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: 8803494
• 负责人: Abbas Bahri
• 金额: $ 5.19万
• 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: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8803494&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Critical; Point; Theory

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.

将在变分法的应用方面做工作 微分方程。 它将侧重于三个领域： 临界点理论和Hamilton系统，椭圆型 曲率问题中的微分方程 微分几何和多重性问题 超线性二阶椭圆边值问题 虽然超线性方程已经研究了一些 有时，某些结果只被部分理解。 之一 重要的是存在无穷多个解 Dirichlet边值问题 着力 以确定适用于所有人的某种全局原因 超线性方程组 这必须通过一个 更抽象的方法，可能使用新的复杂化 想法. 关于哈密顿系统的工作将沿着两个方向进行。 的 首先是确定广义的规则性程度， 具有奇异势的哈密顿系统的解; 二是继续研究温斯坦猜想。 这 后者涉及奇维紧空间上的周期轨道 流形 据信，每个接触矢量场都具有 如果流形的第一个同调群是 零. 以前关于这个猜想的工作已经导致了以下想法： 无穷远临界点和接触形式的伪轨道。 这些反过来又产生了部分解决方案， 猜想 更多的几何性质的工作将集中在卡兹丹- 第五维或更大维球体上的华纳问题。 的 一个基本的问题是决定一个给定的函数何时 与标准度量共形的度量的曲率。 三球的阳性结果不会延续到更高水平 尺寸. 除了对发展中国家作出重要贡献外， 微分几何与偏微分 方程，这项工作可以应用于动力系统， 潜力理论