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\noindent
%Version May 31, 2001
\par
\noindent
%Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 412-268.5734,
tartar@andrew.cmu.edu
\medskip

\noindent
{\it Center for Nonlinear Analysis Summer School, May 31, 2001 - June 9,
2001
\par
\noindent
Multiscale Problems in Nonlinear Analysis}
\bigskip


\noindent
{\bf Weak convergence models for relating various scales}
\medskip

When I was taught Physics in the late 60s, the method for describing the
relations between different scales, from microscopic to macroscopic (and I
only heard much later of the term mesoscopic), was explained in a
traditional way, using a probabilistic language. 
It is still taught that way nowadays. 
\par
In the early 70s, I had initiated a different approach, where instead of
probabilistic methods one uses weak convergences or more generally
convergences of weak type as those appearing in G-convergence which Sergio
S{\eightrm PAGNOLO}\footnote{$^{1}$} 
{Sergio S{\eightrm PAGNOLO}, Italian mathematician.
He works at Universit\'a di Pisa, Pisa, Italy.}\footnote{}
{S{\eightrm PAGNOLO} S., ``Sul limite delle soluzioni di problemi di
Cauchy relativi all'equazione del calore,'' Ann. Scuola Norm. Sup. Pisa
(3) {\bf 21} (1967), 657--699. ``Sulla convergenza di soluzioni di
equazioni paraboliche ed ellitiche,'' Ann. Scuola Norm. Sup. Pisa (3) {\bf
22} (1968), 571--597.}
had developed in the late 60s, helped with the insight of Ennio D{\eightrm
E} G{\eightrm IORGI}\footnote{$^{2}$} 
{Ennio D{\eightrm E} G{\eightrm IORGI}, Italian mathematician, 1928-1996.
He received the W{\eightrm OLF} prize in 1990.
He worked at Scuola Normale Superiore, Pisa, Italy.}\footnote{}
{Ricardo W{\eightrm OLF}, German-born diplomat and philanthropist,
1887-1981. 
The W{\eightrm OLF} foundation was established in 1976 with his wife,
Francisca S{\eightrm UBIRANA}-W{\eightrm OLF}, 1900-1981, ``to promote
science and art for the benefit of mankind''.}, 
and H-convergence, which I had developped with Fran\c{c}ois M{\eightrm
URAT}\footnote{$^{3}$}
{Fran\c{c}ois M{\eightrm URAT}, French mathematician, born in 1947. 
He works at CNRS (Centre National de la Recherche Scientifique) and
Universit\'e Paris VI (Pierre et Marie C{\eightrm URIE}), in Paris,
France.}\footnote{}
{M{\eightrm URAT} F., ``H-convergence,'' {\it S\'eminaire d'analyse
fonctionnelle et num\'erique}, Universit\'e d'Alger, 1977-78. Translated
into English as M{\eightrm URAT} F. \& T{\eightrm ARTAR} L.,
``H-convergence,'' {\it Topics in the mathematical modelling of composite
materials}, 21--43, Progr. Nonlinear Differential Equations Appl., 31,
{\it Birkh\"auser Boston, Boston, MA}, 1997.}\footnote{}
{Pierre C{\eightrm URIE}, French physicist, 1859-1906, and his wife Marie
S{\eightrm LODOWSKA} C{\eightrm URIE}, Polish-born physicist, 1867-1934,
jointly received the N{\eightrm OBEL} prize in Physics in 1903, and she
also received the N{\eightrm OBEL} prize in Chemistry in 1911.}\footnote{}
{Alfred N{\eightrm OBEL}, Swedish industrialist and philanthropist,
1833-1896. 
He created a fund to be used as awards ``for people whose work most
benefited humanity''.}. 
\medskip

When a new idea appears, there are often people who say afterwards that
they had known that already, and that may be true, but it becomes doubtful
when after many years they show that they have not yet understood the
concept correctly. 
This phenomenon is not new, and when G{\eightrm AUSS}\footnote{$^{4}$} 
{Johann Carl Friedrich G{\eightrm AUSS}, German mathematician, 1777-1855.
He worked in G\"ottingen, Germany.}
read about the work of B{\eightrm OLYAI}\footnote{$^{5}$}  
{J\'anos B{\eightrm OLYAI}, Hungarian mathematician, 1802-1860.}
and the work of L{\eightrm OBACHEVSKY}\footnote{$^{6}$}
{Nikolai Ivanovich L{\eightrm OBACHEVSKY}, Russian mathematician,
1792-1856.}
on non Euclidean\footnote{$^{7}$} 
{E{\eightrm UCLID} of Alexandria, mathematician, 325 BCE-265 BCE (BCE =
Before Common Era).}
geometries, he said that he had known that for a long time, but in his
case there is no doubt that he had made similar observations but not
published them, and he had even mentioned that his reputation would suffer
if he admitted in public that he believed in the existence of such a
geometry. 
I had expressed in writing\footnote{$^{8}$}
{T{\eightrm ARTAR} L., ``Probl\`emes de contr\^ole des coefficients dans
des \'equations aux d\'eriv\'ees partielles,'' {\it Control theory,
numerical methods and computer systems modelling (Internat. Sympos., IRIA
LABORIA, Rocquencourt}, 1974), {\it pp. } 420--426. {\it Lecture Notes in
Econom. and Math. Systems, Vol}. 107, {\it Springer, Berlin}, 1975.
Translated into English as M{\eightrm URAT} F. \& T{\eightrm ARTAR} L.,
``On the control of coefficients in partial differential equations,'' {\it
Topics in the mathematical modelling of composite materials}, 1--8, Prog.
Nonlinear Differential Equations Appl., 31, {\it Birkh\"auser Boston, MA},
1997.} in 1974 the new idea of using weak convergence for relating
different scales, which I called microscopic and macroscopic.
Could it be that others had thought of the same idea before, but had been
afraid to mention it, as it was (and still is) against the well accepted
approach which uses a probabilistic language? 
Possibly, but I have received no evidence of that yet, and it may clarify
the discussion to explain who had contributed to this new point of view in
the early 70s, and what the point of view is. 
\medskip

