非线性分析方法

非线性分析方法

张恭庆, 著

出版社:世界图书出版公司北京公司

年代:2014

定价:69.0

书籍简介:

本书从方法论的观点出发系统讲述了与非线性有关的所有最新材料。概述了基本理论和方法,研究了偏微分方程、常微分方程、微分几何和数学物理中的众多有趣的问题,为同时代的研究做足了准备。引用的方法都是精选自力学、物理、工程、几何中的例子。强烈建议这本书作为分析领域的入门书或者参考书。目次:线性化;不动点定理;度理论和应用;最小化方法;拓扑和变分法。读者对象:数学专业的高年级本科生,研究生和相关领域的科研人员。

书籍目录:

1 Linearization 1.1 Differential Calculus in Banach Spaces 1.1.1 Frechet Derivatives and Gateaux Derivatives 1.1.2 Nemytscki Operator 1.1.3 High-Order Derivatives 1.2 Implicit Function Theorem and Continuity Method 1.2.1 Inverse Function Theorem 1.2.2 Applications 1.2.3 Continuity Method 1.3 Lyapunov-Schmidt Reduction and Bifurcation 1.3.1 Bifurcation 1.3.2 Lyapunov-Schmidt Reduction 1.3.3 A Perturbation Problem 1.3.4 Gluing 1.3.5 Transversality 1.4 Hard Implicit Function Theorem 1.4.1 The Small Divisor Problem 1.4.2 Nash-Moser Iteration2 Fixed-Point Theorems 2.1 Order Method 2.2 Convex Function and Its Subdifferentials 2.2.1 Convex Functions 2.2.2 Subdifferentials 2.3 Convexity and Compactness 2.4 Nonexpansive Maps 2.5 Monotone Mappings 2.6 Maximal Monotone Mapping3 Degree Theory and Applications 3.1 The Notion of Topological Degree 3.2 Fundamental Properties and Calculations of Brouwer Degrees 3.3 Applications of Brouwer Degree 3.3.1 Brouwer Fixed-Point Theorem 3.3.2 The Borsuk-Ulam Theorem and Its Consequences 3.3.3 Degrees for S1 Equivariant Mappings 3.3.4 Intersection 3.4 Leray-Schauder Degrees 3.5 The Global Bifurcation 3.6 Applications 3.6.1 Degree Theory on Closed Convex Sets 3.6.2 Positive Solutions and the Scaling Method 3.6.3 Krein-Rutman Theory for Positive Linear Operators 3.6.4 Multiple Solutions 3.6.5 A Free Boundary Problem 3.6.6 Bridging 3.7 Extensions 3.7.1 Set-Valued Mappings 3.7.2 Strict Set Contraction Mappings and Condensing Mappings 3.7.3 Fredholm Mappings4 Minimization Methods 4.1 Variational Principles 4.1.1 Constraint Problems 4.1.2 Euler-Lagrange Equation 4.1.3 Dual Variational Principle 4.2 Direct Method 4.2.1 Fundamental Principle 4.2.2 Examples 4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement 4.3 Quasi-Convexity 4.3.1 Weak Continuity and Quasi-Convexity 4.3.2 Morrey Theorem 4.3.3 Nonlinear Elasticity 4.4 Relaxation and Young Measure 4.4.1 Relaxations 4.4.2 Young Measure 4.5 Other Function Spaces 4.5.1 BV Space 4.5.2 Hardy Space and BMO Space 4.5.3 Compensation Compactness 4.5.4 Applications to the Calculus of Variations 4.6 Free Discontinuous Problems 4.6.1 F-convergence 4.6.2 A Phase Transition Problem 4.6.3 Segmentation and Mumford-Shah Problem 4.7 Concentration Compactness 4.7.1 Concentration Function 4.7.2 The Critical Sobolev Exponent and the Best Constants 4.8 Minimax Methods 4.8.1 Ekeland Variational Principle 4.8.2 Minimax Principle 4.8.3 Applications5 Topological and Variational Methods 5.1 Morse Theory 5.1.1 Introduction 5.1.2 Deformation Theorem 5.1.3 Critical Groups 5.1.4 Global Theory 5.1.5 Applications 5.2 Minimax Principles (Revisited) 5.2.1 A Minimax Principle 5.2.2 Category and Ljusternik-Schnirelmann Multiplicity Theorem 5.2.3 Cap Product 5.2.4 Index Theorem 5.2.5 Applications 5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture 5.3.1 Hamiltonian Operator 5.3.2 Periodic Solutions 5.3.3 Weinstein Conjecture 5.4 Prescribing Gaussian Curvature Problem on S2 5.4.1 The Conformal Group and the Best Constant 5.4.2 The Palais-Smale Sequence 5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2 5.5 Conley Index Theory 5.5.1 Isolated Invariant Set 5.5.2 Index Pair and Conley Index 5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension NotesReferences

内容摘要:

张恭庆编著的《非线性分析方法》内容介绍: The book is the result of many years of revision of the author's lecture notes. Some of the more involved sections were originally used in seminars as introductory parts of some new subjects. However, due to their importance,the materials have been reorganized and supplemented, so that they may be more valuable to the readers.

书籍规格:

书籍详细信息
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9787510075933
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出版地北京出版单位世界图书出版公司北京公司
版次影印本印次1
定价(元)69.0语种英文
尺寸23 × 15装帧平装
页数印数

书籍信息归属:

非线性分析方法是世界图书出版公司北京公司于2014.2出版的中图分类号为 O177.91-34 的主题关于 非线性-分析方法-英文 的书籍。