Universitext For other titles in this series, go to www.springer.com/series/223 Haim Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations 1 C Haim Brezis Distinguished Professor Department of Mathematics Rutgers University Piscataway, NJ 08854 USA [email protected] and Professeur émérite, Université Pierre et Marie Curie (Paris 6) and Visiting Distinguished Professor at the Technion Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczyński, Case Western Reserve University ISBN 978-0-387-70913-0 e-ISBN 978-0-387-70914-7 DOI 10.1007/978-0-387-70914-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938382 Mathematics Subject Classification (2010): 35Rxx, 46Sxx, 47Sxx © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec- tion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) ToFelixBrowder,amentorandclosefriend, whotaughtmetoenjoyPDEsthroughthe eyesofafunctionalanalyst Preface This book has its roots in a course I taught for many years at the University of Paris. It is intended for students who have a good background in real analysis (as expounded, for instance, in the textbooks of G. B. Folland [2],A. W. Knapp [1], and H. L. Royden [1]). I conceived a program mixing elements from two distinct “worlds”:functionalanalysis(FA)andpartialdifferentialequations(PDEs).Thefirst partdealswithabstractresultsinFAandoperatortheory.Thesecondpartconcerns the study of spaces of functions (of one or more real variables) having specific differentiabilityproperties:thecelebratedSobolevspaces,whichlieattheheartof themoderntheoryofPDEs.IshowhowtheabstractresultsfromFAcanbeapplied tosolvePDEs.TheSobolevspacesoccurinawiderangeofquestions,inbothpure andappliedmathematics.TheyappearinlinearandnonlinearPDEsthatarise,for example,indifferentialgeometry,harmonicanalysis,engineering,mechanics,and physics.Theybelongtothetoolboxofanygraduatestudentinanalysis. Unfortunately, FA and PDEs are often taught in separate courses, even though theyareintimatelyconnected.ManyquestionstackledinFAoriginatedinPDEs(for ahistoricalperspective,see,e.g.,J.Dieudonné[1]andH.Brezis–F.Browder[1]). There is an abundance of books (even voluminous treatises) devoted to FA.There arealsonumeroustextbooksdealingwithPDEs.However,asyntheticpresentation intendedforgraduatestudentsisrare.andIhavetriedtofillthisgap.Studentswho are often fascinated by the most abstract constructions in mathematics are usually attractedbytheeleganceofFA.Ontheotherhand,theyarerepelledbythenever- ending PDE formulas with their countless subscripts. I have attempted to present a “smooth” transition from FA to PDEs by analyzing first the simple case of one- dimensionalPDEs(i.e.,ODEs—ordinarydifferentialequations),whichlooksmuch more manageable to the beginner. In this approach, I expound techniques that are possibly too sophisticated for ODEs, but which later become the cornerstones of the PDE theory. This layout makes it much easier for students to tackle elaborate higher-dimensionalPDEsafterward. Apreviousversionofthisbook,originallypublishedin1983inFrenchandfol- lowedbynumeroustranslations,becameverypopularworldwide,andwasadopted asatextbookinmanyEuropeanuniversities.AdeficiencyoftheFrenchtextwasthe vii viii Preface lackofexercises.Thepresentbookcontainsawealthofproblems.Iplantoaddeven moreinfutureeditions.Ihavealsooutlinedsomerecentdevelopments,especially inthedirectionofnonlinearPDEs. Briefuser’sguide 1. Statementsorparagraphsprecededbythebulletsymbol•areextremelyimpor- tant, and it is essential to grasp them well in order to understand what comes afterward. 2. Resultsmarkedbythestarsymbol(cid:2)canbeskippedbythebeginner;theyareof interestonlytoadvancedreaders. 3. In each chapter I have labeled propositions, theorems, and corollaries in a con- tinuousmanner(e.g.,Proposition3.6isfollowedbyTheorem3.7,Corollary3.8, etc.).Onlytheremarksandthelemmasarenumberedseparately. 4. In order to simplify the presentation I assume that all vector spaces are over R. Most of the results remain valid for vector spaces over C. I have added in Chapter11ashortsectiondescribingsimilaritiesanddifferences. 5. Many chapters are followed by numerous exercises. Partial solutions are pre- sentedattheendofthebook.Moreelaborateproblemsareproposedinaseparate sectioncalled“Problems”followedby“PartialSolutionsoftheProblems.”The problems usually require knowledge of material coming from various chapters. I have indicated at the beginning of each problem which chapters are involved. Some exercises and problems expound results stated without details or without proofsinthebodyofthechapter. Acknowledgments DuringthepreparationofthisbookIreceivedmuchencouragementfromtwodear friendsandformercolleagues:Ph.CiarletandH.Berestycki.Iamverygratefulto G.Tronel,M.Comte,Th.Gallouet,S.Guerre-Delabrière,O.Kavian,S.Kichenas- samy,andthelateTh.Lachand-Robert,whosharedtheir“fieldexperience”indealing withstudents.S.Antman,D.Kinderlehrer,andY.Liexplainedtomethebackground and“taste”ofAmericanstudents.C.JoneskindlycommunicatedtomeanEnglish translationthathehadpreparedforhispersonaluseofsomechaptersoftheoriginal French book. I owe thanks toA. Ponce, H.-M. Nguyen, H. Castro, and H. Wang, whocheckedcarefullypartsofthebook.Iwasblessedwithtwoextraordinaryas- sistantswhotypedmostofthisbookatRutgers:BarbaraMiller,whoisretired,and nowBarbaraMastrian.Idonothaveenoughwordsofpraiseandgratitudefortheir constantdedicationandtheirprofessionalhelp.Theyalwaysfoundattractivesolu- tionstothechallengingintricaciesofPDEformulas.Withouttheirenthusiasmand patiencethisbookwouldneverhavebeenfinished.Ithasbeenagreatpleasure,as Preface ix ever,toworkwithAnnKostantatSpringeronthisproject.Ihavehadmanyoppor- tunitiesinthepasttoappreciateherlong-standingcommitmenttothemathematical community. TheauthorispartiallysupportedbyNSFGrantDMS-0802958. HaimBrezis RutgersUniversity March2010