Analytic Theory of Global Bifurcation
Boris Buffoni & John Toland

Preface ix
Chapter 1. Introduction 1
1.1 Example: Bending an Elastic Rod I 2
1.2 Principle of Linearization 5
1.3 Global Theory 6
1.4 Layout 7
PART 1. LINEAR AND NONLINEAR FUNCTIONAL ANALYSIS 9
Chapter 2. Linear Functional Analysis 11
2.1 Preliminaries and Notation 11
2.2 Subspaces 13
2.3 Dual Spaces 14
2.4 Linear Operators 15
2.5 Neumann Series 16
2.6 Projections and Subspaces 17
2.7 Compact and Fredholm Operators 18
2.8 Notes on Sources 20
Chapter 3. Calculus in Banach Spaces 21
3.1 Fréchet Differentiation 21
3.2 Higher Derivatives 27
3.3 Taylors Theorem 31
3.4 Gradient Operators 32
3.5 Inverse and Implicit Function Theorems 35
3.6 Perturbation of a Simple Eigenvalue 38
3.7 Notes on Sources 40
Chapter 4. Multilinear and Analytic Operators 41
4.1 Bounded Multilinear Operators 41
4.2 Faà deBruno Formula 44
4.3 Analytic Operators 45
4.4 Analytic Functions of Two Variables 52
4.5 Analytic Inverse and Implicit Function Theorems 53
4.6 Notes on Sources 57
PART 2. ANALYTIC VARIETIES 59
Chapter 5. Analytic Functions on Fⁿ 61
5.1 Preliminaries 61
5.2 Weierstrass Division Theorem 64
5.3 Weierstrass Preparation Theorem 65
5.4 Riemann Extension Theorem 66
5.5 Notes on Sources 69
Chapter 6. Polynomials 70
6.1 Constant Coefficients 70
6.2 Variable Coefficients 74
6.3 Notes on Sources 77
Chapter 7. Analytic Varieties 78
7.1 F -Analytic Varieties 78
7.2 Weierstrass Analytic Varieties 81
7.3 Analytic Germs and Subspaces 86
7.4 Germs of C -analytic Varieties 88
7.5 One-dimensional Branches 95
7.6 Notes on Sources 99
PART 3. BIFURCATION THEORY 101
Chapter 8. Local Bifurcation Theory 103
8.1 A Necessary Condition 103
8.2 Lyapunov-Schmidt Reduction 104
8.3 Crandall-Rabinowitz Transversality 105
8.4 Bifurcation from a Simple Eigenvalue 109
8.5 Bending an Elastic Rod II 111
8.6 Bifurcation of Periodic Solutions 112
8.7 Notes on Sources 113
Chapter 9. Global Bifurcation Theory 114
9.1 Global One-Dimensional Branches 114
9.2 Global Analytic Bifurcation in Cones 120
9.3 Bending an Elastic Rod III 121
9.4 Notes on Sources 124
PART 4. STOKES WAVES 125
Chapter 10. Steady Periodic Water Waves 127
10.1 Euler Equations 127
10.2 One-dimensional Formulation 131
10.3 Main Equation 137
10.4 A Priori Bounds and Nekrasovs Equation 140
10.5 Weak Solutions Are Classical 146
10.6 Notes on Sources 151
Chapter 11. Global Existence of Stokes Waves 152
11.1 Local Bifurcation Theory 152
11.2 Global Bifurcation from = 1 154
11.3 Gradients, Morse Index and Bifurcation 157
11.4 Notes on Sources 159
Bibliography 161
Index 167