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Luc Tartar
An Introduction to Sobolev
Spaces and Interpolation
Spaces
(cid:65)(cid:66)(cid:67)
Author
LucTartar
DepartmentofMathematicalSciences
CarnegieMellonUniversity
Pittsburgh,PA15213-3890
USA
e-mail:[email protected]
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In memory of Sergei SOBOLEV, 1908–1989
Hepioneeredthestudyofsomefunctionalspaceswhicharecrucialinthestudy
of the partial differential equations of continuum mechanics and physics, and
the first part of these lecture notes is about these spaces, named after him.
In memory of Jacques-Louis LIONS, 1928–2001
He participated in the development of Sobolev spaces, in part with Enrico
MAGENES, applying the general theory of interpolation spaces which he had
developed with Jaak PEETRE, who further simplified the theory so that it
became more easy to use, and the second part of these lecture notes is about
these interpolation spaces.
To Lucia
To my children
Laure, Micha¨el, Andr´e, Marta
Preface
After publishing an introduction to the Navier1–Stokes2,3 equation and
oceanography [18], the revised version of my lecture notes for a graduate
course that I had taught in the spring of 1999, I want to follow with another
set of lecture notes for a graduate course that I had taught in the spring of
2000;thatcoursewasdividedintotwoparts,thefirstpartonSobolev4 spaces,
and the second part on interpolation spaces. The first version had been avail-
able on the Internet, and after a few years, I find it useful to make the text
available to a larger audience by publishing a revised version.
When I was a student at Ecole Polytechnique, which was still in Paris,
France, on the “Montagne Sainte Genevi`eve”,5 I had the chance to have
1 ClaudeLouisMarieHenriNAVIER,Frenchmathematician,1785–1836.Heworked
in Paris, France.
2 SirGeorgeGabrielSTOKES,Irish-bornmathematician,1819–1903.Heworkedin
London, and in Cambridge, England, holding the Lucasian chair (1849–1903).
3 Reverend Henry LUCAS, English clergyman and philanthropist, 1610–1663.
4 Sergei L’vovich SOBOLEV, Russian mathematician, 1908–1989. He worked in
Leningrad, in Moscow, and in Novosibirsk, Russia. I first met him when I was a
student, in Paris in 1969, then at the International Congress of Mathematicians
in Nice in 1970, and conversed with him in French, which he spoke perfectly (all
educatedEuropeansdidlearnFrenchinthebeginningofthe20thcentury).Ionly
met him once more, when I traveled with a French group from INRIA (Institut
National de la Recherche en Informatique et Automatique) in 1976 to Akadem-
gorodoknearNovosibirsk,Russia,whereheworked.ThereisnowaSobolevInsti-
tuteofMathematics oftheSiberianbranchoftheRussianAcademyofSciences,
Novosibirsk, Russia.
5 Genevi`eve, patroness of Paris, c 419 or 422–512.
VIII Preface
Laurent SCHWARTZ6–8 as my main teacher in mathematics in the first year
(1965–1966), and the course contained an introduction9 to his theory of dis-
tributions,10 butIonlyheardaboutSobolevspacesinmysecondyear(1966–
1967), in a seminar organized by Jacques-Louis LIONS11–13 for interested stu-
dents, in addition to his course on numerical analysis. I learnt a little more in
his courses at the university in the following years, and I read a course [13]
that he had taught in 1962 in Montr´eal, Qu´ebec (Canada), and I also read a
book [1] by Shmuel AGMON,14 corresponding to a course that he had taught
at Rice15 University, Houston, TX.
6 Laurent SCHWARTZ, French mathematician, 1915–2002. He received the Fields
Medal in 1950. He worked in Nancy, in Paris, France, at E´cole Polytechnique,
whichwasfirstinParis(whenIhadhimasateacherin1965–1966),andthenin
Palaiseau, France, and at Universit´e Paris VII (Denis Diderot), Paris, France.
7 John Charles FIELDS, Canadian mathematician, 1863–1932. He worked in
Meadville, PA, and in Toronto, Ontario (Canada).
8 DenisDIDEROT,Frenchphilosopherandwriter,1713–1784.HeworkedinParis,
France, and he was the editor-in-chief of the Encyclop´edie. Universit´e Paris 7,
Paris, France, is named after him.
9 Whichmeansthatheonlyconsideredquestionsofconvergenceforsequences,and
hedidnotteachanythingaboutthetopologiesofD orD(cid:1),whichIfirstlearntin
his book [15].
10 LaurentSCHWARTZhasdescribedsomethingabouthisdiscoveryoftheconcept
of distributions in his biography [16].
