非线性泛函分析及其应用（第3卷）变分法及最优化
目 录内容简介
Introduction to the Subject
General Basic Ideas
CHAPTER 37
Introductory Typical Examples
37.1. Real Functions in R
37.2. Convex Functions in R
37.3. Real Functions in RN， Lagrange Multipliers， Saddle Points， and
Critical Points
37.4. One-Dimensional Classical Var…
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General Basic Ideas
CHAPTER 37
Introductory Typical Examples
37.1. Real Functions in R
37.2. Convex Functions in R
37.3. Real Functions in RN， Lagrange Multipliers， Saddle Points， and
Critical Points
37.4. One-Dimensional Classical Var…
查看完整
目 录内容简介
自1932年，波兰数学家Banach发表一部泛函分析专著“Theorie des operations lineaires”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“Methods of Modern MathematicalPhysjcs”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 录内容简介
Introduction to the Subject
General Basic Ideas
CHAPTER 37
Introductory Typical Examples
37.1. Real Functions in R
37.2. Convex Functions in R
37.3. Real Functions in RN， Lagrange Multipliers， Saddle Points， and
Critical Points
37.4. One-Dimensional Classical Variational Problems and Ordinary
Differential Equations， Legendre Transformations， the
Hamilton-Jaeobi Differential Equation， and the Classical
Maximum Principle
37.5. Multidimensional Classical Variational Problems and Elliptic
Partial Differential Equations
37.6. Eigenvalue Problems for Elliptic Differential Equations and
Lagrange Multipliers
37.7. Differential Inequalities and Variational Inequalities
37.8. Game Theory and Saddle Points， Nash Equilibrium Points and
Pareto Optimization
37.9. Duality between the Methods of Ritz and Trefftz， Two-Sided
Error Estimates
37.10. Linear OptimiTation in R N， Lag， range Multipliers， and Duality
37.11. Convex Optimization and Kuhn-Tucker Theory
37.12. Approximation Theory， the Least-Squares Method， Deterministic
and Stochastic Compensation Analysis
37.13. Approximation Theory and Control Problems
37.14. Pseudoinverses， Ill-Posed Problems and Tihonov Regularization
37.15. Parameter Identification
37.16. Chebyshev Approximation and Rational Approximation
37.17. Linear Optimization in Infinite-Dimehsional Spaces， Chebyshev
Approximation， and Approximate Solutions for Partial
Differential Equations
37.18. Splines and Finite Elements
37.19. Optimal Quadrature Formulas
37.20. Control Problems， Dynamic Optimization， and the Bellman
Optimization Principle
37.21. Control Problems， the Pontrjagin Maximum Principle， and the
Bang-Bang Principle
37.22. The Synthesis Problem for Optimal Control
37.23. Elementary Provable Special Case of the Pontrjagin Maximum
Principle
37.24. Control with the Aid of Partial Differential Equations
37.25. Extremal Problems with Stochastic Influences
37.26. The Courant Maximum-Minimum Principle. Eigenvalues，
Critical Points， and the Basic Ideas of the Ljusternik-Schnirelman
Theory
37.27. Critical Points and the Basic Ideas of the Morse Theory
37.28. Singularities and Catastrophe Theory
37.29. Basic Ideas for the Construction of Approximate Methods for
Extremal Problems
TWO FUNDAMENTAL EXISTENCE AND UNIQUENESS
PRINCIPLES
CHAPIER 38
Compactness and Extremal Principles
38.1. Weak Convergence and Weak* Convergence
38.2. Sequential Lower Semicontinuous and Lower Semicontinuous
Functionals
38.3. Main Theorem for Extremal Problems
38.4. Strict Convexity and Uniqueness
38.5. Variants of the Main Theorem
38.6. Application to Quadratic Variational Problems
38.7. Application to Linear Optimization and the Role of Extreme
Points
38.8. Quasisolutions of Minimum Problems
38.9. Application to a Fixed-Point Theorem
38.10. The Palais-Smale Condition and a General Minimum Principle
38.11. The Abstract Entropy Principle
CHAPTER 39
Convexity and Extremal Principles
39.1. The Fundamental Principle of Geometric Functional Analysis
39.2. Duality and the Role of Extreme Points in Linear Approximation
Theory
39.3. Interpolation Property of Subspaces and Uniqueness
39.4. Ascent Method and the Abstract Alternation Theorem
39.5. Application to Chebyshev Approximation
EXTREMAL PROBLEMS WITHOUT SIDE CONDITIONS
CHAPTER 40
Free Local Extrema of Differentiable Functionals and the Calculus
of Variations
40.1. n th Variations， G-Derivative， and F-Derivative
40.2. Necessary and Sufficient Conditions for Free Local Extrema
40.3. Sufficient Conditions by Means of Comparison Functionals and
Abstract Field Theory
40.4. Application to Real Functions in RN
40.5. Application to Classical Multidimensional Variational Problems
in Spaces of Continuously Differentiable Functions
