Fully nonlinear elliptic equations in geometry

几何中的完全非线性椭圆方程

• 批准号: 1620086
• 负责人: Bo Guan
• 金额: $ 25万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620086&HistoricalAwards=false
• 关键词: Fully; nonlinear; elliptic; equations; geometry

Differential equations play a central role in mathematics and its applications, for example, in differential geometry and mathematical physics. Fully nonlinear elliptic and parabolic equations are some of the main equations under current investigations by researchers. Building on the recent progress, in this project the principal investigator will establish general existence and regularity results for these equations on real or complex manifolds and develop technical tools which can be applied to solve other types of equations. Results in this project will yield interesting applications in geometry and help to understand general partial differential equations.A central issue in solving fully nonlinear elliptic and parabolic equations is to establish a priori estimates which present great challenges due to the high nonlinearity of the equations and the general geometry of the manifolds as well as their boundaries. The project's goal is to search for new methods to overcome these difficulties. In particular, it will identify the optimal conditions under which one can solve the equations on general manifolds. The principal investigator also plans to study a very general class of equations on which few results are known, and attack some difficult problems in complex geometry.

微分方程在数学及其应用中起着核心作用，例如在差异几何和数学物理学中。完全非线性的椭圆形和抛物线方程是研究人员当前研究下的一些主要方程式。在最新进展的基础上，在该项目中，主要研究人员将在实际或复杂的流形上为这些方程式建立一般存在和规律性结果，并开发可用于求解其他类型方程的技术工具。该项目中的结果将在几何形状中产生有趣的应用，并有助于了解一般的偏微分方程。解决完全非线性椭圆形和抛物线方程的核心问题是建立一个先验估计值，由于方程式的高非线性以及流派的一般几何形状以及歧管的一般几何形状以及其边界以及其边界以及其边界以及其边界。该项目的目标是寻找克服这些困难的新方法。特别是，它将确定可以在一般歧管上求解方程的最佳条件。首席研究者还计划研究一个非常通用的方程式，其中很少有结果，并攻击了复杂几何形状中的一些困难问题。