Gradient Inequalities : With Applications to Asymptotic Behavior And Stability of Gradient-like Systems梯度不等式
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作者： Sen-zhong Huang著
出 版 社：
出版时间： 2006-6-1字数：版次： 1页数： 184印刷时间： 2006/06/01开本： 24开印次： 1纸张： 胶版纸I S B N ： 9780821840702包装： 精装内容简介
This book presents a survey of the relatively new research field of gradient inequalities and their applications. The exposition emphasizes the powerful applications of gradient inequalities in studying asymptotic behavior and stability of gradient-like dynamical systems. It explains in-depth how gradient inequalities are established and how they can be used to prove convergence and stability of solutions to gradient-like systems. This book will serve as an introduction for further studies of gradient inequalities and their applications in other fields, such as geometry and computer sciences. This book is written for advanced graduate students, researchers and applied mathematicians interested in dynamical systems and mathematical modeling.
目录
Preface
CHAPTER 1 Introduction and overview of the results
The methodology
Convergence results for gradient-like trajectories
Applications to gradient-like systems in Hilbert spaces
Application to the stability problem
Additional remarks
CHAPTER 2 Gradient inequality
Basic properties of gradient maps
Gradient inequality
Finite-dimensional gradient inequality
Infinite-dimensional gradient inequality
Variational gradient inequality
Gradient inequality for monotone gradient maps
Optimal gradient inequality in Hilbert spaces
Remarks on gradient inequalities of type Gk
CHAPTER 3 Abstract convergence results
Gradient-like systems
Three technical lemmas
Convergence in gradient-like systems
Convergence in Hilbert spaces
Convergence in variational problems
CHAPTER 4 Applications to semilinear gradient-like systems in Hilbert spaces
The generic case
The perfect case
Results around the Laplacian operators
Ginzburg-Landau models for superconductivity
Convergence in porous medium models
CHAPTER 5 Applications to the stability problem
Stability of ground states
Convergence and stability of the steepest descent method
The structure of equilibria sets of convergent systems
Numeric test
Bibliography
Index
