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%Version June 1, 2001
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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 Does Nature minimize or conserve energy?}
\medskip

In 1848, S{\eightrm TOKES}\footnote{$^{1}$} 
{Sir George Gabriel S{\eightrm TOKES}, Irish-born mathematician,
1819-1903.
He held the Lucasian chair (endowed by Henry L{\eightrm UCAS}, English
clergyman, 1610-1663) at Cambridge, England UK.}\footnote{}
{``On a difficulty in the theory of sound,'' Philos. Mag. {\bf XXXIII}
(1848), 349--356.
{\it Classic papers in shock compression science}, 71--79, High-press.
Shock Compression Condens. Matter, {\it Springer, New York}, 1998.}
explained a discrepancy which C{\eightrm HALLIS}\footnote{$^{2}$} 
{James C{\eightrm HALLIS}, English Astronomer, 1803-1882. 
He worked at Cambridge, England UK.}\footnote{}
{``On the velocity of sound,'' Philos. Mag. {\bf XXXII} (1848), 494--499.}    
had noticed concerning some solutions of the equations of compressible gas
dynamics, which P{\eightrm OISSON}\footnote{$^{3}$} 
{Sim\'eon Denis P{\eightrm OISSON}, French mathematician, 1781-1840.
He worked in Paris, France.}\footnote{}
{``M\'emoire sur la th\'eorie du son,'' J. \'Ec. Polytech. {\bf 14}
(1808), 319--392.
{\it Classic papers in shock compression science}, 3--65, High-press.
Shock Compression Condens. Matter, {\it Springer, New York}, 1998.}
had obtained in an implicit form in 1808, by showing that solutions could
approach a discontinuity in finite time; by using conservation of mass and
conservation of momentum he had then correctly derived the jump conditions
that discontinuous solutions must satisfy, and these are now interpreted
as meaning that the discontinuous functions satisfy a partial differential
equation in the sense of distributions, as developped by Laurent
S{\eightrm CHWARTZ}\footnote{$^{4}$}
{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.}, 
following the pioneer work of Sergei S{\eightrm OBOLEV}\footnote{$^{5}$} 
{Sergei L'vovich S{\eightrm OBOLEV}, Russian mathematician, 1908-1989. 
He worked in Akademgorodok, Novosibirsk, Russia.} 
and of Jean L{\eightrm ERAY}\footnote{$^{6}$}
{Jean L{\eightrm ERAY}, French mathematician, 1906-1998. 
He received the W{\eightrm OLF} prize in 1979.
He held a chair (Th\'eorie des \'equations diff\'erentielles et
fonctionnelles, 1947-1978) at Coll\`ege de France, Paris,
France.}\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''.}.
After S{\eightrm TOKES}, R{\eightrm IEMANN}\footnote{$^{7}$} 
{Georg Friedrich Bernhard R{\eightrm IEMANN}, German mathematician,
1826-1866.
He worked in G\"ottingen, Germany.}\footnote{}
{``\"Uber die Fortpssanzung ebener Luftwellen von endlicher
Schwingungsweite,'' Abhandlungen der Gesell\-schaft der Wissenschafte zu
G\"ottingen, Mathematisch-Physikalische Klasse 8, 43 (1860). Gesammelte
Werke, 1876, p144.}
derived independently the jump conditions in his thesis in 1860, but
instead of being called the S{\eightrm TOKES}-R{\eightrm IEMANN}
conditions, the jump conditions are now named after R{\eightrm
ANKINE}\footnote{$^{8}$} 
{William John Macquorn R{\eightrm ANKINE}, Scottish engineer, 1820-1872.
He worked in Glasgow, Scotland UK.}\footnote{}
{``On the thermodynamic theory of waves of finite longitudinal
disturbance,'' Philos. Trans. {\bf 160} (1870), part II, 277--288. 
{\it Classic papers in shock compression science}, 133--147, High-press.
Shock Compression Condens. Matter, {\it Springer, New York}, 1998.}
and H{\eightrm UGONIOT}\footnote{$^{9}$}
{Pierre Henri H{\eightrm UGONIOT}, French engineer, 1851-1887.}\footnote{}
{``Sur la propagation du mouvement dans les corps et sp\'ecialement dans
les gaz parfaits, I, II'' J. \'Ec. Polytech. {\bf 57} (1887), 3--97, {\bf
58} (1889), 1--125. {\it Classic papers in shock compression science},
161--243, 245--358, High-press. Shock Compression Condens. Matter, {\it
Springer, New York}, 1998.}.
