Nonlinear Functional Analysis and its Applications: III: Variational Methods and Optimization

✍ Scribed by E. Zeidler

• Publisher: Springer
• Year: 1984
• Tongue: English
• Leaves: 685
• Edition: 1985
• Category: Library

✦ Synopsis

As long as a branch of knowledge offers an abundance of problems, it is full of vitality. David Hilbert Over the last 15 years I have given lectures on a variety of problems in nonlinear functional analysis and its applications. In doing this, I have recommended to my students a number of excellent monographs devoted to specialized topics, but there was no complete survey-type exposition of nonlinear functional analysis making available a quick survey to the wide range of readers including mathematicians, natural scientists, and engineers who have only an elementary knowledge of linear functional analysis. I have tried to close this gap with my five-part lecture notes, the first three parts of which have been published in the Teubner-Texte series by Teubner-Verlag, Leipzig, 1976, 1977, and 1978. The present English edition was translated from a completely rewritten manuscript which is significantly longer than the original version in the Teubner-Texte series. The material is organized in the following way: Part I: Fixed Point Theorems. Part II: Monotone Operators. Part III: Variational Methods and Optimization. Parts IV jV: Applications to Mathematical Physics. The exposition is guided by the following considerations: (a) What are the supporting basic ideas and what intrinsic interrelations exist between them? (/3) In what relation do the basic ideas stand to the known propositions of classical analysis and linear functional analysis? ( y) What typical applications are there? Vll Preface viii Special emphasis is placed on motivation.

