\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small Sixth Mississippi State
Conference on Differential Equations and Computational
Simulations, {\em Electronic Journal of Differential Equations},
Conference 15 (2007), pp. 221--228.\newline ISSN: 1072-6691. URL:
http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document} \setcounter{page}{221}
\title[\hfilneg EJDE-2006/Conf/15\hfil Louis Nirenberg and Klaus Schmitt]
{Louis Nirenberg and Klaus Schmitt: The joy of differential
equations}
\author[J. Mawhin\hfil EJDE/Conf/15 \hfilneg]
{Jean Mawhin}
\address{Jean Mawhin\newline
D\'epartement de Math\'ematiques\\
Universit\'e Catholique de Louvain\\
1348 Louvain-la-Neuve, Belgium}
\email{mawhin@math.ucl.ac.be}
\thanks{Published February 28, 2007.}
\subjclass[2000]{01A70, 01A60,34-03, 35-03}
\keywords{Nirenberg; Schmitt; differential equations}
\begin{abstract}
A tribute to Louis Nirenberg for his 80th birthday anniversary and
to Klaus Schmitt for his 65th birthday anniversary.
\end{abstract}
\maketitle
\section{Introduction}
The life of a research mathematician is like the one of a gold
digger: a constant and often harassing work, from time to time the
discovery of a nugget, and for mathematicians like for diggers,
the size of nuggets may be somewhat different. But, for
mathematicians like for diggers, the joy lies not only in the
discovery, but mostly in the quest.
The two mathematicians we celebrate today have devoted their life
to this quest, and I know them enough to say that this quest gave
them a lot of joy. Their successes, their beautiful results, have
been in turn a source of joy and inspiration for many other
mathematicians, who have climbed on their shoulders to find their
own nuggets. The common denominator of the amazingly diverse
production of Louis Nirenberg and of the many elegant results of
Klaus Schmitt is differential equations, giving me the title of
this afterdinner speech.
\section{Louis Nirenberg}
\emph{A tout seigneur tout honneur}, as we say in French, let me
start by the inhuman task of describing for you the achievements
of Louis Nirenberg, the results of about fifty-five years of
cruising in the mathematical sky at the highest altitude.
Louis Nirenberg was born February 28 1925. Can you do anything
else than mathematics when your birth place is Hamilton. Louis
refused taking any advantage of this, and decided to study physics
at McGill. But nobody can escape to his predestination. The young
physicist started to work, on the atomic bomb, at the National
Research Council in Montr\'eal, and had for colleagues Richard
Courant's older son Ernst and his Canadian wife. He asked her to
consult her father-in-law about the best place in New York to
study theoretical physics. Courant's answer was an interview at
New York University, followed by an assistantship in mathematics.
This is the reason why Canada does not have today the world
leadership in nuclear weapons, and why its military threatening of
United States only exists in the imagination of Michael Moore.
Then comes a first keyword in Louis's career: \textbf{fidelity}.
He entered the Mathematics Department of New York University as a
graduate student in 1945 and he never left it since for another
institution. There should be big celebrations in Courant this
year. In addition, more than one quarter of his papers have
appeared in the \emph{Communications in Pure and Applied
Mathematics}, with a record of four in 1953, his first year of
publication. They include his PhD thesis, defended in 1949, with
Jim Stoker as adviser and Kurt Friedrichs as model. The thesis
contains the complete positive solution of \emph{Weyl's problem in
differential geometry} : given a Riemannian metric on the unit
sphere, with positive Gauss curvature, can you embed this 2-sphere
isometrically into 3-space as a convex surface ? So we must retain
\textbf{geometry} as another keyword for Nirenberg's
contributions, an area that he revisited regularly during his life
with results on surfaces with curvature of fixed sign, on the
rigidity of closed surfaces, on the intrinsic norms on complex
manifolds (with Chern and Levine) and on the characterization of
convex bodies.
Many historians claim that the whole career of a scientist can be
read in his first publications like in a cristal ball : Louis
Nirenberg's proof of Weyl's problem consists in a reduction to a
partial differential equation, solved, by a continuity method,
through the obtention of new a priori estimates. This equation is
elliptic, and \textbf{ellipticity} is definitely another keyword
of Nirenberg's mathematical work. More than one third of his
articles contain the work `elliptic' just in their title, and a
much larger fraction deals with elliptic equations or systems.
There is hardly one aspect of those equations he has not
considered.
