INDEXTHEORIES IN NONLINEAR ANALYSI
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广东广州
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作者刘春根
出版社科学出版社
ISBN9787030595669
出版时间2019-06
装帧其他
开本16开
定价168元
货号1201901982
上书时间2024-09-06
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目录
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 ABrief 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 Inequalicies
6 Maslov Type Index for Lagrangian Paths
6.1 Lagrangian Paths
6.2 Maslov Type Index for a Pair of Lagrangian Paths
6.3 Hormander 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 (ω,L0,L1)-Signature
7.4 The Mixed (,L0,L1)-Concavity
8 Applications of P-Index
8.1 The Existence of P-Solution of Nonlinear Hamiltonian 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
8.3 The Minimal Period Porblem for P-Symmetric Solutions
9 Applications of L-Index
9.1 The Existence of L-Solutions of Nonlinear Hamiltonian Systems
9.2 The Minimal Period Problem for Brake Solutions
9.3 Brake Subhrdrmonic Solutions of First Order Hamiltonian 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 Conseauences 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
内容摘要
在Maslov型指标理论的基础上，此书系统介绍近年来的指标理论一些新的发展。Maslov型指标理论适合于研究闭弦理论（周期解），近几年，开弦理论得到了很大的发展，此专著所介绍的指标理论适合于研究开弦理论。很典型的开弦有两种，其一是在辛流形中以拉格朗日子流形为边值的哈密顿系统，例如有名的闸轨道问题（Seifert猜测）。另一类是我们称之为哈密顿系统P边值问题，例如很多时滞微分方程（组）的周期解问题可以转化为这类问题，还很天体力学多体问题中由于具有对称性，很多对称解正是这类问题的解。在此书中我们介绍L-指标理论和P-指标理论，就是适合这两类开弦问题的研究，它是Maslov型指标理论的新的发展，有比较广泛的应用，例如哈密顿系统对称解的研究，很多时候需要应用这些指标理论来解决一些比较困难的问题，例如多重性问题，稳定性问题等等。同时我们也在此书中介绍一个方法，把各种指标理论很好地联系起来，这对更进一步的应用起很重要的作用。为了展示这些指标理论的应用，在这书中我们给出了多个方面的应用，这些例子涉及非线性哈密顿系统，时滞微分方程（组），偏微分方程（波方程）边值问题。尤其是波方程的指标理论的研究方法，鲜于在其它文献中出现，是相当新的研究成果。这书的内容，主要源于作者与其合作者近几年在指标理论方面研究工作，是这几年在这方面研究成果的一个系统化的总结。
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