However, when S{\eightrm TOKES} edited his complete works in
1880\footnote{$^{10}$}
{S{\eightrm TOKES} George Gabriel Sir, {\it Mathematical and physical
papers}, Reprinted from the original journals and transactions, with
additional notes by the author. Cambridge, University Press, 1880-1905.}, 
he did not reproduce there his 1848 proof of the jump conditions, and
instead he apologized for having made a mistake, because he had been
(wrongly) convinced by Lord R{\eightrm AYLEIGH}\footnote{$^{11}$}  
{John William S{\eightrm TRUTT}, 3$^{\rm rd}$ baron R{\eightrm AYLEIGH},
English physicist, 1842-1919.
He received the N{\eightrm OBEL} prize in Physics in 1904.
He held the C{\eightrm AVENDISH} Professorship (named after Henry
C{\eightrm AVENDISH}, English physicist, 1731-1810) at Cambridge, England
UK.}\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''.}
and T{\eightrm HOMSON}\footnote{$^{12}$} 
{William T{\eightrm HOMSON}, Irish-born physicist, 1824-1907. 
In 1892 he was made baron K{\eightrm ELVIN} of Largs, and thereafter known
as Lord K{\eightrm ELVIN}.
He worked in Glasgow, Scotland UK.}
(later to become Lord K{\eightrm ELVIN}), that his discontinuous solutions
were not physical: they did not conserve energy. 
\par
So in the third part of the 19$^{\rm th}$ Century, good physicists were
adamant: energy is conserved!
\medskip
   
K{\eightrm ELVIN}, R{\eightrm AYLEIGH} and S{\eightrm TOKES} must have
understood later that heat is a form of energy, and that the missing
energy in S{\eightrm TOKES}'s discontinuous solutions of (isentropic) gas
dynamics is transformed into heat, which does make the temperature of the
gas increase, but since that possibility is not allowed in the
mathematical model this energy is ``apparently lost''; in other terms,
these three great scientists had not grasped yet why one needs a notion of
``internal energy'', although it seems that W{\eightrm
ATT}\footnote{$^{13}$} 
{James W{\eightrm ATT}, Scottish engineer, 1736-1819.} 
and C{\eightrm ARNOT}\footnote{$^{14}$} 
{Sadi Nicolas L\'eonard C{\eightrm ARNOT}, French engineer, 1796-1832.}
had understood much earlier (and independently) that mechanical energy may
be transformed into heat and heat may be transformed into mechanical
energy, but with inherent limitations about the proportion of mechanical
energy which can be recovered after it has been transformed into heat.
\par
So in the fourth part of the 19$^{\rm th}$ Century, good physicists knew
that energy is conserved, if one is careful to count all its
forms\footnote{$^{15}$}
{Soon after, E{\eightrm INSTEIN} was going to observe that mass is also a
form of energy, with the famous relation $e = m\,c^{2}$; energy over mass
has the dimension of the square of a velocity, and the only intrinsic
velocity is the velocity of Light $c$ (299,792,458 meter/second), so once
mass is understood to be energy, it is natural to compare this energy to
$m\,c^{2}$, but physicists seem to have accepted without criticism that
the proportionality factor must be 1 for all ``particles'', or maybe they
consider that relation as a definition of ``mass''.}\footnote{}
{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.}, 
including heat. 
\medskip

One way for solving an equation of the form $-\Delta\,u = f$ with boundary
conditions, an equation named after L{\eightrm APLACE}\footnote{$^{16}$} 
{Pierre-Simon L{\eightrm APLACE}, French mathematician, 1749-1827. 
He was made count in 1806 and marquis in 1817.
He worked in Paris, France.}
or P{\eightrm OISSON}, is to invoke the D{\eightrm
IRICHLET}\footnote{$^{17}$} 
{Johann Peter Gustav L{\eightrm EJEUNE} D{\eightrm IRICHLET}, German
mathematician, 1805-1859.
He worked in Berlin, and then in G\"ottingen, Germany.}
principle, which consists in noticing that if a function $u$ of class
$C^{2}$ minimizes the functional $J$ defined by $J(v) = {1\over
2}\int_{\Omega} |grad(v)|^{2}\,dx -\int_{\Omega} f\,v\,dx$ among all
smooth functions having a given boundary value, then $u$ satisfies
$-\Delta\,u = f$ in $\Omega$. 
The principle was named after D{\eightrm IRICHLET} by R{\eightrm IEMANN},
who had heard it from him, but it had been used before by G{\eightrm
AUSS}\footnote{$^{18}$}
{Johann Carl Friedrich G{\eightrm AUSS}, German mathematician, 1777-1855.
He worked in G\"ottingen, Germany.} 
and by G{\eightrm REEN}\footnote{$^{19}$}
{George G{\eightrm REEN}, English mathematician, 1793-1841.
He was a miller and never held any academic position.},
and the same principle is also named after T{\eightrm HOMSON}. 
\par
Could it be that any of these scientists believed that Nature is
minimizing energy? 