✦ Table of Contents

Front cover
Leonhard Euler (1707-1783)
Title page
Date-line
Dedication
Preface
Contents
Introduction to the Subject
General Basic Ideas
CHAPTER 37. Introductory Typical Examples
§37.1. Real Functions in $\mathbb{R}^1$
§37.2. Convex Functions in $\mathbb{R}^1$
§37.3. Real Functions in $\mathbb{R}^N$, Lagrange Multipliers, Saddle Points, and Critical Points
§37.4. One-Dimensional Classical Variational Problems and Ordinary Differential Equations, Legendre Transformations, the Hamilton-Jacobi 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 Optimization in $\mathbb{R}^N$, Lagrange 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-Dimensional 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
CHAPTER 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 DifTerentiable 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 $\mathbb{R}^N$
§40.5. Application to Classical Multidimensional Variational Problems in Spaces of Continuously DifTerentiable 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 $\mathbb{R}^N$
§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. Ljusternik-Schnirelman Theory and the Existence of Several Eigenvectors
§44.1. The Courant Maximum-Minimum Principle
§44.2. The Weak and the Strong Ljusternik 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
§44.8. The Main Theorem for Eigenvalue Problems in Finite- Dimensional B-Spaces
§44.9. Application to Eigenvalue Problems for Quasilinear Elliptic Differential Equations
§44.10. Application to Eigenvalue Problems for Abstract Hammerstein Equations with Symmetric Kernel Operators
§44.11. Application to Hammerstein Integral Equations
§44.12. The Mountain Pass Theorem
CHAPTER 45. Bifurcation for Potential Operators
§45.1. Krasnoselskii's Theorem
§45.2. The Main Theorem
§45.3. Proof of the Main Theorem
EXTREMAL PROBLEMS WITH GENERAL SIDE CONDITIONS
CHAPTER 46. Differentiable Functionals on Convex Sets
§46.1. Variational Inequalities as Necessary and Sufficient Extremal Conditions
§46.2. Quadratic Variational Problems on Convex Sets and Variational Inequalities
§46.3. Application to Partial Differential Inequalities
§46.4. Projections on Convex Sets
§46.5. The Ritz Method
§46.6. The Projected Gradient Method
§46.7. The Penalty Functional Method
§46.8. Regularization of Linear Problems
§46.9. Regularization of Nonlinear Problems
CHAPTER 47. Convex Functionals on Convex Sets and Convex Analysis
§47.1. The Epigraph
§47.2. Continuity of Convex Functionals
§47.3. Subgradient and Subdifferential
§47.4. Subgradient and the Extremal Principle
§47.5. Subgradient and the G-Derivative
§47.6. Existence Theorem for Subgradients
§47.7. The Sum Rule
§47.8. The Main Theorem of Convex Optimization
§47.9. The Main Theorem of Convex Approximation Theory
§47.10. Generalized Kuhn-Tucker Theory
§47.11. Maximal Monotonicity, Cyclic Monotonicity, and Subgradients
§47.12. Application to the Duality Mapping
CHAPTER 48. General Lagrange Multipliers (Dubovickii-Miljutin Theory)
§48.1. Cone and Dual Cone
§48.2. The Dubovickii-Miljutin Lemma
§48.3. The Main Theorem on Necessary and Sufficient Extremal Conditions for General Side Conditions
§48.4. Application to Minimum Problems with Side Conditions in the Form of Equalities and Inequalities
§48.5. Proof of Theorem 48.B
§48.6. Application to Control Problems (Pontrjagin's Maximum Principle)
§48.7. Proof of the Pontrjagin Maximum Principle
§48.8. The Maximum Principle and Classical Calculus of Variations
§48.9. Modifications of the Maximum Principle
§48.10. Return of a Spaceship to Earth
SADDLE POINTS AND DUALITY
CHAPTER 49. General Duality Principle by Means of Lagrange Functions and Their Saddle Points
§49.1. Existence of Saddle Points
§49.2. Main Theorem of Duality Theory
§49.3. Application to Linear Optimization Problems in B-Spaces
CHAPTER 50. Duality and the Generalized Kuhn-Tucker Theory
§50.1. Side Conditions in Operator Form
§50.2. Side Conditions in the Form of Inequalities
CHAPTER 51. Duality, Conjugate Functionals, Monotone Operators and Elliptic Differential Equations
§51.1. Conjugate Functionals
§51.2. Functionals Conjugate to Differentiable Convex Functionals
§51.3. Properties of Conjugate Functionals
§51.4. Conjugate Functionals and the Lagrange Function
§51.5. Monotone Potential Operators and Duality
§51.6. Applications to Linear Elliptic Differential Equations, Trefftz's Duality
§51.7. Application to Quasilinear Elliptic Differential Equations
CHAPTER 52. General Duality Principle by Means of Perturbed Problems and Conjugate Functionals
§52.1. The $S$-Functional, Stability, and Duality
§52.2. Proof of Theorem 52.A
§52.3. Duality Propositions of Fenchel-Rockafellar Type
§52.4. Application to Linear Optimization Problems in Locally Convex Spaces
§52.5. The Bellman Differential Inequality and Duality for Nonconvex Control Problems
§52.6. Application to a Generalized Problem of Geometrical Optics
CHAPTER 53. Conjugate Functionals and Orlicz Spaces
§53.1. Young Functions
§53.2. Orlicz Spaces and Their Properties
§53.3. Linear Integral Operators in Orlicz Spaces
§53.4. The Nemyckii Operator in Orlicz Spaces
§53.5. Application to Hammerstein Integral Equations with Strong Nonlinearities
§53.6. Sobolev-Orlicz Spaces
VARIATIONAL INEQUALITIES
CHAPTER 54. Elliptic Variational Inequalities
§54.1. The Main Theorem
§54.2. Application to Coercive Quadratic Variational Inequalities
§54.3. Semicoercive Variational Inequalities
§54.4. Variational Inequalities and Control Problems
§54.5. Application to Bilinear Forms
§54.6. Application to Control Problems with Elliptic Differential Equations
§54.7. Semigroups and Control of Evolution Equations
§54.8. Application to the Synthesis Problem for Linear Regulators
§54.9. Application to Control Problems with Parabolic Differential Equations
CHAPTER 55. Evolution Variational Inequalities of First Order in H-Spaces
§55.1. The Resolvent of Maximal Monotone Operators
§55.2. The Nonlinear Yosida Approximation
§55.3. The Main Theorem for Inhomogeneous Problems
§55.4. Application to Quadratic Evolution Variational Inequalities of First Order
CHAPTER 56. Evolution Variational Inequalities of Second Order in H-Spaces
§56.1. The Main Theorem
§56.2. Application to Quadratic Evolution Variational Inequalities of Second Order
CHAPTER 57. Accretive Operators and Multivalued First-Order Evolution Equations in B-Spaces
§57.1. Generalized Inner Products on B-Spaces
§57.2. Accretive Operators
§57.3. The Main Theorem for Inhomogeneous Problems with $m$-Accretive Operators
§57.4. Proof of the Main Theorem
§57.5. Application to Nonexpansive Semigroups in B-Spaces
§57.6. Application to Partial Differential Equations
Appendix
References
List of Symbols
List of Theorems
List of the Most Important Definitions
Index
Back cover