This manuscript, prepared to defend my Habilitation thesis , summarizes a selection of my research results obtained in the eld of homogenization… [610107]

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

Abstract
This manuscript, prepared to defend my Habilitation thesis , summarizes a selection of my
research results obtained in the eld of homogenization theory after defending my Ph.D. thesis.
The main motivation behind this endeavour is to brie
y describe the state of the art in the
eld of homogenization theory, to give an overview of my contributions in this broad research
area and to discuss some open problems and several perspectives I see for my future scientic
and academic career.
The thesis relies on some of my original contributions to the applications of the homoge-
nization theory, contained in twenty-ve articles already published or submitted for publication
in international journals with a broad audience, including not only mathematicians, but also
physicists, engineers, and scientists from various applied elds. Many of the results in the pu-
blications 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 ne-
cessary, to give more details that are not present in the corresponding published papers, making
my main results accessible to an audience with strong, general mathematical background, but
not necessarily experts in the specic eld of homogenization theory.
The thesis is based on some of the most relevant results I obtained during the last fteen
years of research conducted, alone or in collaboration, in four major areas: multiscale analysis
of reaction-diffusion processes in porous media, upscaling in unilateral problems, multiscale
modeling of composite media with imperfect interfaces, and mathematical models in biology
and in engineering. Thus, the homogenization theory and its applications represent the core of
my scientic work done during these last fteen years.
Apart from two short abstracts, one in Romanian and another one in English, the thesis
comprises two parts and a comprehensive bibliography. The rst part, structured into ve
chapters, is devoted to the presentation of my main scientic achievements since the completion
of my Ph.D. thesis. After a brief introductory chapter presenting the state of the art in
the eld of homogenization theory and offering the general framework and a motivation for
my post-doctoral research work in this area, the second chapter is divided in two distinct
sections, summarizing my main contributions to the homogenization of reactive
ows in porous
media. More precisely, some original results for upscaling in stationary nonlinear reactive
ows
in porous media and, also, results on nonlinear adsorption phenomena in porous media are
presented. The chapter relies on the papers [11], [12], [21], and [23]. The third chapter is
devoted to the homogenization of some relevant unilateral problems in perforated domains.
Homogenization results for Signorini's type problems and for elliptic problems with mixed
boundary conditions in perforated media are presented. The chapter is based on the articles
[7], [8], [13], [14], and [20]. The fourth chapter contains some recent results about homogenized
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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 [22], [23], [24], [27], [29], [30],
and [31]. The last chapter of this rst part summarizes the most important results I achieved,
alone or in collaboration, in the eld of heat transfer in composite materials with imperfect
interfaces and is based on the papers [4], [5], [6], [25], [26], and [28].
For the denitions of the basic notions in homogenization theory and for well-known general
results of functional analysis we shall use throughout this thesis, we refer to [1], [2], [3], [9],
[10], [15], [16], [17], [18], and [19].
The second part of this thesis presents some career evolution and development plans. After
a brief review of my scientic and academic background, further research directions and several
future plans on my scientic and academic career are presented. I shall discuss some short,
medium and long term development plans. A brief description of some open questions I would
like to study in the future will be made, as well.
The thesis ends by a comprehensive bibliography, illustrating the state of the art in this
vast eld of homogenization theory and its applications.
My major original contributions contained in this habilitation thesis can be summarized as
follows:
performing a rigorous study of nonlinear reaction-diffusion processes in porous media,
including diffusion, chemical reactions and different types of adsorption rates;
obtaining new homogenization results for unilateral problems in perforated media;
elaborating new mathematical models for ionic transport phenomena in periodic charged
media;
getting original homogenization results for calcium dynamics in living cells;
deriving new nonlinear mathematical models for carcinogenesis in human cells;
performing a rigorous multiscale analysis of some relevant thermal diffusion processes in
composite structures;
rening the study of diffusion problems with dynamical boundary conditions;
obtaining new mathematical models for diffusion problems with
ux jump.
The results included in this thesis have been obtained alone or in close collaboration with
several academic and research institutions from Romania or from abroad. I am grateful to
all my co-authors, Professor C. Conca, Professor F. Murat, Professor J. I. D az, Professor
A. Li~ n an, Professor H. I. Ene, Dr. A. C ap at  ^ n a, Dr. I. T  ent ea, and Dr. R. Bunoiu, for a
nice collaboration, for their important contribution to our papers, and for useful advices and
interesting discussions. I hope that all these results might open new and promising perspectives
for further developments and future collaborations with well-known scientists from Romania
and from abroad.
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[4]R. Bunoiu, C. Timofte, Homogenization of a thermal problem with
ux jump , Networks
and Heterogeneous Media, 11(4), 545{562, 2016.
[5]R. Bunoiu, C. Timofte, On the homogenization of a two-conductivity problem with
ux
jump , in press, Communication in Mathematical Sciences, 2016.
[6]R. Bunoiu, C. Timofte, Diffusion problems with
ux jump , in preparation, 2016.
[7]A. Capatina, H. I. Ene, C. Timofte, Homogenization results for elliptic problems in pe-
riodically perforated domains with mixed-type boundary conditions , Asymptotic Analysis,
80(1-2), 45-56, 2012.
[8]A. Capatina, C. Timofte, Homogenization results for micro-contact elasticity problems ,
Journal of Mathematical Analysis and Applications, 441 (1), 462-474, 2016.
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[13] 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.
[14] C. Conca, C. Timofte, Interactive oscillation sources in Signorini's type problems , Contem-
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[20] C. Timofte, Upscaling of variational inequalities arising in nonlinear problems with uni-
lateral constraints , Z. Angew. Math. Mech., 87(6), 406-412, 2007.
[21] C. Timofte, Homogenization results for climatization problems , Annali dell'Universita di
Ferrara Sez. VII (N.S.), 53(2), 437-448, 2007.
[22] C. Timofte, Upscaling in dynamical heat transfer problems in biological tissues , Acta Phys-
ica Polonica B, 39(11), 2811-2822, 2008.
[23] C. Timofte, Homogenization results for enzyme catalyzed reactions through porous media ,
Acta Mathematica Scientia, 29B (1), 74-82, 2009.
[24] C. Timofte, Homogenization results for dynamical heat transfer problems in heterogeneous
biological tissues , Bulletin of the Transilvania University of Bra sov, 2(51), 143-148, 2009.
[25] C. Timofte, Multiscale analysis in nonlinear thermal diffusion problems in composite struc-
tures , Cent. Eur. J. Phys., 8, 555-561, 2010.
[26] C. Timofte, Multiscale analysis of diffusion processes in composite media , Comp. Math.
Appl., 66, 1573-1580, 2013.
[27] C. Timofte, Homogenization results for ionic transport in periodic porous media , Comp.
Math. Appl., 68, 1024-1031, 2014.
[28] C. Timofte, Multiscale modeling of heat transfer in composite materials , Romanian Journal
of Physics, 58(9-10), 1418-1427, 2013.
[29] C. Timofte, Multiscale analysis of a carcinogenesis model , Biomath Communications, 2
(1), 2015.
[30] C. Timofte, Homogenization results for the calcium dynamics in living cells , Math. Com-
put. Simulat., in press, 2016, doi:10.1016/j.matcom.2015.06.01 2015.
[31] C. Timofte, Homogenization results for a carcinogenesis model , in press, Mathematics and
Computers in Simulation, 2016, doi: 10.1016/j.matcom.2016.06.008..
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