Mathematical Sciences: Critical Point Theory and Differential Equations

本课程将讨论变分方法在微分方程中的应用。它将集中在三个领域：临界点理论和哈密顿系统，在微分几何的曲率问题中出现的椭圆微分方程，以及超线性二阶椭圆边值问题的多重性问题。虽然超线性方程已经研究了一段时间，但某些结果只是部分理解。其中一个重要的问题是零边值狄利克雷问题的无穷多个解的存在性。将努力确定一些适用于所有超线性方程的全局原因。这将不得不通过一种更抽象的方法来实现，可能会使用新的复杂化思想。对哈密顿系统的研究将遵循两个方向。首先确定具有奇异势的哈密顿系统广义解的正则度；二是继续对温斯坦猜想的研究。后者涉及奇维紧化流形上的周期轨道。如果流形的第一个同调群为零，则认为每个接触向量场都有这样一个轨道。先前关于这一猜想的工作已经导致了无穷远处的临界点和接触形式的伪轨道的想法。这些结果反过来又产生了该猜想的部分解。更几何性质的工作将集中在五维或更大的球体上的Kazdan - Warner问题上。最基本的问题是确定一个给定函数的曲率何时为标准度规的保形。对于三球的肯定结果并不适用于更高的维度。除了对微分几何和偏微分方程领域做出基本贡献外，这项工作还可以应用于动力系统和势理论。

Work will be done in the application of variational methods to differential equations. It will focus on three areas: critical point theory and Hamiltonian systems, on elliptic differential equations arising in curvature problems of differential geometry and on multiplicity questions for superlinear second order elliptic boundary value problems. Although superlinear equations have been studied for some time, certain results are only partially understood. One of the important ones is the existence of infinitely many solutions of the Dirichlet problem with null boundary values. Efforts will be made to determine some global reason applicable to all superlinear equations. This will have to be achieved through a more abstract approach, possible using new complexification ideas. Work on Hamiltonian systems will follow two directions. The first is to determine the degree of regularity of generalized solutions to Hamiltonian systems with singular potentials; the second is to continue studies on the Weinstein conjecture. This latter concerns periodic orbits on odd-dimensional compact manifolds. It is believed that every contact vector-field has such an orbit if the first homology group of the manifold is zero. Previous work on this conjecture has led to ideas of critical points at infinity and pseudo-orbits of contact forms. These, in turn, have produced partial solutions to the conjecture. Work of a more geometric nature will center on the Kazdan - Warner problem on spheres of dimension five or greater. The fundamental question is one of deciding when a given function is the curvature of a metric conformal to the standard one. Positive results for the three-sphere do not carry over to higher dimensions. In addition to making fundamental contributions to the fields of differential geometry and partial differential equations, this work can be applied to dynamical systems and potential theory.