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  • 非线形泛函分析及其应用,第2A卷,线性单调算子[平装]
  • 共1个商家     51.70元~51.70
  • 作者:宰德勒(作者)
  • 出版社:世界图书出版公司;第1版(2009年8月1日)
  • 出版时间:
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  • ISBN:9787510005206

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    《非线形泛函分析及其应用,第2A卷,线性单调算子》的写作起点很低,具备本科数学水平就可以读。

    作者简介

    作者:(德国)宰德勒

    目录

    Preface to Part II/A
    INTRODUCTION TO THE SUBJECT
    CHAPTER 18
    Variational Problems, the Ritz Method, and
    the Idea of Orthogonality
    18.1. The Space C(G) and the Variational Lemma
    18.2. Integration by Parts
    18.3. The First Boundary Value Problem and the Ritz Method
    18.4. The Second and Third Boundary Value Problems and
    the Ritz Method
    18.5. Eigenvalue Problems and the Ritz Method
    18.6. The H61der Inequality and its Applications
    18.7. The History of the Dirichlet Principle and Monotone Operators
    18.8. The Main Theorem on Quadratic Minimum Problems
    18.9. The Inequality of Poincar6-Friedrichs
    18.10. The Functional Analytic Justification of the Dirichlet Principle
    18.11. The Perpendicular Principle, the Riesz Theorem, and
    the Main Theorem on Linear Monotone Operators
    18.12. The Extension Principle and the Completion Principle
    18.13. Proper Subregions
    18.14. The Smoothing Principle
    18.15. The Idea of the Regularity of Generalized Solutions and
    the Lemma of Weyl
    18.16. The Localization Principle
    18.17. Convex Variational Problems, Elliptic Differential Equations,
    and Monotonicity
    18.18. The General Euler-Lagrange Equations
    18.19. The Historical Development of the 19th and 20th Problems of
    Hilbert and Monotone Operators
    18.20. Sufficient Conditions for Local and Global Minima and
    Locally Monotone Operators

    CHAPTER 19
    The Galerkin Method for Differential and Integral Equations,
    the Friedrichs Extension, and the Idea of Self-Adjointness
    19.1. Elliptic Differential Equations and the Galerkin Method
    19.2. Parabolic Differential Equations and the Galerkin Method
    19.3. Hyperbolic Differential Equations and the Galerkin Method
    19.4. Integral Equations and the Galerkin Method
    !9.5. Complete Orthonormal Systems and Abstract Fourier Series
    19.6. Eigenvalues of Compact Symmetric Operators
    (Hilbert-Schmidt Theory)
    19.7. Proof of Theorem 19.B
    19.8. Self-Adjoint Operators
    19.9. The Friedrichs Extension of Symmetric Operators
    19.10. Proof of Theorem 19.C
    19.11. Application to the Poisson Equation
    19.12. Application to the Eigenvalue Problem for the Laplace Equation
    19.13. The Inequality of Poincar6 and the Compactness
    Theorem of Rellich
    19.14. Functions of Self-Adjoint Operators
    19.15. Application to the Heat Equation
    19.16. Application to the Wave Equation
    19.17. Semigroups and Propagators, and Their Physical Relevance
    19.18. Main Theorem on Abstract Linear Parabolic Equations
    !9.19. Proof of Theorem 19.D
    !9.20. Monotone Operators and the Main Theorem on
    Linear Nonexpansive Semigroups
    19.21. The Main Theorem on One-Parameter Unitary Groups
    19.22. Proof of Theorem 19.E
    19.23. Abstract Semilinear Hyperbolic Equations
    19.24. Application to Semilinear Wave Equations
    19.25. The Semilinear Schr6dinger Equation
    19.26. Abstract Semilinear Parabolic Equations, Fractional Powers of
    Operators, and Abstract Sobolev Spaces
    19.27. Application to Semilinear Parabolic Equations
    19.28. Proof of Theorem 19.1
    19.29. Five General Uniqueness Principles and Monotone Operators
    19.30. A General Existence Principle and Linear Monotone Operators

    CHAPTER 20
    Difference Methods and Stability
    20.1. Consistency, Stability, and Convergence
    20.2. Approximation of Differential Quotients
    20.3. Application to Boundary Value Problems for
    Ordinary Differential Equations
    20.4. Application to Parabolic Differential Equations
    20.5. Application to Elliptic Differential Equations
    20.6. The Equivalence Between Stability and Convergence
    20.7. The Equivalence Theorem of Lax for Evolution Equations
    LINEAR MONOTONE PROBLEMS

