Y2(
a2
ij+n∑
k=1a2
ik∂χ2j
∂yk)
dy,
in terms of the functions χ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n, weak solutions
of the local problems
−divy(A1(y)(∇yχ1k+ek)) = 0, y∈Y1,
A1(y)(∇yχ1k+ek)·ν= 0, y∈Γ,
−divy(A2(y)(∇yχ2k+ek)) = 0, y∈Y2,
A2(y)(∇yχ2k+ek)·ν= 0, y∈Γ.
In this case, the exchange at the interface leads to the modification of the limiting diffusion
matrix, but the insulation is not enough strong to impose the existence of two different limit
concentrations.
Theorem 4.19 For the case γ=−1, i.e. for very fast interfacial exchange of calcium
between the cytosol and the endoplasmic reticulum (i.e. for weak contact resistance), at the
limit, we also obtain u1=u2=u0and, in this case, the effective concentration field u0
satisfies:
∂u0
∂t−div(A0∇u0) =θf(u0) + (1 −θ)g(u0)in(0,T)×Ω,
u0(0,x) =u0
1(x) +u0
2(x)inΩ.(4.51)
The effective coefficients are given by:
A0,ij=∫
Y1(
a1
ij+n∑
k=1a1
ik∂w1j
∂yk)
dy+∫
Y2(
a2
ij+n∑
k=1a2
ik∂w2j
∂yk)
dy,
wherew1k∈H1
per(Y1)/R, w2k∈H1
per(Y2)/R,k= 1,…,n, are the weak solutions of the cell
problems
−divy(A1(y)(∇yw1k+ek)) = 0, y∈Y1,
−divy(A2(y)(∇yw2k+ek)) = 0, y∈Y2,
(A1(y)∇yw1k)·ν= (A2(y)∇yw2k)·ν, y ∈Γ,
(A1(y)∇yw1k)·ν+h0(y)(w1k−w2k) =−A1(y)ek·ν, y ∈Γ.
It is important to notice that the diffusion coefficients depend now on h0. A similar result
holds true for the case in which his given by (4.46). Let us notice that in this case the
homogenized matrix is no longer constant, but it depends on the solution u0. A similar effect
was noticed in [6].
Remark 4.20 For simplicity, we address here only the relevant cases γ=−1,0,1. For the
caseγ∈(−1,1), we get, at the limit, the macroscopic problem (4.50), while for γ >1, we
obtain a problem similar to (4.49), but without the exchange term H(u2−u1)or|Γ|h(u1,u2),
respectively. Finally, for the case γ <−1, we obtain, at the limit, a standard composite
medium without any barrier resistance. It is worth mentioning that in this case we get w1k=
w2konΓ, fork= 1,…,n .
Remark 4.21 The conditions imposed on the nonlinear functions f,g, andhcan be relaxed.
For instance, we can consider that fandgare maximal monotone graphs, verifying suitable
growth conditions (see [75]). Also, as in [179], [195] and [223], we can work with more general
functionsh.
94
Chapter 5
Multiscale modeling of composite
media with imperfect interfaces
In the last two decades, the study of the macroscopic properties of heterogeneous composite
materials with imperfect contact between their constituents has been a subject of major inter-
est for engineers, mathematicians, physicists (see [26] and [152]). In particular, the problem
of thermal transfer in heterogeneous media with imperfect interfaces has attracted the atten-
tion of a broad category of researchers, due to the fact that the macroscopic properties of a
composite can be strongly influenced by the imperfect bonding between its components (for a
review of the literature on imperfect interfaces in heterogeneous media, we refer to [165] and
[176]). This imperfect contact can be generated by various causes: the presence of impurities
at the boundaries, the presence of a thin interphase, the interface damage, chemical processes.
The homogenization theory was successfully applied for modeling the behavior of such het-
erogeneous materials, with inhomogeneities at a length scale which is much smaller than
the characteristic dimensions of the system, leading to appropriate macroscopic continuum
models, obtained by averaging the rapid oscillations of the material properties. Besides, such
effective models have the advantage of avoiding extensive numerical computations arising
when dealing with the small scale behavior of the system.
The homogenization of a thermal problem in a two-component composite with interfacial
barrier, with jump of the temperature and continuity of the flux, was studied for the first
time in the pioneering work [103], where the asymptotic expansion method was used. Many
mathematical studies were performed since then, in order to rigorously justify the convergence
results. Various mathematical methods were used: the energy method in [97] and [177], the
two-scale convergence method in [106], and more recently the unfolding method for periodic
homogenization in [96] and [218], to cite just a few of them. The main common point of all
these studies is the fact that at the interface between the two components the flux of the
temperature is continuous, the temperature field has a jump and the flux is proportional to
this jump. Several cases are studied, following the order of magnitude with respect to the
small parameter εcharacterizing the size of the two constituents of the resistance generated by
the imperfect contact between the constituents, leading to completely different macroscopic
95
96 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
problems. In some cases, an effect of the imperfect conditions is observed in the coefficients
of the homogenized matrix, via the local problems; in other cases, there is no effect at all in
the homogenized problem.
Such problems were addressed mainly in two geometrical settings. For the case when
both components of the composite material are connected, we refer to [161], [194], [193],
[192], [215] and [216]. The case in which only one phase is connected, while the other one is
disconnected was considered, for instance, in [97], [177], and [96].
For similar homogenization problems of parabolic or hyperbolic type, we refer the reader
to [94] 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],
[98], [108], [110], [109], [142], [189], and [223].
In this chapter, we shall present some recent homogenization results for diffusion prob-
lems in composite media with imperfect imperfect interfaces. We start by presenting the
results obtained in [218] for a thermal diffusion problem in a bi-connected structure. In the
same geometry, we shall present some homogenization results obtained in [215] and [216]
for diffusion problems with dynamical boundary conditions. We remark that, using similar
techniques, we can analyze the asymptotic behavior of the solution of a system of coupled
partial differential equations appearing in the modeling of an elasticity problem in a periodic
structure formed by two interwoven and connected components with imperfect contact at the
interface (see [108]) or of a dynamic coupled thermoelasticity problem in composite media
with imperfect interfaces and various geometries (see [110] and [109]). We end this chapter by
presenting some recent results obtained in [48], [49], and [50] for diffusion problems involving
jumps both in the solution and in the flux.
This chapter is based on the papers [48], [49], [50], [215], [216], and [218].
5.1 Multiscale analysis in thermal diffusion problems in com-
posite structures
In [218], we have analyzed, using the periodic unfolding method, the effective thermal transfer
in a periodic composite material formed by two constituents, separated by an imperfect
interface. Our results were set in the framework of thermal transfer, but they remain true for
more general reaction-diffusion processes. We assumed that we have nonlinear sources acting
in each media and that at the interface between the two constituents the flux is continuous,
but the temperature field has a jump. We were interested in describing the asymptotic beha-
vior, as the small parameter which characterizes the sizes of the two constituents tends to
zero, of the temperature field in the periodic composite. The imperfect contact between
the constituents generates a contact resistance and, depending on the magnitude of this
resistance, a threshold phenomenon arises. So, depending on the rate exchange between the
two phases, three important cases are considered and three different types of limit problems
are obtained from the same type of micromodel. The results in [218] constitute a gene-
HEAT TRANSFER IN COMPOSITE MATERIALS 97
ralization of those obtained in [103], [190], [215] and [216]. For heat conduction problems in
a periodic material with a different geometry, we refer to [96] 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 [75] and [190]). Similar problems have been addressed,
using different techniques, formal or not, in [26], [27], [165] and [103].
5.1.1 Problem setting
We place ourselves in the same setting as in Section 4.3. So, we consider an open bounded
material body Ω in Rn, withn≥3, with a Lipschitz-continuous boundary ∂Ω. We assume
that Ω is formed by two constituents, Ωε
1and Ωε
2, representing two materials with different
thermal characteristics, separated by an imperfect interface Γε. We assume that both phases
Ωε
1and Ωε
2= Ω\Ωε
1are connected, but only Ωε
1reaches the external fixed boundary ∂Ω.
Here,εrepresents a small parameter related to the characteristic size of the two constituents.
