Mathematical Sciences: Global Differential Geometry and Nonlinear Partial Differential Equations

数学科学：全局微分几何和非线性偏微分方程

• 批准号: 8702742
• 负责人: Vladimir Oliker
• 金额: $ 4.93万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702742&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Global; Differential; Geometry

Vladimir Oliker will continue his research in the area of global differential geometry and related problems in nonlinear partial differential equations. He will study existence, uniqueness and regularity problems for compact and noncompact complete hypersurfaces with prescribed curvature functions. The emphasis is placed on the analytic approach based on a study of corresponding questions for Monge-Ampere equations. As a part of this research variational problems for nonlinear elliptic equations will also be investigated. Earlier results on the realization of a given function as a curvature function have been based on pointwise growth conditions. Oliker will attempt to replace these with integral estimates. He will also look for corresponding results in Lorenz space. These investigations will encounter the added difficulty that the unit sphere is not compact.

弗拉基米尔·奥利克（Vladimir Oliker）将继续在非线性偏微分方程中的全球微分几何形状和相关问题领域继续进行研究。他将研究具有规定的曲率功能的紧凑和非紧密和非伴随的完整超曲面的存在，独特性和规律性问题。重点是基于对Monge-Ampere方程相应问题的研究的分析方法。作为非线性椭圆方程的研究变异问题的一部分，也将研究。 关于给定函数作为曲率函数实现的早期结果是基于点的生长条件。奥利克（Oliker）将尝试用积分估计来替换它们。他还将在洛伦兹空间中寻找相应的结果。这些调查将遇到额外的困难，即单位球体不紧凑。