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Partial Differential Equations PDF

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Graduate Texts in Mathematics 214 Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooksingraduatecourses,theyarealsosuitableforindividualstudy. Forfurthervolumes: http://www.springer.com/series/136 Ju¨rgen Jost Partial Differential Equations Third Edition 123 Ju¨rgenJost MaxPlanckInstitute forMathematicsintheSciences Leipzig,Germany ISSN0072-5285 ISBN978-1-4614-4808-2 ISBN978-1-4614-4809-9(eBook) DOI10.1007/978-1-4614-4809-9 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012951053 MathematicsSubjectClassification:35-XX,35-01,35JXX,35KXX,35LXX,35AXX,35BXX,35DXX ©SpringerScience+BusinessMediaNewYork2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Thisisthethirdeditionofmytextbookintendedforstudentswhowishtoobtainan introductionto the theory of partial differentialequations (PDEs, for short). Why istherea newedition?Theanswerissimple:Iwantedto improvemybook.Over the years, I have received much positive feedback from readers from all over the world. Nevertheless, when looking at the book or using it for courses or lectures, I alwaysfindsome topicsthatare important,butnotyetcontainedin thebook,or I see places where the presentation could be improved. In fact, I also found two errorsinSect.6.2,andseveralothercorrectionshavebeenbroughttomyattention byattentiveandcarefulreaders. So, what is new? I have completely reorganized and considerably extended Chap.7 on hyperbolic equations. In particular, it now also contains a treatment of first-order hyperbolic equations. I have written a new Chap.9 on the relations betweendifferenttypesofPDEs.Ihaveinsertedmaterialontheregularitytheoryfor semilinearellipticequationsandsystemsinvariousplaces.Inparticular,thereisa newSect.14.3thatshowshowtousetheHarnackinequalitytoderivethecontinuity ofboundedweaksolutionsofsemilinearellipticequations.Suchequationsplayan importantroleingeometricanalysisandelsewhere,andIthereforethoughtthatsuch anadditionshouldserveausefulpurpose.Ihavealsoslightlyrewritten,reorganized, orextendedmostothersectionsofthebook,withadditionalresultsinsertedhereand there. Butletmenowdescribethebookinamoresystematicmanner.Asanintroduc- tionto themoderntheoryofPDEs, itdoesnotoffera comprehensiveoverviewof thewholefieldofPDEs,buttriestoleadthereadertothemostimportantmethods and central results in the case of elliptic PDEs. The guiding question is how one canfindasolutionofsuchaPDE.Suchasolutionwill,ofcourse,dependongiven constraintsand, in turn, if the constraintsare of the appropriatetype, be uniquely determinedbythem.We shallpursuea numberofstrategiesforfindinga solution ofaPDE;theycanbeinformallycharacterizedasfollows: 0. Write down an explicit formula for the solution in terms of the given data (constraints). This may seem like the best and most natural approach, but this v vi Preface is possible only in rather particular and special cases. Also, such a formula mayberathercomplicated,sothatitisnotveryhelpfulfordetectingqualitative propertiesof a solution. Therefore,mathematicalanalysis has developedother, morepowerful,approaches. 1. Solve a sequence of auxiliary problems that approximate the given one and show that their solutions converge to a solution of that original problem. Differentialequationsare posedin spacesoffunctions,andthose spacesareof infinitedimension.Thestrengthofthisstrategyliesincarefullychoosingfinite- dimensionalapproximatingproblemsthatcanbesolvedexplicitlyornumerically and that still share importantcrucial features with the original problem. Those featureswillallowustocontroltheirsolutionsandtoshowtheirconvergence. 2. Start anywhere, with the required constraints satisfied, and let things flow towardsasolution.Thisisthediffusionmethod.Itdependsoncharacterizinga solutionofthePDEunderconsiderationasanasymptoticequilibriumstatefora diffusionprocess.ThatdiffusionprocessitselffollowsaPDE,withanadditional independentvariable.Thus,wearesolvingaPDEthatismorecomplicatedthan theoriginalone.Theadvantageliesinthefactthatwecansimplystartanywhere andletthePDEcontroltheevolution. 