我们将线性泛函分析与非线性泛函分析结合，尤其强调将抽象的泛函分析中的部分重要定理与一些具体的来自物理、几何中的经典变分问题结合起来，使研究生既能体会到泛函分析的高度抽象性和统一性，又能体会到它应用的广泛性，尤其是通过变分最小即物理中的最小作用原理，可以领悟到泛函分析与大自然的和谐统一，内容包含十章及八个附录。十章内容是：来自物理、几何中的经典变分问题；线性泛函分析中的基本定理；广义函数与Sob olev空间；泛函极值的一阶和二阶条件；Ekeland变分原理及其应用；P ontryagin最大值原理及其应用；凸泛函的共轭理论及其应用；极小极大原理及其应用；多体问题的周期解；几个著名的不动点定理及其应用。

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1981.9-1985.6：重庆大学应用数学本科生 1985.9-1988.1：重庆大学应用数学硕士生 1988.9-1991.6：南开大学陈省身数学研究所博士生1988.2-1988.8：重庆大学应用数学助教 1991.6-1992.12：重庆大学应用数学讲师 1993.1-1997.7：重庆大学应用数学副教授 1997.7-2002.1：重庆大学应用数学教授 2002.2-2005.7：扬州大学数学教授，博导 2005.7- ：四川大学数学教授，博导2002年获教育部跨世纪人才基金，已主持5项国家自然科学基金项目，还参加了一项由北京大学张恭庆院士主持的国家自然科学基金重点项目，另主持了1项教育部优秀年轻教师基金和2项博士点基金。曾应邀在英国 Warwick 大学及美国 Princeton 大学、 Michigan 州立大学等近10 所国外大学及国内多所著名大学做过学术报告。

Contents
1 Several Classical Examples of Variational Calculus 1
1.1 Isoperimetric Problem and Brachistochrone Problem and Other Classical Extremal Value Problems 1
1.2 Definitions and Notations 9
Exercises 10
2 Introduction to Banach Space and Hilbert Space 11
2.1 Banach Spaces and Some Basic Properties 11
2.2 Hahn-Banach Extension Theorem and Separation Theorem 17
2.3 Hilbert Spaces, Riesz Representation Theorem, and Lax-Milgram Theorem 24
Recollections of Some Classic Theorems from Linear Functional Analysis. 29
3 Generalized Functions and Sobolev Spaces 33
3.1 Generalized Functions 33
3.2 Several Common Inequalities 46
3.3 Sobolev Spaces 52
Exercises and Questions 76
4 First-and Second-Order Conditions for Extrema of Functionals 79
4.1 Fréchet Derivative and Gateaux Derivative 79
4.2 Euler-Lagrange Equation 93
4.3 Generalization of the Weierstrass Theorem to Infinite Dimensions and the Dirichlet Principle 114
4.4 Second Variations: Necessary Conditions of Legendre and Jacobi 131
4.5 Sufficient Conditions for a Weak Extremum 143
Exercises and Questions 144
5 Ekeland’s Variational Principle and Its Applications 147
5.1 Ekeland’s Variational Principle 147
5.2 Generalization of Ekeland’s Variational Principle 151
5.3 Applications of the Ekeland’s Variational Principle 155
Exercises and Questions 161
6 Pontryagin’s Maximum Principle and Its Applications 163
6.1 Introduction 163
6.2 Pontryagin Maximum Principle 164
6.3 Application of Pontryagin’s Maximum Principle to Variational Problems 168
6.4 Ekeland’s Variational Principle Applied to Pontryagin’s Maximum Principle 171
Exercises and Questions 172
7 Conjugate Convex Functionals and Their Applications 173
7.1 A Brief Introduction to Fenchel-Moreau Convex Conjugate 173
7.2 Hamilton’s Duality and Clarke’s Duality 184
Exercises and Questions 190
8 Minimax Principle and Its Applications 191
8.1 Pseudo-Gradient Vector Fields and Deformation Lemmas 194
8.2 A General Minimax Theorem 203
8.3 Ambrosetti-Rabinowitz Mountain Pass Theorem and Some Variants Forms 205
8.4 Applications of the Mountain Pass Theorem to Elliptic Boundary Value Problems 211
Exercises and Questions 228
9 Periodic Solutions of Many-Body Problems 229
9.1 The Property of Least Action for Kepler’s Orbits 233
9.2 Euler and Lagrange Solutions of the Three-Body Problem and Their Variational Characterizations 241
9.2.1 Euler Collinear Central Configurations of the Three-Body Problem 242
9.2.2 The Lagrange Equilibrium Points in the Three-Body Problem 245
9.2.3 Homographic Solutions of the Three-Body Problem 246
9.2.4 Planar Circular Restricted Three-Body Problem 247
9.2.5 Palais’s Symmetry Principle 249
9.2.6 The Variational Minimizing Property of the Lagrange Solution to the Three-Body Problem 250
9.3 The “8” Shape Solution of the Planar Three-Body Problem 255
9.4 New Periodic Solutions of Planar Three-Body Problems 261
9.5 Non-planar and Non-collision Periodic Solutions of the N-Body Problem in Three-Dimensional Space 266
9.6 Introduction to Saari’s Conjecture 274
Exercises and Questions 276
10 Several Famous Fixed Point Theorems and Their Applications 277
10.1 Banach Contraction Mapping Principle and Its Applications 277
10.2 Brouwer’s Fixed Point Theorem, Ky Fan’s Inequality, and Nash Equilibrium 284
10.3 Schauder Fixed Point Theorem and Its Applications 301
10.4 Leray-Schauder Fixed Point Theorem 306
10.5 A Brief Introduction to the Poincaré-Birkhoff Fixed Point Theorem 309
Exercises and Questions 310
Appendix A: Zorn’s Lemma 313
Appendix B: Some Important Theorems in Lebesgue Measurable Functions and Their Integrals 317
Appendix C: Eberlein-Shmul’yan Theorem 331
Appendix D: Ascoli-Arzelà Theorem and Kolmogorov-Riesz-Fréchet Theorem 341
Appendix E: Eigenvalues and Eigenfunctions of the Laplace Operator 349
Appendix F: Regularity of Weak Solutions 363
Appendix G: Mountain Pass Lemma via Ekeland’s Principle 379
Appendix H: Minimal Period Solutions for Super-quadratic Convex Hamiltonian Systems 389
References 403