When I had worked with Fran\c{c}ois M{\eightrm URAT} on an academic
problem of optimization that had been proposed by Jacques-Louis L{\eightrm
IONS}\footnote{$^{9}$} 
{Jacques-Louis L{\eightrm IONS}, French mathematician, 1928-2001. 
He received the Japan prize in 1991.
He held a chair (Analyse math\'ematique des syst\`emes et de leur
contr\^ole, 1973-1998) at Coll\`ege de France, Paris, France.
I first had him as a teacher at \'Ecole Polytechnique in 1966/67, when
\'Ecole Polytechnique was still in Paris, and I did research under his
direction, until my thesis in 1971.}\footnote{}
{{\it Contr\^ole optimal de syst\`emes gouvern\'es par des \'equations aux
d\'eriv\'ees partielles}, Dunod - Gauthier-Villars, Paris, 1968 xiii+426
pp. {\it Optimal control of systems governed by partial differential
equations} Die Grundlehren der mathematischen Wissenschaften, Band 170
Springer-Verlag, New York-Berlin 1971 xi+396 pp.}, 
we used weak convergence in the usual way that we had learned from him in
Functional Analysis; there were sequences converging in $L^{2}(\Omega)$
weak\footnote{$^{10}$} 
{The notion of a weakly converging sequence is due F. R{\eightrm IESZ},
but I do not know if he had introduced the precise weak topology (which is
not metrizable).}\footnote{}
{Frigyes (Frederic) R{\eightrm IESZ}, Hungarian mathematician, 1880-1956.
He worked in Budapest, Hungary.}, 
and sequences converging in $L^{\infty}(\Omega)$ weak $\star$, and there
was also a different topology involved; then we found the work that Sergio
S{\eightrm PAGNOLO} had done on G-convergence, with some ideas from Ennio
D{\eightrm E} G{\eightrm IORGI}, and what we had done was slightly
different (so the term H-convergence was coined later), but at that point
there were only classical ideas in Functional Analysis, as G-convergence
is just the weak convergence of the inverses of some operators anyway,
i.e. the convergence of G{\eightrm REEN}\footnote{$^{11}$} 
{George G{\eightrm REEN}, English mathematician, 1793-1841.
He was a miller and never held any academic position.}
kernels (and G is a reminder for G{\eightrm REEN}), the difficult part of
the work being to prove that the limit of G{\eightrm REEN} functions of a
sequence of elliptic problems is the G{\eightrm REEN} function of another
elliptic problem. 
Then we found the work of Henri S{\eightrm ANCHEZ}-P{\eightrm
ALENCIA}\footnote{$^{12}$}
{Enrique Evariste S{\eightrm ANCHEZ}-P{\eightrm ALENCIA}, Spanish-born
mathematician, born in 1941.
He works at CNRS (Centre National de la Recherche Scientifique) and
Universit\'e Paris VI (Pierre et Marie C{\eightrm URIE}), Paris,
France.}\footnote{}
{S{\eightrm ANCHEZ}-P{\eightrm ALENCIA} Enrique, ``Solutions p\'eriodiques
par rapport aux variables d'espace et applications,'' C. R. Acad. Sci.
Paris S\'er. A-B {\bf 271} (1970) A1129--A1132.}, 
who was doing partially formal computations in a periodic setting, 
and that helped us realize that what we had been doing was related to
properties of mixtures. 
\par
That connection with questions in Continuum Mechanics was illuminating for
me, and it gave me a way to understand what I had been told in my Physics
courses about microscopic/macroscopic relations. 
Then we proved the Div-Curl lemma, which appeared important for the
question of whether or not one always needs quantities like internal
energy, which only exist at the macroscopic level.
I also observed that the Div-Curl lemma explains why there is an
equipartition of energy between the kinetic part and the potential part of
the energy for oscillating solutions of the wave equation, and that gave
an interpretation radically different from anything that I had been taught
in my Physics courses, where the principle is related to counting degrees
of freedom; the application of the same idea to M{\eightrm
AXWELL}\footnote{$^{13}$} 
{James C{\eightrm LERK} M{\eightrm AXWELL}, Scottish physicist, 1831-1879.
He held the first C{\eightrm AVENDISH} Professorship of Physics (named
after Henry C{\eightrm AVENDISH}, English physicist, 1731-1810) at
Cambridge, England UK.}
equation gives more than just the equipartition of energy between the
electric part and the magnetic part of the energy, and the application to
nonlinear equations suggests to question the way that physicists have used
for discussing equipartition.
\par  
During the year 1974/75 which I spent at the University of Wisconsin in
Madison, I had clarifying discussions with Joel R{\eightrm
OBBIN}\footnote{$^{14}$}
{Joel W. R{\eightrm OBBIN}, American mathematician, born in 1941.
He works at University of Wisconsin, Madison WI.}
and it became clear why the classical weak convergence is only adapted to
coefficient of differential forms, and why one needs other types of weak
convergence.
\par
I had derived a new proof for H-convergence based on repeated use of the
Div-Curl lemma, and as I had mentioned my proof to Jacques-Louis
L{\eightrm IONS} at a meeting in Marseille, France, in the Fall 1975, his
article is the first mention for it\footnote{$^{15}$}
{L{\eightrm IONS} J.-L., ``Asymptotic behaviour of solutions of
variational inequalities with highly oscillating coefficients,'' {\it
Applications of methods of functional analysis to problems in mechanics
(Joint Sympos., IUTAM/IMU, Marseille, 1975)}, pp. 30--55. Lecture Notes in
Math., 503. {\it Springer, Berlin}, 1976.};
I only wrote about it in the Fall of 1976\footnote{$^{16}$}
{T{\eightrm ARTAR} L., ``Quelques remarques sur l'homog\'en\'eisation,''
{\it Functional analysis and numerical analysis} ({\it Tokyo and Kyoto,}
1976), 469--481, {\it Japan Soc. Promotion of Sci., Tokyo,} 1978.}, 
and it was rediscovered independently by Leon S{\eightrm
IMON}\footnote{$^{17}$}
{Leon M. S{\eightrm IMON}, Australian-born mathematician, born in 1945.
He works at S{\eightrm TANFORD} University, Stanford CA.}\footnote{}
{He was told about my previous work by the referee of his article, ``On
$G$-convergence of elliptic operators,'' Indiana Univ. Math. J. {\bf 28}
(1979), no. 4, 587--594.}\footnote{}
{Leland S{\eightrm TANFORD}, American businessman, 1824-1893.}. 
My method, which Jacques-Louis L{\eightrm IONS} subsequently refered to as
the {\it energy method}, which others have called the {\it duality
method}, and for which I prefer the {\it method of oscillating test
functions}, follows precisely what one does in a physical problem for
identifying effective\footnote{$^{18}$} 
{For a while I have talked about homogenized equations, but after having
learned from George P{\eightrm APANICO\-LAOU} that physicists use the term
effective coefficients, I started using the qualificative effective
instead of homogen\-ized.}\footnote{}
{George C. P{\eightrm APANICOLAOU}, Greek-born mathematician, born in
1943.
He works at S{\eightrm TANFORD} University, Stanford CA.}
coefficients, and it suggests that the topology adapted to a physical
quantity is found by understanding how one measures it, or more precisely
how one identifies what it is from the measurement of other quantities;
differential forms are defined by their integrals on manifolds, and
therefore the topology adapted to them is the weak topology which looks at
the limits of these integrals, but other physical quantities have
different interpretations and other topologies are adapted for them. 
\medskip

