Mathematical Sciences: Partial Differential Equations in Riemannian and CR Geometry

数学科学：黎曼和 CR 几何中的偏微分方程

• 批准号: 9101832
• 负责人: John Lee
• 金额: $ 4.03万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1994-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101832&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Differential; Equations

Professor Lee will study nonlinear partial differential equations related to problems in geometry. One of the goals will be to find complete Einstein metrics on the ball with prescribed conformal infinity. A second area of research will be the space of CR structures on a three dimensional manifold. The work on partial differential equations will involve an attempt to extend the idea of the maximum principle to systems. There have been numerous semi- successful attempts at this in the past and any progress made by the principal investigator in this direction should find immediate application. This award will support research in the cross disciplinary area of global analysis which studies geometry through partial differential equations. One of the interesting applications of the research involves the inverse problem of measuring the conductivity properties of a material from electrical measurements made on the boundary of the material. It is possible to reformulate this problem as one in differential geometry and then uses methods from this well developed field to attempt to answer questions concerning the inverse problem.

李教授将研究与几何问题有关的非线性偏微分方程。其中一个目标将是在球上找到完整的爱因斯坦度量，具有规定的正形无穷大。第二个研究领域将是三维流形上CR结构的空间。偏微分方程的工作将包括尝试将极大原理的思想推广到系统中。过去在这方面有许多半成功的尝试，首席研究员在这方面取得的任何进展都应立即得到应用。该奖项将支持通过偏微分方程研究几何的全球分析跨学科领域的研究。该研究的一个有趣的应用涉及通过在材料边界上进行的电测量来测量材料的导电性的反问题。将这个问题重新表述为微分几何中的一个问题是可能的，然后使用这个发达领域的方法来尝试回答有关逆问题的问题。