Our goal in [218] was to describe the effective behavior of the solution ( uε
1,uε
2) of the
following coupled system of equations:
−div (Aε
1∇uε
1) +α(uε
1) =fin Ωε
1,
−div (Aε
2∇uε
2) +β(uε
2) =fin Ωε
2,
Aε
1∇uε
1·ν=Aε
2∇uε
2·νon Γε,
Aε
1∇uε
1·ν=εγh(uε
1,uε
2) on Γε,
uε
1= 0 on∂Ω.(5.1)
Here,νis the unit outward normal to Ωε
1andf∈L2(Ω).
Thus, we consider that the flux is continuous across the boundary Γε, but, since the
interface between the two phases is not perfect, the continuity of temperatures is replaced
by a Biot boundary condition. We assume that the functions α=α(r) andβ=β(r)
are continuous, monotonously non-decreasing with respect to rand such that α(0) = 0
andβ(0) = 0. Moreover, we suppose that there exist C≥0 and an exponent q, with
0≤q<n/ (n−2), such that
|α(r)| ≤C(1 +|r|q) (5.2)
and
|β(r)| ≤C(1 +|r|q). (5.3)
We also assume, as in Section 4.3, that
h(uε
1,uε
2) =hε
0(x)(uε
2−uε
1),
wherehε
0(x) =h0(x
ε)
andh0(y) is aY-periodic, smooth real function with h0(y)≥δ >0.
Moreover, we consider that
H=∫
Γh0(y) dσ̸= 0.
Let us notice that we can deal with the more general case in which the nonlinear functions
αandβare multi-valued maximal monotone graphs, as in [75].
98 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
For results concerning the well posedness of problem (5.1), we refer to [15], [13], [190],
[189], [215] and [216]. Since it is impossible to solve this system, the microstructure must be
homogenized in order to obtain a new model that describes its macroscopic properties. Using
the periodic unfolding method, we can describe the asymptotic behavior of the solution of
system (5.1). This behavior depends on the values of the parameter γ, i.e. on the contact
resistance between the two constituents. There are three interesting cases to be considered:
γ= 1,γ= 0, andγ=−1.
In the most interesting case, γ= 1, we obtain at the limit a new nonlinear system (see
(5.5)). At a macroscopic scale, the composite medium can be represented by a continuous
model, which conceives it as the superimposition of two interpenetrating continuous media,
coexisting at every point of the domain. For the other two cases, we obtain at the limit only
one equation (see (5.6) and (5.7)).
5.1.2 The main results
We describe now the effective behavior of the solutions of the microscopic model (5.1) for the
above mentioned three cases.
Forγ= 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 u1∈H1
0(Ω),u2∈H1(Ω),bu1∈L2(Ω;H1
per(Y1)),bu2∈L2(Ω;H1
per(Y2)) such that,
passing to a subsequence, for ε→0, we have:
Tε
1(uε
1)→u1strongly in L2(Ω,H1(Y1)),
Tε
1(∇uε
1)⇀∇u1+∇ybu1weakly inL2(Ω×Y1),
Tε
2(uε
2)⇀u 2weakly inL2(Ω,H1(Y2)),
Tε
2(∇uε
2)⇀∇u2+∇ybu2weakly inL2(Ω×Y2).(5.4)
Therefore, the main convergence result in this case is stated in the following theorem, proven
in [218].
Theorem 5.1 Forγ= 1, the solution (uε
1, uε
2)of system (5.1) converges, as ε→0, in
the sense of (5.4), to the unique solution (u1, u2), withu1,u2∈H1
0(Ω), of the following
macroscopic problem:
{
−div(A1∇u1) +θα(u1)−H(u2−u1) =θfinΩ,
−div(A2∇u2) + (1 −θ)β(u2) +H(u2−u1) = (1 −θ)finΩ.(5.5)
In (5.5),A1andA2are the homogenized matrices, defined by:
A1
ij=∫
Y1(
aij+aik∂χ1j
∂yk)
dy,
A2
ij=∫
Y2(
aij+aik∂χ2j
∂yk)
dy
HEAT TRANSFER IN COMPOSITE MATERIALS 99
andχ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n , are the weak solutions of the cell
problems
−∇ y·((A1(y)∇yχ1k) =∇yA1(y)ek, y∈Y1,
(A1(y)∇yχ1k)·ν=−A1(y)ek·ν, y ∈Γ,
−∇ y·((A2(y)∇yχ2k) =∇yA2(y)ek, y∈Y2,
(A2(y)∇yχ2k)·ν=−A2(y)ek·ν, y ∈Γ.
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 diffusion in partially fissured media (see [31] and [103]) or in the case of
electrical activity of the heart (see [15], [13] and [189]).
Following the same techniques as those used in Section 4.3, we can deal with the other
two relevant cases, namely γ= 0 andγ=−1.
Theorem 5.2 Forγ= 0, i.e. for high contact resistance, we get, at the macroscale, only
one temperature field. So, u1=u2=u0∈H1
0(Ω)andu0satisfies:
−div(A0∇u0) +θα(u0) + (1 −θ)β(u0) =finΩ. (5.6)
Here, the effective matrix A0is given by:
A0
ij=∫
Y1(
aij+aik∂χ1j
∂yk)
dy+∫
Y2(
aij+aik∂χ2j
∂yk)
dy,
in terms of the functions χ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n, weak solutions
of the cell problems
−∇ y·((A1(y)∇yχ1k) =∇yA1(y)ek, y∈Y1,
(A1(y)∇yχ1k)·ν=−A1(y)ek·ν, y ∈Γ,
−∇ y·((A2(y)∇yχ2k) =∇yA2(y)ek, y∈Y2,
(A2(y)∇yχ2k)·ν=−A2(y)ek·ν, y ∈Γ.
Let us notice that in this case, the insulation provided by the interface is sufficient to modify
the limiting diffusion matrix, but it is not strong enough to force the existence of two different
limit phases.
Theorem 5.3 For the case γ=−1, i.e. for weak contact resistance, we also get, at the
limit,u1=u2=u0∈H1
0(Ω)and, in this case, the effective temperature field u0satisfies:
−div(A0∇u0) +θα(u0) + (1 −θ)β(u0) =finΩ. (5.7)
100 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
The macroscopic coefficients are given by:
A0,ij=∫
Y1(
aij+aik∂w1j
∂yk)
dy+∫
Y2(
aij+aik∂w2j
∂yk)
dy,
wherew1k∈H1
per(Y1)/R, w2k∈H1
per(Y2)/R,k= 1,…,n, are the weak solutions of the cell
problems
−∇ y·(A1(y)∇yw1k) =∇yA1(y)ek, y∈Y1,
−∇ y·(A2(y)∇yw2k) =∇yA2(y)ek, y∈Y2,
(A1(y)∇yw1k)·ν= (A2(y)∇yw2k)·ν y∈Γ,
(A1(y)∇yw1k)·ν+h0(y)(w1k−w2k) =−A1(y)ek·ν y∈Γ.
In this case, as expected, the effective coefficients depend on h0.
5.2 Diffusion problems with dynamical boundary conditions
Recent theoretical and advanced computational studies investigating the effective behavior
of composite materials are based on a model which considers the composite material, despite
of its discrete structure, as the coupling of two continuous superimposed domains. Homoge-
nization theory will allow us to justify such a model and to give a meaning to the effective
properties of highly heterogeneous materials, modeled by equations with rapidly oscillating
periodic coefficients.
In this section, we shall present some homogenization results obtained by using the test
function method of L. Tartar in [215] and generalized later, using the periodic unfolding
method, in [216]. The aim of these papers was to analyze the asymptotic behavior of the so-
lution of a nonlinear problem arising in the modeling of thermal diffusion in a two-component
composite material. We consider, at the microscale, a periodic structure formed by two ma-
terials with different thermal properties. We assume that we have nonlinear sources and that
at the interface between the two materials the flux is continuous and depends in a dynamical
nonlinear way on the jump of the temperature field. As usual, we are interested in describ-
ing the asymptotic behavior of the temperature field in the periodic composite as the small
parameter which characterizes the sizes of our two regions tends to zero. We prove that
the effective behavior of the solution of this system is governed by a new system, similar to
Barenblatt’s model, with additional terms capturing the effect of the interfacial barrier, of
the dynamical boundary condition, and of the nonlinear sources.