\par
I do not think so.
In those days the equation may have arisen from a question of Gravitation
(using N{\eightrm EWTON}\footnote{$^{20}$}
{Sir Isaac N{\eightrm EWTON}, English mathematician, 1643-1727.
He held the Lucasian chair at Cambridge, England UK.}'s 
law of attraction), and in that case the potential energy for a finite
number of masses must have been used and the limit for a density of masses
must have been derived, probably first by L{\eightrm APLACE} and by
P{\eightrm OISSON}; the same ideas were certainly applied to Magnetism,
and to Electricity, but all these people must have known the basic truth
of Classical Mechanics that total energy (kinetic plus potential) is
conserved; they must have known that a local strict minimum of the
potential energy gives rise to a stable equilibrium, but they could not
have mistaken the minimization of potential energy as a physical process;
in other words, they must have known that the D{\eightrm IRICHLET}
Principle is a useful mathematical trick\footnote{$^{21}$}
{Specialists of Numerical Analysis know that any unphysical principle is
useful if it permits to compute quickly and accurately something that one
is interested in, but there is no reason to mistake such a principle for
reality.}. 
\par
So it seems better to consider that Nature does not minimize energy, but
that the mathematical problem of minimizing a suitable functional may give
information on the solutions of some physical processes which are not
about minimizing anything. 
\medskip

Although Nature does not minimize energy, one may well arrive at something
that looks like what Nature has created in a particular area by looking at
the mathematical point of view of minimizing a particular functional, but
one should not be fooled by this analogy and jump to conclusion too fast.
If $A$ implies $B$ and one observes something which looks like $B$, then
one must be an optimist to deduce that it must be because $A$ is true, and
physicists seem to be of an optimistic nature, but mathematicians would
probably consider as intellectually challenged a student who thinks that
if $A$ implies $B$ and $B$ is true then $A$ is true and they may think of
showing the deluded student a large number of counter-examples (thinking
that such a student would not even be convinced by seeing only one
counter-example), so mathematicians seem to be of a pessimistic nature. 
If one observes a microstructure and one discovers that such a
microstructure appears when one minimizes a particular functional, and if
one claims that it is the reason why this particular microstructure
occurs, one might be making precisely the mistake that mathematicians are
warned against. 
\medskip

Although not related to a physical process, the D{\eightrm IRICHLET}
Principle has helped discover interesting mathematical structures.
W{\eightrm EIERSTRASS}\footnote{$^{22}$}
{Karl Theodor Wilhelm W{\eightrm EIERSTRASS}, German mathematician,
1815-1897.
He worked in Berlin, Germany.} 
pointed out that the functional $J$ might not attain its minimum, and this
question may have been at the onset of his study of compactness. 
I think that the complete solution of the D{\eightrm IRICHLET} Principle
was one in the famous list of problems which H{\eightrm
ILBERT}\footnote{$^{23}$}
{David H{\eightrm ILBERT}, German mathematician, 1862-1943.
He worked in G\"ottingen, Germany.} 
proposed in 1900 at the International Congress of Mathematicians in Paris,
France; it might have been the motivation of Sergei S{\eightrm OBOLEV} for
introducing his famous spaces, and for that purpose he may have considered
the question of Electrostatics, a simplification of M{\eightrm
AXWELL}\footnote{$^{24}$}
{James C{\eightrm LERK} M{\eightrm AXWELL}, Scottish physicist, 1831-1879.
He held the first C{\eightrm AVENDISH} Professorship of Physics at
Cambridge, England UK.}
equation; it might have been the motivation of Laurence C. Y{\eightrm
OUNG}\footnote{$^{25}$}
{Laurence Chisholm Y{\eightrm OUNG}, English-born mathematician,
1905-2000.
He worked at University of Wisconsin, Madison WI.
His father, William Henry Y{\eightrm OUNG} (English-born mathematician,
1863-1942, who worked in Lausanne, Switzerland), and his mother, Grace
C{\eightrm HISHOLM}-Y{\eightrm OUNG} (English-born mathematician,
1868-1944), worked on many problems together, and some classical theorems
attributed to Y{\eightrm OUNG} were their joint works.}\footnote{}
{Y{\eightrm OUNG} L. C., ``Generalized curves and the existence of an
attained absolute minimum in the calculus of variations,'' C. R. Soc. Sci.
Lett. Varsovie, Classe III {\bf 30} (1937), 212--234. ``Generalized
surfaces in the calculus of variations, I, II. Ann. of Math. (2) {\bf 43},
(1942) 84--103, 530--544. {\it Lectures on the Calculus of Variation and
Optimal Control Theory,} W. B. Saunders Co., Philadelphia-London-Toronto,
Ont. 1969 xi+331 pp.}
for analyzing minimization problems without solutions and introducing a
class of generalized solutions, for which one uses now the term {\it
Y{\eightit OUNG} measures}.
