Index theory in nonlinear analysis

✍ Scribed by Chungen Liu

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 346
• Category: Library

✦ Synopsis

This book provides detailed information on index theories and their applications, especially Maslov-type index theories and their iteration theories for non-periodic solutions of Hamiltonian systems. It focuses on two index theories: L-index theory (index theory for Lagrangian boundary conditions) and P-index theory (index theory for P-boundary conditions). In addition, the book introduces readers to recent advances in the study of index theories for symmetric periodic solutions of nonlinear Hamiltonian systems, and for selected boundary value problems involving partial differential equations.

✦ Table of Contents

Foreword
Preface
Contents
1 Linear Algebraic Aspects
1.1 Linear Symplectic Spaces
1.2 Symplectic Matrices
1.3 Lagrangian Subspaces
1.4 Linear Hamiltonian Systems
1.5 Eigenvalues of Symplectic Matrices
2 A Brief Introduction to Index Functions
2.1 Maslov Type Index i1(γ)
2.2 ω-Index iω(γ)
3 Relative Morse Index
3.1 Relative Index via Galerkin Approximation Sequences
3.2 Relative Morse Index via Orthogonal Projections
3.3 Morse Index via Dual Methods
3.3.1 The Definition of Index Pair in Case 1 and 2
3.3.2 The Definition of Index Pair in Case 3
3.4 Saddle Point Reduction for the General Cases
4 The P-Index Theory
4.1 P-Index Theory
4.2 Relative Index via Saddle Point Reduction Method
4.3 Galerkin Approximation for the (P, ω)-Boundary Problem of Hamiltonian Systems
4.4 (P, ω)-Index Theory from Analytical Point of View
4.5 Bott-Type Formula for the Maslov Type P-Index
4.6 Iteration Theory for P-Index
4.6.1 Splitting Numbers
4.6.2 Abstract Precise Iteration Formulas
4.6.3 Iteration Inequalities
5 The L-Index Theory
5.1 Definition of L-Index
5.1.1 The Properties of the L-Indices
5.1.2 The Relations of iL(γ) and i1(γ)
5.1.3 L-Index for General Symplectic Paths
5.2 The (L,L')-Index Theory
5.3 Understanding the Index iP(γ) in View of the Lagrangian Index iLL'(γ)
5.4 The Relation with the Morse Index in Calculus Variations
5.5 Saddle Point Reduction Formulas
5.6 Galerkin Approximation Formulas for L-Index
5.7 Dual L-Index Theory for Linear Hamiltonian Systems
5.8 The (L,ω)-Index Theory
5.9 The Bott Formulas of L-Index
5.10 Iteration Inequalities of L-Index
5.10.1 Precise Iteration Index Formula
5.10.2 Iteration Inequalities
6 Maslov Type Index for Lagrangian Paths
6.1 Lagrangian Paths
6.2 Maslov Type Index for a Pair of Lagrangian Paths
6.3 Hörmander Index Theory
7 Revisit of Maslov Type Index for Symplectic Paths
7.1 Maslov Type Index for Symplectic Paths
7.2 The ω-Index Function for P-Index
7.3 The Concavity of Symplectic Paths and (, L0, L1)-Signature
7.4 The Mixed (L0,L1)-Concavity
8 Applications of P-Index
8.1 The Existence of P-Solution of NonlinearHamiltonian Systems
8.2 The Existence of Periodic Solutions for Delay Differential Equations
8.2.1 M-Boundary Problem of a Hamiltonian System
8.2.2 Delay Differential Systems
8.2.3 Poisson Structure
8.2.4 First Order Delay Hamiltonian Systems
8.2.5 Second Order Delay Hamiltonian Systems
8.2.6 Background and Related Works
8.2.7 Main Results
8.2.7.1 Asymptotically Linear Delay Differential Systems
8.2.7.2 First Order Delay Hamiltonian Systems
8.2.7.3 Second Order Delay Hamiltonian Systems
8.3 The Minimal Period Problem for P-Symmetric Solutions
9 Applications of L-Index
9.1 The Existence of L-Solutions of NonlinearHamiltonian Systems
9.2 The Minimal Period Problem for Brake Solutions
9.3 Brake Subharmonic Solutions of First OrderHamiltonian Systems
10 Multiplicity of Brake Orbits on a Fixed Energy Surface
10.1 Brake Orbits of Nonlinear Hamiltonian Systems
10.1.1 Seifert Conjecture
10.1.2 Some Related Results Since 1948
10.1.3 Some Consequences of Theorem 1.2 and Further Arguments
10.2 Proofs of Theorems 1.2 and 1.9
11 The Existence and Multiplicity of Solutions of Wave Equations
11.1 Variational Setting and Critical Point Theories
11.1.1 Critical Point Theorems in Case 1 and Case 2
11.1.2 Critical Point Theorems in Case 3
11.2 Applications: The Existence and Multiplicity of Solutions for Wave Equations
11.2.1 One Dimensional Wave Equations
11.2.2 n-Dimensional Wave Equations
Bibliography
Index