A new point of view had been created, and since that time I have had no
hint that this basic idea should be changed, for understanding in a better
way some questions relating different scales in Continuum Mechanics or
Physics. 
I am not saying that I could see in 1975 how some particular questions of
contemporary Continuum Mechanics or Physics could fit into my program of
research, as I understood some of these questions only in the late 80s. 
I am not saying either that I see clearly now how all questions of
contemporary Continuum Mechanics or Physics can be explained in that point
of view, as there are some new mathematical objects that must be defined,
but it seems that everything will eventually fit into the framework that I
have advocated, which does not postulate any probabilistic rules for the
way Nature evolves.
\medskip

When I met John M. B{\eightrm ALL}\footnote{$^{19}$} 
{John MacLeod B{\eightrm ALL}, British mathematician, born in 1948. 
He holds the Sedleian chair (founded in 1618 by Sir William S{\eightrm
EDLEY}) at Oxford, England UK.}
in the Fall of 1975, I was puzzled that he had been taught that elastic
materials minimize their potential energy, but it was obvious to me from
my new understanding of the relations between scales that the
strain-stress relation of an elastic material should be stable by weak
convergence, and I will only correct that statement now by saying that I
was thinking in terms of what I have described as science-fiction elastic
materials, which only have an elastic range and do not show any onset of
plasticity or apparition of cracks, whatever forces are applied to it (and
I have used the term science-fiction because I think now that no such
material will ever exist). 
When in 1977, Clifford T{\eightrm RUESDELL}\footnote{$^{20}$}
{Clifford Ambrose T{\eightrm RUESDELL}, III, American mathematician,
1919-2000.
He worked at Johns H{\eightrm OPKINS} University, Baltimore
MD.}\footnote{}
{Johns H{\eightrm OPKINS}, American financier and philanthropist,
1795-1873.}
had questioned my framework\footnote{$^{21}$}
{``Weak convergence in nonlinear partial differential equations,'' {\it
Existence Theory in Nonlinear Elasticity,} 209--218. {\it The University
of Texas at Austin,} 1977. ``Nonlinear constitutive relations and
homogenization,'' {\it Contemporary developments in continuum mechanics
and partial differential equations} ({\it Proc. Internat. Sympos., Inst.
Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro,} 1977), {\it pp.}
472--484. {\it North-Holland Math. Studies,} 30, {\it North-Holland,
Amsterdam-New York,} 1978.}, 
I had thought that it was due to a difference in generation that he did
not see why I was proposing that, and by politeness I had not argued with
him on that point. 
Younger mathematicians have seemed to have trouble with that idea too (and
a few have obviously not been educated in a way where one respects older
people), and I should explain now that if this was not the case, then
macroscopic measurements could show the same average strains but different
average stresses, so that such a material would certainly not be called
elastic. 
I have learned from Ekhard S{\eightrm ALJE}\footnote{$^{22}$}
{Ekhard K. H. S{\eightrm ALJE}, German-born physicist. 
He works at University of Cambridge, England UK.}\footnote{}
{ S{\eightrm ALJE} Ekhard K., {\it Phase transitions in ferroelastic and
co-elastic crystals: an introduction for mineralogists, material
scientists and physicists} 296pp. Cambridge University Press, 1993. ISBN:
0521429366.}
the terms co-elastic or ferroelastic for describing some nonelastic
materials of that type, whose strain-stress relations show hysteresis
effects; they can be studied in the program that I was describing in the
late 70s, either by neglecting time or by considering the evolution
problem, for which the mathematical tools are obviously still inadequate
(and one must have to question a modelization using an equilibrium
equation of state anyway). 
In the way some people have applied the approach by $\Gamma$-convergence
of Ennio D{\eightrm E} G{\eightrm IORGI}, they have not seen any
hysteresis phenomenon, and they have thought of talking about an elastic
material, probably because they had implicitely postulated that a material
can only exist in the global minimum of its potential energy; this has
also happened because they have only considered the weak topology for the
strain (while I have advocated utilising both the strain and the stress,
and adding the entropies which are valid), and I do not think that it is a
defect of Ennio D{\eightrm E} G{\eightrm IORGI}'s $\Gamma$-convergence
approach, because he had purposely introduced the notion with a general
topology, and I believe that it is only the result of having chosen the
wrong topology that the real hysteresis effect had vanished; in order to
understand which is the right topology (although there could be more than
one), I would prefer to follow my approach, but I do not know the answer
of that question at the moment. 
\medskip