\par
In the Summer 1978, I taught a course\footnote{$^{26}$}
{T{\eightrm ARTAR} L. C., ``Compensated compactness and applications to
partial differential equations, {\it Nonlinear analysis and mechanics:
Heriot-Watt Symposium, Vol. IV}, pp. 136--212, Res. Notes in Math., 39,
{\it Pitman, Boston, Mass.-London}, 1979.}
at H{\eightrm ERIOT}\footnote{$^{27}$}
{George H{\eightrm ERIOT}, Scottish goldsmith, 1563-1624.}-W{\eightrm ATT} 
University in Edinburgh, Scotland, on Compensated Compactness, a theory
which I had partly developed with Fran\c{c}ois M{\eightrm
URAT}\footnote{$^{28}$}
{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{}
{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.};
in that course, I used Y{\eightrm OUNG} measures as a way to describe the
constraints that were obtained by applying my Compensated Compactness
Method, but I did not know at the time that Laurence C. Y{\eightrm OUNG}
had introduced them, and I used the term {\it parametrized measures} which
I had heard in the seminar of Robert P{\eightrm ALLU} {\eightrm DE}
{\eightrm LA} B{\eightrm ARRI\`ERE}\footnote{$^{29}$}
{Robert P{\eightrm ALLU} {\eightrm DE} {\eightrm LA} B{\eightrm
ARRI\`ERE}, French mathematician. 
He worked at Universit\'e Paris VI (Pierre et Marie C{\eightrm URIE}),
Paris, France.}
at IRIA (Institut de Recherche en Informatique et Automatique,
Rocquencourt, France), where the idea in control theory was attributed to
G{\eightrm HOUILA}-H{\eightrm OURI}\footnote{$^{30}$}
{Alain G{\eightrm HOUILA}-H{\eightrm OURI}, French mathematicien,
-196?.}\footnote{}
{G{\eightrm HOUILA}-H{\eightrm OURI} Alain, ``Sur la g\'en\'eralisation de
la notion de commande d'un syst\`eme guidable, Rev. Fran\c{c}aise
Informat. Recherche Op\'erationnelle {\bf 1} (1967) no. 4, 7--32.}, 
but they had been used in control theory before by Jack W{\eightrm
ARGA}\footnote{$^{31}$} 
{Jack W{\eightrm ARGA}, American mathematician, born in 1922.
He works at Northeastern University, Boston MA.}\footnote{}
{W{\eightrm ARGA} J., ``Relaxed variational problems,'' ``Necessary
conditions for minimum in relaxed variational problems,'' J. Math. Anal.
Appl. {\bf 4} (1962) 111--128, 129--145.} 
and called relaxed controls. 
Although I was the first to use parametrized/Y{\eightrm OUNG} measures in
a context of partial differential equations, I knew of some earlier
work\footnote{$^{32}$} 
{B{\eightrm ERLIOCCHI} Henri \& L{\eightrm ASRY} Jean-Michel,
``Int\'egrandes normales et mesures param\'etr\'ees en calcul des
variations, Bull. Soc. Math. France {\bf 101} (1973), 129--184.}
on variational problems involving no derivatives by Henri B{\eightrm
ERLIOCCHI}\footnote{$^{33}$} 
{Henri B{\eightrm ERLIOCCHI}, French mathematician. 
He works at Universit\'e Paris XIII, Villetaneuse, France.}
and Jean-Michel L{\eightrm ASRY}\footnote{$^{34}$}
{Jean-Michel L{\eightrm ASRY}, French mathematician. 
He works at Universit\'e Paris IX Dauphine, Paris, France.}. 
In my course, it was not the Y{\eightrm OUNG} measures which were the
important concept, and from my work on Homogenization with Fran\c{c}ois
M{\eightrm URAT} I already knew about their limitations; in physical
terms, Y{\eightrm OUNG} measures are just a precise mathematical way for
talking about one point statistics, without falling into the usual trap of
using probabilities for describing the laws of Nature.
\medskip

Although M{\eightrm AXWELL} had found his equation in his quest for
unifying Electricity and Magnetism, his system of equations describes
waves which propagate with the speed of Light\footnote{$^{35}$}
{Human eye is designed to ``see'' electromagnetic waves with wavelength
0.4 to 0.7 micron, corresponding to visible light with colour ranging from
violet to red.}.
Many systems of partial differential equations (of hyperbolic type) are
used now as mathematical models of physical processes where energy is
conserved, and waves explain an important phenomenon, that energy can be
globally conserved but may nevertheless flow away, transported by waves;
for the wave equation I had read some works in this direction in the early
70s by various combinations of authors, Peter L{\eightrm
AX}\footnote{$^{36}$}
{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.}, 
Cathleen M{\eightrm ORAWETZ}\footnote{$^{37}$} 
{Cathleen S{\eightrm YNGE} M{\eightrm ORAWETZ}, Canadian-born
mathematician, born in 1923.