    CHAPTER 21
    Auxiliary Tools and the Convergence of the Galerkin
    Method for Linear Operator Equations
    21.1. Generalized Derivatives
    21.2. Sobolev Spaces
    21.3. The Sobolev Embedding Theorems
    21.4. Proof of the Sobolev Embedding Theorems
    21.5. Duality in B-Spaces
    21.6. Duality in H-Spaces
    21.7. The Idea of Weak Convergence
    21.8. The Idea of Weak* Convergence
    21.9. Linear Operators
    21.10. Bilinear Forms
    21.11. Application to Embeddings
    21.12. Projection Operators
    21.13. Bases and Galerkin Schemes
    21.14. Application to Finite Elements
    21.15. Riesz-Schauder Theory and Abstract Fredholm Alternatives
    21.16. The Main Theorem on the Approximation-Solvability of Linear
    Operator Equations, and the Convergence of the Galerkin Method
    21.17. Interpolation Inequalities and a Convergence Trick
    21.18. Application to the Refined Banach Fixed-Point Theorem and
    the Convergence of Iteration Methods
    21.19. The Gagliardo-Nirenberg Inequalities
    21.20. The Strategy of the Fourier Transform for Sobolev Spaces
    21.21. Banach Algebras and Sobolev Spaces
    21.22. Moser-Type Calculus Inequalities
    21.23. Weakly Sequentially Continuous Nonlinear Operators on
    Sobolev Spaces

    CHAPTER 22
    Hilbert Space Methods and Linear Elliptic Differential Equations
    22.1. Main Theorem on Quadratic Minimum Problems and the
    Ritz Method
    22.2. Application to Boundary Value Problems
    22.3. The Method of Orthogonal Projection, Duality, and a posteriori
    Error Estimates for the Ritz Method
    22.4. Application to Boundary Value Problems
    22.5. Main Theorem on Linear Strongly Monotone Operators and
    the Galerkin Method
    22.6. Application to Boundary Value Problems
    22.7. Compact Perturbations of Strongly Monotone Operators,
    Fredholm Alternatives, and the Galerkin Method
    22.8. Application to Integral Equations
    22.9. Application to Bilinear Forms
    22.10. Application to Boundary Value Problems
    22.11. Eigenvalue Problems and the Ritz Method
    22.12. Application to Bilinear Forms
    22.13. Application to Boundary-Eigenvalue Problems
    22.14. Garrding Forms
    22.15. The Garding Inequality for Elliptic Equations
    22.16. The Main Theorems on Garding Forms
    22.17. Application to Strongly Elliptic Differential Equations of Order 2m
    22.18. Difference Approximations
    22.19. Interior Regularity of Generalized Solutions
    22.20. Proof of Theorem 22.H
    22.21. Regularity of Generalized Solutions up to the Boundary
    22.22. Proof of Theorem 22.I

    CHAPTER 23
    Hilbert Space Methods and Linear Parabolic Differential Equations
    23.1. Particularities in the Treatment of Parabolic Equations
    23.2. The Lebesgue Space Lp(0, T; X) of Vector-Valued Functions
    23.3. The Dual Space to Lp(O, T;X)
    23.4. Evolution Triples
    23.5. Generalized Derivatives
    23.6. The Sobolev Space Wp(0, T; V, H)
    23.7. Main Theorem on First-Order Linear Evolution Equations and
    the Galerkin Method
    23.8. Application to Parabolic Differential Equations
    23.9. Proof of the Main Theorem

    CHAPTER 24
    Hilbert Space Methods and Linear Hyperbolic
    Differential Equations
    24.1. Main Theorem on Second-Order Linear Evolution Equations
    and the Galerkin Method
    24.2. Application to Hyperbolic Differential Equations
    24.3. Proof of the Main Theorem

    序言

    自1932年,波兰数学家Banach发表第一部泛函分析专著“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。”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。例如上世纪50年代,F.Riesz和Sz.-Nagy的《泛函分析讲义》(中译版,科学出版社,1985),就是那个时代的一部具有代表性的著作;而60年代,N.Dunford和J.Schwartz的三大卷“Linear Operators”则是泛函分析学发展到那个时代的主要成果和应用的一个较全面的总结。泛函分析一经产生,它的发展就受到量子力学的强有力的推动,上世纪70年代,M.Reed和B.Simon的4卷“Methods 0f M0dern Mathematical Physics”是泛函分析对于量子力学应用的一个很好的总结。

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