More precisely, in the same geometry as the one described in Section 5.1, we shall be
interested in analyzing the asymptotic behavior of the solutions of the following nonlinear
system:
−div (Aε
1∇uε
1) +α(uε
1) =fin Ωε
1×(0,T),
−div (Aε
2∇uε
2) =f, in Ωε
2×(0,T),
Aε
1∇uε
1·ν=Aε
2∇uε
2·νon Γε×(0,T),
Aε
1∇uε
1·ν+aε∂
∂t(uε
1−uε
2) =εg(uε
2−uε
1) on Γε×(0,T),
uε
1= 0 on∂Ω×(0,T),
uε
1(0,x)−uε
2(0,x) =c0(x),on Γε.(5.8)
HEAT TRANSFER IN COMPOSITE MATERIALS 101
Here,νis the exterior unit normal to Ωε
1,f∈L2(0,T;L2(Ω)),c0∈H1
0(Ω) anda > 0.
The function αis continuous, monotonously non-decreasing and such that α(0) = 0 and the
functiongis continuously differentiable, monotonously non-decreasing and with g(0) = 0. We
shall suppose that there exist a positive constant Cand an exponent q, with 0 ≤q<n/ (n−2),
such that
|α(v)| ≤C(1 +|v|q), dg
dv ≤C(1 +|v|q). (5.9)
As particular examples of such functions we can consider, for instance, the following important
practical ones: α(v) =δv
1 +γv, δ,γ > 0 (Langmuir kinetics), α(v) =|v|p−1v,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], [46] and [189]. Our goal is to obtain the asymptotic behavior, when
ε→0, of the solution of problem (5.8). Using Tartar’s method of oscillating test functions
(see [207]), coupled with monotonicity methods and results from the theory of semilinear
problems (see [46] and [75]), we can prove that the asymptotic behavior of the solution of our
problem is governed by a new nonlinear system, similar to the famous Barenblatt’s model
(see [31] and [103]), with additional terms capturing the effect of the interfacial barrier, of
the dynamical boundary condition and of the nonlinear sources. Our results constitute a
generalization of those obtained in [31] and [103], by considering nonlinear sources, nonlinear
dynamical transmission conditions and different techniques in the proofs. Similar problems
have been considered, using different techniques, in [13] and [189], for studying electrical
conduction in biological tissues.
Using well-known extension results (see, for instance, [71], [62] and [189]) and suitable test
functions, we can take the limit in the variational formulation of problem (5.8) and obtain
the effective behavior of the solution of our microscopic model. Therefore, the main result
can be formulated as follows:
Theorem 5.4 (Theorem 2.1 in [215]) One can construct two extensions Pεuε
1andPεuε
2
of the solutions uε
1anduε
2of problem (4.8) such that Pεuε
1⇀ u 1, Pεuε
2⇀ v, weakly in
L2(0,T;H1
0(Ω)), where
a|Γ|∂
∂t(u1−u2)−div(A1∇u1) +θα(u1)− |Γ|g(u2−u1) =
=θfinΩ×(0,T),
a|Γ|∂
∂t(u2−u1)−div(A2∇u2)+|Γ|g(u2−u1) =
= (1−θ)finΩ×(0,T),
u1(0,x)−u2(0,x) =c0(x)onΩ.(5.10)
Here,A1andA2are the homogenized matrices, defined by:
A1
ij=∫
Y1(
aij+aik∂χ1j
∂yk)
dy,
A2
ij=∫
Y2(
aij+aik∂χ2j
∂yk)
dy,
102 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
in terms of the functions χ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n , weak solutions
of the cell problems
−∇ y·((A1(y)∇yχ1k) =∇yA1(y)ek, y∈Y1,
(A1(y)∇yχ1k)·ν=−A1(y)ek·ν, y ∈Γ,
−∇ y·((A2(y)∇yχ2k) =∇yA2(y)ek, y∈Y2,
(A2(y)∇yχ2k)·ν=−A2(y)ek·ν, y ∈Γ.
Thus, in the limit, we obtain a system similar to the so-called Barenblatt model . Alternatively,
such a system is similar to the so-called bidomain model , appearing in the context of elec-
trical 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. The resulting macroscopic model
describes the composite material as the superimposition of two interpenetrating continuous
media, coexisting at every point of the domain. Also, note that the above model is a de-
generate parabolic system, as the time derivatives involve the unknown v−u, instead of the
unknownsuandvoccurring in the second-order conduction term.
These results were generalized in [216] by using the periodic unfolding method, which
allows us to avoid the use of extension operators and, hence, to deal with more general media
and by considering more general nonlinearities α(x,uε) andg(x,vε−uε)).
5.3 Homogenization of a thermal problem with flux jump
In [48], our goal was to analyze, through homogenization techniques, the effective thermal
transfer in a periodic composite material formed by two constituents, one connected and the
other one disconnected, separated by an imperfect interface where both the temperature and
the flux exhibit jumps. This mathematical model is not restricted to the thermal transfer,
but can be used in other contexts, too. Transmission problems involving jumps in the so-
lutions or in the fluxes are encountered in various domains, such as linear elasticity, theory
of semiconductors, the study of photovoltaic systems or problems in media with cracks (see,
for instance, [26], [39], [47], [134] and [157]). Formal methods of averaging were widely used
in the literature to deal with such imperfect transmission problems. Still, obtaining rigorous
results based on the homogenization theory is a difficult task in many cases. Some results
were nevertheless obtained for problems with flux jump, by using homogenization techniques.
We mention here the results obtained in [197] for problems arising in the combustion theory
and in [113] for a problem corresponding to the Gouy-Chapman-Stern model for an electric
double layer.
The main novelty brought by us in [48] consists in allowing the presence, apart from the
discontinuity in the temperature field, of a jump in the thermal flux across the imperfect
interface Γε, given by the function Gε. Two different representative cases were studied,
HEAT TRANSFER IN COMPOSITE MATERIALS 103
following the conditions imposed on Gε(stated explicitly in Section 5.3.1). Let us mention
that such functions were already encountered in a different context, more precisely in [57]
and [56] for the case of the perforated domains with non homogeneous Neumann boundary
conditions on the perforations. After passage to the limit with the unfolding method, we
obtain here two different unfolded problems (stated in Theorem 5.10 and Theorem 5.15),
corresponding to the above mentioned cases for the flux jump function Gε. In both situations,
the homogenized matrix Ahomis constant and it depends on the function describing the jump
of the solution. Moreover, for the first case studied here, we notice in the right-hand side the
presence of a new source term distributed all over the domain Ω and depending on the flux
jump function. For the second case, we notice that the influence of the jump in the flux is
captured by the correctors only and so this jump plays no role in the homogenized problem;
nevertheless, in Remark 5.18 we mention a case when the homogenized problem depends on
this jump, too.
5.3.1 Setting of the problem
We consider a composite material occupying an open bounded set Ω in Rn, withn≥2, with
a Lipschitz-continuous boundary ∂Ω. We assume that Ω is formed by two parts denoted
Ωε
1and Ωε
2, occupied by two materials with different thermal characteristics, separated by
an imperfect interface Γε. We also assume that the phase Ωε
1is connected and reaches the
external fixed boundary ∂Ω and that Ωε
2is disconnected: it is the union of domains of size
ε, periodically distributed in Ω with periodicity ε. More precisely, let Y= (0,1)nbe the
reference cell in Rn. We assume that Y1andY2are two non-empty disjoint connected open
subsets ofYsuch thatY2⊂YandY=Y1∪Y2. We also suppose that Γ = ∂Y2is Lipschitz
continuous and that Y2is connected. In fact, our results can be extended to the case in which
the setY2has a finite number of connected components, as in [96]. For each k∈ZN, we denote
Yk=k+YandYk
α=k+Yα, forα= 1,2. For each ε, we define, Zε={
k∈ZN:εYk
2⊂Ω}
and we set Ωε
2=∪
k∈Zε(
εYk
2)
and Ωε
1= Ω\Ωε
2. The boundary of Ωε
2is denoted by Γεand
νis the unit outward normal to Ωε
1.
Our goal is to describe the asymptotic behavior, as ε→0, of the solution uε= (uε
1,uε
2)
of the following problem:
−div (Aε∇uε
1) =fin Ωε
1,
−div (Aε∇uε
2) =fin Ωε
2,
−Aε∇uε
1·ν=hε
ε(uε
1−uε
2)−Gεon Γε,
−Aε∇uε
2·ν=hε
ε(uε
1−uε
2) on Γε,
uε
1= 0 on∂Ω.(5.11)
Remark 5.5 We notice that
Aε∇uε
2·ν−Aε∇uε
1·ν=−Gε, (5.12)
which clearly shows that the flux of the solution exhibits a jump across Γε.