She works at the C{\eightrm OURANT} Institute of Mathematical Sciences at
New York University, New York NY.
Her father, John Lighton S{\eightrm YNGE} (Irish mathematician, 1897-1995,
who worked in Dublin, Ireland), had been from 1946 to 1948 the Head of the
Mathematics Department at the C{\eightrm ARNEGIE} Institute ot Technology
in Pittsburgh PA, later to become C{\eightrm ARNEGIE} M{\eightrm ELLON}
University.}\footnote{}
{Andrew C{\eightrm ARNEGIE}, Scottish-born businessman and philanthropist,
1835-1919.}\footnote{}
{Andrew William M{\eightrm ELLON}, American financier and  philanthropist,
1855-1937.}, 
Ralph P{\eightrm HILLIPS}\footnote{$^{38}$}
{Ralph Saul P{\eightrm HILLIPS}, American mathematician, 1913-1998.
He worked at S{\eightrm TANFORD} University, Stanford CA.}\footnote{}
{Leland S{\eightrm TANFORD}, American businessman, 1824-1893.},
and Walter S{\eightrm TRAUSS}\footnote{$^{39}$}
{Walter Alexander S{\eightrm TRAUSS}, American mathematician, born in
1937. 
He works at B{\eightrm ROWN} University, Providence RI.}.
\medskip

There is another way for the energy to be conserved while it may seem to
disappear.\footnote{}
{Nicholas B{\eightrm ROWN} Jr., American merchant, 1769-1841.}\footnote{}
{L{\eightrm AX} Peter D. \& M{\eightrm ORAWETZ} Cathleen S. \& P{\eightrm
HILLIPS} Ralph S., ``The exponential decay of solutions of the wave
equation in the exterior of a star-shaped obstacle,'' Bull. Amer. Math.
Soc. {\bf 68} (1962) 593--595. ``Exponential decay of solutions of the
wave equation in the exterior of a star-shaped obstacle,'' Comm. Pure
Appl. Math. {\bf 16} (1963) 477--486.}\footnote{}
{M{\eightrm ORAWETZ} Cathleen S., ``The decay of solutions of the exterior
initial-boundary value problem for the wave equation,'' Comm. Pure Appl.
Math. {\bf 14} (1961) 561--568. ``The limiting amplitude principle,''
Comm. Pure Appl. Math. {\bf 15} (1962) 349--361.}\footnote{}
{S{\eightrm TRAUSS} Walter A., ``Local exponential decay of a group of
conservative nonlinear operators,'' J. Functional Analysis {\bf 6} (1970)
152--156.}
It may happen from a nonlinear effect, like for the creation of shocks in
quasilinear systems, and more precisely for (hyperbolic) systems of
conservation laws; this question generalizes the particular situations
which S{\eightrm TOKES} and R{\eightrm IEMANN} had studied, and it was put
into a contemporary mathematical framework by Peter L{\eightrm AX} in the
50s\footnote{$^{40}$} 
{L{\eightrm AX} Peter D., ``Weak solutions of nonlinear hyperbolic
equations and their numerical computation,'' Comm. Pure Appl. Math. {\bf
7}, (1954). 159--193. ``Hyperbolic systems of conservation laws. II. Comm.
Pure Appl. Math. {\bf 10} (1957) 537--566.}. 
A good way to learn about this area now is to look at the recent
book\footnote{$^{41}$}
{D{\eightrm AFERMOS} Constantine M., {\it Hyperbolic conservation laws in
continuum physics}, Grundlehren der Mathematischen Wissenschaften, 325.
Springer-Verlag, Berlin, 2000. xvi+443 pp. ISBN 3-540-64914-X} 
of my good friend Constantine D{\eightrm AFERMOS}\footnote{$^{42}$}
{Constantine M. D{\eightrm AFERMOS}, Greek-born mathematician, born in
1941. 
He works at B{\eightrm ROWN} University, Providence RI.}, 
but it should be noted that it was a book\footnote{$^{43}$} 
{C{\eightrm OURANT} R. \& F{\eightrm RIEDRICHS} K. O., {\it Supersonic
Flow and Shock Waves} Interscience Publishers, Inc., New York, N. Y. 1948.
xvi+464 pp. Applied Mathematical Sciences, Vol. 21. Springer-Verlag, New
York-Heidelberg, 1976. xvi+464 pp.}
by C{\eightrm OURANT} and F{\eightrm RIEDRICHS}\footnote{$^{44}$}
{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.}
who had stemmed interest\footnote{$^{45}$} 
{It is the historical section in this book which triggered my interest in
reading more about the origin of ideas, some of them I had learned as a
student without any precise attribution. 
