Partial Differential Equations

416 Pages·2012·2.556 MB·English

by  Jürgen Jost

Graduate Texts in Mathematics 214
Graduate Texts in Mathematics
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SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
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AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
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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
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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
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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