Lecture Notes of (cid:51) the Unione Matematica Italiana EditorialBoard FrancoBrezzi(EditorinChief) PersiDiaconis DipartimentodiMatematica DepartmentofStatistics UniversitàdiPavia StanfordUniversity ViaFerrata1 Stanford,CA94305-4065,USA 27100Pavia,Italy e-mail:[email protected], e-mail:[email protected] [email protected] JohnM.Ball NicolaFusco MathematicalInstitute DipartimentodiMatematicaeApplicazioni 24-29StGiles’ UniversitàdiNapoli“FedericoII”,viaCintia OxfordOX13LB ComplessoUniversitariodiMonteS.Angelo UnitedKingdom 80126Napoli,Italy e-mail:[email protected] e-mail:[email protected] AlbertoBressan CarlosE.Kenig DepartmentofMathematics DepartmentofMathematics PennStateUniversity UniversityofChicago UniversityPark 1118E58thStreet,UniversityAvenue StateCollege Chicago PA.16802,USA IL60637,USA e-mail:[email protected] e-mail:[email protected] FabrizioCatanese FulvioRicci MathematischesInstitut ScuolaNormaleSuperiorediPisa Universitätstraße30 PiazzadeiCavalieri7 95447Bayreuth,Germany 56126Pisa,Italy e-mail:[email protected] e-mail:[email protected] CarloCercignani GerardVanderGeer DipartimentodiMatematica Korteweg-deVriesInstituut PolitecnicodiMilano UniversiteitvanAmsterdam PiazzaLeonardodaVinci32 PlantageMuidergracht24 20133Milano,Italy 1018TVAmsterdam,TheNetherlands e-mail:[email protected] e-mail:[email protected] CorradoDeConcini CédricVillani DipartimentodiMatematica EcoleNormaleSupérieuredeLyon UniversitàdiRoma“LaSapienza” 46,alléed’Italie PiazzaleAldoMoro2 69364LyonCedex07 00133Roma,Italy France e-mail:[email protected] e-mail:[email protected] TheEditorialPolicycanbefoundatthebackofthevolume. 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] LibraryofCongressControlNumber:(cid:50)(cid:48)(cid:48)(cid:55)(cid:57)(cid:50)(cid:53)(cid:51)(cid:54)(cid:57) MathematicsSubjectClassification((cid:50)(cid:48)(cid:48)(cid:48)(cid:41)(cid:58)(cid:51)(cid:53)(cid:45)(cid:88)(cid:88)(cid:44)(cid:52)(cid:54)(cid:45)(cid:120)(cid:120)(cid:44)(cid:52)(cid:54)(cid:66)(cid:55)(cid:48)(cid:44)(cid:52)(cid:54)(cid:77)(cid:51)(cid:53) ISSNprintedition:(cid:49)(cid:56)(cid:54)(cid:50)(cid:45)(cid:57)(cid:49)(cid:49)(cid:51) ISSNelectronicedition:(cid:49)(cid:56)(cid:54)(cid:50)(cid:45)(cid:57)(cid:49)(cid:50)(cid:49) ISBN-10 (cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:49)(cid:52)(cid:56)(cid:50)-0SpringerBerlinHeidelbergNewYork ISBN-13 (cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:49)(cid:52)(cid:56)(cid:50)(cid:45)(cid:56)SpringerBerlinHeidelbergNewYork DOI(cid:49)(cid:48)(cid:46)(cid:49)(cid:48)(cid:48)(cid:55)/(cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:49)(cid:52)(cid:56)(cid:51)(cid:45)(cid:53) Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember(cid:57), (cid:49)(cid:57)(cid:54)(cid:53),initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:176)c Springer-VerlagBerlinHeidelberg(cid:50)(cid:48)(cid:48)(cid:55) Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsandSPiusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:(cid:49)(cid:50)(cid:48)(cid:51)(cid:54)(cid:55)(cid:54)(cid:57) (cid:86)(cid:65)(cid:52)(cid:49)(cid:47)(cid:51)(cid:49)(cid:48)(cid:48)/SPi (cid:53)(cid:52)(cid:51)(cid:50)(cid:49)(cid:48) 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.