Peter L{\eightrm AX}\footnote{$^{23}$}  
{Peter D. L{\eightrm AX}, Hungarian-born mathematician, born in 1925. 
He received the W{\eightrm OLF} prize in 1987. 
He works at the C{\eightrm OURANT} Institute of Mathematical Sciences at
New York University, New York NY.}\footnote{}
{Richard C{\eightrm OURANT}, German-born mathematician, 1888-1972. 
He worked at New York University, New York NY.} 
has noticed that {\eightrm VON} N{\eightrm EUMANN}\footnote{$^{24}$} 
{J\'anos (John) {\eightrm VON} N{\eightrm EUMANN}, Hungarian-born
mathematician, 1903-1957. 
He worked at the Institute for Advanced Study, Princeton NJ.}
had thought that a numerical scheme which presented oscillations is
converging to the right quantity only in a weak topology (and I think that
Peter L{\eightrm AX} showed that it was not the case), but this remark
does not show that my approach was well known. 
Actually, the usual weak topologies have been used implicitly for a long
time, even before the notion of weak convergence had been introduced by F.
R{\eightrm IESZ} and the spaces named after B{\eightrm
ANACH}\footnote{$^{25}$} 
{Stefan B{\eightrm ANACH}, Polish mathematician, 1892-1945.
He worked in Lvov, then in Poland, now in Ukraine.}, 
F{\eightrm R\'ECHET}\footnote{$^{26}$} 
{Maurice Ren\'e F{\eightrm R\'ECHET}, French mathematician, 1878-1973.
He worked in Paris, France.}, 
H{\eightrm ILBERT}\footnote{$^{27}$}
{David H{\eightrm ILBERT}, German mathematician, 1862-1943.
He worked in G\"ottingen, Germany.}, 
L{\eightrm EBESGUE}\footnote{$^{28}$} 
{Henri L\'eon L{\eightrm EBESGUE}, French mathematician, 1875-1941. 
He held a chair (Math\'ematiques, 1921-1941) at Coll\`ege de France,
Paris, France.}
or S{\eightrm OBOLEV}\footnote{$^{29}$}  
{Sergei L'vovich S{\eightrm OBOLEV}, Russian mathematician, 1908-1989. 
He worked in Akademgorodok, Novosibirsk, Russia.}
had been introduced, because when one replaces a discrete repartition of
mass by a (measurable) density of mass, one says that they are very near
in a weak topology (and on bounded sets such weak topologies are often
metrizable). 
\par
It is only when something nonlinear is introduced that my point of view
starts being relevant, and although we had been taught that nonlinear
mappings are not continuous with respect to weak convergence, we found a
way to use weak convergences in nonlinear settings, and to consider other
types of weak convergence when necessary. 
\medskip

Considering sequences is just a mathematical artifice used for identifying
the right topology, and it is just like the usual mathematical idea of
completing a space; although real numbers seemed well known to all, they
have only been defined correctly at the time of C{\eightrm
ANTOR}\footnote{$^{30}$} 
{Georg Ferdinand Ludwig Philipp C{\eightrm ANTOR}, German mathematician,
1845-1918.
He worked in Halle, Germany.}
and D{\eightrm EDEKIND}\footnote{$^{31}$}
{Julius Wihelm Richard D{\eightrm EDEKIND}, German mathematician,
1831-1916.
He worked in Brunswick, Germany.}, 
and in one approach a real number is just a C{\eightrm
AUCHY}\footnote{$^{32}$} 
{Augustin Louis C{\eightrm AUCHY}, French mathematician, 1789-1857. 
He was named baron in 1838.
He worked in Paris, France.} 
sequence of rationals numbers (more precisely an equivalence class of such
C{\eightrm AUCHY} sequences), and that intellectual exercise is useful
when one only knows rational numbers and one wants to prove that real
numbers exist, which have the properties that we are used to; after that
one may safely work with real numbers. 
Engineers and physicists may think that mathematicians are a little crazy
to spend some time proving such things that they find obvious; after
R{\eightrm USSELL}\footnote{$^{33}$} 
{Bertrand Arthur William, 3$^{\rm rd}$ earl R{\eightrm USSELL}, Welsh
mathematician, 1972-1970.
He received the N{\eightrm OBEL} Prize in Literature in 1950.
He worked in Cambridge, England UK.}
had found his famous paradox, mathematicians realized that one even has to
define what a set is; from time to time physicists discard some old
theories that they do not believe in anymore, and soon they will have to
discard some of the silly games that they have invented in the 20$^{\rm
th}$ Century, but mathematicians do that on a continuous basis, being more
careful about what they know and what they only conjecture is true;
mathematicians are not supposed to mistake the information coming from an
experiment with the properties of solutions of an equation even when this
equation is supposed to be a model for a physical effect which the
experiment is supposed to test. 
\par
Part of what I have been doing in my research work in the 70s and 80s was
to give precise mathematical definitions of what mixtures are, and of
course it depends upon what properties of the mixture one is interested
in; then, after having given precise mathematical definitions of how one
relates different scales, one can start discussing about some precise
questions. 
Once the definitions are clearly set, many will say that they are not
surprised, but some will prefer to continue using the vague probabilistic
language that they have used before as an incantation. 
Good physicists do not need the definitions of mathematicians in order to
go forward and if D{\eightrm IRAC}\footnote{$^{34}$}  
{Paul Adrien Maurice D{\eightrm IRAC}, English physicist, 1902-1984. 
He received the N{\eightrm OBEL} prize in Physics in 1933.
He held the Lucasian chair (named after Henry L{\eightrm UCAS}, English
clergyman, 1610-1663) at Cambridge, England UK.}
has his name attached to a measure (which mathematicians do not call a
function), it is not for having been the first to use the notation of
functions for a point mass, but for having done strangely efficient formal
computations with his ``functions'', not being afraid to use their
derivatives, and such a bold move was only explained to mathematicians by
the theory of distributions of Laurent S{\eightrm
CHWARTZ}\footnote{$^{35}$} 
{Laurent S{\eightrm CHWARTZ}, French mathematician, born in 1915. 
He received the F{\eightrm IELDS} medal in 1950.
He worked at Ecole Polytechnique, Palaiseau, France, and I had him as a
teacher in 1965/66 (when Ecole Polytechnique was in Paris).}\footnote{}
{John Charles F{\eightrm IELDS}, Canadian mathematician, 1863-1932.}; 
however, it would be wrong to believe that any formal computation made by
a physicist or an engineer must be right, even if it seems useful in a
practical problem. 
\par
In his forthcoming book on composites, Graeme M{\eightrm
ILTON}\footnote{$^{36}$} 
{Graeme Walter M{\eightrm ILTON}, Australian-born physicist, born in 1956.
He works at University of Utah, Salt Lake City UT.}\footnote{}  
{{\it The Theory of Composites}, Cambridge Monographs on Applied and
Computational Mathematics, 6, Cambridge University Press; ISBN:
0521781256.}, 
who is the world expert on questions relative to bounds on effective
properties for composite materials, will probably have used his physical
intuition instead of precise mathematical definitions, but I suppose that
many of the results will easily be understood with minor adaptations in
the mathematical framework that I have developed, while some might not fit
so easily. 
\par
It is useful to observe that some others who have been initially trained
as mathematicians often do not follow a mathematical approach, even in an
area where precise mathematical definitions have been given, and prefer to
choose the nonmathematician mode where one does not prove everything that
one writes, but nevertheless do not emphasize which of their results are
only conjectures, as their former mathematical training would dictate; one
should then use a detective mode in order to decide if one should trust
the statements which they write, or check for gaps in their ``proofs'' or
simply look for their use of terms which have never been defined correctly
(my observation is that they often also misattribute the ideas of other
mathematicians, perhaps because they resent the fact that others have
continued working as mathematicians do).
\medskip

