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Applied Mathematical Sciences Volume 115 Editors S.S.Antman J.E.Marsden DepartmentofMathematics ControlandDynamicalSystems, and 107-81 InstituteforPhysical CaliforniaInstituteofTechnology ScienceandTechnology Pasadena,CA91125 UniversityofMaryland USA CollegePark,MD20742-4015 [email protected] USA [email protected] L.Sirovich LaboratoryofAppliedMathematics DepartmentofBiomathematical Sciences MountSinaiSchoolofMedicine NewYork,NY10029-6574 [email protected] Advisors L.Greengard P.Holmes J.Keener J.Keller R.Laubenbacher B.J.Matkowsky A.Mielke C.S.Peskin K.R.Sreenivasan A.Stevens A.Stuart Forfurthervolumes: http://www.springer.com/series/34 Michael E. Taylor Partial Differential Equations I Basic Theory Second Edition ABC MichaelE.Taylor DepartmentofMathematics UniversityofNorthCarolina ChapelHill,NC27599 USA [email protected] ISSN0066-5452 ISBN978-1-4419-7054-1 e-ISBN978-1-4419-7055-8 DOI10.1007/978-1-4419-7055-8 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2010937758 Mathematics Subject Classification (2010): 35A01, 35A02, 35J05, 35J25, 35K05, 35L05, 35Q30, 35Q35,35S05 (cid:2)c SpringerScience+BusinessMedia,LLC1996,2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork, NY 10013, USA), except for brief excerpts inconnection with reviews orscholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To mywifeanddaughter,JaneHawkins andDianeTaylor Contents ContentsofVolumesIIandIII ................................................ xi Preface............................................................................ xiii 1 BasicTheoryofODEandVectorFields................................. 1 1 Thederivative .......................................................... 3 2 FundamentallocalexistencetheoremforODE....................... 9 3 Inversefunctionandimplicitfunctiontheorems...................... 12 4 Constant-coefficientlinearsystems;exponentiationofmatrices .... 16 5 Variable-coefficientlinearsystemsofODE:Duhamel’sprinciple... 26 6 Dependenceofsolutionsoninitialdataandonotherparameters.... 31 7 Flowsandvectorfields................................................. 35 8 Liebrackets............................................................. 40 9 Commutingflows;Frobenius’stheorem.............................. 43 10 Hamiltoniansystems................................................... 47 11 Geodesics............................................................... 51 12 Variationalproblemsandthestationaryactionprinciple............. 59 13 Differentialforms...................................................... 70 14 Thesymplecticformandcanonicaltransformations................. 83 15 First-order,scalar,nonlinearPDE..................................... 89 16 Completelyintegrablehamiltoniansystems.......................... 96 17 Examplesofintegrablesystems;centralforceproblems.............101 18 Relativisticmotion.....................................................105 19 Topologicalapplicationsofdifferentialforms........................110 20 Criticalpointsandindexofavectorfield.............................118 A Nonsmoothvectorfields...............................................122 References..............................................................125 2 TheLaplaceEquationandWaveEquation .............................127 1 Vibratingstringsandmembranes......................................129 2 Thedivergenceofavectorfield.......................................140 3 Thecovariantderivativeanddivergenceoftensorfields.............145 4 TheLaplaceoperatoronaRiemannianmanifold ....................153 5 Thewaveequationonaproductmanifoldandenergyconservation 156 6 Uniquenessandfinitepropagationspeed .............................162 7 Lorentzmanifoldsandstress-energytensors .........................166 8 Moregeneralhyperbolicequations;energyestimates................172 viii Contents 9 Thesymbolofadifferentialoperatorandageneral Green–Stokesformula.................................................176 10 TheHodgeLaplacianonk-forms.....................................180 11 Maxwell’sequations...................................................184 References..............................................................194 3 FourierAnalysis,Distributions, andConstant-CoefficientLinearPDE ...................................197 1 Fourierseries...........................................................198 2 Harmonicfunctionsandholomorphicfunctionsintheplane........209 3 TheFouriertransform..................................................222 4 Distributionsandtempereddistributions..............................230 5 Theclassicalevolutionequations .....................................244 6 Radialdistributions,polarcoordinates,andBesselfunctions........263 7 ThemethodofimagesandPoisson’ssummationformula...........273 8 Homogeneousdistributionsandprincipalvaluedistributions .......278 9 