Cathleen M{\eightrm ORAWETZ} had the charge of editing the book when she
was a graduate student; a few years ago she told me that other historical
informations had to be discarded due to a restriction in space.}
about these questions in the academic community.
\par
More generally, there are situations where the information goes to
infinity, but instead of infinity in the space variable $x$, it is
infinity in the dual space of the F{\eightrm OURIER}\footnote{$^{46}$}
{Jean-Baptiste Joseph F{\eightrm OURIER}, French mathematician, 1768-1830.
He was prefect in Grenoble under N{\eightrm APOL\'EON}.}\footnote{}
{Napol\'eon B{\eightrm ONAPARTE}, French general and statesman, 1769-1821;
he proclaimed himself emperor, under the name N{\eightrm APOL\'EON} I,
1804-1814 (and 100 days in 1815).} 
variable $\xi$, i.e. energy flows into smaller wavelengths (higher
frequencies). 
\medskip
In elastic structures, at least the two aspects occur, and one part of the
energy is lost away through the support, and another part of the energy is
trapped into high frequency vibrations, and this often happens because of
a small scale in some direction, as one uses thin plates and beams in
buildings; in real buildings there is an added damping due to the
viscoelastic properties of concrete, which one may interpret as resulting
from its microstructure by an Homogenization process, as I heard from
Henri S{\eightrm ANCHEZ}-P{\eightrm ALENCIA}\footnote{$^{47}$}
{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, {\it Nonhomogeneous media
and vibration theory} Lecture Notes in Physics, 127. Springer-Verlag,
Berlin-New York, 1980. ix+398 pp. ISBN 3-540-10000-8.}, 
and his explanation is based on a coupling between a (linearly) elastic
material and a viscous fluid trapped in cavities inside\footnote{$^{48}$} 
{Indeed, the water content in concrete is known to be related to its
viscoelastic properties; in particular the viscoelastic properties of
concrete change with time, and this aging may be due to excessive
drying.}.
However, I then heard of another scenario in a talk of Joseph K{\eightrm
ELLER}\footnote{$^{49}$} 
{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.},
at a meeting at IMA (Institut for Mathematics and its Applications,
Minneapolis MN) in 1985; he mentioned the effect of inclusions (of
different elastic materials), together with a small coupling due to taking
into account the nonlinear geometric effect in Elasticity, as a way to
explain an origin of damping and the flow of energy to higher frequencies.
\par
It is possible that the energy trapped in high frequency vibrations
transforms more easily into heat, but it is possible that one has been
calling heat too many different kinds of energy, and this is one
difficulty for putting Thermodynamics into a sound basis. 
In the traditional description, heat is taken away by conduction,
diffusion, convection of the fluid surrounding the structure and radiation
from the boundary (according to the blackbody radiation formula of
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.}, 
which implies a total radiated energy following the law\footnote{$^{51}$} 
{The absolute temperature $T$ is 273.15 + temperature in degrees
C{\eightrm ELSIUS}.}\footnote{}
{Anders C{\eightrm ELSIUS}, Swedish astronomer, 1701-1744.
He worked in Uppsala, Sweden.}\footnote{} 
{C{\eightrm ELSIUS} had put the boiling point of water at 0 and the
freezing point at 100 degrees. 
This was reversed by L{\eightrm INN\'E} who invented the scale that is
used now, under C{\eightrm ELSIUS}'s name.}\footnote{}
{Carolus L{\eightrm INNAEUS} (Carl {\eightrm VON} L{\eightrm INN\'E}),
Swedish naturalist, 1707-1778.}
in $T^{4}$ which S{\eightrm TEFAN}\footnote{$^{52}$}
{Josef S{\eightrm TEFAN}, Austrian physicist, 1835-1893. 
He worked in Vienna, Austria.}
had found earlier); part of the energy trapped in high frequency is also
transmitted as waves to the fluid surrounding the structure, some being
audible\footnote{$^{53}$} 
{Human ear is designed to ``hear'' acoustic waves with frequency 20 to
20,000 Hertz.}\footnote{}
{Heinrich Rudolf H{\eightrm ERTZ}, German physicist, 1857-1894.
He worked in Bonn, Germany.}.
\par
It must be noted that in some situations physicists call noise these
particular high frequency oscillations which others are trying to
describe, and of course calling it noise will not help explaining what it
is, and postulating that it follows some rules which probabilists like is
certainly not the way to understand the phenomenon. 
\medskip

I like to call science-fiction elastic materials those inexistant
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. 