In the early 70s, after my work with Fran\c{c}ois M{\eightrm URAT}, I was
quite puzzled when I discovered that a text book by L{\eightrm
ANDAU}\footnote{$^{37}$}
{Lev Davidovich L{\eightrm ANDAU}, Russian physicist, 1908-1968. 
He received the N{\eightrm OBEL} prize in Physics in 1962. 
He worked in Moscow, Russia.}
and L{\eightrm IFSCHITZ}\footnote{$^{38}$}
{Evgenii Mikhailovich L{\eightrm IFSCHITZ}, Russian physicist, 1915-1985.}
contained a section giving a formula for the conductivity of a mixture. 
The reason why it was puzzling was that the results that I had obtained
with Fran\c{c}ois M{\eightrm URAT} (and probably the results that Antonio
M{\eightrm ARINO}\footnote{$^{39}$} 
{Antonio M{\eightrm ARINO}, Italian mathematician.
He works at Universit\'a di Pisa, Pisa, Italy.}
and Sergio S{\eightrm PAGNOLO} had obtained too) showed that one cannot
deduce the effective conductivity of a mixture from the sole knowledge of
the proportions of materials used, except in dimension\footnote{$^{40}$}
{It is related to the rule used in electricity that resistances in series
must be added.
For resistances in parallel the conductivities, inverse of the
resistivities, must be added, and in more than one dimension one does not
know in advance where the electric current will flow, but if one knew one
could consider current tubes where resistances are in series inside the
tubes, and the tubes are themselves in parallel.} 
1 (in other words, the shape of the pieces play a role).
Of course, they could have said that their formula was only an
approximation (it was easy to guess why they implicitly considered their
mixture to be isotropic), they could have told how it compared with
experimental measurements, and from a theoretical point of view they could
have explained why, if one mixed in equal parts two (isotropic) conductors
of conductivity $\alpha$ and $\beta$, their formula was not symmetric in
$\alpha$ and $\beta$. 
These are details which annoy mathematicians, who are taught to be precise
and only write statements which are proved to be true, but the real
difficulty lies elsewhere.
\par
L{\eightrm ANDAU} and L{\eightrm IFSCHITZ} talked of a scenario where one
first grinds two (or more) different materials into fine powders, then one
pours in given proportions these powders into a container before shaking
it thoroughly (in order to mix the components of the mixture so that no
preferred direction can be discovered in the end), and finally one
compresses the result (in order to avoid having too much air as a third
component).
There is obviously not yet a mathematical formulation for expressing in
how many different ways this scenario can be performed so that one may try
to answer the question about the conductivity of the mixture. 
If one talks about grinding, a specialized engineer may understand
different things, depending if he/she is a specialist of grinding grains
or of grinding minerals with various degrees of hardness, but that is an
impossible problem for a mathematician at the moment, as one is talking
about going beyond the elastic range and one cannot even stop once a crack
appears, and one must admit that the models about cracks which
mathematicians play with are problematic anyway.
Avoiding to even try to understand what these processes are and forcing
probabilistic ideas on the problem is just a way to sweep the dirt under
the rug, which cannot result in the cleaning process that mathematicians
are responsible for, and one should keep in mind that not everything can
be transformed into a sound mathematical framework\footnote{$^{41}$} 
{There is a paradox which states that ``for every integer $n$ there is a
property $P$, which can be expressed in less than one hundred English
words, and such that $n$ has the property $P$ but no smaller integer has
that property''; if it was not true there would exist a smallest integer
$N$ which does not have such a property, but that would be a
contradiction, because $N$ would be the first integer not to have a
property such that ..., and one can check that this sentence
characterizing $N$ as the first integer having a property $Q$ can be
written with less than one hundred English words; however, there are only
a finite number of words in English and therefore a finite number of
sentences which use less than one hundred English words, and therefore a
finite number of properties for an infinite number of integers and there
must be $m\ne n$ with the same property for $m$ and $n$, which contradicts
the statement for the largest of the two integers.
What the paradox shows is that there is absolutely no way to transform the
fake mathematical jargon used in the first part of the argument into a
sound mathematical formulation.}.
\par
It is clear that in front of a problem that they do not understand some
people like to pretend that it cannot be understood, and many feel better
by inventing a probabilistic game, without realizing that they can only
postulate that it is related to the initial problem, of which they may
understand near to nothing.
In 1981, at a meeting at New York University, Joseph K{\eightrm
ELLER}\footnote{$^{42}$} 
{Joseph Bishop K{\eightrm ELLER}, American mathematician, born in 1923. 
He received the W{\eightrm OLF} prize for 1996/97. 
He works at S{\eightrm TANFORD} University, Stanford CA.}
talked about random waves, and he mentioned that at some time there had
been many articles written about the distribution of wavelengths of the
waves at the surface of the sea, but there were no measurements; a few
years later some measurements were made by satellites, and they showed
that all the articles were wrong; I guess that the authors had not
considered a realistic version of the equations of hydrodynamic and tried
to deduce something about the waves (which is too hard a problem at the
moment), but they must have postulated that one of their preferred
probabilistic game applied to the problem, which ended up not being the
case. 
\medskip

