非线性泛函分析及其应用 第3卷 《变分法及最优化》

图书标准信息

• 作者 [德]宰德勒 著
• 出版社 世界图书出版公司
• 出版时间 2009-08
• 版次 1
• ISBN 9787510005220
• 定价 79.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 662页
• 正文语种 英语

【内容简介】

　　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。”

【目录】

IntroductiontotheSubject
GeneralBasicIdeas
CHAPTER37
IntroductoryTypicalExamples
37.1.RealFunctionsinR
37.2.ConvexFunctionsinR
37.3.RealFunctionsinRN，LagrangeMultipliers，SaddlePoints，and
CriticalPoints
37.4.One-DimensionalClassicalVariationalProblemsandOrdinary
DifferentialEquations，LegendreTransformations，the
Hamilton-JaeobiDifferentialEquation，andtheClassical
MaximumPrinciple
37.5.MultidimensionalClassicalVariationalProblemsandElliptic
PartialDifferentialEquations
37.6.EigenvalueProblemsforEllipticDifferentialEquationsand
LagrangeMultipliers
37.7.DifferentialInequalitiesandVariationalInequalities
37.8.GameTheoryandSaddlePoints，NashEquilibriumPointsand
ParetoOptimization
37.9.DualitybetweentheMethodsofRitzandTrefftz，Two-Sided
ErrorEstimates
37.10.LinearOptimiTationinRN，Lag，rangeMultipliers，andDuality
37.11.ConvexOptimizationandKuhn-TuckerTheory
37.12.ApproximationTheory，theLeast-SquaresMethod，Deterministic
andStochasticCompensationAnalysis
37.13.ApproximationTheoryandControlProblems
37.14.Pseudoinverses，Ill-PosedProblemsandTihonovRegularization
37.15.ParameterIdentification
37.16.ChebyshevApproximationandRationalApproximation
37.17.LinearOptimizationinInfinite-DimehsionalSpaces，Chebyshev
Approximation，andApproximateSolutionsforPartial
DifferentialEquations
37.18.SplinesandFiniteElements
37.19.OptimalQuadratureFormulas
37.20.ControlProblems，DynamicOptimization，andtheBellman
OptimizationPrinciple
37.21.ControlProblems，thePontrjaginMaximumPrinciple，andthe
Bang-BangPrinciple
37.22.TheSynthesisProblemforOptimalControl
37.23.ElementaryProvableSpecialCaseofthePontrjaginMaximum
Principle
37.24.ControlwiththeAidofPartialDifferentialEquations
37.25.ExtremalProblemswithStochasticInfluences
37.26.TheCourantMaximum-MinimumPrinciple.Eigenvalues，
CriticalPoints，andtheBasicIdeasoftheLjusternik-Schnirelman
Theory
37.27.CriticalPointsandtheBasicIdeasoftheMorseTheory
37.28.SingularitiesandCatastropheTheory
37.29.BasicIdeasfortheConstructionofApproximateMethodsfor
ExtremalProblems
TWOFUNDAMENTALEXISTENCEANDUNIQUENESS
PRINCIPLES

CHAPIER38
CompactnessandExtremalPrinciples
38.1.WeakConvergenceandWeak*Convergence
38.2.SequentialLowerSemicontinuousandLowerSemicontinuous
Functionals
38.3.MainTheoremforExtremalProblems
38.4.StrictConvexityandUniqueness
38.5.VariantsoftheMainTheorem
38.6.ApplicationtoQuadraticVariationalProblems
38.7.ApplicationtoLinearOptimizationandtheRoleofExtreme
Points
38.8.QuasisolutionsofMinimumProblems
38.9.ApplicationtoaFixed-PointTheorem
38.10.ThePalais-SmaleConditionandaGeneralMinimumPrinciple
38.11.TheAbstractEntropyPrinciple