104 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
The function f∈L2(Ω) is given. Let hbe aY–periodic function in L∞(Γ) such that
there exists h0∈Rwith 0<h0<h(y) a.e. on Γ. We set
hε(x) =h(x
ε)
a.e. on Γε.
Forα,β∈R, with 0<α≤β, letM(α,β,Y ) be the set of all the matrices A∈(L∞(Y))N×N
with the property that, for any ξ∈RN,α|ξ|2≤(A(y)ξ, ξ)≤β|ξ|2, almost everywhere in Y.
For aY-periodic smooth symmetric matrix A∈ M (α,β,Y ), we set
Aε(x) =A(x
ε)
a.e. in Ω.
Letgbe aY-periodic function that belongs to L2(Γ). We define
gε(x) =g(x
ε)
a.e. on Γε.
For the given function Gεin (5.11), we consider the following two relevant forms (see [56]):
Case 1 :Gε=εg(x
ε)
, ifMΓ(g)̸= 0.
Case 2 :Gε=g(x
ε)
, ifMΓ(g) = 0.
Here, MΓ(g) =1
|Γ|∫
Γg(y) dydenotes the mean value of the function gon Γ.
In order to write the variational formulation of problem (5.11), we introduce, for every
positiveε<1, the Hilbert space
Hε=Vε×H1(Ωε
2). (5.13)
The spaceVε={
v∈H1(Ωε
1), v= 0 on∂Ω}
is endowed with the norm ∥v∥Vε=∥∇v∥L2(Ωε
1),
for anyv∈Vεand the space H1(Ωε
2) is equipped with the usual norm. On the space Hε, we
consider the scalar product
(u,v)Hε=∫
Ωε
1∇u1∇v1dx+∫
Ωε
2∇u2∇v2dx+1
ε∫
Γε(u1−u2)(v1−v2) dσx (5.14)
whereu= (u1,u2) andv= (v1,v2) belong to Hε. The norm generated by the scalar product
(5.14) is given by
∥v∥2
Hε=∥∇v1∥2
L2(Ωε
1)+∥∇v2∥2
L2(Ωε
2)+1
ε∥v1−v2∥2
L2(Γε). (5.15)
The variational formulation of problem (5.11) is the following one: find uε∈Hεsuch that
a(uε,v) =l(v),∀v∈Hε, (5.16)
where the bilinear form a:Hε×Hε→Rand the linear form l:Hε→Rare given by
a(u,v) =∫
Ωε
1Aε∇u1∇v1dx+∫
Ωε
2Aε∇u2∇v2dx+∫
Γεhε
ε(u1−u2)(v1−v2) dσx
HEAT TRANSFER IN COMPOSITE MATERIALS 105
and
l(v) =∫
Ωε
1fv1dx+∫
Ωε
2fv2dx+∫
ΓεGεv1dσx,
respectively.
We state now an existence and uniqueness result and some necessary a priori estimates
for the solution of the variational problem (5.16).
Theorem 5.6 For anyε∈(0,1), the variational problem (5.16)has a unique solution uε∈
Hε. Moreover, there exists a constant C > 0, independent of ε, such that
∥∇uε
1∥L2(Ωε
1)≤C,∥∇uε
2∥L2(Ωε
2)≤C (5.17)
and
∥uε
1−uε
2∥L2(Γε)≤Cε1/2. (5.18)
In order to obtain the macroscopic behavior of the solution of problem (5.16), we shall use
the unfolding operators Tε
1andTε
2and the boundary unfolding operator Tε
bdefined in Section
4.3.1. As already mentioned, the main feature of these operators is that they map functions
defined on the oscillating domains Ωε
1, Ωε
2and, respectively, Γε, into functions defined on the
fixed domains Ω ×Y1, Ω×Y2and Ω ×Γ, respectively. The following result was proven, for
our geometry, in [96].
Lemma 5.7 Ifuε= (uε
1,uε
2)is a sequence in Hε, then
1
ε|Y|∫
Ω×Γ|Tε
1(uε
1)− Tε
2(uε
2)|2dxdσy≤∫
Γε|uε
1−uε
2|2dσx.
Moreover, if φ∈ D(Ω), then, forεsmall enough, we have
ε∫
Γεhε(uε
1−uε
2)φdσx=∫
Ω×Γh(y) (Tε
1(uε
1)− Tε
2(uε
2))Tε
α(φ) dxdσy,
withα= 1orα= 2.
We also recall here some general compactness results obtained in [96] for bounded sequences
inHε.
Lemma 5.8 Letuε= (uε
1,uε
2)be a bounded sequence in Hε. Then, there exists a constant
C > 0, independent of ε, such that
∥Tε
1(∇uε
1)∥L2(Ω×Y1)≤C,
∥Tε
2(∇uε
2)∥L2(Ω×Y2)≤C,
∥Tε
2(uε
1)− Tε
1(uε
2)∥L2(Ω×Γ)≤Cε.
106 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Theorem 5.9 Letuε= (uε
1,uε
2)be a bounded sequence in Hε. Then, up to a subsequence,
still denoted by ε, there exist u1∈H1
0(Ω),u2∈L2(Ω),bu1∈L2(
Ω,H1
per(Y1))
andbu2∈
L2(
Ω,H1(Y2))
such that
Tε
1(uε
1)−→u1strongly in L2(
Ω,H1(Y1))
,
Tε
1(∇uε
1)⇀∇u1+∇ybu1weakly inL2(Ω×Y1),
Tε
2(uε
2)⇀u 2weakly inL2(Ω,H1(Y2)),
Tε
2(∇uε
2)⇀∇ybu2weakly inL2(Ω×Y2),
euε
α⇀|Yα|
|Y|uαweakly inL2(Ω), α = 1,2,
where MΓ(bu1) = 0 for almost every x∈Ωand˜·denotes the extension by zero of a function
to the whole of the domain Ω. Moreover, we have u1=u2and
1
ε[Tε
1(uε
1)− M Γ(Tε
1(uε
1))]⇀y Γ∇u1+bu1weakly inL2(
Ω,H1(Y1))
,
withyΓ=y− M Γ(y)and
1
ε[Tε
2(uε
2)− M Γ(Tε
2(uε
2))]⇀bu2weakly inL2(
Ω,H1(Y2))
.
5.3.2 The macroscopic models
Our goal now is to pass to the limit, with ε→0, in the variational formulation (5.16) of the
problem (5.11). We start by emphasizing again that by applying the general results stated in
Theorem 5.9 to the solution uε= (uε
1,uε
2) of the variational problem (5.16), which is bounded
inHε, we obtain, at the macroscale, u1=u2. In what follows, we shall denote their common
value byu. We notice that ubelongs toH1
0(Ω).
Moreover, using the priori estimates (5.17)-(5.18) and the above mentioned general com-
pactness results, we know that there exist u∈H1
0(Ω),bu1∈L2(Ω,H1
per(Y1)),bu2∈L2(Ω,H1(Y2))
such that MΓ(bu1) = 0 and up to a subsequence, for ε→0, we have:
Tε
1(uε
1)→ustrongly in L2(Ω,H1(Y1)),
Tε
1(∇uε
1)⇀∇u+∇ybu1weakly inL2(Ω×Y1),
Tε
2(uε
2)⇀u weakly inL2(Ω,H1(Y2)),
Tε
2(∇uε
2)⇀∇ybu2weakly inL2(Ω×Y2),
euε
α⇀|Yα|
|Y|uweakly inL2(Ω), α = 1,2.(5.19)
Moreover, one has
Tε
1(uε
1)− Tε
2(uε
2)
ε⇀bu1−¯u2weakly inL2(Ω×Γ), (5.20)
HEAT TRANSFER IN COMPOSITE MATERIALS 107
where ¯u2∈L2(Ω,H1(Y2)) is defined by
¯u2=bu2−yΓ∇u−ξΓ,
for someξΓ∈L2(Ω).