I am not opposed to probabilists, and Probability is just a part of
Analysis, and I would advocate a probabilistic proof in a case where it
would be simpler than a non probabilist one, but I prefer not to use a
probabilistic approach because of the wrong ideology which often comes
with using probabilities (which has nothing to do with Probability, but is
about faith in dogmas); it is mostly because of a misunderstanding of
Physics (and Probability) that probabilistic ideas have become so popular
in some circles, but they have not been used correctly. 
\par
Some people try to prove that B{\eightrm OLTZMANN}\footnote{$^{43}$}  
{Ludwig B{\eightrm OLTZMANN}, Austrian physicist, 1844-1906.
He worked in Graz and Vienna, Austria, and also in Leipzig, Germany.}
equation is a good model, and for that they try to deduce it from some
Hamiltonian\footnote{$^{44}$} 
{Sir William Rowan H{\eightrm AMILTON}, Irish mathematician, 1805-1865.
He worked in Dublin, Ireland.}
framework of particles interacting with forces depending upon their
distances; this project is necessarily doomed, as I had learned a long
time ago in a book by Clifford T{\eightrm RUESDELL}; if two mathematical
observers were looking at the solutions of the Hamiltonian system with a
large number of interacting particles, one looking at time flowing forward
and the other looking at time flowing backward and reversing the
velocities, they would be looking at the same equation but they could not
both deduce that this implies that what they observe is near the solution
of B{\eightrm OLTZMANN} equation, because of the H-theorem of B{\eightrm
OLTZMANN}, unless they were observing a situation near a global Maxwellian
distribution at equilibrium; in other words there is an irreversible
process which has been postulated in order to write B{\eightrm OLTZMANN}
equation, and the equation cannot be used for studying how irreversibility
occurs. 
I have suggested that obtaining the right effective equation could show an
effect like the apparition of nonlocal terms by Homogenization, and in
this case the two observers would describe their limiting equation, one
with an integral term from $-\infty$ to $t$ and the other with an integral
term from $t$ to $+\infty$, and these two equations could well describe
the same solutions; it would be by getting rid of the nonlocal term and
replacing an integro-differential equation by a differential equation that
some irreversibility would be introduced into the model. 
Of course, integral equations can be studied without probabilistic ideas,
and if in some instance the integral equation follows from a probabilistic
game, there is no reason to prefer the probabilistic game; in the same way
if one talks about numerical methods, finite differences, finite elements,
spectral methods, wavelet transforms, one is just making a list of tools
that are useful to know, and none of them is always better than the others
whatever the problem is, each method has advantages and defects and a good
numerical analyst should learn what they are in order to choose the most
efficient method for the problem that he/she considers at a given time
(but some postulate that their method is better than the others). 
\par
A celebrated tool in probability is the ``Brownian\footnote{$^{45}$} 
{Robert B{\eightrm ROWN}, Scottish botanist, 1773-1858.}
motion'', which was developed by W{\eightrm IENER}\footnote{$^{46}$} 
{Norbert W{\eightrm IENER}, American mathematician, 1894-1964. 
He worked at MIT (Massachussetts Institute of Technology), Cambridge MA.}
after the initial work of B{\eightrm ACHELIER}\footnote{$^{47}$} 
{Louis B{\eightrm ACHELIER}, French mathematician, 1870-1946. 
He worked in Besan\c{c}on, France.}
in 1900 and E{\eightrm INSTEIN}\footnote{$^{48}$} 
{Albert E{\eightrm INSTEIN}, German-born physicist, 1879-1955. 
He received the N{\eightrm OBEL} prize in Physics in 1921.
He worked at the Institute for advanced Study, Princeton NJ.}
in 1905, and it is associated with the heat equation $u_{t}-u_{xx} = 0$
and uses random walks which jumps in position, but this is quite different
from what B{\eightrm ROWN} had observed, which were jumps in velocity,
which are more related to the F{\eightrm OKKER}\footnote{$^{49}$}
{Adriaan Dani\"{e}l F{\eightrm OKKER}, Dutch physicist and composer,
1887-1972. 
He worked in Leiden, The Netherlands. 
He wrote music under the pseudonym Arie D{\eightrm E} K{\eightrm
LEIN}.}-P{\eightrm LANCK}\footnote{$^{50}$}
{Max Karl Ernst Ludwig P{\eightrm LANCK}, German physicist, 1858-1947.
He received the N{\eightrm OBEL} prize in Physics in 1918. 
He worked in Berlin, Germany.}
equation $f_{t}+v.f_{x}-f_{vv} = 0$ and to the O{\eightrm
RNSTEIN}\footnote{$^{51}$}
{Leonard Salomon O{\eightrm RNSTEIN}, Dutch physicist, 1880-1941. 
He worked in Utrecht, The Netherlands.}-U{\eightrm
HLENBECK}\footnote{$^{52}$} 
{George Eugene U{\eightrm HLENBECK}, Dutch physicist, 1900-1988.
He worked in Utrecht, The Netherlands.}
process. 
Some people like to derive F{\eightrm OKKER}-P{\eightrm LANCK} equation
from B{\eightrm OLTZMANN} equation, and this is illogical, because
F{\eightrm OKKER}-P{\eightrm LANCK} equation is precisely about the
grazing collisions which are not well described by B{\eightrm OLTZMANN}
equation, at least as long as one does not know how to avoid G{\eightrm
RAD}\footnote{$^{53}$}
{Harold G{\eightrm RAD}, American physicist, 1923-1987. 
He worked at the C{\eightrm OURANT} Institute of Mathematical Sciences at
New York University, New York NY.}'s 
angular cut-off hypothesis (there is a recent tendency to wrongly use the
term F{\eightrm OKKER}-P{\eightrm LANCK} equation for a different equation
with no velocity variable, often calling it forward K{\eightrm
OLMOGOROV}\footnote{$^{54}$} 
{Andrei Nikolaevich K{\eightrm OLMOGOROV}, Russian mathematician,
1903-1987. 
He received the W{\eightrm OLF} prize in 1980.
He worked in Moscow, Russia.}
equation too).
Except for the work of B{\eightrm ACHELIER}, which was related to finance
(in 1900!), the physical processes where diffusion equations are used seem
to have a first step where a scattering effect occurs (the mathematical
analysis of which I have not seen), a second step where a large time
behaviour creates a diffusion in velocity and a third step which creates a
diffusion in space by letting a characteristic velocity tend to $\infty$. 
\par
Whatever the physical origin of the heat equation is, one may try to solve
it in any way one likes, including the use of ``Brownian motion'', but I
have learned other tools which I find more general; for example, one could
use the explicit L{\eightrm AX}-F{\eightrm RIEDRICHS}\footnote{$^{55}$}
{Kurt Otto F{\eightrm RIEDRICHS}, German-born mathematician, 1901-1982.
He worked at the C{\eightrm OURANT} Institute of Mathematical Sciences at
New York University, New York NY.}
 finite difference scheme ${1\over \Delta\,t}\bigl(
U_{i}^{n+1}-{U_{i-1}^{n}+U_{i+1}^{n}\over 2}
\bigr)-{U_{i-1}^{n}+U_{i+1}^{n}-2U_{i}^{n}\over (\Delta\,x)^{2}} = 0$,
where $\Delta\,x$ and $\Delta\,t$ are mesh sizes in space and time and
$U_{i}^{n}$ is an approximation for $U(i\,\Delta\,x,n\,\Delta\,t)$; one
needs a stability condition $2\Delta\,t\le(\Delta\,x)^{2}$, which is
exactly what one needs to have $U_{i}^{n+1} =
a\,U_{i-1}^{n}+b\,U_{i}^{n}+c\,U_{i+1}^{n}$ with $a,b,c\ge 0$, which
permits an ``interpretation'' in terms of expected value for a
probabilistic game, which seems to be the basic idea for the ``Brownian
motion''; however there are also standard Hilbertian methods, which permit
also to approach by discretization many other partial differential
equations for which no probability game is known. 
\medskip