In the late 70s, I made the mistake of assuming that such materials could
exist in principle, although I had failed to find a good class of
strain-stress relations for them; of course, one has to impose that the
evolution problem is well posed (and the mathematical methods for studying
this question still seem quite inadequate), and I also expected this class
to be stable by Homogenization\footnote{$^{54}$}
{``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.}. 
A few years later, Gianni D{\eightrm AL} M{\eightrm ASO}\footnote{$^{55}$}  
{Gianni D{\eightrm AL} M{\eightrm ASO}, Italian mathematician.
He works at SISSA (International School for Advanced Studies) in Trieste,
Italy.}
had mentioned to me that one can say something about the stored energy
functional of the limiting material by using the method of
$\Gamma$-convergence which Ennio D{\eightrm E} G{\eightrm
IORGI}\footnote{$^{56}$} 
{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.}
had developed, and until a few years ago, I did find that it was an
advantage of the $\Gamma$-convergence approach that it could talk about
the stored energy functional of such a limiting material, whose
strain-stress relation I had been unable to characterize; however, I
believe now that no such material exist (which has to satisfy some
invariance properties), and that the $\Gamma$-convergence result has
answered another question, which may not be very physical. 
\par
I like to call ultra-science-fiction elastic materials those different
inexistant materials which instantaneously discover a minimum of their
potential energy; this is usually the class which is studied by those who
pretend to study Elasticity but never mention the term stress.
\par
Although ideal (science-fiction) elastic materials are conservative, it is
clear that real materials are slightly dissipative, but the way a real
material tends to dissipate its kinetic energy and apparently stop in an
equilibrium configuration is certainly not by looking for a global minimum
of the potential energy; of course, if there is only one equilibrium
solution satisfying the desired constraints it will be the global minimum,
and an example\footnote{$^{57}$} 
{K{\eightrm NOPS} R. J. \& S{\eightrm TUART} C. A., ``Quasiconvexity and
uniqueness of equilibrium solutions in nonlinear elasticity,'' Arch.
Rational Mech. Anal. {\bf 86} (1984), no. 3, 233--249.} 
of such a situation has been shown in Elasticity by Robin K{\eightrm
NOPS}\footnote{$^{58}$} 
{Robin J. K{\eightrm NOPS}, English mathematician, born in 1932. 
He works at H{\eightrm ERIOT}-W{\eightrm ATT} University in Edinburgh,
Scotland UK.}
and Charles S{\eightrm TUART}\footnote{$^{59}$}
{Charles A. S{\eightrm TUART}, Scottish-born mathematician. 
He works at EPFL (\'Ecole Polytechnique F\'ed\'erale de Lausanne) in
Lausanne, Switzerland.}.
\medskip

I have left aside an argument which is sometimes used for explaining the
interest in configurations of lowest energy, which is to invoke
Statistical Physics; only systems in thermal equilibrium are supposed to
exist in this framework, and the state of a system is indexed by the
absolute temperature $T$, and the rule says that there is a
``probability'' to find the system in a state of energy $w$, which is
proportional to $exp\bigl( -{w\over k\,T} \bigr)$, where $k$ is the
B{\eightrm OLTZMANN}\footnote{$^{60}$}
{Ludwig B{\eightrm OLTZMANN}, Austrian physicist, 1844-1906.
He worked in Graz and Vienna, Austria, and also in Leipzig, Germany.}
constant\footnote{$^{61}$}
{$k = 1.3807\;10^{-23}$ J{\eightrm OULE}/K{\eightrm ELVIN}, and of course
the J{\eightrm OULE} is the unit for energy, N{\eightrm
EWTON}$\times$meter, or
kilogram$\times$meter$^{2}\times$second$^{-2}$.}\footnote{} 
{James Prescot J{\eightrm OULE}, English physicist, 1818-1889.}.
Of course, the basic rule of this game makes no sense but for large
systems whose parts are connected enough to interact, and it is worth
observing that this rule has been invented by analogy with the equilibrium
solutions of B{\eightrm OLTZMANN} equation, where the parts are identical
``particles'' which collide at a sufficient rate in order to arrive at a
thermal equilibrium quickly\footnote{$^{62}$}
{However, specialists of Plasma Physics have observed that in their
experiments lighter electrons tend to settle quickly to some temperature,
while heavier ions tend to settle quickly to another temperature, and
their experiments do not last long enough for these two temperatures to
come together. 
It would be very silly either for a specialist of meteorology to use a one
temperature model for the atmosphere.
These obvious considerations are but one of the many known defects of
using equilibrium models for deducing some properties of the real world;
one should be aware that this is the point of view which has been taught
for generations, that there is an equation of state valid all the time,
although it has been deduced from the sole knowledge of equilibria, and
part of the scientific problem is not only to ascertain if some equations
have a solution and how to select the physical solution when there are
many, but to question the validity of these equations that one has studied
and construct better physical models of reality.}; 
those who like this framework sometime argue that when the temperature $T$
tends to 0, a system tends to occupy only the positions of lowest energy
(with equal probabilities if there are a few of them), and therefore an
expansion near zero temperature is mentioned, where one first considers
the global minima of the energy $w$. 