LetWper(Y1) ={v∈H1
per(Y1)|MΓ(v) = 0}. We consider the space
V=H1
0(Ω)×L2(Ω;Wper(Y1))×L2(
Ω,H1(Y2))
,
endowed with the norm
∥V∥2
V=∥∇v+∇ybv1∥2
L2(Ω×Y1)+∥∇v+∇y¯v2∥2
L2(Ω×Y2)+∥bv1−¯v2∥2
L2(Ω×Γ),
for allV= (v,bv1,v2)∈ V.
For the passage to the limit, we have to distinguish between two cases, following the form
of the function Gε.
Case 1 :Gε=εg(x
ε)
, ifMΓ(g)̸= 0.
Theorem 5.10 The unique solution uε= (uε
1,uε
2)of the variational problem (5.16) con-
verges, in the sense of (5.19), to the unique solution (u,bu1,¯u2)∈ Vof the following unfolded
limit problem:
1
|Y|∫
Ω×Y1A(y)(∇u+∇ybu1)(∇φ+∇yΦ1) dxdy+
1
|Y|∫
Ω×Y2A(y)(∇u+∇y¯u2)(∇φ+∇yΦ2) dxdy+
1
|Y|∫
Ω×Γh(y)(bu1−¯u2)(Φ1−Φ2) dxdσy=∫
Ωf(x)φ(x) dx+|Γ|
|Y|MΓ(g)∫
Ωφ(x) dx,(5.21)
for allφ∈H1
0(Ω),Φ1∈L2(Ω,H1
per(Y1))andΦ2∈L2(Ω,H1(Y2)).
Proof. In order to obtain the limit problem (5.21), we first unfold the variational
formulation (5.16) and we get
1
|Y|∫
Ω×Y1Tε
1(Aε)Tε
1(∇uε
1)Tε
1(∇v1) dx+1
|Y|∫
Ω×Y2Tε
2(Aε)Tε
2(∇uε
2)Tε
2(∇v2) dx+
1
|Y|∫
Ω×Γh(y)Tε
1(uε
1)− Tε
2(uε
2)
εTε
1(v1)− Tε
2(v2)
εdσx=
1
|Y|∫
Ω×Y1Tε
1(f)Tε
1(v1) dx+1
|Y|∫
Ω×Y2Tε
2(f)Tε
2(v2) dx+1
ε1
|Y|∫
Ω×ΓTε
b(Gε)Tε
b(v1) dσx.
Then, forα= 1,2, we choose in this unfolded problem the admissible test functions
vα=φ(x) +εωα(x)ψα(x
ε)
, (5.22)
108 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
withφ,ω α∈ D(Ω),ψ1∈H1
per(Y1),ψ2∈H1(Y2) and for which we obviously have
Tε
α(vα)→φ(x) strongly in L2(Ω×Yα) (5.23)
and
Tε
α(∇vα)→ ∇φ(x) +∇yΦαstrongly in L2(Ω×Yα), (5.24)
where Φ α(x,y) =ωα(x)ψα(y).
Then, the passage to the limit with ε→0 is classical, by using the above convergences
and the ideas in [95]. The only term which requires more attention is in the right-hand side,
the integral term involving the function Gε. For this term, we have:
1
ε1
|Y|∫
Ω×ΓTε
b(Gε)Tε
b(v1) dσx=1
|Y|∫
Ω×ΓTε
b(
g(x
ε))
Tε
b(
φ(x) +εω1(x)ψ1(x
ε))
dσx=
1
|Y|∫
Ω×Γg(y)Tε
b(φ)(x,y) dxdσy+ε1
|Y|∫
Ω×Γg(y)Tε
b(ω1)(x,y)Tε
b(ψ1)(x,y) dxdσy→
|Γ|
|Y|MΓ(g)∫
Ωφ(x) dx.
By the density of D(Ω)⊗H1
per(Y1) inL2(Ω,H1
per(Y1)) and of D(Ω)⊗H1(Y2) inL2(Ω,H1(Y2)),
we get (5.21).
We notice that our limit problem (5.21) is similar with the one obtained in [96] (see relation
(3.43)), the only difference being the right-hand side, in which an extra term involving the
functiongarises. More precisely, our right-hand side writes
∫
ΩF(x)φ(x) dx,
with
F(x) =f(x) +|Γ|
|Y|MΓ(g).
The extra term is, in fact, just a real constant and this allows us to prove the uniqueness of
the solution of problem (5.21) exactly as in [96], since the presence of this constant term does
not change the linearity nor the continuity of its right-hand side. Thus, due to the uniqueness
of (u,bu1,¯u2)∈ V, all the above convergences hold true for the whole sequence, which ends
the proof of the theorem.
Corollary 5.11 (see [48]) The function u∈H1
0(Ω)defined by (5.19) is the unique solution
of the following homogenized equation:
−div(Ahom∇u) =f+|Γ|
|Y|MΓ(g)inΩ, (5.25)
whereAhomis the homogenized matrix whose entries are given, for i,j= 1,…,n , by
Ahom
ij=1
|Y|∫
Y1(
aij−n∑
k=1aik∂χj
1
∂yk)
dy+1
|Y|∫
Y2(
aij−n∑
k=1aik∂χj
2
∂yk)
dy, (5.26)
HEAT TRANSFER IN COMPOSITE MATERIALS 109
in terms of χj
1∈H1
per(Y1)andχj
2∈H1(Y2),j= 1,…,n , the weak solutions of the following
cell problems:
−divy(A(y)(∇yχj
1+ej)) = 0, y∈Y1,
−divy(A(y)(∇yχj
2+ej)) = 0, y∈Y2,
(A(y)∇yχj
1)·ν= (A(y)∇yχj
2)·ν, y ∈Γ,
−(A(y)(∇yχj
1+ej))·ν+h(y)(χj
1−χj
2) = 0, y∈Γ.
MΓ(χj
1) = 0.(5.27)
whereνdenotes the outward normal to Y1.
Remark 5.12 The right scaling εin front of the function gεprescribed at the interface Γε
leads in the limit to the presence of a new source term distributed all over the domain Ω.
Remark 5.13 It is possible to study our initial problem (5.11) also for a nonzero function
gwith mean-value MΓ(g)equal to zero. But, in this situation, there is no contribution of g
in the right-hand side of the homogenized equation and, thus, the limit problem is the same
as in the case with no gat all in the microscopic problem.
Remark 5.14 We remark that the homogenized matrix Ahomdepends on the function h. So,
the effect of the two jumps involved in our microscopic problem is recovered in the homoge-
nized problem, in the right-hand side and also in the left-hand side (through the homogenized
coefficients).
Case 2 :Gε(x) =g(x
ε)
, ifMΓ(g) = 0.
Theorem 5.15 The unique solution uε= (uε
1,uε
2)of the variational problem (5.16) con-
verges, in the sense of (5.19), to the unique solution (u,bu1,¯u2)∈ Vof the following unfolded
limit problem:
1
|Y|∫
Ω×Y1A(y)(∇u+∇ybu1)(∇φ+∇yΦ1) dxdy+
1
|Y|∫
Ω×Y2A(y)(∇u+∇y¯u2)(∇φ+∇yΦ2) dxdy+
1
|Y|∫
Ω×Γh(y)(bu1−¯u2)(Φ1−Φ2) dxdσy=
∫
Ωf(x)φ(x) dx+1
|Y|∫
Ω×Γg(y)Φ1(x,y) dxdσy, (5.28)
for allφ∈H1
0(Ω),Φ1∈L2(Ω,H1
per(Y1)),Φ2∈L2(Ω,H1(Y2)).
In order to get the problem (5.28), we pass to the limit in the unfolded form of the
variational formulation (5.16) with the same test functions (5 .22) as in Theorem 5.7????,
110 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
which verify (5 .23) and (5.24). The only difference is that now the limit of the term involving
the function Gεis different. More precisely, we have:
1
ε1
|Y|∫
Ω×ΓTε
b(Gε)Tε
b(v1) dσx=1
ε1
|Y|∫
Ω×ΓTε
b(
g(x
ε))
Tε
b(
φ(x) +εω1(x)ψ1(x
ε))
dσx=
1
ε1
|Y|∫
Ω×Γg(y)Tε
b(φ)(x,y) dxdσy+1
|Y|∫
Ω×Γg(y)Tε
b(ω1)(x,y)Tε
b(ψ1)(x,y) dxdσy=
1
ε|Γ|
|Y|MΓ(g)∫
Ωφ(x) dx+1
|Y|∫
Ω×Γg(y)ω1(x)ψ1(y) dxdσy.
