解析函数空间上的算子理论和Landau-Lifshitz型方程

摘要

本文对解析函数空间上的算子理论和Landau-Lifshitz型方程进行了研究。文章描述了Toeplitz算子和复合算子理论的发展概貌，讨论了Dirichlet空间上某些Toeplitz算子的Fredholm性质，计算符号在C1中的Toeplitz算子的本质谱，而且对Dirichlet空间上复合算子的Fredholm性质和可逆性做了刻划。讨论了单位圆盘的Hardy-Orlicz空间上由解析自映射诱导的复合算子的有界性，可逆性，紧性和Fredholm性质，并且计算了这种算子的谱。文章同时对Landau-Lifshitz方程的提出和理论发展做了介绍，然后证明Landau-Lifshitz型方程(用()=√-△代替-▽((-△)-1/2▽·)))全空间问题弱解的存在性，并研究了Landau-Lifshitz型方程在周期边界条件情形解的存在性问题。

目录

文摘

英文文摘

CHAPTER1:Introduction

CHAPTER 2: Dirichlet space on bounded connected domain and its operators

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2.1.1 Toeplitz operators with symbols in C1(M)

2.1.2 An index formula of Toeplitz operators

2.2 Composition operators of Dirichlet spaces on connected domains

2.2.1 Invertibility of composition operators

2.2.2 Freholmness of Composition Operators

CHAPTER 3:Composition operators on Hardy-Orlicz spaces

3.1 Composition operators on Hardy-Orlicz spaces

3.1.1 Composition operators on spaces Hψ generated by log-convex ψ-functions

3.1.2 The case of spaces Hψ generated by convex ψ-functions

CHAPTER 4: The simplified Landau-Lifshitz equation in the whole space

4.1 The simplified Landau-Lifshitz equation in the whole space

4.1.1 The linear equation

4.1.2 The penalized Equation

4.1.3 The convergence process

CHAPTER 5: The Landau-Lifshitz type equation with periodic boundary condition

5.1 The Landau-Lifshitz type equation with periodic boundary condition

5.1.1 The simplified Landau-Lifshitz equation

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