Mathematical Sciences: Critical Point Theory and Differential Equations

这项研究的不同方向被一个共同的主题所束缚，即在解决数学分析问题时使用变分方法。关注的领域包括动力系统、非线性偏微分方程式和几何。这些方法，无论是解析的还是拓扑学的，都植根于以前的工作；特别是在紧致性假设不可用的变分问题上。要研究的主题之一是符号动力学，其中由Birkhoff-Lewis定理提供的周期轨道的贡献将动力系统中使用的框架与变分理论中使用的框架联系起来。一个特别的目标是将已有的工作推广到双曲不动点的稳定流形和不稳定流形横交的情况。这种情况经常出现在动力学系统的研究中，例如在受限三体问题中。将开展关于三体型碰撞和周期轨道存在的相关工作。我们计划证明无穷多个广义解的存在性，并证明这些解具有良好的性质，即人们可以在每个解上附加一个Morse指数，该指数描述了它们在变分问题的水平集上诱导的拓扑差异。科学中的变分方法可以追溯到十九世纪，当时我们所知的许多物理宇宙的数学框架已经形成。最小作用量和能量最小化原则仍然适用于描述自然现象的许多模型。本工作继续研究由变分方法研究复杂运动所产生的数学结果和方程。

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.