40.6. Accessory Quadratic Variational Problems and Sufficient
Eigenvalue Criteria for Local Extrema
40.7. Application to Necessary and Sufficient Conditions for Local
Extrema for Classical One-Dimensional Variational Problems
CHAPTER 41
Potential Operators
41.1. Minimal Sequences
41.2. Solution of Operator Equations by Solving Extremal Problems
41.3. Criteria for Potential Operators
41.4. Criteria for the Weak Sequential Lower Semicontinuity of
Functionals
41.5. Application to Abstract Hammerstein Equations with Symmetric
Kernel Operators
41.6. Application to Hammerstein Integral Equations
CHAPTER 42
Free Minima for Convex Functionals， Ritz Method and the
Gradient Method
42.1. Convex Functionals and Convex Sets
42.2. Real Convex Functions
42.3. Convexity of F， Monotonicity of F， and the Definiteness of the
Second Variation
42.4. Monotone Potential Operators
42.5. Free Convex Minimum Problems and the Ritz Method
42.6. Free Convex Minimum Problems and the Gradient Method
42.7. Application to Variational Problems and Quasilinear Elliptic
Differential Equations in Sobolev Spaces
EXTREMAL PROBLEMS WITH SMOOTH SIDE CONDITIONS
CHAPTER 43
Lagrange Multipliers and Eigenvalue Problems
43.1. The Abstract Basic Idea of Lagrange Multipliers
43.2. Local Extrema with Side Conditions
43.3. Existence of an Eigenvector Via a Minimum Problem
43.4. Existence of a Bifurcation Point Via a Maximum Problem
43.5. The Galerkin Method for Eigenvalue Problems
43.6. The Generalized Implicit Function Theorem and Manifolds in
B-Spaces
43.7. Proof of Theorem 43.C
43.8. Lagrange Multipliers
43.9. Critical Points and Lagrange Multipliers
43.10. Application to Real Functions in RN
43.11. Application to Information Theory
43.12. Application to Statistical Physics. Temperature as a Lagrange
Multiplier
43.13. Application to Variational Problems with Integral Side Conditions
43.14. Application to Variational Problems with Differential Equations
as Side Conditions
CHAPTER 44
Ljustemik-Schnirelman Theory and the Existence of
Several Eigenvectors
44.1. The Courant Maximum-Minimum Principle
44.2. The Weak and the Strong Ljustemik Maximum-Minimum
Principle for the Construction of Critical Points
44.3. The Genus of Symmetric Sets
44.4. The Palais-Smale Condition
44.5. The Main Theorem for Eigenvalue Problems in Infinite-
Dimensional B-spaces
44.6. A Typical Example
44.7. Proof of the Main Theorem
……
CHAPTER 45
CHAPTER 46
CHAPTER 47
CHAPTER 48
CHAPTER 49
CHAPTER 50
CHAPTER 51
CHAPTER 52
CHAPTER 53
CHAPTER 54
CHAPTER 55
CHAPTER 56
CHAPTER 57
Index
^ 收 起
General Basic Ideas
CHAPTER 37
Introductory Typical Examples
37.1. Real Functions in R
37.2. Convex Functions in R
37.3. Real Functions in RN， Lagrange Multipliers， Saddle Points， and
Critical Points
37.4. One-Dimensional Classical Variational Problems and Ordinary
Differential Equations， Legendre Transformations， the
Hamilton-Jaeobi Differential Equation， and the Classical
Maximum Principle
37.5. Multidimensional Classical Variational Problems and Elliptic
Partial Differential Equations
37.6. Eigenvalue Problems for Elliptic Differential Equations and
Lagrange Multipliers
37.7. Differential Inequalities and Variational Inequalities
37.8. Game Theory and Saddle Points， Nash Equilibrium Points and
Pareto Optimization
37.9. Duality between the Methods of Ritz and Trefftz， Two-Sided
Error Estimates
37.10. Linear OptimiTation in R N， Lag， range Multipliers， and Duality
37.11. Convex Optimization and Kuhn-Tucker Theory
37.12. Approximation Theory， the Least-Squares Method， Deterministic
and Stochastic Compensation Analysis
37.13. Approximation Theory and Control Problems
37.14. Pseudoinverses， Ill-Posed Problems and Tihonov Regularization
37.15. Parameter Identification
37.16. Chebyshev Approximation and Rational Approximation
37.17. Linear Optimization in Infinite-Dimehsional Spaces， Chebyshev
Approximation， and Approximate Solutions for Partial
Differential Equations
37.18. Splines and Finite Elements
37.19. Optimal Quadrature Formulas
37.20. Control Problems， Dynamic Optimization， and the Bellman
Optimization Principle
37.21. Control Problems， the Pontrjagin Maximum Principle， and the
Bang-Bang Principle
37.22. The Synthesis Problem for Optimal Control
37.23. Elementary Provable Special Case of the Pontrjagin Maximum
Principle
37.24. Control with the Aid of Partial Differential Equations
37.25. Extremal Problems with Stochastic Influences
37.26. The Courant Maximum-Minimum Principle. Eigenvalues，
Critical Points， and the Basic Ideas of the Ljusternik-Schnirelman