\medskip

I do not think that there is much Physics in a model where there is no
time. 
In Classical Mechanics, one often invokes N{\eightrm
OETHER}\footnote{$^{63}$}
{Emmy Amalie N{\eightrm OETHER}, German-born mathematician, 1882-1935. 
Until 1933 she worked in G\"ottingen, Germany, and then in Bryn Mawr
PA.}'s 
theorem which links symmetries and conservation laws, and it is precisely
the invariance by translation in time which is related to the conservation
of total energy, and therefore one should not be surprised that the
partisans of Nature minimizing energy live in a world where time does not
exist.
Some may pretend that time is not so important for their models because
they are only interested in stationary solutions, but they should then
certainly avoid mentioning the question of stability of the equilibrium
model that they study, because the only possible meaning of stability is
related to the precise evolution equation that is satisfied, at least
since the work of L{\eightrm YAPUNOV}\footnote{$^{64}$}
{Aleksandr Mikha\u{\i}lov L{\eightrm YAPUNOV}, Russian mathematician,
1857-1918.}, 
and I had read in the correspondence\footnote{$^{65}$} 
{{\it La correspondance entre Henri Poincar\'e et G\"osta Mittag-Leffler}
Publications des Archives Henri-Poinca\-r\'e. Birkh\"auser Verlag, Basel,
1999. 421 pp. ISBN 3-7643-5992-7.} 
between M{\eightrm ITTAG}-L{\eightrm EFFLER}\footnote{$^{66}$}  
{Magnus G\"osta M{\eightrm ITTAG}-L{\eightrm EFFLER}, Swedish
mathematician, 1846-1927.
He worked in Stockholm, Swe\-den.}
and P{\eightrm OINCAR\'E}\footnote{$^{67}$}  
{Jules Henri P{\eightrm OINCAR\'E}, French mathematician, 1854-1912. 
He worked in Paris, france.}
that a different notion of stability had been used at the beginning; it
would be very silly for a partisan of Nature minimizing energy, to use the
term stability and mistake the gradient flow corresponding to that
minimization for the evolution equation governing the system. 
\medskip

I do not think that there is much Physics either in a model where there is
no space. 
Once one knows that energy is conserved and may change its form, the
problem is to understand where it is located, in space and time or in the
dual frequency space. 
It is precisely when one wants to localize elastic energy in space that
one discovers the C{\eightrm AUCHY}\footnote{$^{68}$} 
{Augustin Louis C{\eightrm AUCHY}, French mathematician, 1789-1857.
He was made baron in 1838.
He worked in Paris, France.}
stress tensor; one should notice that this stress tensor is natural in the
physical Eulerian\footnote{$^{69}$} 
{Leonhard E{\eightrm ULER}, Swiss-born mathematician, 1707-1783. 
He worked in St. Petersburg, Russia.}
point of view, and that in the less realistic mathematical
Lagrangian\footnote{$^{70}$} 
{Giuseppe Lodovico L{\eightrm AGRANGIA} (Joseph Louis L{\eightrm
AGRANGE}), Italian-born mathematician, 1736-1813.
He was made count in 1808.
He worked in Paris, France.}
point of view (also studied by E{\eightrm ULER}) it corresponds to the
P{\eightrm IOLA}\footnote{$^{71}$}
{Gabrio P{\eightrm IOLA}, Italian physicist, 1794-1850.
He worked in Milan, Italy.}-K{\eightrm IRCHHOFF}\footnote{$^{72}$} 
{Gustav Robert K{\eightrm IRCHHOFF}, German physicist, 1824-1887.
He worked in Berlin, Germany.}
stress tensor.
\medskip

The mathematical difficulty for keeping track of all the energy is that
one needs to develop efficient mathematical tools for analyzing where the
energy has gone, in space-time as well as in the dual variables. 
For doing that one certainly needs to use the relation between different
scales which I have developed in the early 70s, based on the use of
various types of weak convergence, like H-convergence\footnote{$^{73}$}
{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.}, 
developped with Fran\c{c}ois M{\eightrm URAT}, which extends the notion of
G-convergence\footnote{$^{74}$}
{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.}, 
which Sergio S{\eightrm PAGNOLO}\footnote{$^{75}$} 
{Sergio S{\eightrm PAGNOLO}, Italian mathematician.
He works at Universit\'a di Pisa, Pisa, Italy.}
had developed in the late 60s, helped with the insight of Ennio D{\eightrm
E} G{\eightrm IORGI}.
\bigskip


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