11 Jacques-Louis LIONS, French mathematician, 1928–2001. He received the Japan
Prizein1991.HeworkedinNancyandinParis,France,holdingachair(analyse
math´ematiquedessyst`emesetdeleurcontrˆole,1973–1998)atColl`egedeFrance,
Paris, France. I first had him as a teacher at Ecole Polytechnique in 1966–1967,
and I did research under his direction, until my thesis in 1971. The laboratory
dedicatedtofunctionalanalysisandnumericalanalysiswhichheinitiated,funded
byCNRS(CentreNationaldelaRechercheScientifique)andUniversit´eParisVI
(Pierre et Marie Curie), is now named after him, the Laboratoire Jacques-Louis
Lions.
12 Pierre CURIE, French physicist, 1859–1906, and his wife Marie SKL(cid:1)ODOWSKA-
CURIE, Polish-born physicist, 1867–1934, jointly received the Nobel Prize in
Physics in 1903, and she also received the Nobel Prize in Chemistry in 1911.
They worked in Paris, France. Universit´e Paris 6, Paris, France, is named after
them.
13 Alfred NOBEL, Swedish industrialist and philanthropist, 1833–1896. He created
a fund to be used as awards for people whose work most benefited humanity.
14 ShmuelAGMON,Israelimathematician,bornin1922.HeworkedatTheHebrew
University, Jerusalem, Israel.
15 William Marsh RICE, American financier and philanthropist, 1816–1900.
Preface IX
Ifirstread aboutinterpolation spaces(in aHilbert16,17 setting) in abook
that Jacques-Louis LIONS had written with Enrico MAGENES18 [14], and
then he gave me his article with Jaak PEETRE19 to read for the theory in
a Banach20,21 setting, and later he asked me to solve some problems about
interpolation for my thesis in 1971, and around that time I did read a few
articles on interpolation, although I can hardly remember in which of the
many articles of Jaak PEETRE I may have read about some of his results.
For the purpose of this course, I also consulted a book by BERGH22,23 &
LO¨FSTRO¨M24 [2].
I also learnt in other courses, by Jacques-Louis LIONS or others, in sem-
inars, and the usual process went on, learning, forgetting, inventing a new
proof or reinventing one, when asked a question by a fellow researcher or a
student, so that for many results in this course I can hardly say if I have read
them or filled the gaps in statements that I had heard, and my memory may
be inaccurate on some of these details. Some of the results may have been
obtained in my own research work, which is concerned with partial differen-
tial equations from continuum mechanics or physics, and my personal reason
for being interested in the subject of this course is that some of the ques-
tions studied have appeared in a natural way in a few practical problems. Of
course, although a few problems of continuum mechanics or physics have led
to some of the mathematical questions described in this course, many have
beenaddedfortheusualreasonthatmathematiciansaresupposedtodiscover
general structures hidden behind particular results, and describe something
16 David HILBERT, German mathematician, 1862–1943. He worked in K¨onigsberg
(then in Germany, now Kaliningrad, Russia) and in G¨ottingen, Germany. The
term Hilbert space was coined by his student VON NEUMANN, when he worked
on his mathematical foundation of quantum mechanics.
17 J´anos (John) VON NEUMANN, Hungarian-born mathematician, 1903–1957. He
worked in Berlin, in Hamburg, Germany, and at IAS (Institute for Advanced
Study), Princeton, NJ.
18 Enrico MAGENES, Italian mathematician, born in 1923. He worked in Pavia,
Italy.
19 JaakPEETRE,Estonian-bornmathematician,bornin1935.HeworkedinLund,
Sweden.
20 Stefan BANACH, Polish mathematician, 1892–1945. He worked in Lw´ow (then
in Poland, now Lvov, Ukraine). There is a Stefan Banach International Mathe-
matical Center in Warsaw, Poland. The term Banach space was introduced by
FRE´CHET.
21 Maurice Ren´e FRE´CHET, French mathematician, 1878–1973. He worked in
Poitiers, in Strasbourg and in Paris, France. I do not know who introduced the
term Fr´echet space.
22 J¨oran BERGH, Swedish mathematician, born in 1941. He has worked in Lund,
and at Chalmers University of Technology, G¨oteborg, Sweden.
23 William CHALMERS Jr., Swedish merchant, 1748–1811.
24 J¨orgen LO¨FSTRO¨M, Swedish mathematician, born in 1937. He worked at
Chalmers University of Technology, G¨oteborg, Sweden.
X Preface
moregeneralafterhavingdoneasystematicstudy,akintoacleaningprocess.
For those who do not yet know much about continuum mechanics or physics,
I recommend looking first at more classical descriptions of the problems, for
example by consulting the books which have been prepared under the direc-
tion of Robert DAUTRAY25 and Jacques-Louis LIONS [4–12]. For those who
alreadyknowsomethingaboutcontinuummechanicsorphysics,Irecommend
looking at my other lecture notes for reading about the defects which I know
about classical models, because other authors rarely mention these defects
even when they have heardabout them: Isupposethat itis the resultof hav-
ing been raised as the son of a (Calvinist) Protestant minister that I learnt
and practiced the point of view that one should not follow the path of the
majoritywhenreasonclearlypointstoadifferentdirection.However,although
I advocate using reason for criticizing without concessions the points of view
that are taught in order to find better “truths”, one should observe that this
approach is more suited to mathematicians than to engineers or physicists;
actually,notall“mathematicians”havebeentrainedwellenoughforfollowing
thatpath,andthatmightexplainwhysomepeopleinitiallytrainedasmathe-
maticianswriteinexactstatements,whichtheyoftendonotchangeevenafter
being told about their mistakes, which others repeat then without knowing
that they propagate errors; if their goal had not been to mislead others, a
better strategy would have been to point out that some statements were only
conjectures.