3. Solveanoptimizationproblemandidentifyanoptimalstateasasolutionofthe PDE. This is a powerfulmethod for a large class of elliptic PDEs, namely,for thosethatcharacterizetheoptimaofvariationalproblems.Infact,inapplications inphysics,engineering,oreconomics,mostPDEsarisefromsuchoptimization problems. The method depends on two principles. First, one can demonstrate the existenceof an optimalstate fora variationalproblemunderrathergeneral conditions. Second,the optimality of a state is a powerfulpropertythat entails many detailed features: If the state is not very good at every point, it could be improvedandthereforecouldnotbeoptimal. 4. Connectwhatyouwanttoknowtowhatyouknowalready.Thisisthecontinuity method.Theideaisthatifyoucanconnectyourgivenproblemcontinuouslywith another,simpler,problemthatyoucanalreadysolve,thenyoucanalsosolvethe former.Ofcourse,thecontinuationofsolutionsrequirescarefulcontrol. Thevariousexistenceschemeswillleadustoanother,moretechnical,butequally important,question,namely,theoneabouttheregularityofsolutionsofPDEs.Ifone writesdownadifferentialequationforsomefunction,thenonemightbeinclinedto assumeexplicitlyorimplicitlythata solutionsatisfies appropriatedifferentiability propertiessothattheequationismeaningful.Theproblem,however,withmanyof the existence schemes described above is that they often only yield a solution in some function space that is so large that it also contains nonsmooth and perhaps evennoncontinuousfunctions.Thenotionofasolutionthushastobeinterpretedin somegeneralizedsense.Itisthetaskofregularitytheorytoshowthattheequation in question forces a generalized solution to be smooth after all, thus closing the circle.Thiswillbethesecondguidingproblemofthisbook. The existence and the regularity questions are often closely intertwined. Reg- ularity is often demonstrated by deriving explicit estimates in terms of the given Preface vii constraintsthatanysolutionhastosatisfy,andtheseestimatesinturncanbeused forcompactnessargumentsinexistenceschemes.Suchestimatescanalsooftenbe usedtoshowtheuniquenessofsolutions,and,ofcourse,theproblemofuniqueness isalsofundamentalinthetheoryofPDEs. After this informaldiscussion, let us now describe the contentsof this book in morespecificdetail. Our starting point is the Laplace equation, whose solutions are the harmonic functions.The field of elliptic PDEs is then naturallyexploredas a generalization of the Laplace equation, and we emphasize various aspects on the way. We shall developamultitudeofdifferentapproaches,whichinturnwillalsoshednewlight onourinitialLaplaceequation.Oneoftheimportantapproachesistheheatequation method,where solutions of elliptic PDEs are obtainedas asymptotic equilibria of parabolicPDEs.Inthissense,onechaptertreatstheheatequation,sothatthepresent textbook definitely is not confined to elliptic equations only. We shall also treat thewaveequationastheprototypeofahyperbolicPDEanddiscussitsrelationto theLaplaceandheatequations.Ingeneral,thebehaviorofsolutionsofhyperbolic differential equations can be rather different from that of elliptic and parabolic equations,andweshallusefirst-orderhyperbolicequationstoexhibitsometypical phenomena.Inthecontextoftheheatequation,anotherchapterdevelopsthetheory of semigroups and explains the connection with Brownian motion. There exist many connections between different types of differential equations. For instance, the density functionof a system of ordinarydifferentialequationssatisfies a first- orderhyperbolicequation.Suchequationscanbe studiedby semigrouptheory,or onecanaddasmallregularizingelliptictermtoobtainaso-calledviscositysolution. OthermethodsforobtainingtheexistenceofsolutionsofellipticPDEs,likethe differencemethod,whichisimportantforthenumericalconstructionofsolutions, the Perron method;and the alternating method of H.A. Schwarz are based on the maximum principle. We shall present several versions of the maximum principle thatarealsorelevanttoapplicationstononlinearPDEs. In any case, it is an important guiding principle of this textbook to develop methodsthatare also usefulfor the study of nonlinearequations,as those present theresearchperspectiveofthefuture.MostofthePDEsoccurringinapplicationsin thesciences,economics,andengineeringareofnonlineartypes.Oneshouldkeepin mind,however,that,becauseofthemultitudeofoccurringequationsandresulting phenomena, there cannot