The \textbf{regularity} problem for example. Royal paths exist to
prove the existence and uniqueness of weak solutions of elliptic
problems, but a much more difficult question is to know how
regular is a weak solution. Today, no graduate student ignores
Nirenberg's \emph{method of differences} for proving interior and
boundary regularity, or the Agmon-Douglis-Nirenberg's extension to
general elliptic systems! For \emph{Navier-Stokes equation},
essential questions about regularity are still open seventy years
after Leray pioneering work (and the nugget worth one million
dollars !). Among the best results today is
Caffarelli-Kohn-Nirenberg's estimate of the Hausdorf measure of
the set of singularities. It is the study, with Joseph Kohn, of
\emph{regularity problems in several complex variables} connected
with boundary regularity for the $\overline \partial$-Neumann
problem, which led Kohn and Nirenberg to create and developed the
fruitful concept of \emph{pseudo-differential operator},
generalizing and unifying the singular integral and partial
differential operators. Their importance in analysis needs not to
be emphasized. Incidentally, Friedrichs found the name for the new
baby.
It is hard to be a democrat when you are an analyst. Your success
depends much more on creating and using \textbf{inequalities} than
equalities. In this respect, Louis is one of the tycoons of this
mathematical wild liberalism. We all know, teach and use the
\emph{ Gagliardo-Nirenberg}, the \emph{John-Nirenberg} and the
\emph{ Caffarelli-Kohn-Nirenberg} inequalities, which extend and
unify so many fundamental ones. Louis has expressed himself, at
several occasions, his love of inequalities :
\begin{quote}
Friedrichs was a great lover of inequalities and that affected me
very much. The point of view was that the inequalities are more
interesting than the equalities.
I love inequalities. If somebody
shows me a new inequality, I say: Oh, that's beautiful, let me
think about it, and I may have some ideas connected to it.
\end{quote}
Those inequalities play a fundamental role in partial differential
equations for the obtention of \emph{a priori estimates} for the
possible solutions or through some \emph{operator inequalities}.
Descartes is famous, in his philosophy of existence, for his
sentence
\begin{quote}
Je pense, donc je suis -- I am thinking, hence I do exist.
\end{quote}
The philosophy of the solutions of partial differential equations
could be summarized by something like
\begin{quote}
I am limited, hence I do exist.
\end{quote}
You can now wonder how such an strong advocate of inequalities was
able to build so many tight professional and personal relations
with French people. You all know the motto of France:
\begin{quote}
Libert\'e, \'egalit\'e, fraternit\'e -- Liberty, equality, fraternity
\end{quote}
His cult of inequalities did not prevent Louis to be elected a
foreign member of the French Academy of Science and promoted
Doctor \emph{honoris causa} of the University of Paris ! Much
more, Louis has won over the heart of Nanette, a charming French
lady! I bed that his fights for liberty and his cult of fraternity
have won over his unlimited taste for inequality. When you observe
that ninety percents of Louis's papers are written in
collaboration, you can add fraternity to the keywords describing
his personality. Any afterdinner speech being uncomplete without
one bad play upon words, Liberty gives me the opportunity to
mention the \emph{free boundary value problems} and Louis's
fundamental contributions, with David Kinderlehrer, to the
regularity of their solutions.
\textbf{Positivity} is another feature of Louis's character :
almost each of us has received from him, one day or another, a
positive encouragement or a stimulating remark. His forty-five PhD
students (several ones are here) would all testify in this
direction. In elliptic and parabolic differential equations,
positivity is an essential manifestation of the \emph{maximum
principle}, and Louis's virtuosity in using this beautiful
instrument is unequalled. Do I need to recall the
\emph{Gidas-Ni-Nirenberg symmetry theorem} and its variations, in
particular in joint work with Henri Berestycki and others. Louis
Nirenberg is the Paganini of the maximum principle, and the fact
is caught for ever in an AMS movie, a deserved reward to a movies
lover. Positivity can also occurs in variational problems, for
example in Brezis-NirenbergÕs discussion of positive solutions of
\emph{ nonlinear elliptic equations of the Yamabe type}, which
have inspired so many other mathematicians by opening the way to
problems with lack of compactness. This is only one of the many
contributions of Nirenberg in critical point theory, advocated and
developed in inspiring survey papers.