Theory
37.27. Critical Points and the Basic Ideas of the Morse Theory
37.28. Singularities and Catastrophe Theory
37.29. Basic Ideas for the Construction of Approximate Methods for
Extremal Problems
TWO FUNDAMENTAL EXISTENCE AND UNIQUENESS
PRINCIPLES
CHAPIER 38
Compactness and Extremal Principles
38.1. Weak Convergence and Weak* Convergence
38.2. Sequential Lower Semicontinuous and Lower Semicontinuous
Functionals
38.3. Main Theorem for Extremal Problems
38.4. Strict Convexity and Uniqueness
38.5. Variants of the Main Theorem
38.6. Application to Quadratic Variational Problems
38.7. Application to Linear Optimization and the Role of Extreme
Points
38.8. Quasisolutions of Minimum Problems
38.9. Application to a Fixed-Point Theorem
38.10. The Palais-Smale Condition and a General Minimum Principle
38.11. The Abstract Entropy Principle
CHAPTER 39
Convexity and Extremal Principles
39.1. The Fundamental Principle of Geometric Functional Analysis
39.2. Duality and the Role of Extreme Points in Linear Approximation
Theory
39.3. Interpolation Property of Subspaces and Uniqueness
39.4. Ascent Method and the Abstract Alternation Theorem
39.5. Application to Chebyshev Approximation
EXTREMAL PROBLEMS WITHOUT SIDE CONDITIONS
CHAPTER 40
Free Local Extrema of Differentiable Functionals and the Calculus
of Variations
40.1. n th Variations， G-Derivative， and F-Derivative
40.2. Necessary and Sufficient Conditions for Free Local Extrema
40.3. Sufficient Conditions by Means of Comparison Functionals and
Abstract Field Theory
40.4. Application to Real Functions in RN
40.5. Application to Classical Multidimensional Variational Problems
in Spaces of Continuously Differentiable Functions
40.6. Accessory Quadratic Variational Problems and Sufficient
Eigenvalue Criteria for Local Extrema
40.7. Application to Necessary and Sufficient Conditions for Local
Extrema for Classical One-Dimensional Variational Problems
CHAPTER 41
Potential Operators
41.1. Minimal Sequences
41.2. Solution of Operator Equations by Solving Extremal Problems
41.3. Criteria for Potential Operators
41.4. Criteria for the Weak Sequential Lower Semicontinuity of
Functionals
41.5. Application to Abstract Hammerstein Equations with Symmetric
Kernel Operators
41.6. Application to Hammerstein Integral Equations
CHAPTER 42
Free Minima for Convex Functionals， Ritz Method and the
Gradient Method
42.1. Convex Functionals and Convex Sets
42.2. Real Convex Functions
42.3. Convexity of F， Monotonicity of F， and the Definiteness of the
Second Variation
42.4. Monotone Potential Operators
42.5. Free Convex Minimum Problems and the Ritz Method
42.6. Free Convex Minimum Problems and the Gradient Method
42.7. Application to Variational Problems and Quasilinear Elliptic
Differential Equations in Sobolev Spaces
EXTREMAL PROBLEMS WITH SMOOTH SIDE CONDITIONS
CHAPTER 43
Lagrange Multipliers and Eigenvalue Problems
43.1. The Abstract Basic Idea of Lagrange Multipliers
43.2. Local Extrema with Side Conditions
43.3. Existence of an Eigenvector Via a Minimum Problem
43.4. Existence of a Bifurcation Point Via a Maximum Problem
43.5. The Galerkin Method for Eigenvalue Problems
43.6. The Generalized Implicit Function Theorem and Manifolds in
B-Spaces
43.7. Proof of Theorem 43.C
43.8. Lagrange Multipliers
43.9. Critical Points and Lagrange Multipliers
43.10. Application to Real Functions in RN
43.11. Application to Information Theory
43.12. Application to Statistical Physics. Temperature as a Lagrange
Multiplier
43.13. Application to Variational Problems with Integral Side Conditions
43.14. Application to Variational Problems with Differential Equations
as Side Conditions
CHAPTER 44
Ljustemik-Schnirelman Theory and the Existence of
Several Eigenvectors
44.1. The Courant Maximum-Minimum Principle
44.2. The Weak and the Strong Ljustemik Maximum-Minimum
Principle for the Construction of Critical Points
44.3. The Genus of Symmetric Sets
44.4. The Palais-Smale Condition
44.5. The Main Theorem for Eigenvalue Problems in Infinite-
Dimensional B-spaces
44.6. A Typical Example
44.7. Proof of the Main Theorem
……
CHAPTER 45
CHAPTER 46
CHAPTER 47
CHAPTER 48
CHAPTER 49
CHAPTER 50
CHAPTER 51
CHAPTER 52
CHAPTER 53
CHAPTER 54
CHAPTER 55
CHAPTER 56
CHAPTER 57
Index
^ 收 起
目 录内容简介
自1932年，波兰数学家Banach发表一部泛函分析专著“Theorie des operations lineaires”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“Methods of Modern MathematicalPhysjcs”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。”
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