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非线形泛函分析及其应用:线性单调算子(第2A卷)
自1932年,波兰数学家Banach发表第一部泛函分析专著“Theoriedesoperationslineaires”以来,这一学科取得了巨大的发展,它在其他领域的应用也是相当成功。如今,数学的很多领域没有了泛函分析恐怕寸步难行,不仅仅在数学方面,在理论物理方面的作用也具有同样的意义,M.Reed和B.Simon的“MethodsofModernMathematicalPhysjcs”在前言中指出:“自1926年以来,物理学的前沿已与日俱增集中于量子力学,以及奠定于量子理论的分支:原子物理、核物理固体物理、基本粒子物理等,而这些分支的中心数学框架就是泛函分析。”所以,讲述泛函分析的文献已浩如烟海。而每个时代,都有这个领域的代表性作品。
PrefacetoPartII/AINTRODUCTIONTOTHESUBJECTCHAPTER18VariationalProblems,theRitzMethod,andtheIdeaofOrthogonality18.1.TheSpaceC(G)andtheVariationalLemma18.2.IntegrationbyParts18.3.TheFirstBoundaryValueProblemandtheRitzMethod18.4.TheSecondandThirdBoundaryValueProblemsandtheRitzMethod18.5.EigenvalueProblemsandtheRitzMethod18.6.TheH61derInequalityanditsApplications18.7.TheHistoryoftheDirichletPrincipleandMonotoneOperators18.8.TheMainTheoremonQuadraticMinimumProblems18.9.TheInequalityofPoincar6-Friedrichs18.10.TheFunctionalAnalyticJustificationoftheDirichletPrinciple18.11.ThePerpendicularPrinciple,theRieszTheorem,andtheMainTheoremonLinearMonotoneOperators18.12.TheExtensionPrincipleandtheCompletionPrinciple18.13.ProperSubregions18.14.TheSmoothingPrinciple18.15.TheIdeaoftheRegularityofGeneralizedSolutionsandtheLemmaofWeyl18.16.TheLocalizationPrinciple18.17.ConvexVariationalProblems,EllipticDifferentialEquations,andMonotonicity18.18.TheGeneralEuler-LagrangeEquations18.19.TheHistoricalDevelopmentofthe19thand20thProblemsofHilbertandMonotoneOperators18.20.SufficientConditionsforLocalandGlobalMinimaandLocallyMonotoneOperatorsCHAPTER19TheGalerkinMethodforDifferentialandIntegralEquations,theFriedrichsExtension,andtheIdeaofSelf-Adjointness19.1.EllipticDifferentialEquationsandtheGalerkinMethod19.2.ParabolicDifferentialEquationsandtheGalerkinMethod19.3.HyperbolicDifferentialEquationsandtheGalerkinMethod19.4.IntegralEquationsandtheGalerkinMethod!9.5.CompleteOrthonormalSystemsandAbstractFourierSeries19.6.EigenvaluesofCompactSymmetricOperators(Hilbert-SchmidtTheory)19.7.ProofofTheorem19.B19.8.Self-AdjointOperators19.9.TheFriedrichsExtensionofSymmetricOperators19.10.ProofofTheorem19.C19.11.ApplicationtothePoissonEquation19.12.ApplicationtotheEigenvalueProblemfortheLaplaceEquation19.13.TheInequalityofPoincar6andtheCompactnessTheoremofRellich19.14.FunctionsofSelf-AdjointOperators19.15.ApplicationtotheHeatEquation19.16.ApplicationtotheWaveEquation19.17.SemigroupsandPropagators,andTheirPhysicalRelevance19.18.MainTheoremonAbstractLinearParabolicEquations!9.19.ProofofTheorem19.D!9.20.MonotoneOperatorsandtheMainTheoremonLinearNonexpansiveSemigroups19.21.TheMainTheoremonOne-ParameterUnitaryGroups19.22.ProofofTheorem19.E19.23.AbstractSemilinearHyperbolicEquations19.24.ApplicationtoSemilinearWaveEquations19.25.TheSemilinearSchr6dingerEquation19.26.AbstractSemilinearParabolicEquations,FractionalPowersofOperators,andAbstractSobolevSpaces19.27.ApplicationtoSemilinearParabolicEquations19.28.ProofofTheorem19.119.29.FiveGeneralUniquenessPrinciplesandMonotoneOperators19.30.AGeneralExistencePrincipleandLinearMonotoneOperatorsCHAPTER20DifferenceMethodsandStability20.1.Consistency,Stability,andConvergence20.2.ApproximationofDifferentialQuotients20.3.ApplicationtoBoundaryValueProblemsforOrdinaryDifferentialEquations20.4.ApplicationtoParabolicDifferentialEquations20.5.ApplicationtoEllipticDifferentialEquations20.6.TheEquivalenceBetweenStabilityandConvergence20.7.TheEquivalenceTheoremofLaxforEvolutionEquationsLINEARMONOTONEPROBLEMSCHAPTER21AuxiliaryToolsandtheConvergenceoftheGalerkinMethodforLinearOperatorEquations21.1.GeneralizedDerivatives21.2.SobolevSpaces21.3.TheSobolevEmbeddingTheorems21.4.ProofoftheSobolevEmbeddingTheorems21.5.DualityinB-Spaces21.6.DualityinH-Spaces21.7.TheIdeaofWeakConvergence21.8.TheIdeaofWeak*Convergence21.9.LinearOperators21.10.BilinearForms21.11.ApplicationtoEmbeddings21.12.ProjectionOperators21.13.BasesandGalerkinSchemes21.14.ApplicationtoFiniteElements21.15.Riesz-SchauderTheoryandAbstractFredholmAlternatives21.16.TheMainTheoremontheApproximation-SolvabilityofLinearOperatorEquations,andtheConvergenceoftheGalerkinMethod21.17.InterpolationInequalitiesandaConvergenceTrick21.18.ApplicationtotheRefinedBanachFixed-PointTheoremandtheConvergenceofIterationMethods21.19.TheGagliardo-NirenbergInequalities21.20.TheStrategyoftheFourierTransformforSobolevSpaces21.21.BanachAlgebrasandSobolevSpaces21.22.Moser-TypeCalculusInequalities21.23.WeaklySequentiallyContinuousNonlinearOperatorsonSobolevSpacesCHAPTER22HilbertSpaceMethodsandLinearEllipticDifferentialEquations22.1.MainTheoremonQuadraticMinimumProblemsandtheRitzMethod22.2.ApplicationtoBoundaryValueProblems22.3.TheMethodofOrthogonalProjection,Duality,andaposterioriErrorEstimatesfortheRitzMethod22.4.ApplicationtoBoundaryValueProblems22.5.MainTheoremonLinearStronglyMonotoneOperatorsandtheGalerkinMethod22.6.ApplicationtoBoundaryValueProblems22.7.CompactPerturbationsofStronglyMonotoneOperators,FredholmAlternatives,andtheGalerkinMethod22.8.ApplicationtoIntegralEquations22.9.ApplicationtoBilinearForms22.10.ApplicationtoBoundaryValueProblems22.11.EigenvalueProblemsandtheRitzMethod22.12.ApplicationtoBilinearForms22.13.ApplicationtoBoundary-EigenvalueProblems22.14.GarrdingForms22.15.TheGardingInequalityforEllipticEquations22.16.TheMainTheoremsonGardingForms22.17.ApplicationtoStronglyEllipticDifferentialEquationsofOrder2m22.18.DifferenceApproximations22.19.InteriorRegularityofGeneralizedSolutions22.20.ProofofTheorem22.H22.21.RegularityofGeneralizedSolutionsuptotheBoundary22.22.ProofofTheorem22.ICHAPTER23HilbertSpaceMethodsandLinearParabolicDifferentialEquations23.1.ParticularitiesintheTreatmentofParabolicEquations23.2.TheLebesgueSpaceLp(0,T;X)ofVector-ValuedFunctions23.3.TheDualSpacetoLp(O,T;X)23.4.EvolutionTriples23.5.GeneralizedDerivatives23.6.TheSobolevSpaceWp(0,T;V,H)23.7.MainTheoremonFirst-OrderLinearEvolutionEquationsandtheGalerkinMethod23.8.ApplicationtoParabolicDifferentialEquations23.9.ProofoftheMainTheoremCHAPTER24HilbertSpaceMethodsandLinearHyperbolicDifferentialEquations24.1.MainTheoremonSecond-OrderLinearEvolutionEquationsandtheGalerkinMethod24.2.ApplicationtoHyperbolicDifferentialEquations24.3.ProofoftheMainTheorem
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