Critical point theory is one of the keys to attack \textbf{
nonlinearity}. A pioneer in nonlinear elliptic equations (the
topics of his first published paper), Louis has returned, at
various stages of his career, to \emph{fully nonlinear elliptic
equations} to make striking breakthroughs, like the ones in a
series of papers with Caffarelli and Spruck on
\emph{Monge-Amp\`ere and related equations}. His invited lecture
at the International Congress of Mathematicians of 1962 in
Stockholm, contains two sentences that I always offer as a guide
to my students dealing with nonlinear problems. The first one is:
\begin{quote}
Most results for nonlinear problems are still obtained via linear
ones, i.e. despite the fact that the problems are nonlinear not
because of it.
\end{quote}
The second one comments a result of Moser:
\begin{quote}
The nonlinear character of the equations is used in an essential
way, indeed he obtains results because of the nonlinearity not
despite it.
\end{quote}
Both aspects have been masterfully explored by Louis Nirenberg,
but there is some Dr Jekyll and Mr Hyde aspect we cannot hide:
this master of nonlinearity never hesitated to betray the club in
making striking contributions to linear problems, forcing us to
retain \textbf{linearity} as another keyword. Louis probably could
argue that some of his fundamental contributions to linear
elliptic equations, in particular his \emph{generalized Schauder
and Sobolev estimates}, were motivated by solving new nonlinear
problems. But how can he justify the necessary and sufficient
conditions for \emph{local solvability of general linear partial
differential equations} obtained with Fran\c{c}ois Treves ?
Another keyword of Louis's mathematical Wonderland is
\textbf{holomorphy}. One important question is the study of
\emph{almost complex structures} : how to recognize the
Cauchy-Riemann operators when given in some arbitrary coordinate
system. For higher dimensions, necessary integrability conditions
are needed. Newlander and Nirenberg have shown them to be
sufficient as well. Another question is the existence of
\emph{deformations of complex structures}, families of
diffeomorphic complex manifolds, differentiable in the parameter.
The answer is given by a theorem of Kodaira-Nirenberg-Spencer.
Needless to say that some exciting partial differential equations
are hidden in the proof.
\textbf{Topology} must be retained as an other keyword of Louis's
mathematical activity. In recent work with Haim Brezis, he created
a big scandal among topologists when developing -- motivated by
some nonlinear models of physics -- a \emph{degree theory for
mappings which need not be continuous} ! They live instead in the
VMO space of functions with \emph{vanishing mean oscillation}, a
close relative of the BMO space of functions with \emph{bounded
mean oscillation}, invented, for other purposes, by Fritz John and
Louis Nirenberg, and widely used in many parts of analysis. The
BMO space is an enlargement of the space of essentially bounded
measurable functions, and the VMO space a subspace of BMO, the
closure of the space of continuous functions. But Louis's links
with topology are somewhat older. Already in 1970, he introduced
the use of \emph{stable homotopy} in generalizing Landesman-Lazer
conditions for bounded nonlinear perturbations of elliptic
operators with positive index. In terms of classical degree,
instead of questioning the continuity of the map, he was
questioning the equal dimensions of the underlying spaces. This
paper has had a smaller lineage as most other ones, because, as
Louis frankly observed :
\begin{quote}
So far, not natural example has come up in which it has been used.
\end{quote}
When I am depressed, I like to remember that I gave one, neither
natural nor elliptic, just a simple system of two second order
ordinary differential equations with three nonlinear boundary
conditions. Other important contributions of Brezis-Nirenberg were
inspired by Landesman-Lazer's paper.
Incidentally, I am not sure everybody is aware of the role played
by Louis in advertising, within the PDE world, Landesman-Lazer's
result. In French language, we like to oppose the \emph{savoir
faire}, the ability, to the \emph{faire savoir}, the advertising.
I think that the marvelous saga of the Landesman-Lazer problem is
a wonderful combination of Landesman-Lazer ``savoir faire'' with
Louis Nirenberg's ``faire savoir''. And this is far from an unique
example of the \textbf{generosity} (another keyword) and
enthusiasm of Louis for presenting, in his marvellous lectures and
in survey papers, the results of other mathematicians. This is
often an opportunity for a silver nugget to be changed in a golden
one. In such a lecture, Louis always starts by apologizing that he
is not really an expert in the area he is presenting. Each
evening, before sleeping, I ask God to make me a non-expert like
Louis Nirenberg. To punish me for this arrogance, God has changed
me into an afterdinner speaker, transforming this personal
punishment into a collective one.
