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Full Description
Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue.
Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.
Contents
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 Frechet Differentiation 21 3.2 Higher Derivatives 27 3.3 Taylor's 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 Faa 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 Nekrasov's 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
