Mathematical Sciences: Critical Point Theory and Differential Equations

将进行将变分法应用于微分方程的工作。 它将重点关注三个领域：临界点理论和哈密顿系统、微分几何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。 尽管超线性方程已被研究了一段时间，但某些结果仅被部分理解。 其中重要的一点是具有零边界值的狄利克雷问题存在无限多个解。 我们将努力确定一些适用于所有超线性方程的全局原因。 这必须通过更抽象的方法来实现，可能使用新的复杂化思想。 哈密​​顿系统的工作将遵循两个方向。 首先是确定奇异势哈密顿系统广义解的正则程度；二是继续温斯坦猜想的研究。 后者涉及奇维紧流形上的周期轨道。 人们相信，如果流形的第一同调群为零，则每个接触矢量场都具有这样的轨道。 先前关于这个猜想的工作已经产生了无穷远临界点和接触形式伪轨道的想法。 这些反过来又为这个猜想提供了部分答案。 更具几何性质的工作将集中在五维或更高维度球体上的卡兹丹-华纳问题。 基本问题是确定给定函数何时是与标准函数共形的度量曲率。 三球体的积极结果不会延续到更高的维度。 除了对微分几何和偏微分方程领域做出基础贡献外，这项工作还可以应用于动力系统和势论。

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.