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Problem Books in Mathematics Edited by P. Winkler For other titles published in this series, go to www.springer.com/series/714 Alexander Komech • Andrew Komech Principles of Partial Differential Equations Alexander Komech Andrew Komech Faculty of Mathematics Department of Mathematics Vienna University Texas A&M University 1090 Vienna College Station, TX 77843 Austria USA [email protected] [email protected] Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA [email protected] ISBN 978-1-4419-1095-0 e-ISBN 978-1-4419-1096-7 DOI 10.2007/978-1-4419-1096-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009932894 Mathematics Subject Classification (2000): 32-XX, 35-00, 35-01 ©SpringerScience + BusinessMedia,LLC2009 Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Cover art: Olga Rozmakhova, [email protected] Printedonacid-freepaper Springer is part of Springer Science+Business Media (www.springer.com) Preface ThisbookisintendedtogivethereaderanopportunitytomastersolvingPDEprob- lems.Ourmaingoalwastohaveaconcisetextthatwouldcovertheclassicaltools of PDE theory that are used in today’s science and engineering, such as charac- teristics, the wave propagation,the Fourier method,distributions, Sobolevspaces, fundamentalsolutions, and Green’s functions. While introductoryFourier method – based PDE books do not give an adequate description of these areas, the more advancedPDEbooksarequitetheoreticalandrequireahighlevelofmathematical backgroundfromareader.Thisbookwaswrittenspecificallytofillthisgap,satis- fyingthe demandofthewiderangeofenduserswhoneedthe knowledgeofhow tosolvethePDEproblemsandatthesametimearenotgoingtospecializeinthis areaofmathematics.Arguably,thisistheshortestPDEcourse,whichstretchesfar beyondcommon,Fouriermethod–basedPDEtexts.Forexample,[Hab03],which is a commonthoroughtextbookonpartialdifferentialequations,teachesa similar setoftoolswhilebeingaboutfivetimeslonger. Thebookisproblem-oriented.Thetheoreticalpartisrigorousyetshort.Some- timeswereferthereadertotextbooksthatgivewidercoverageofthetheory.Yet,im- portanttheoreticaldetailsarepresentedwithcare,whilethehintsgivethereaderan opportunitytorestoretheargumentstothefullrigor.Manyexamplesfromphysics are intended to keep the book intuitive for the reader and to illustrate the applied natureofthesubject. Thebookwillbeusefulforanyhigher-levelundergraduatecourseandforself- study for both graduateand higher-levelundergraduatestudents, and for any spe- cialtyinsciences.ItsRussianversionhasbeenastandardproblem-solvingmanual atMoscowStateUniversitysince1988,andisalsousedbystudentsofSt.Peters- burgUniversityandNovosibirskUniversities.ItsSpanishversionisusedatMorelia UniversityinMexico,whiletheEnglishdrafthasalreadybeenusedinViennaUni- versityandatTexasA&MUniversity. Forfurtherreadingwerecommend[Str92],[Eva98],and[EKS99]. Mu¨nchen, AlexanderKomech August2007 AndrewKomech v Acknowledgements The first authoris indebtedto MargaritaKorotkinaforthe fortunatesuggestionto write this book, to A.F. Filippov, A.S. Kalashnikov, M.A. Shubin, T.D. Ventzel, andM.I.Vishikforcheckingthefirstversionofthemanuscriptandfortheadvice. Both authors are grateful to H. Spohn (Technische Universita¨t, Mu¨nchen) and to E.Zeidler(Max-PlanckInstituteforMathematics,Leipzig)fortheirhospitalityand supportduringtheworkonthebook. BothauthorsweresupportedbyInstituteforInformationTransmissionProblems (RussianAcademyofSciences).ThefirstauthorwassupportedbytheDepartment ofMechanicsandMathematicsofMoscowStateUniversity,bytheAlexandervon HumboldtResearchAward,FWFGrantP19138-N13,andtheGrantsofRFBR.The secondauthorwassupportedbyTexasA&MUniversityandbytheNationalScience FoundationunderGrantsDMS-0621257andDMS-0600863. vii Contents 1 Hyperbolicequations.Methodofcharacteristics................... 1 1 Derivationofthed’Alembertequation ......................... 1 2 Thed’Alembertmethodforinfinitestring ...................... 7 3 Analysisofthed’Alembertformula ........................... 12 4 Second-orderhyperbolicequationsintheplane ................. 19 5 Semi-infinitestring ......................................... 30 6 Finitestring ............................................... 44 7 Waveequationwithmanyindependentvariables ................ 46 8 Generalhyperbolicequations................................. 56 2 TheFouriermethod ............................................ 