Some Questions in Nonlinear Differential Equations

从变分法的工具和方法将开发和应用于各种问题的动力系统，偏微分方程和几何。 对于动力学系统，将建立哈密顿系统的各种连接轨道的存在性。 有相关的应用几何中存在的最小异宿，同宿，和混沌测地线的n-环面和其他流形将被研究。 对于偏微分方程，我们将展示在相变模型中出现的一些方程是如何允许大量不同类型的平衡态的。我们的项目的目标是开发新的极小化和极大极小化方法，并将其应用于动力系统，几何和偏微分方程等领域。 在动力系统中的应用包括建立渐近于简单基本状态的轨道的存在性，如平衡点和周期轨道。 相关的问题发生在几何学中，基本的周期性状态是周期测地线，我们寻求连接状态也是测地线。 对于偏微分方程，我们将研究一类相变模型，我们试图了解什么样的平衡状态是可能的。

Tools and methods from the calculus of variations will be developed and applied to a variety of problems in dynamical systems, partial differential equations and geometry. For dynamical systems, the existence of various kinds of connecting orbits for Hamiltonian systems will be established. There are related applications to geometry where the existence of minimal heteroclinic, homoclinic, and chaotic geodesics for the n-torus and other manifolds will be studied. For partial differential equations, it will be shown how some equations that arise in phase transition models admit a large number of different kinds of equilibrium states.The goal of our project is to develop new minimization and minimax methods and apply them to areas such as dynamical systems, geometry, and partial differential equations. The applications to dynamical systems include establishing the existence of orbits asymptotic to simple basic states like equilibrium points and periodic orbits. Related problems occur in geometry where the basic periodic states are periodic geodesics and we seek connecting states that are also geodesics. For partial differential equations, a class of phase transition models will be studied where we seek to understand what kinds of equilibrium states are possible.