The point of view of describing the relation between different scales by
using convergences of weak type is adapted to working with the partial
differential equations of Continuum Mechanics or Physics, and the
identification of the adapted topology usually helps solving the important
question of adding variables which are necessary for describing the
oscillating behaviour of solutions, and the effective equation itself. 
Forgetting about these added variables automatically gives a model where
something is missing, and for correcting the defect of the reduced system
many advocate the use of probabilities, but this approach does not seem to
lead to the discovery of a better new equation. 
By avoiding the use of probabilities, one may discover why some phenomenon
occurs, and when one discovers that some of the rules of Quantum Mechanics
can be obtained without probabilities, one understands that it is not the
use of probabilities which is wrong, but the postulate that one cannot
explain what one observes without the use of probabilities. 
If a probabilistic game can be found for describing the solution of an
integral equation, why forget that there are elementary results in
Functional Analysis which can give the same existence result in a quicker
way.
\medskip

One should first understand what the physical problems are about, and one
will probably be led to discover that there are some microstructures which
appear in a natural way, and if Nature needs these microstructures it has
to be for a good reason, which is certainly that they offer a much better
solution than a boring smooth solution showing no microstructures. 
I believe that the microstructures which one observes in real problems
show some kind of optimality, but one should certainly not rush at
pretending that one knows which functional is being minimized, because
that may not be exactly what Nature is doing anyway.
\medskip

It is the goal of Homogenization to describe the oscillations (or
concentration effects) in solutions of partial differential equations, and
doing so it is expected that some effective equations will be discovered,
which govern the new important quantities that one will have identified
for describing these oscillating solutions, and that one will have
identified which topologies of weak type are adapted to the problem at
hand. 
It is the goal of $\Gamma$-convergence to understand similar questions for
sequences converging in adequate topologies and trying to minimize adapted
functionals, and it is important to discover interesting classes of
functionals and which are the interesting topologies to consider in
combination with these functionals. 
I would prefer to see such mathematical questions which may be relevant
for explaining something from outside Mathematics, for example problems
from Continuum Mechanics or Physics, although I know that in the process
of developing a mathematical theory one must necessarily study a certain
number of academic problems (which one should not mistake with reality).
There are issues which at the moment are better studied in one aspect than
in the other, and Homogenization and $\Gamma$-convergence are two points
of view about the study of microstructures which do not coincide, and may
be complementary in certain situations, but cannot be antagonistic (except
maybe for those who are more interested in politics and ideology than in
Science). 
With the restricted point of view which has been used in the applications
of $\Gamma$-convergence, it is not the same than Homogenization, but once
the full strength of Ennio D{\eightrm E} G{\eightrm IORGI}'s method will
have been used by paying more attention to the choice of the topology, it
may happen that the difference with Homogenization (in the way
Fran\c{c}ois M{\eightrm URAT} and I have developped it first, where no
restrictions like periodicity are imposed) may almost disappear.
\medskip

