Partial Differential Equations I

2011·4.91 MB·English

by  Michael E. Taylor

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
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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 Fre´chetspaces;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