Fully nonlinear partial differential equations and related problems in geometry

全非线性偏微分方程及几何中的相关问题

• 批准号: 0805899
• 负责人: Bo Guan
• 金额: $ 13.19万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805899&HistoricalAwards=false
• 关键词: Fully; nonlinear; partial; differential; equations

Abstract:The primary goal of the work described in this proposal is to understand some nonlinear elliptic equations which arise from problems in geometry and analysis.The research will be focused on geometric problems in four areas:Plateau-type problems for hypersurfaces in Euclidean and hyperbolic spaces, complete conformal metrics determined by functions of Ricci tensor on Riemannian manifolds with boundary, problems in isometric embedding of 2-dimensional Riemannian manifolds of positive or negative curvature, and complex Monge-Ampre equations.These problems involve solving some fully nonlinear partial differential equations. Because of their origin in geometry, these equations are closely related and therefore share many common technical issues. Progress in this project may lead to solutions to others problems in the theory of nonlinear partial differential equations on manifolds and geometric analysis.The theory and methods of geometric analysis and fully nonlinear equations play important roles in understanding difficult problems in pure and applied mathematics, such as the Poincare conjecture, the positive mass theorem and Penrose inequality in general relativity, and applications in image processing, optical reflector designs, optimal mass transport, mathematical biology and physics.The proposal concerns some very typical problems in differential geometry and highly nonlinear partial differential equations.The research will advance the understanding of these problems and develop techniques that will be useful in studying fully nonlinear equations which arise from other problems in mathematics and physical sciences.

摘要：这项工作的主要目的是了解一些几何和分析问题所产生的非线性椭圆型方程。这些研究将集中在四个领域：欧氏空间和双曲空间中超曲面的平台型问题，由黎曼流形上的Ricci张量函数确定的完备共形度量，正负曲率的二维黎曼流形的等距嵌入问题，以及复Monge-Ampre方程。这些问题涉及求解一些完全非线性的偏微分方程组。由于它们起源于几何学，这些方程密切相关，因此共享许多共同的技术问题。几何分析和完全非线性方程的理论和方法在理解纯数学和应用数学中的困难问题，如广义相对论中的庞加莱猜想、正质量定理和彭罗斯不等式，以及在图像处理、光学反射镜设计、最优质量传输、数学、生物学和物理学。这项建议涉及微分几何和高度非线性偏微分方程中的一些非常典型的问题。这项研究将促进对这些问题的理解，并开发出有助于研究由数学和物理科学中的其他问题引起的完全非线性方程的技术。