Nonlinear Elliptic Equations and Systems

非线性椭圆方程和系统

• 批准号: 0100819
• 负责人: Yanyan Li
• 金额: $ 13.5万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100819&HistoricalAwards=false
• 关键词: Nonlinear; Elliptic; Equations; Systems

The first part of the proposed work concerns Monge-Ampere equations, including extensions of classical results of Jorgens, Calabi and Pogorelov which state that entire solutions to such equations are quadratic polynomials. Dirichlet problem for Monge-Ampere equations in exterior domains of Euclidean space will also be investigated. Related problems concerning the affine Bernstein problem will be studied. The second part of the proposed work concerns best Sobolev inequality on Riemannian manifolds and the Yamabe problem on manifolds with boundary.This includes efforts in establishing a new form of best Sobolev inequalities on Riemannian manifolds as well as existence and compactness results concerning the Yamabe problem on manifolds with boundary. Sharp pointwise estimates to blow up solutions play important roles in such studies.The role of mathematical analysis is not so much to create the equations as it is to create qualitative and quantitative information about the solutions. This may include answers to questions about existence, uniqueness, smoothness, and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.

第一部分的拟议工作涉及蒙日安培方程，包括扩展经典的结果Jorgens，卡拉比和Pogorelov其中指出，整个解决方案，这样的方程是二次多项式。我们还将研究欧氏空间外区域上Monge-Ampere方程的Dirichlet问题。仿射伯恩斯坦问题的相关问题将被研究。第二部分是关于黎曼流形上的最佳Sobolev不等式和带边流形上的Yamabe问题，包括建立黎曼流形上最佳Sobolev不等式的一种新形式，以及带边流形上Yamabe问题的存在性和紧性结果。在这类研究中，精确的逐点估计爆破解起着重要的作用。数学分析的作用与其说是创建方程，不如说是创建关于解的定性和定量信息。这可能包括对存在、唯一、平滑和增长等问题的回答。此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。