Mathematical Sciences: Problems Related to Differential Geometry and Mathematical Physics

数学科学：与微分几何和数学物理相关的问题

• 批准号: 8703027
• 负责人: Patricio Aviles
• 金额: $ 1.63万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-15 至 1988-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703027&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Related; Differential

Patricio Aviles will continue his work on partial differential equations connected with problems on differential geometry and mathematical physics. This work concerns four areas: the Yamabe problem in complete Riemannian manifolds; liquid crystal problems; harmonic maps between two manifolds; singularities of positive solutions of semi-linear elliptic equations. The compact Yamabe problem was solved by Schoen. Aviles will try to extend his work with McOwen where they made some progress in dealing with the case of complete manifolds. The liquid crystal work involves minimal surface theory and concerns an investigation of the size of singular sets of limiting configurations. The third topic focusses on the Dirichlet problem for harmonic maps between complete manifolds of negative curvature. Interest in positive solutions of semi-linear elliptic equations is motivated by study of the Yangs-Mills equations. Aviles is going to study non-isolated singularities.

帕特里西奥·阿维莱斯将继续研究与微分几何和数学物理问题有关的偏微分方程式。这项工作涉及四个领域：完备黎曼流形上的Yamabe问题；液晶问题；两个流形之间的调和映射；半线性椭圆型方程正解的奇性。舒恩解决了紧凑的Yamabe问题。Aviles将尝试扩展他与McOwen的工作，在处理完全流形的情况下，他们取得了一些进展。液晶工作涉及极小表面理论，并涉及极限构型奇异集的大小的研究。第三个主题是关于负曲率完备流形之间调和映射的Dirichlet问题。对半线性椭圆型方程正解的兴趣源于对杨氏方程的研究。阿维莱斯将研究非孤立奇点。