片断：
Introduction
Partialdifferentialequationsisamany-facetedsubject.Createdtodescribethe
mechanicalbehaviorofobjectssuchasvibratingstringsandblowingwinds,it
hasdevelopedintoabodyofmaterialthatinteractswithmanybranchesofmath-
ematics,suchasdifferentialgeometry,complexanalysis,andhannonicanalysis,
aswellasaubiquitousfactorinthedescriptionandelucidationofproblemsin
mathemaricalphysics.
Thisworkisintendedtoprovideacourseofstudyofsomeofthemajoraspects
ofPDE.Itisaddressedtoreaderswithabackgroundinthebasicintroductorygrad-
uatemathematicscoursesinAmericanuniversities:elementaryrealandcomplex
analysis,differentialgeometry,andmeasuretheory.
Chapter1providesbackgroundmaterialonthetheoryofordinarydifferential
equations(ODE).Thisincludesbothverybasicmaterial-ontopicssuchasthe
existenceanduniquenessofsolutionstoODEandexplicitsolutionstoequations
withconstantcoefficientsandrelationstolinearalgebra-andmoresophisticated
results--onflowsgeneratedbyvectorfields,connectionswithdifferentialgeom-
etry,thecalculusofdifferentialforms,stationaryactionprinciplesinmechanics,
andtheirrelationtoHamiltoniansystems.Wediscussequationsofrelativistic
motionaswellasequationsofclassicalNewtonianmechanics.Therearealso
applicationstotopologicalresults,suchasdegreetheory,theBrouwerfixed-point
theorem,andtheJordan-Brouwerseparationtheorem.Inthischapterwealsotreat
scalarfirst-orderPDE,viaHamilton-Jacobitheory.
Chapters2through6constituteasurveyofbasiclinearPDE.Chapter2begins
withthederivationofsomeequationsofcontinuummechanicsinafashionsimilar
tothederivationofODEinmechanicsinChapter1,viavariationalprinciples.We
obtainequationsforvibratingstringsandmembranes;theseequationsarenot
necessarilylinear,andhencetheywillalsoprovidesourcesofproblemslater,
whennonlinearPDEistakenup.FurthermaterialinChapter2centersaroundthe
Laplaceoperator,whichonEuclideanspaceR"is
andthelinearwaveequation,
WealsoconsidertheLaplaceoperatoronageneralRiemannianmanifoldand
thewaveequationonageneralLorentzmanifold.Wediscussbasicconsequences
ofGreen'sformula,includingenergyconservatiohandfinitepropagationspeed
forsolutionstolinearwaveequations.WealsodiscussMaxwell'sequationsfor
electromagneticfieldsandtheirrelationwithspecialrelativity.Beforewecan
establishgeneralresultsonthesolvabilityoftheseequations,itisnecessaryto
developsomeanalyticaltechniques.Thisisdoneinthenextcoupleofchapters.
Chapter3isdevotedtoFourieranalysisandthetheoryofdistributions.These
topicsarecrucialforthestudyoflinearPDE.Wegiveanumberofbasicapplica-
tionstothestudyoflinearPDEwithconstantcoefficients.Amongtheseapplica-
tionsareresultsonhannonicandholomorphicfunctionsintheplane,includinga
shorttreatmentofelementarycomplexfunctiontheory.Wederiveexplicitformu-
lasforsolutionstoLaplaceandwaveequationsonEuclideanspace,andalsothe
heatequation,
Wealsoproducesolutionsoncertainsubsets,suchasrectangularregions,usingthe
methodofimages.WeincludematerialonthediscreteFouriertransfonn,gemane
tothediscreteapproximationofPDE,andonthefastevaluationofthistransform,
theFFT.Chapter3isthefirstchaptertomakeextensiveuseoffunctionalanalysis.
BasicresultsonthistopicarecompiledinAppendixA,OutlineofFunctional
Analysis.
Sobolevspaceshaveproventobeaveryeffectivetoolintheexistencetheory
ofPDE,andinthestudyofregularityofsolutions.InChapter4weintroduce
Sobolevspacesandstudysomeoftheirbasicproperties.Werestrictattention
toL-Sobolevspaces,suchasH(R),whichconsistsofL-functionswhose
derivativesoforderk(definedinadistributionalsense,inChapter3)belongto
L2(R),whenkisapositiveinteger.Wealsoreplacekbyageneralrealnumber
s.TheL-Sobolevspaces,whichareveryusefiilfornonlinearPDE,aretreated
later,inChapter13.
Chapter5isdevotedtothestudyoftheexistenceandregularityofsolutionsto
linearellipticPDE,onboundedregions.WebeginwiththeDirichletproblemfor
theLaplaceoperator,
andthentreattheNeumannproblemandvariousotherboundaryproblems,in-
cludingsomethatapplytoelectromagneticfields.Wealsostudygeneralboundary
problemsforlinearellipticoperators,givingaconditionthatguaranteesregularity
andsolvability(perhapsgivenafinitenumberoflinearconditionsonthedata).
AlsoinChapter5aresomeapplicationstootherareas,suchasaproofoftheRie-
mannmappingtheorem,firstforsmoothsimplyconnecteddomainsinthecomplex
planeC,then,afteratreatmentoftheDirichletproblemfortheLaplaceoperator
ondomainswithroughboundary,forgeneralsimplyconnecteddomainsinC.We
alsodevelopHodgetheoryandapplyittoDeRhamcohomology,extendingthe
studyoftopologicalapplicationsofdifferentialfonnsbeguninChapter1.
|
内容加载中，请稍后...
商品评论（0条）