It is quite common to experience the existence of microstructures in some
fluid flows for example, but it is a very arduous task to propose a model
that describes accurately the important effects occuring in these flows. 
A first observation concerns the structure of the interface in front of a
rainstorm, as I had observed many times after a hot Summer day in the
French countryside, many years before I thought of becoming a
mathematician; one knows that a storm is coming, although the air is
still, perhaps because the pressure is higher than usual, and then one
hears the leaves of the trees moving, while the branches stay still; soon
after the small branches start to move too, followed by the large branches
a little after and the whole trees may be in motion when the rain arrives.  
It clearly suggests that the classical idea of a sharp interface with some
partial differential equations being satisfied on each side and with some
boundary conditions being imposed on the ``interface'' might not be so
efficient for describing the effects occuring in that living layer, with
small vortices on the dry side and large vortices on the wet side.  
A second observation concerns the structure of the ``wind'', as I had
observed twenty years ago, on a week end where I went sailing with some
friends, between La Rochelle and Ile\footnote{$^{56}$}
{Ile is the French word for island, and R\'e was still an island at the
time, but a bridge connects it to the continent now.}
de R\'e, but the morning had provided us with what one calls ``calme
plat'' in French: there was no wind, and the surface of the sea was
extremely smooth and only showing a long swell (``houle'' in French),
which combined with the steady movement sustained by the small engine of
the boat to produce a beginning of seasickness; fortunately, it did not
last too long, because after a while we saw what one calls ``une ris\'ee''
in French (my English dictionary translates as ``light squall''), the wind
waiting for us! 
It is an amazing fact to come from the windless side with a smooth sea
surface to the place where the wind is, with the surface of the sea all
wrinkled with wavelengths of the order of 5 to 10 centimeters, and when
one crosses the transition line (which seemed stationary, but it might
have been moving at a much slower pace that the boat, which was carried by
its small engine), the sails inflated, and sailing started. 
Again, I will not dare to propose a functional, for which minimizing
sequences show the precise microstructures which Nature uses in such a
marvelous way, because that is probably not the reason they occur, and
considering that these microstructures are optimal should be thought as an
approximation. 
\medskip

It is useful to familiarize oneself with partial differential equations
from Continuum Mechanics and Physics, in order to understand physical
meanings which can be attached to a given partial differential equation,
even if one is only interested in its mathematical properties. 
An equation may have lots of properties, some of them being more or less
intuitive depending upon one's training, and the same equation may arise
as a model in various applications, but some quantities may be more
natural to introduce in one model than in another; for example, the
equation $u_{t} - (u^{m})_{xx} = 0$ is used as a model in porous media for
$m>1$ (with $u\ge 0$ denoting a density of mass), and the quantity $M =
\int_{R} u(x,t)\,dx$ is independent of $t$ and is the total mass, and the
quantity $\int_{R} x\,u(x,t)\,dx$ is also independent of $t$ and is
$M\,x_{*}$ where $x_{*}$ is the abscissa of the center of gravity of the
distribution of mass; however the case $m = 1$ is the heat equation, and
in the interpretation that $u$ is a temperature (out of equilibrium), no
physical meaning has been attached to $\int_{R} u(x,t)\,dx$ and $\int_{R}
x\,u(x,t)\,dx$, although these quantities are independent of $t$.
Using physical examples in Partial Differential Equations is a pedagogical
approach similar to using pictures in Geometry\footnote{$^{57}$}  
{I have been told that Jean D{\eightrm IEUDONN\'E} was once giving a
course where he was proving some abstract result of Geometry, and as a
member of Bourbaki, he was doing that without using any drawing, of
course. 
At some point, he could not figure out the next step in the proof, a
difficult situation that every teacher has experienced; he went nearer to
the blackboard so that what he was doing could not be seen by everyone (as
he was tall and strong he could hide a reasonable part of the blackboard),
and he drew a picture; after he had understood how to proceed from the
hint given by the picture, he erased it and he finished his proof. 
Of course, such a behaviour should be avoided for obvious pedagogical
reasons; if pictures are useful hints for the teacher why not explain how
to use them to the students.}, 
and it does not give the physical situations more value than hints: a
drawing is not a proof but it is often a sufficient hint for a trained
person for telling how to write a proof, if needed.\footnote{}
{Jean Alexandre Eug\`ene D{\eightrm IEUDONN\'E}, French mathematician,
1906-1992.
He worked in Paris and Nice, France.}
\footnote{}
{Nicolas Bourbaki is the pseudonym of a group of mathematicians, mostly
French; those who chose the name certainly knew about a French general
named B{\eightrm OURBAKI}.}\footnote{}
{Charles Denis Sauter B{\eightrm OURBAKI}, French general, 1816-1897; of
Greek ancestry, he declined an offer of the throne of Greece in 1862.} 
\par
Of course, I imagine that the students have already heard about the
corresponding physical theories, and would like to understand what they
have been taught in their Physics course in a more mathematical way; this
is often not the case nowadays, as Mathematics and Physics are often
taught in a too ideological manner, and very few students have the
motivation of acquiring a vast knowledge, and they may have problems
understanding examples which could have been useful if they had learned
more. 
\par
Although I agree with the motto of Hugues de Saint
Victor\footnote{$^{58}$} 
{Hugo von Blankenburg, German-born theologian, 1096-1141.
He worked at the monastery of Saint Victor in Paris, France.} 
``Learn everything, and you will see afterward that nothing is useless'',
one must certainly take the time to understand how the various pieces that
one has learned fit together before learning a little more.
\bigskip


\vfill

\eject

\end