I have decided to write my lecture notes with some information given
in footnotes about the people who have participated in the creation of the
knowledgerelatedtothesubjectofthecourse,andIhavementionedin[18]a
fewreasonsfordoingthat:Ihadgreatteachers26 likeLaurentSCHWARTZand
Jacques-Louis LIONS, and I have met many mathematicians, for whom I use
theirfirstnamesinthetext,butIhavetriedtogivesomesimplebiographical
data for all people quoted in the text in order to situate them both in time
and in space, the famous ones as well as the almost unknown ones; I have
seen so many ideas badly attributed and I have tried to learn more about the
mathematicians who have introduced some of the ideas which I was taught
whenIwasastudent,andtobeasaccurateaspossibleconcerningtheworkof
all.27 Another reason is that I enjoy searching for clues, even about questions
that might be thought irrelevant for my goals; I might be stopped by a word,
25 Ignace Robert DAUTRAY (KOUCHELEVITZ), French physicist, born in 1928.
26 Although I immediately admired their qualities, like pedagogical skill, I later
became aware of some of their defects, the discussion of which I shall postpone
until I decide to publish all the letters that I wrote to them.
27 Although I have never read much, it would be quite inefficient for me to change
mymethodofworkforthemoment,becausetoomanypeoplehaverecentlyshown
a tendency to badly quote their sources. In some cases, information that I had
proven something in the 1970s has been ignored, for the apparent reason that I
had told that to people who wanted to avoid mentioning my name, the strange
thing being that instead of trying to find someone who would have done similar
Preface XI
wondering about its etymology, or by a new name, wondering about who this
person was, or even by a name which has been attached to a well-known
institution and I want to discover who was that forgotten person in honor of
whom the institution is named; the Internet has given me the possibility to
find such answers, sometimes as the result of many searches which had only
given small hints, and I hope that I shall be told about all the inaccuracies
that are found in my text.
I was glad to learn a few years ago the motto of Hugo of Saint Victor28
“Learn everything, and you will see afterward that nothing is useless”, and
to compare it with what I had already understood in my quest about how
creation of knowledge occurs. I have often heard people say about a famous
physicist from the past, that luck played an important role in his discovery,
but the truth must be that if he had not known beforehand all the aspects of
his problem he would have missed the importance of the new hint that had
occurred, and so this instance of “luck” reminds me of the saying “aide toi,
le ciel t’aidera” (God helps those who help themselves). Those who present
chance as an important factor in discovery probably wish that every esoteric
subject that they like be considered important and funded, but that is not at
allwhatthequotedmottoisabout.Myreasonsforpublishinglecturenotesis
totellthereaderssomeofwhatIhaveunderstood;thetechnicalmathematical
aspectsofthecourseareonething,thescientificquestionsbehindthetheories
are another, but there is more than that, a little difficult to express in words:
I will have succeeded if many become aware, and go forward on the path of
discovery, not mistaking research and development, knowing when and why
theydooneortheother,andkeepingahighergoalinmindwhenforpractical
reasons they decide to obey the motto of the age for a while, “publish or
perish”.
When I was a graduate student in Paris, my advisor invited me a few
times to join a dinner held for a visitor, who had usually talked in the Lions–
Schwartzseminar,whichmeteveryFridayatIHP(InstitutHenriPoincar´e29).
It was before Universit´e de Paris split into many smaller universities, which
happened in 1970 or 1971, and I had heard my advisor mention a special
fund from ZAMANSKI,30 the dean of “Facult´e des Sciences”. The buildings
for sciences were then known as “Halle aux Vins”, because they were being
built on a place previously used for the wine market, and it was only after
all the wine merchants had moved to Bercy, on the other bank of the river
work before me they sometimes preferred to quote one of their friends who had
used the result in the 1990s, without any mention of an author for it.
28 HugoVONBLANKENBURG,German-borntheologian,1096–1141.Heworkedat
the monastery of Saint Victor in Paris, France.
29 Jules Henri POINCARE´, French mathematician, 1854–1912. He worked in Paris,
France.ThereisanInstitutHenriPoincar´e(IHP),dedicatedtomathematicsand
theoretical physics, part of Universit´e Paris VI (Pierre et Marie Curie), Paris,
France.
30 MarcZAMANSKI,Frenchmathematician,1915–1996.HeworkedinParis,France.
