Mathematical Sciences: Monge-Ampere Type Equations and Related Problems in Differential Geometry

数学科学：蒙日-安培型方程及微分几何中的相关问题

• 批准号: 9626722
• 负责人: Bo Guan
• 金额: $ 9.5万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626722&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Monge; Ampere; Type

9626722 Guan This project pursues geometric partial differential equations. Analysis of many problems in differential geometry often leads to questions concerning nonlinear partial differential equations. The investigator plans to study Monge-Ampere type equations with an emphasis on the regularity of solutions in the degenerate case; the Christoffel-Minkowski problems and related Hessian equations; the Dirichlet problem for hypersurfaces of prescribed curvatures; curvature evolution equations for hypersurfaces. Unlike linear partial differential equations there is little general theory regarding nonlinear partial differential equations. Yet many problems arising in various geometric settings can only be formulated as nonlinear differential equations. And the proposed research represents an attempt to pursue these equations from a geometric perspective - the main advantage of the geometric approach seems to be that one is able to make fairy strong, but geometrically meaningful, assumptions thereby simplifying the equations.

小行星9626722 这个项目追求几何偏微分方程。在分析微分几何中的许多问题时，常常会涉及到非线性偏微分方程的问题。研究人员计划研究Monge-Ampere型方程，重点是退化情况下的解的正则性; Christoffel-Minkowski问题和相关的Hessian方程;规定曲率的超曲面的Dirichlet问题;超曲面的曲率演化方程。 与线性偏微分方程不同，关于非线性偏微分方程的一般理论很少。然而，许多问题中出现的各种几何设置只能制定为非线性微分方程。这项研究代表了一种从几何角度追求这些方程的尝试--几何方法的主要优点似乎是，人们能够做出非常强的、但在几何上有意义的假设，从而简化方程。