This book provides a wide view of the calculus of variations as it plays an essential role in various areas of mathematics and science. Containing many examples, open problems, and exercises with complete solutions, the book would be suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods. The first part of the book is devoted to explaining the notion of (infinite-dimensional) manifolds and contains many examples. An introduction to Morse theory of Banach manifolds is provided, along with a proof of the existence of minimizing functions under the Palais-Smale condition. The second part, which may be read independently of the first, presents the theory of harmonic maps, with a careful calculation of the first and second variations of the energy. Several applications of the second variation and classification theories of harmonic maps are given.
Author(s): Hajime Urakawa
Series: Translations of Mathematical Monographs
Publisher: AMS
Year: 1993
Language: English
Pages: 271
Contents......Page 8
Preface to the English Edition......Page 10
Preface......Page 12
1. The aims of this book......Page 16
2. Methods of variations and field theories......Page 19
3. Examples of the method of variations......Page 22
4. A guide to the further study of the calculus of variations......Page 34
Exercises......Page 35
< Coffee Break > Classical mechanics......Page 36
1. Continuity, differentiation, and integration......Page 40
2. C^k-manifolds......Page 54
3. Finite-dimensional C^infty-manifolds......Page 66
4. Examples of manifolds......Page 77
Exercises......Page 94
1. Critical points of a smooth function......Page 98
2. Minimum values of smooth functions......Page 109
3. The condition (C)......Page 117
4. An application to closed geodesics......Page 130
< Coffee Break > The isoperimetric problem and Oueen Dido......Page 132
1. What is a harmonic mapping?......Page 136
2. An alternative expression for the first variation......Page 147
3. Examples of harmonic mappings......Page 155
Exercises......Page 164
< Coffee Break . Soap films and minimal surfaces (Plateau's problem)......Page 165
1. The second variation formula......Page 168
2. Instability theorems......Page 177
3. Stability of holomorphic mappings......Page 186
Exercises......Page 198
1. Existence, construction, and classification problems......Page 200
2. The case of the unit sphere......Page 204
3. The case of symmetric spaces......Page 229
4. Proof of the Eells-Sampson theorem via the variational method......Page 236
Solutions to Exercises......Page 242
References......Page 258
Subject Index......Page 264
