非线性泛函分析.pdf

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1、NonlinearFunctionalAnalysisGeraldTeschlGeraldTeschlFakult•atf•urMathematikNordbergstrae15Universit•atWien1090Wien,AustriaE-mail:Gerald.Teschl@univie.ac.atURL:http://www.mat.univie.ac.at/~gerald/1991Mathematicssubjectclassication.46-01,47H10,47H11,58Fxx,76D05Abstract.Thismanuscriptpr

2、ovidesabriefintroductiontononlinearfunc-tionalanalysis.WestartoutwithcalculusinBanachspaces,reviewdierentiationandintegration,derivetheimplicitfunctiontheorem(usingtheuniformcon-tractionprinciple)andapplytheresulttoproveexistenceanduniquenessofsolutionsforordinarydierentialequations

3、inBanachspaces.Nextweintroducethemappingdegreeinbothnite(Brouwerdegree)andinnitedimensional(Leray-Schauderdegree)Banachspaces.Severalapplicationstogametheory,integralequations,andordinarydierentialequationsarediscussed.Asanapplicationweconsiderpartialdierentialequationsandproveexi

4、stenceanduniquenessforsolutionsofthestationaryNavier-Stokesequa-tion.Finally,wegiveabriefdiscussionofmonotoneoperators.Keywordsandphrases.Mappingdegree,xed-pointtheorems,dierentialequations,Navier{Stokesequation.TypesetbyLATEXandMakeindex.Version:May19,2010Copyrightc1998-2010byGeral

5、dTeschlContentsPrefaceiiiChapter1.AnalysisinBanachspaces1x1.1.DierentiationandintegrationinBanachspaces1x1.2.Contractionprinciples5x1.3.Ordinarydierentialequations7Chapter2.TheBrouwermappingdegree11x2.1.Introduction11x2.2.Denitionofthemappingdegreeandthedeterminantformula13x2.3.Ext

6、ensionofthedeterminantformula17x2.4.TheBrouwerxed-pointtheorem23x2.5.Kakutani'sxed-pointtheoremandapplicationstogametheory25x2.6.Furtherpropertiesofthedegree28x2.7.TheJordancurvetheorem30Chapter3.TheLeray{Schaudermappingdegree31x3.1.ThemappingdegreeonnitedimensionalBanachspaces31x3

7、.2.Compactoperators32x3.3.TheLeray{Schaudermappingdegree33x3.4.TheLeray{SchauderprincipleandtheSchauderxed-pointtheorem35x3.5.Applicationstointegralanddierentialequations36iiiContentsChapter4.ThestationaryNavier{Stokesequation39x4.1.Introductionandmotivation39x4.2.AninsertonSobolevs

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