The world would not turn round if this combination of exceptional
personality and outstanding achievements had not been recognized
and crowned by distinctions and awards. I mention only a few, to
avoid starting another lecture and becoming a tormentor for
Louis's simplicity. The Bocher, Crafoord and Steele prizes, the
National Medal of Science, fellowship from all important academies
in the world and honorary degrees from many prestigious
universities. They show the universal appreciation of a man whose
human qualities are at the level of his mathematical
accomplishments.
\section{Klaus Schmitt}
If my feelings for Louis Nirenberg are much like the ones of a son
for a father (although he by no means needs to recognize any
paternity), my feelings for Klaus Schmitt are the ones to a
brother, because of the proximity of our ages and the many
memories we share. Klaus Schmitt is born on May 14 1940, in
Rimbach (Germany), near Heidelberg, and I do not know of any
mathematician whose name is Rimbach. Klaus has discovered America
when he was still a teenager and, in terms of fidelity, he has
followed the example of Louis, just replacing New York by Salt
Lake City. Klaus is a member of the Department of Mathematics of
the University of Utah since 1967.
Even if he looks quite healthy and is a true sportman, one must
describe Klaus' professional life in terms of a impressive series
of mathematical diseases. In each case, using the strength of his
scientific vocation and the force of his character, Klaus has been
able, by observing and analyzing his own case (an interesting
fixed point problem) to contribute to a better knowledge of the
diseases he was suffering. Let me tell you the hight lights of
this fight.
Klaus got the first disease is a city of Nebraska, officially
called Lincoln but better known locally as Jackson city. He was
trapped into some type of sect or gang, usually called a research
group to escape the local police and FBI. This first disease is
characterized by an immodest use of \textbf{upper and lower
solutions (or sub- and super solutions)}, to study nonlinear
boundary value problems. In the language of western movies, the
method can be described by saying that if you are able to miss
your target to the left and to miss it to the right, then you are
able to catch it. It is a technique widely used by militars, and I
would not be surprised that Klaus got some financial support from
them. The target here is the solution of an ordinary, retarded or
partial differential equation of elliptic or parabolic type. The
symptoms of the illness, as observed on Klaus, can take very
different forms, with periods of invariance, crises of convexity,
or periodic returns. This is why we owe to Klaus elegant and
fruitful extensions of the method to systems, based upon
\emph{convexity properties}. As the method of upper and lower
solutions is connected to the maximum principle and the maximum
principle to positivity, Klaus proved also in this period a few
striking results on the existence and multiplicity of
\emph{positive solutions}, a topics to which he has remained
faithful all his life. He has been very early a distinguished
ambassador in the West for \emph{Krasnosel'skii's fixed point
theorem in cones}, and proved several extensions.
The first signs of (uncomplete) recovering appeared in the late
seventies, unfortunately due to the emergence of another disease,
called \textbf{bifurcation}. The main symptom there is the
apparition of continuous branches at some of your eigenvalues, and
the main question about their evolution is to know if they will
grow to infinity or connect another eigenvalue (the famous
\emph{Rabinowitz dilemma} in the diagnostics). The second case is
of course better if you want to continue to travel or if you live
in a small country. In the case of Klaus, the apparition of the
symptoms was detected even in the absence of
\emph{differentiability properties}, but he was able to use the
bifurcation branches in the most efficient way to prove the
existence and multiplicity of solutions for some nonlinear
elliptic partial differential equations which had escaped the
upper and lower solutions butchery. A result which remain
unsurpassed in this area is a beautiful discussion, with Renate
Schaaf, of how the multiplicity of the solutions depends upon the
space dimension for \emph{nonlinear periodic perturbations of the
Laplacian} with Dirichlet boundary conditions. During some period
of fever, Klaus has even discovered some \emph{ghost solutions},
which occur when you discretize a continuous problem without
enough care. It was a joint work with Heinz-Otto Peitgen, who
never recovered and became the apostle of mathematics seen as a
fine art.
A very invasive virus, called the \textbf{p-Laplacian}, can affect
visitors to Chile, in particular in Santiago, or people who have
close scientific contacts with the Man\'asevich family. This
variant of the well known Laplacian virus is characterized by
degenerescence properties, which make difficult or impossible the
use of the classical variational or topological treatments
prescribed for the Laplacian. Again, this illness has left traces
both in Klaus' body and in mathematical knowledge, in the form of
\emph{variational identities} and \emph{mountain pass solutions}.