Then, since MΓ(g) = 0, by using the density of D(Ω)⊗H1
per(Y1) inL2(Ω,H1
per(Y1)) and of
D(Ω)⊗H1(Y2) inL2(Ω,H1(Y2)), we get the unfolded limit problem (5.28).
Due to the uniqueness of ( u,bu1,¯u2)∈ V, which is proven by the Lax-Milgram theorem,
all the above convergences hold true for the whole sequence, which ends the proof of the
theorem.
Remark 5.16 Let us point out that the term1
|Y|∫
Ω×Γg(y)Φ1(x,y) dxdσyin (5.28) repre-
sents the main difference with respect to the unfolded equation (5.21), where the term involving
gis a nonzero constant, recovered explicitly in the right-hand side of the homogenized equa-
tion. This cannot be the case here, since this term involves now explicitly both variables xand
y. We have to understand the contribution in the homogenized problem of this nonstandard
term generated by the discontinuity of the flux in the initial problem. Actually, it will be
seen in the next theorem that, apart from the classical solutions χj
1andχj
2of the cell prob-
lems (5.27), we are led to introduce two additional scalar terms η1andη2, verifying a new
imperfect transmission cell problem (see (5.38)).
Theorem 5.17 The function u∈H1
0(Ω)defined in (5.19) is the unique solution of the
following homogenized equation
−div(Ahom∇u) =finΩ, (5.29)
whereAhomis the homogenized matrix whose entries are given in (5.26). Moreover, we have
bu1(x,y) =−N∑
j=1∂u
∂xj(x)χj
1(y) +η1(y),
¯u2(x,y) =−N∑
j=1∂u
∂xj(x)χj
2(y) +η2(y),
whereχj
1andχj
2are defined by (5.27) and the function (η1,η2)is the unique solution of the
cell problem
−divy(A(y)∇η1) = 0 inY1,
−divy(A(y)∇η2) = 0 inY2,
−A(y)∇η1·ν=h(y)(η1−η2)−g(y)onΓ,
−A(y)∇η2·ν=h(y)(η1−η2)onΓ,
MΓ(η1) = 0.
HEAT TRANSFER IN COMPOSITE MATERIALS 111
Proof. By choosing φ= 0 in (5.28), we obtain:
1
|Y|∫
Ω×Y1A(y)(∇u+∇ybu1)∇yΦ1dxdy+1
|Y|∫
Ω×Y2A(y)(∇u+∇y¯u2)∇yΦ2dxdy+
1
|Y|∫
Ω×Γh(y)(bu1−¯u2)(Φ1−Φ2) dxdσy=1
|Y|∫
Ω×Γg(y)Φ1(x,y) dxdσy. (5.30)
Let us point out again that the presence of the term1
|Y|∫
Ω×Γg(y)Φ1(x,y) dxdσyin this
equation represents the main difference with respect to the previous case.
By choosing now suitable test functions Φ 1and Φ 2in (5.30), we obtain
−divy(A(y)∇ybu1) = div y(A(y)∇u) a.e. in Ω ×Y1, (5.31)
−divy(A(y)∇y¯u2) = div y(A(y)∇u) a.e. in Ω ×Y2, (5.32)
−A(y)(∇u+∇y¯u2)·ν=h(y)(bu1−¯u2) a.e. on Ω ×Γ, (5.33)
−A(y)(∇u+∇ybu1)·ν=h(y)(bu1−¯u2)−g(y) a.e. on Ω ×Γ. (5.34)
We point out here that we also have a discontinuity type condition:
A(y)(∇u+∇y¯u2)·ν−A(y)(∇u+∇ybu1)·ν=−g(y) a.e. on Ω ×Γ. (5.35)
In the classical case with jump in the solution and with continuity of the flux, the use of
the standard correctors χj
1andχj
2defined in (5.27) is enough in order to express the functions
bu1and ¯u2in terms of the function ∇u. The presence of the function gin relations (5.34) and
(5.35) suggests us to search bu1and ¯u2in the following nonstandard form:
bu1(x,y) =−n∑
j=1∂u
∂xj(x)χj
1(y) +η1(y), (5.36)
¯u2(x,y) =−n∑
j=1∂u
∂xj(x)χj
2(y) +η2(y), (5.37)
whereχj
1andχj
2are defined by (5.27) and the functions η1,η2have to be found. To this
end, we introduce (5.36) and (5.37) in (5.36)-(5.37) and we obtain:
−divy(A(y)∇η1) = 0 inY1,
−divy(A(y)∇η2) = 0 inY2,
−A(y)∇η1·ν=h(y)(η1−η2)−g(y) on Γ,
−A(y)∇η2·ν=h(y)(η1−η2) on Γ,
MΓ(η1) = 0.(5.38)
We obviously have
A(y)∇η2·ν−A(y)∇η1·ν=−g(y) (5.39)
and then we notice that the new local problem (5.42) is an imperfect transmission problem,
involving both the discontinuities in the solution and in the flux, given in terms of handg,
respectively.
112 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
By the Lax-Milgram theorem, the problem (5.38) has a unique solution in the space
H=Wper(Y1)×H1(Y2),
endowed with the scalar product
(η,ζ)H= (∇η1,∇ζ1)L2(Y1)+ (∇η2,∇ζ2)L2(Y2)+ (η1−η2,ζ1−ζ2)L2(Γ).
By choosing now Φ 1= Φ 2= 0 in (5.28), we get:
1
|Y|∫
Ω×Y1A(y)(∇u+∇ybu1)∇φdxdy+1
|Y|∫
Ω×Y2A(y)(∇u+∇y¯u2)∇φdxdy=
∫
Ωf(x)φ(x) dx. (5.40)
Integrating it by parts with respect to x, we obtain:
−divx(1
|Y|∫
Y1A(y)(∇u+∇ybu1) dy+1
|Y|∫
Y2A(y)(∇u+∇y¯u2) dy)
=f(x) in Ω.
By using here the particular form (5.36) and (5.37) of the functions bu1and ¯u2and the
definition of the matrix Ahom, we get:
−divx(Ahom∇u) =f+ div x(1
|Y|∫
Y1A(y)∇η1(y)dy+1
|Y|∫
Y2A(y)∇η2(y)dy)
in Ω,
(5.41)
which leads immediately to the homogenized problem (5.29). We notice that this problem
does not involve the function g, because the second term of the right-hand side in (5.41)
actually vanishes.
Remark 5.18 All the above results can be extended to the case in which Aεis a sequence of
matrices in M(α,β, Ω)such that
Tε
α(Aε)→Astrongly in L1(Ω×Y), (5.42)
for some matrix A=A(x,y)inM(α,β, Ω×Y). The heterogeneity of the medium modeled by
such a matrix induces different effects in our limit problem. In both cases, since the correctors
χj
αdepend also on x, the new homogenized matrix Ahom
xis no longer constant, but it depends
onx. A more interesting effect arises in the second case. As we have seen in Theorem 4.9, if
the matrixAdepends only on the variable y, the functions ηαare independent of xand there
is no contribution of the term containing gin the decoupled form of the limit problem. So,
the limit equation is the same as that corresponding to the case with no jump on the flux in
the microscopic problem. Now, the dependence of Aonxprevails this phenomenon to occur,
and, hence, the function gbrings an explicit contribution in the homogenized problem, which
becomes
−divx(Ahom
x∇u) =
f+divx(1
|Y|∫
Y1A(x,y)∇η1(x,y)dy+1
|Y|∫
Y2A(x,y)∇η2(x,y)dy)
inΩ.
A similar effect was observed in the homogenization of the Neumann problem in perforated
domains (see [57]).
HEAT TRANSFER IN COMPOSITE MATERIALS 113
5.4 Other homogenization problems in composite media with
imperfect interfaces
Problems involving jumps in the solution can be encountered in various other situations. For
instance, our goal in [108] 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 field of material and structural engineering.
