Analytic Theory of Global Bifurcation: An Introduction (Princeton Series in Applied Mathematics Book 9)

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 Chapter 1. Introduction 1.1 Example: Bending an Elastic Rod I 1.2 Principle of Linearization 1.3 Global Theory 1.4 Layout PART 1. LINEAR AND NONLINEAR FUNCTIONAL ANALYSIS Chapter 2. Linear Functional Analysis 2.1 Preliminaries and Notation 2.2 Subspaces 2.3 Dual Spaces 2.4 Linear Operators 2.5 Neumann Series 2.6 Projections and Subspaces 2.7 Compact and Fredholm Operators 2.8 Notes on Sources Chapter 3. Calculus in Banach Spaces 3.1 Fréchet Differentiation 3.2 Higher Derivatives 3.3 Taylor’s Theorem 3.4 Gradient Operators 3.5 Inverse and Implicit Function Theorems 3.6 Perturbation of a Simple Eigenvalue 3.7 Notes on Sources Chapter 4. Multilinear and Analytic Operators 4.1 Bounded Multilinear Operators 4.2 Faà de Bruno Formula 4.3 Analytic Operators 4.4 Analytic Functions of Two Variables 4.5 Analytic Inverse and Implicit Function Theorems 4.6 Notes on Sources PART 2. ANALYTIC VARIETIES Chapter 5. Analytic Functions on Fn 5.1 Preliminaries 5.2 Weierstrass Division Theorem 5.3 Weierstrass Preparation Theorem 5.4 Riemann Extension Theorem 5.5 Notes on Sources Chapter 6. Polynomials 6.1 Constant Coefficients 6.2 Variable Coefficients 6.3 Notes on Sources Chapter 7. Analytic Varieties 7.1 F-Analytic Varieties 7.2 Weierstrass Analytic Varieties 7.3 Analytic Germs and Subspaces 7.4 Germs of C-analytic Varieties 7.5 One-dimensional Branches 7.6 Notes on Sources PART 3. BIFURCATION THEORY Chapter 8. Local Bifurcation Theory 8.1 A Necessary Condition 8.2 Lyapunov-Schmidt Reduction 8.3 Crandall-Rabinowitz Transversality 8.4 Bifurcation from a Simple Eigenvalue 8.5 Bending an Elastic Rod II 8.6 Bifurcation of Periodic Solutions 8.7 Notes on Sources Chapter 9. Global Bifurcation Theory 9.1 Global One-Dimensional Branches 9.2 Global Analytic Bifurcation in Cones 9.3 Bending an Elastic Rod III 9.4 Notes on Sources PART 4. STOKES WAVES Chapter 10. Steady Periodic Water Waves 10.1 Euler Equations 10.2 One-dimensional Formulation 10.3 Main Equation 10.4 A Priori Bounds and Nekrasov’s Equation 10.5 Weak Solutions Are Classical 10.6 Notes on Sources Chapter 11. Global Existence of Stokes Waves 11.1 Local Bifurcation Theory 11.2 Global Bifurcation from ? = 1 11.3 Gradients, Morse Index and Bifurcation 11.4 Notes on Sources Bibliography Index