65 9 Derivationoftheheatequation ............................... 65 10 Mixedproblemfortheheatequation .......................... 67 11 TheSturm–Liouvilleproblem ............................... 68 12 Eigenfunctionexpansions.................................... 74 13 TheFouriermethodfortheheatequation....................... 78 14 Mixedproblemforthed’Alembertequation .................... 83 15 TheFouriermethodfornonhomogeneousequations ............. 86 16 TheFouriermethodfornonhomogeneousboundaryconditions .... 93 17 TheFouriermethodfortheLaplaceequation ................... 95 3 DistributionsandGreen’sfunctions ..............................105 18 Motivation ................................................105 19 Distributions ..............................................109 20 Operationsondistributions ..................................110 21 Differentiationofjumpsandtheproductrule ...................115 22 Fundamentalsolutionsofordinarydifferentialequations..........118 23 Green’sfunctiononaninterval ...............................121 24 Solvabilityconditionfortheboundaryvalueproblems............125 25 TheSobolevfunctionalspaces................................128 26 Well-posednessofthewaveequationintheSobolevspaces .......130 ix x Contents 27 Solutionstothewaveequationinthesenseofdistributions........131 4 FundamentalsolutionsandGreen’sfunctionsinhigherdimensions..133 28 FundamentalsolutionsoftheLaplaceoperatorinRn .............133 29 Potentialsandtheirproperties ................................137 30 ComputingpotentialsviatheGausstheorem....................143 31 Methodofreflections .......................................144 32 Green’sfunctionsin2Dviaconformalmappings ................149 A Classificationofthesecond-orderequations.......................155 References.........................................................159 Index .............................................................161 Chapter 1 Hyperbolic equations. Method of characteristics 1 Derivationofthe d’Alembert equation Thed’Alembertequation,alsocalledtheone-dimensionalwaveequation, ∂2u ∂2u (x,t)=a2 +f(x,t), x [0,l], t>0, (1.1) ∂t2 ∂x2 ∈ where a>0 is a constant, describes small transversal oscillations of a stretched stringorlongitudinaloscillationsofanelasticrod. Letusgiveabriefderivationofthisequation.Foramorerigorousargument,see [Vla84,SD64,TS90]. Transversaloscillationsofastring We assume that a string of length l is u stretchedwiththeforceT.Wechoosethe direction of the axis Ox along the string initsequilibriumconfiguration.Letx=0 u(x,t) correspond to the left end of the string. Then the right end of the string is given 0 x l x byx=l.SeeFig.1.1.Wechoosetheaxis Ou normal to Ox, and only consider the transversaloscillationsofthestring,such Fig.1.1 thateachpointxmovesonlyinthedirec- tion of the axis Ou. We denote by u(x,t) thedisplacementofthepointxofthestringatamomentt. Alexander Komech and Andrew Komech, Principles of Partial Differential Equations, 1 Problem Books in Mathematics, DOI 10.2007/978-1-4419-1096-7_1, © Springer Science + Business Media, LLC 2009 2 1 Hyperbolicequations.Methodofcharacteristics We assume that the angles between the string and the axis Ox are small (see Fig.1.2): α, β 1. | | | |≪ Fr u α β F l 0 x x+∆x x Fig.1.2 Let us prove that u(x,t) satisfies equation (1.1). To do so, we write Newton’s SecondLawforapieceofthestringfromxtox+∆x,andtakeitsprojectiononto theaxis0u: a m=F . (1.2) u u Herea ∂2u(x,t);m=µ ∆x,whereµisthelineardensityofthestring,thatis, u≈ ∂t2 · themassofitsunitlength(weassumethatthestringisuniform),and F (F) +(F) +f˜(x,t)∆x. (1.3) u l u r u ≈ ByF andF wedenotedtheforcewhichactsontheregion[x,x+∆x]fromtheleft l r andtherightpartofthestring,and(F) ,(F) standfortheirprojectionsontothe l u r u axisOu. f˜(x,t)isthedensityofthetransversalexternalforces.Forexample,inthe gravitationalfieldoftheEarth,ifthestringishorizontalandtheaxis0uisdirected upward,then f˜(x,t)= gµ,whereg 9.8m/s2. − ≈ Substitutinga ,mandF into(1.2),weobtain u u ∂2u µ∆x (F) +(F) +f˜(x,t)∆x. (1.4) ∂t2 ≈ l u r u Further, for an elastic string the force of tension T at each point is tangent to the stringandhasthesamemagnitude(see[Vla84]).Then (F) = Tsinβ, (F) =Tsinα (1.5) l u r u − and(1.4)takestheform ∂2u µ∆x Tsinβ+Tsinα+f˜(x,t)∆x. (1.6) ∂t2 ≈− Since we consider the “small” oscillations, such that α and β 1, with the | | | |≪ precisionuptohigherpowersofαandβ,

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