For the case of perfect contact between the two elastic media, the continuity of the displace-
ments and the tractions across their common boundary is assumed. This idealized contact
condition can be relaxed by allowing a discontinuity in the displacement fields across the im-
perfect interface between the two elastic media, the jump in displacements being proportional
to the traction vector. In such a model, called a spring type interface model in the literature,
the imperfect interface conditions are equivalent to the effect produced by a very thin and
soft (i.e. very compliant) elastic interphase between the two media. Another interesting
imperfect interface condition arises in the case of a thin and stiff interphase, characterized
by a jump of the traction vector across the interface between the two media (see [39] and
[134]). In both cases, the imperfect contact can be generated by various causes (the presence
of a thin interphase, chemical processes, the presence of impurities at the boundaries, the
interface damage, etc.). Let us notice that by imposing such imperfect interface conditions,
we are allowed to deal with only two-phases media, instead of considering a threephase model,
consisting of two constituents and an interphase formed by a third material, with perfect in-
terface conditions between them (see [ ?]). Imperfect contact problems are of huge importance
for studying composite materials, which might contain coated particles or fibers and cannot
be modeled with the aid of continuous displacements and tractions across the boundaries.
For more details concerning the corresponding mechanical models, the interested reader is
referred to [39], [134], [165], [170], and [ ?].
In [108], we assumed that on the interface between the two media there is a jump in
the displacement vector. The order of magnitude of this jump with respect to the small
parameter εdefines the macroscopic elastostatic equations and our analysis reveals three
different important cases. More precisely, we obtain, at the macroscale, one or two equations,
with different stiffness tensors: (i) if the intensity of the jump is of order ε−1, we obtain only
one equation at the macroscale, with the stiffness tensor depending on the jump coefficient;
(ii) if the intensity of the jump is of order ε, we get a system of two coupled equations
with classical stiffness tensors; (iii) if the intensity of the jump is of order one, we obtain at
the macroscale only one equation, with no influence of the jump in the macroscopic tensor.
The convergence of the homogenization process is proven in all the cases. Our setting is
relevant for dealing with contact problems for multiphase composites with an interfacial
resistance that may arise from the presence of impurities at the boundaries between the
phases or from chemical reactions between the constituents. An example of such a material
114 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
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 different techniques, formal or not, in [104], [160], [165], and [175]. Recently, using the
periodic unfolding method, some elasticity problems for media with open and closed cracks
were studied in [61]. For other related elasticity problems, see [120] 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 [109]. The homogenized problem, derived via the periodic un-
folding method, comprises new coupling terms involving the macroscopic displacement and
temperature fields, 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], [103],
[104] and [115]. In [110], a similar model was considered, but in a different geometry and with
different scalings of the temperature-displacement tensors of the two constituents, leading to
different homogenized results. More precisely, the domain Ω was considered to be the union
of a connected part Ωε
1and a disconnected one Ωε
2and the temperature-displacement tensor
was supposed to be of order of unity in the connected part of the medium and, respectively,
of orderεin 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 different specific heat coefficient in the equation for the
macroscopic temperature field coming from the disconnected part. Moreover, let us mention
the presence of new coupling terms in the macroscopic system and the different functional
setting compared to the one used in [109].
Capatina-Timofte JMAA … to mention ………
In …., we have been concerned with the derivation of macroscopic models for some elastic-
ity problems in periodically perforated domains with rigid inclusions of the same size as the
period. This periodic structure is occupied by a linearly elastic body which is considered to
be clamped along a part of its outer boundary. On the rest of the exterior boundary, surface
tractions are given. The body is subjected to the action of given volume forces. Several
nonlinear conditions on the boundary of the rigid inclusions are considered. More precisely,
we study 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 extend, via the periodic unfolding method, some of the results contained in [12] and
[14], by considering general nonlinearities in the condition imposed on the boundary of the
inclusions. Also, we establish an homogenization result for a Signorini problem with Tresca
friction. The difficulties of this problem come from the fact that the unilateral condition
generates a convex cone of admissible displacements, and, especially, from the fact that the
friction condition involves a nonlinear functional containing the norm of the tangential dis-
placement on the boundary of the rigid inclusions. As shown in Section4, the macroscopic
problem is different from the one addressed in [5]. In particular, for the frictionless contact
case, we regain a result obtained, under more restrictive assumptions, in [14]. This frictionless
problem was also addressed in [15], by the two-scale convergence method, for more general
geometric structures of the inclusions on which the Signorini conditions act.
115
116 HOMOGENIZATION RESULTS FOR HETEROGENEOUS MEDIA
Part II
Career Evolution and Development
Plans
117
Chapter 6
Scienti c and academic background
and research perspectives
In this chapter, I shall briefly present my scientific and academic career. The autonomy and
the visibility of my scientific activity performed after obtaining my Ph.D. in 1996 will be
emphasized. Also, some further research directions I plan to follow will be presented.
6.1 Scientific 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
fluid flow 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 fluid 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 benefited 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 field of homogenization theory: University of Cantabria,
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
119
120
(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´ ıaz (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 scientific research has been mainly devoted to the following fields, in
which I published more than 80 papers (see the list of publications): homogenization theory;
multiscale modeling; reaction-diffusion 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 Differential 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 gave … talks at national and 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,
such as SIAM Journal of Applied Mathematics, Networks and Heterogeneous Media, etc.
•I am member of editorial board of international journals (Biomath Communications,
Abstract and Applied Analysis);
•I was member of the scientific committee for several international conferences (MMSC
2016, BIOMATH 2016, SVCS 2014, 2015, 2016).
121
6.2 Further research directions
I shall briefly describe here the perspectives I see for my research in the next years. A few of
them are already ongoing works.
Basically, I plan to continue my work in the broad field of homogenization theory and
to perform a rigorous multiscale analysis of some relevant nonlinear phenomena in hetero-
geneous media, with applications in biology and engineering. I aim at developing new ideas
and methods in the field of multiscale analysis, the focus being on obtaining a better un-
derstanding of some aspects of the modeling of heterogeneous media. 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 homogenization study for periodic structured
materials with imperfect interfaces. Also, I think at elaborating new mathematical models
for electromagnetic periodic composites and at analyzing nonlinear transmission problems in
composites with various other geometries than those already considered.
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. I
shall rigorously prove, via the periodic unfolding method, the formal results obtained in […20]
for doughball gap junction model. Homogenization results were obtained in the literature
mainly for the syncytial model. In the doughball gap junction model, gap junctions are
considered to be thin conductors between cells, coupling them electrically and chemically. 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
fluxes across junctions. There are very few results for junctional nonlinearities in islets
and many aspects of such models need further investigations. A realistic comparison of the
syncytial and doughball models will be made, as well.
I.2. Homogenization results for skin electropermeabilization. In an ongoing
project, which is a collaboration with Professor Daniele Andreucci and Professor Micol Amar
from Sapienza University of Rome, Italy, we aim at studying, via homogenization techniques,
some suitable mathematical models for the evolution of thermal and electrical fields in bio-
logical tissues. Transdermal drug delivery represents an alternative to standard drug delivery
methods of injection or oral administration. The outermost layer of the epidermis acts as a
barrier, limiting the penetration of drugs through the skin. To overcome this barrier, innova-
tive 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 [..]). We need to control the electric pulse parameters in order to
122
maximize the amount of electropermeabilized tissue in the targeted area and to minimize the
damage produced to the surrounding tissue. Apart from the amount of electropermeabilized
tissue, it is important to take into account the thermal effects produced in the skin by the
electrical pulses in order to design useful electropermeabilization protocols. The problem is
complex, involving a very complicated geometry and the nonlinear coupling of a diffusion
equation for the drug molecules, of a heat equation, and of an equation for the electric po-
tential. 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 effective behavior, as the period of the microstructure tends to zero,
of the solutions of this coupled system of partial differential equations. We shall analyti-
cally investigate the effect of various parameters on the effective temperature field in the
tissue exposed to permeabilizing electric pulses. The results can be used for designing skin
electropermeabilization protocols for cancer treatment planning.
I.3. Mathematical models for carcinogenesis in living cells. I shall be concerned
with the carcinogenic effects produced in the human cells by Benzo-[a]-pyrene molecules
(BP), which are reactive toxic molecules found in coal tar, cigarette smoke, charbroiled food,
etc. I plan to generalize the results obtained in [..] and […]. The microscopic mathematical
model, including reaction-diffusion processes and binding and cleaning mechanisms, will be
homogenized in order to reduce its complexity and to make it numerically treatable and not
so computationally expensive. We shall consider that BP molecules enter in the cytosol inside
of a human cell. There, they diffuse freely, but they cannot enter in the nucleus. Also, they
bind to the surface of the endoplasmic reticulum (ER), where chemical reactions take place,
BP molecules being chemically activated to Benzo-[a]-pyrene-7,8-diol-9,10-epoxide molecules
(DE). These molecules can unbind from the surface of the ER and they can diffuse again in
the cytosol, entering in the nucleus. Natural cleaning mechanisms occurring in the cytosol are
taken into account, too. For describing the binding-unbinding process at the surface of the
ER, we shall consider various nonlinear functions, with various scalings, leading to different
homogenized models. We shall deal with the case of general nonlinear (even discontinuous)
isotherms, similar to those used in [9], [24], and [31], and of multiple metabolisms BP →DE.