CHAPTER39
ConvexityandExtremalPrinciples
39.1.TheFundamentalPrincipleofGeometricFunctionalAnalysis
39.2.DualityandtheRoleofExtremePointsinLinearApproximation
Theory
39.3.InterpolationPropertyofSubspacesandUniqueness
39.4.AscentMethodandtheAbstractAlternationTheorem
39.5.ApplicationtoChebyshevApproximation
EXTREMALPROBLEMSWITHOUTSIDECONDITIONS

CHAPTER40
FreeLocalExtremaofDifferentiableFunctionalsandtheCalculus
ofVariations
40.1.nthVariations，G-Derivative，andF-Derivative
40.2.NecessaryandSufficientConditionsforFreeLocalExtrema
40.3.SufficientConditionsbyMeansofComparisonFunctionalsand
AbstractFieldTheory
40.4.ApplicationtoRealFunctionsinRN
40.5.ApplicationtoClassicalMultidimensionalVariationalProblems
inSpacesofContinuouslyDifferentiableFunctions
40.6.AccessoryQuadraticVariationalProblemsandSufficient
EigenvalueCriteriaforLocalExtrema
40.7.ApplicationtoNecessaryandSufficientConditionsforLocal
ExtremaforClassicalOne-DimensionalVariationalProblems

CHAPTER41
PotentialOperators
41.1.MinimalSequences
41.2.SolutionofOperatorEquationsbySolvingExtremalProblems
41.3.CriteriaforPotentialOperators
41.4.CriteriafortheWeakSequentialLowerSemicontinuityof
Functionals
41.5.ApplicationtoAbstractHammersteinEquationswithSymmetric
KernelOperators
41.6.ApplicationtoHammersteinIntegralEquations

CHAPTER42
FreeMinimaforConvexFunctionals，RitzMethodandthe
GradientMethod
42.1.ConvexFunctionalsandConvexSets
42.2.RealConvexFunctions
42.3.ConvexityofF，MonotonicityofF，andtheDefinitenessofthe
SecondVariation
42.4.MonotonePotentialOperators
42.5.FreeConvexMinimumProblemsandtheRitzMethod
42.6.FreeConvexMinimumProblemsandtheGradientMethod
42.7.ApplicationtoVariationalProblemsandQuasilinearElliptic
DifferentialEquationsinSobolevSpaces
EXTREMALPROBLEMSWITHSMOOTHSIDECONDITIONS

CHAPTER43
LagrangeMultipliersandEigenvalueProblems
43.1.TheAbstractBasicIdeaofLagrangeMultipliers
43.2.LocalExtremawithSideConditions
43.3.ExistenceofanEigenvectorViaaMinimumProblem
43.4.ExistenceofaBifurcationPointViaaMaximumProblem
43.5.TheGalerkinMethodforEigenvalueProblems
43.6.TheGeneralizedImplicitFunctionTheoremandManifoldsin
B-Spaces
43.7.ProofofTheorem43.C
43.8.LagrangeMultipliers
43.9.CriticalPointsandLagrangeMultipliers
43.10.ApplicationtoRealFunctionsinRN
43.11.ApplicationtoInformationTheory
43.12.ApplicationtoStatisticalPhysics.TemperatureasaLagrange
Multiplier
43.13.ApplicationtoVariationalProblemswithIntegralSideConditions
43.14.ApplicationtoVariationalProblemswithDifferentialEquations
asSideConditions

CHAPTER44
Ljustemik-SchnirelmanTheoryandtheExistenceof
SeveralEigenvectors
44.1.TheCourantMaximum-MinimumPrinciple
44.2.TheWeakandtheStrongLjustemikMaximum-Minimum
PrinciplefortheConstructionofCriticalPoints
44.3.TheGenusofSymmetricSets
44.4.ThePalais-SmaleCondition
44.5.TheMainTheoremforEigenvalueProblemsinInfinite-
DimensionalB-spaces
44.6.ATypicalExample
44.7.ProofoftheMainTheorem
……
CHAPTER45
CHAPTER46
CHAPTER47
CHAPTER48
CHAPTER49
CHAPTER50
CHAPTER51
CHAPTER52
CHAPTER53
CHAPTER54
CHAPTER55
CHAPTER56
CHAPTER57
Index