exist a unified theory of nonlinear (elliptic) PDEs, in contrastto thelinear case.Thus,there arealso nouniversallyapplicablemethods, andweaiminsteadatdoingjusticetothismultitudeofphenomenabydeveloping verydiversemethods. Thus, after the maximum principle and the heat equation, we shall encounter variationalmethods,whoseideaisrepresentedbytheso-calledDirichletprinciple. For that purpose, we shall also develop the theory of Sobolev spaces, including fundamentalembedding theorems of Sobolev, Morrey, and John–Nirenberg.With the help of such results, one can show the smoothness of the so-called weak solutionsobtainedbythevariationalapproach.Wealsotreattheregularitytheoryof theso-calledstrongsolutions,aswellasSchauder’sregularitytheoryforsolutionsin viii Preface Ho¨lderspaces.Inthiscontext,wealsoexplainthecontinuitymethodthatconnects an equation that one wishes to study in a continuous manner with one that one understands already and deduces solvability of the former from solvability of the latterwiththehelpofaprioriestimates. ThefinalchapterdevelopstheMoseriterationtechnique,whichturnedouttobe fundamentalinthetheoryofellipticPDEs.Withthattechniqueonecanextendmany properties that are classically known for harmonic functions (Harnack inequality, localregularity,maximumprinciple)tosolutionsofalargeclassofgeneralelliptic PDEs.TheresultsofMoserwillalsoallowusto provethefundamentalregularity theoremofdeGiorgiandNashforminimizersofvariationalproblems. Attheendofeachchapter,webrieflysummarizethemainresults,occasionally suppressing the precise assumptions for the sake of saliency of the statements. I believe that this helps in guiding the reader through an area of mathematics that doesnotallowaunifiedstructuralapproach,butratherderivesitsfascinationfrom themultitudeanddiversityofapproachesandmethodsandconsequentlyencounters thedangerofgettinglostinthetechnicaldetails. Some words about the logical dependencebetween the various chapters: Most chapters are composed in such a manner that only the first sections are necessary forstudyingsubsequentchapters.Thefirst—ratherelementary—chapter,however, is basic for understanding almost all remaining chapters. Section 3.1 is useful, although not indispensable, for Chap.4. Sections 5.1 and 5.2 are important for Chaps.7and8.Chapter9,whichpartlyhassomesurveycharacter,connectsvarious previous chapters. Sections 10.1–10.4are fundamental for Chaps.11 and 14, and Sect.11.1willbeemployedinChaps.12and14.Withthoseexceptions,thevarious chapters can be read independently. Thus, it is also possible to vary the order in whichthechaptersarestudied.Forexample,itwouldmakesensetoreadChap.10 directlyafterChap.2,inordertoseethevariationalaspectsoftheLaplaceequation (inparticular,Sect.10.1)andalsothetransformationformulaforthisequationwith respecttochangesoftheindependentvariables.Inthiswayoneisnaturallyledtoa largerclassofellipticequations.Inanycase,itisusuallynotveryefficienttoread amathematicaltextbooklinearly,andthereadershouldrathertryfirsttograspthe centralstatements. This bookcan be utilized for a one-yearcourse on PDEs, and if time doesnot allow all the materialto be covered,one couldomitcertain sectionsandchapters, forexample,Sect.4.3andthefirstpartofSect.4.4andChap.12.Also,Chap.9will not be needed for the rest of the book. Of course, the lecturer may also decide to omitChap.14ifheorshewishestokeepthetreatmentatamoreelementarylevel. ThisbookisbasedonvariousgraduatecoursesthatIhavegivenatBochumand Leipzig.I thankAntje Vandenbergforgenerallogisticsupport,and ofcoursealso all the people who had helped me with the previous editions. They are listed in thepreviousprefaces,butIshouldrepeatmythankstoLutzHabermannandKnut Smoczykherefortheirhelpwiththefirstedition. Preface ix Concerning corrections for the present edition, I would like to thank Andreas Scha¨ferforaverydetailedandcarefullycompiledlistofcorrections.Also,Ithank Lei Ni for pointing out that the statement of Lemma5.3.2 needed a qualification. Finally,IthankmysonLeonardoJostforadiscussionthatleadstoanimprovement ofthepresentationinSect.11.3.IamalsogratefultoTimHealeyandhisstudents RobertKeslerandAaronPalmerforalertingmetoanerrorinSect.13.1. Leipzig,Germany Ju¨rgenJost

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