Ellipticoperators.......................................................286 10 Localsolvabilityofconstant-coefficientPDE ........................289 11 ThediscreteFouriertransform ........................................292 12 ThefastFouriertransform.............................................301 A ThemightyGaussianandthesublimegammafunction..............306 References..............................................................312 4 SobolevSpaces..............................................................315 1 SobolevspacesonRn..................................................315 2 Thecomplexinterpolationmethod....................................321 3 Sobolevspacesoncompactmanifolds................................328 4 Sobolevspacesonboundeddomains .................................331 5 TheSobolevspacesHs.(cid:2)/ ...........................................338 0 6 TheSchwartzkerneltheorem..........................................345 7 Sobolevspacesonroughdomains.....................................349 References..............................................................351 5 LinearEllipticEquations..................................................353 1 ExistenceandregularityofsolutionstotheDirichletproblem ......354 2 Theweakandstrongmaximumprinciples............................364 3 TheDirichletproblemontheballinRn ..............................373 4 TheRiemannmappingtheorem(smoothboundary).................379 5 TheDirichletproblemonadomainwitharoughboundary.........383 6 TheRiemannmappingtheorem(roughboundary)...................398 7 TheNeumannboundaryproblem .....................................402 8 TheHodgedecompositionandharmonicforms......................410 9 NaturalboundaryproblemsfortheHodgeLaplacian................421 10 Isothermalcoordinatesandconformalstructuresonsurfaces .......438 11 Generalellipticboundaryproblems...................................441 12 Operatorpropertiesofregularboundaryproblems...................462 Contents ix A Spacesofgeneralizedfunctionsonmanifoldswithboundary.......471 B TheMayer–VietorissequenceindeRhamcohomology..............475 References..............................................................478 6 LinearEvolutionEquations...............................................481 1 Theheatequationandthewaveequationonboundeddomains.....482 2 Theheatequationandwaveequationonunboundeddomains ......490 3 Maxwell’sequations...................................................496 4 TheCauchy–Kowalewskytheorem ...................................499 5 Hyperbolicsystems ....................................................504 6 Geometricaloptics.....................................................510 7 Theformationofcaustics..............................................518 8 Boundarylayerphenomenafortheheatsemigroup..................535 A SomeBanachspacesofharmonicfunctions..........................541 B Thestationaryphasemethod ..........................................543 References..............................................................545 A OutlineofFunctionalAnalysis............................................549 1 Banachspaces..........................................................549 2 Hilbertspaces ..........................................................556 3 Frećhetspaces;locallyconvexspaces.................................561 4 Duality..................................................................564 5 Linearoperators........................................................571 6 Compactoperators.....................................................579 7 Fredholmoperators ....................................................593 8 Unboundedoperators ..................................................596 9 Semigroups.............................................................603 References..............................................................615 B Manifolds,VectorBundles,andLieGroups.............................617 1 Metricspacesandtopologicalspaces.................................617 2 Manifolds...............................................................622 3 Vectorbundles..........................................................624 4 Sard’stheorem..........................................................626 5 Liegroups ..............................................................627 6 TheCampbell–Hausdorffformula ....................................630 7 RepresentationsofLiegroupsandLiealgebras......................632 8 RepresentationsofcompactLiegroups...............................636 9 RepresentationsofSU(2)andrelatedgroups.........................641 References..............................................................647 Index..............................................................................649

Description:

The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations.The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.