About the same period, Klaus also suffered from \textbf{variational
inequalities}, a disease he probably got from contacts with
Vietnamese colleagues or students. Be careful, some of them are
here. Those variational inequalities are innocent looking symptoms
of severe \emph{obstacle problems}, which can become particularly
serious when they come in contact with p-Laplacians. Klaus
striking observations during this most recent period of activity
has shed much light to this very active domain of research, and
his book with Le is on the shelve of all practitioners in this
area. One can be especially concerned by the fact that Klaus has
taken the risk of combining upper and lower solutions,
p-Laplacians and variational inequalities. Such a coalition can
create new mutations and could overcome the strongest patients,
but I remain convinced that Klaus once more, and the mathematical
community, will take benefit of this attack.
Klaus' generosity is well known among the mathematical community,
and corroborated by the large number of collaborators and students
(many of them are here). Klaus has shared with them, without the
slightest restriction, his successive favorite diseases, and
enjoyed with proudness and pleasure the apparition and development
of new forms of illnesses in those young organisms. Klaus has
tried various remedies to escape to other diseases, even some
interest in \emph{ecology}. You will not be surprised that he has
developed a strong interest for questions of \emph{permanence}.
But I must tell you a few words about another disease which may
look less professional than the previous ones, but is far to be
unrelated. The \textbf{Hirschen disease} is a mysterious illness
spread in the German Black Forest, but concentrated in an obscure
hamlet called Oberwolfach Walke. Its main symptom is a permanent
and intense feeling of thirst, especially in the evening. The only
known remedy is to alternate white wine with kirchwasser during a
sufficiently long period. Klaus' reason of catching the Hirschen
disease is of course the many conferences he has co-organized in
Oberwolfach, and the many ones he was invited to. Needless to say
that the consequences of the Hirschen disease are specially
dramatic if you live in Utah, where Joseph SmithÕs questionable
ideas on alcoholic beverages are still dominant. But, once more
and for our delight, Klaus has successfully solved this problem in
the form of a striking mathematical result, already called in
literature the \emph{Schmitt-Smith theorem}, whose statement is as
deep as beautiful:
\begin{quote}
A dry state needs not to be locally dry.
\end{quote}
Many proofs by pictures exist for this theorem, and experimental
proofs have been given during gorgeous hikes in the mountains or
friendly parties in Klaus and Claudia's house.
I first met Klaus, in Salt Lake City, in 1973 and cherish a
friendship and collaboration of more than thirty years. It
started through some reprints request in the early seventies, when
Klaus was running in Utah a pioneering seminar on the use of
degree in nonlinear ordinary differential equations, a most
unusual topics in those years among ode people. Klaus had been
pleased to discover that some young Belgian mathematician had
published a few papers on similar topics. Having always be
responsive to some forms of beauty in mathematics, Klaus was
anxious to meet this Belgian whose first name was Jean. He
enthusiastically arranged a visit to Salt Lake City at the
occasion of Jean's first trip to the US. Unverified rumors claim
that a romantic dinner with candles had even been planned. Klaus
had enough self-control to hide his surprise and disappointment
when Jean came out of the plane with É his wife and children, but
he never reproached Jean to be Jean. On the contrary, at each of
Jean's many visits to Salt Lake, Klaus initiated him, besides
beautiful new mathematics, to new areas, like cross-country ski,
mountain biking or cow-boy bars. Each time Jean was so excited
that, back home, he immediately bought a pair of skis (forgetting
that Belgium has little or no snow), then a mountain-bike
(forgetting that Belgium has no mountains), then a pair of guns
(forgetting that Belgium has no Wild West; only Flemishs in the
North and Walloons in the South).
Klaus of course was always happy to visit Germany and, as a
Humboldt Prize Winner, has spent several sabbaticals in his
native country. He has been a visiting professor in most important
German universities, and one can say that he is a legend there. He
visited many other places in the world, very often with his
charming wife Claudia, who succeeded like no other one in
distracting Klaus from his mathematical diseases. She is a
wonderful host if you happen to visit Utah and her beautiful smile
shines like the sun on the Rocky Mountains.
\section*{Conclusion}
Louis Nirenberg said, in a recent interview:
\begin{quote}
One of the wonders of mathematics is you go somewhere in the world
and you meet other mathematicians, and it's like one big family.
This large family is a wonderful joy.
\end{quote}
The joy of such a meeting was offered to all of us in Mississipi
State, at the occasion of this double celebration. We unanimously
ask Louis and Klaus to give us many further opportunities of
sharing this wonderful joy with them.
\end{document}