Principles of Partial Differential Equations

164 Pages·3.263 MB·English

by  Komech Al., Komech An.

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
[e-mail] [e-mail]
Series Editor:
Peter Winkler
Department of Mathematics
Dartmouth College
Hanover, NH 03755
USA
[e-mail]
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, [e-mail]
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,