Mathematical Sciences: Critical Point Theory and Differential Equations

数学科学：临界点理论和微分方程

• 批准号: 9202044
• 负责人: Abbas Bahri
• 金额: $ 10.7万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-15 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9202044&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Critical; Point; Theory

The various directions of this research are bound by a common theme, the use of variational methods in addressing problems of mathematical analysis. Areas of concern include dynamical systems, nonlinear partial differential equations and geometry. The methods, both analytical and topological, are rooted in previous work; in particular on variational problems in which compactness assumptions are not available. Among the topics to be investigated is symbolic dynamics, in which the contribution of the periodic orbits provided by the Birkhoff-Lewis theorem connects the framework used in dynamical systems with that used in variational theory. A particular goal is to carry over existing work to the case where the stable and unstable manifolds of a hyperbolic fixed point intersect transversely. This situation arises frequently in the study of dynamical systems, for example in the restricted three-body problem. Related work will be carried out on collisions and existence of periodic orbits of three-body type. It is planned to establish the existence of infinitely many generalized solutions and to demonstrate that these solutions have good properties, i.e. that one can attach to each of them a Morse index which describes the difference of topology that they induce in the level sets of the variational problem. Variational methods in science date back to the nineteenth century when the mathematical framework of much of what we know as the physical universe was laid out. The principles of least action and energy minimization still apply to many of the models used in describing natural phenomena. The present work continues investigations into the mathematical consequences and equations which arise from variational approaches to the study of complex motions.

这项研究的各个方向都受到一个 共同的主题，使用变分方法在解决 数学分析问题。 关注的领域包括 动力系统，非线性偏微分方程， 几何 分析和拓扑的方法， 植根于以前的工作，特别是在变分问题， 这些紧凑性假设是不可用的。 其中要研究的主题是符号动力学， 其中的贡献的周期轨道所提供的 Birkhoff-Lewis定理将动力学中使用的框架 系统与变分理论中使用的系统相同。 特定目标 是把现有的工作进行到稳定的情况下， 双曲不动点交的不稳定流形 横向。 这种情况经常出现在研究 动力学系统，例如在限制性三体系统中 问题. 将就碰撞和存在的 周期性轨道的三体类型。 计划在 建立无穷多个广义解的存在性 并证明这些溶液具有良好的性能， 即，可以将莫尔斯索引附加到它们中的每一个，该莫尔斯索引 描述了它们在 变分问题的水平集。 科学中的变分方法可以追溯到19世纪 世纪我们所知的数学框架 物理宇宙的布局。 最小原则 作用和能量最小化仍然适用于许多模型 用于描述自然现象。 目前的工作仍在继续 对数学结果和方程的研究 它产生于变分方法来研究复杂的 动议