I shall also generalize a carcinogenesis model, introduced in [21], involving free receptors on
the surface of the ER (see, also, Section …).
II.1. Homogenization of a two-conductivity problem with
ux jump. This
is an ongoing joint work with Dr. Renata Bunoiu from the University of Lorraine-Metz,
France. We shall continue our study on the homogenization of a thermal diffusion problem
in a highly heterogeneous medium formed by two constituents, separated by an imperfect
interface (see Section …). In this case, the order of magnitude of the thermal conductivity of
the material occupying the domain Ωε
2is of orderε2, while the conductivity of the material
occupying the domain Ωε
1is supposed to be of order one. Our problem presents various
sources of singularities: the geometric one related to the interspersed periodic distribution
of the components, the material one related to the conductivities and the ones generated by
123
the presence of an imperfect interface between the two materials. The case Gε= 0, which
corresponds to a continuous flux, proportional to the jump of the temperature field across the
imperfect interface, has attracted, in the last two decades, the interest of a broad category
of researchers (see, e.g., [194], [193], [192], [98], [132], [ ?], and [ ?]). After passing to the limit
with respect to the small parameter ε, a regularised model of diffusion is obtained, which
in fact is a special case of the double-porosity model, introduced in [198] in the frame of
the heat transfer and in [ ?] in the context of the flow in porous media. We shall consider
the caseGε̸= 0, which corresponds to a discontinous flux as well. We shall study the two
representative cases for the jump function Gε, similar to those stated in Section …., which
both lead to different modified reguralized models of diffusion.
Also, in this context, one of our objectives is to obtain new homogenization results for
reaction-diffusion problems in periodic composite media which exhibit at the interfaces be-
tween their components jumps of the solution and of the flux. Such problems are relevant in
the the context of thermal diffusion in composites, in the theory of semiconductors, in linear
elasticity or in reaction-diffusion problems in biological tissues. We plan to apply our results
to the study of calcium dynamics in biological tissues modeled as media with imperfect in-
terfaces. 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 finding the effective 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
field. 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 effective properties.
II.4. Multiscale modeling of thermoelastic microstructured materials. The
prediction of the macroscopic behavior of thermoelastic microstructured materials is a sub-
ject of topical interest for a broad category of researchers. The growing interest in such a
problem is justified by the increased need of designing advanced composite materials, with
124
useful mechanical and thermodynamical properties. In particular, the problem of multiscale
modeling of thermoelastic composites with imperfect interfaces has attracted a lot of inter-
est in the last years, due to the great importance of such heterogeneous materials in many
engineering applications. For instance, there are important applications of the interphase
effects on the thermoelastic response of polymer nanocomposite materials. The case in which
the strain-stress law is viscoelastic and the case in which we consider thermal effects in the
history of a composite material will be treated, as well.
6.3 Future plans
In the next years, in order to disseminate my results, I plan to publish them in well-known
international journals and to attend several prestigious international conferences. My re-
search activity in the field 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 fields, such as biology or geology. Also, I would like to give
talks at foreign universities, to take part in the organization of scientific events in the field
of homogenization and to extend the editorial activities for applied mathematics scientific
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 field 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 field of appled 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.
………..
(a) Rezumatul (minimum 4.000 de caractere, maximum 6.000 de caractere) prezint sinteza
tezei de abilitare. (b) Realizri tiinifice i profesionale i planuri de evoluie i dezvoltare a carierei.
Aceasta parte are trei seciuni. (i) In prima seciune, de minimum 150.000 de caractere si de
maximum 300.000 de caractere (inclusiv formule), se prezint realizrile tiinifice, profesionale i
academice, pe direcii tematice disciplinare sau interdisciplinare. Cele mai importante lucrri
(maxim 10) vor fi incluse in dosarul de abilitare. Realizrile personale sunt prezentate n
contextul stadiului actual al cercetrii tiinifice din domeniul tematic al specialitii, pe plan
internaional. (ii) n a doua seciune, de maximum 25.000 de caractere, se prezint planuri de
125
evoluie i dezvoltare a propriei cariere profesionale, tiinifice i academice, respectiv direcii de
cercetare/predare/aplicaii practice.
(iii) A treia seciune prezint referine bibliografice asociate coninutului primelor dou seciuni.
Rezumatul tezei de abilitare se redacteaz n dou versiuni, n limbile romn i englez.
126
List of publications
This habilitation thesis is based on the following publications:
[1]C. Conca, J. I. D´ ıaz, C. Timofte ,Effective chemical processes in porous media , Math.
Models Methods Appl. Sci. (M3AS), 13(10), 1437-1462, 2003.
[2]C. Conca, F. Murat, C. Timofte ,A generalized strange term in Signorini’s type problems ,
ESAIM: Mod´ el. Math. Anal. Num´ er. (M2AN), 37(5), 773-806, 2003.
[3]C. Conca, J. I. D´ ıaz, A. Li˜ n´ an, C. Timofte ,Homogenization in chemical reactive flows ,
Electronic Journal of Differential Equations, 40, 1-22, 2004.
[4]C. Conca, C. Timofte ,Interactive oscillation sources in Signorini’s type problems ,
Contemporary Mathematics, 362, 381-392, American Mathematical Society Book Se
ries, Providence, Rhode Island, 2004.
[5] C. Timofte ,Upscaling of variational inequalities arising in nonlinear problems with
unilateral constraints , Z. Angew. Math. Mech., 87(6), 406-412, 2007.
[6] C. Timofte ,Homogenization results for climatization problems , Annali dell’Universita
di Ferrara Sez. VII (N.S.), 53(2), 437-448, 2007.
[7] C. Timofte ,Upscaling in dynamical heat transfer problems in biological tissues , Acta
Physica Polonica B, 39(11), 2811-2822, 2008.
[8] C. Timofte ,Homogenization results for enzyme catalyzed reactions through porous media ,
Acta Mathematica Scientia, 29B (1), 74-82, 2009.
[9] C. Timofte ,Homogenization results for dynamical heat transfer problems in heteroge-
neous biological tissues , Bulletin of the Transilvania University of Bra¸ sov, 2(51), 143-148,
2009.
[10] C. Timofte ,Multiscale analysis in nonlinear thermal diffusion problems in composite
structures , Cent. Eur. J. Phys., 8, 555-561, 2010.
[11] A. Capatina, H.I. Ene, C. Timofte ,Homogenization results for elliptic problems in
periodically perforated domains with mixed-type boundary conditions , Asymptotic Analy
sis,80(1-2), 45-56, 2012.
[12] C. Timofte ,Multiscale analysis of diffusion processes in composite media , Comp. Math.
Appl., 66(9), 1573-1580, 2013.
127
128
[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, in press, 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 flux jump , Networks
and Heterogeneous Media, in press, 2016. Accepted on 04.02.2016. https://hal.inria.fr/hal-
01272936.
[22] R. Bunoiu, C. Timofte ,On the homogenization of a two-conductivity problem with flux
jump , to appear in Communication in Mathematical Sciences, 2016.
[23] R. Bunoiu, C. Timofte ,On the homogenization of a diffusion problem with flux 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.
Bibliography
[1] G. Allaire, Homogenization and two-scale convergence , SIAM Journal on Mathematical
Analysis, 23(6), 1482-1518, 1992.
[2] G. Allaire, M. Briane, Multiscale convergence and reiterated homogenization, Proc. R.
Soc. Edinburgh, Sect. A, 126, 297-342, 1996.
[3] G. Allaire, R. Brizzi, A. Mikeli´ c, A. Piatnitski, Two-scale expansion with drift approach to
the Taylor dispersion for reactive transport through porous media , Chemical Engineering
Science, 65, 2292-2300, 2010.
[4] G. Allaire, Z. Habibi, Second order corrector in the homogenization of a conductive-
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