Fully Nonlinear Elliptic Equations and Related Geometric Problems

完全非线性椭圆方程及相关几何问题

• 批准号: 1313218
• 负责人: Bo Guan
• 金额: $ 20.91万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1313218&HistoricalAwards=false
• 关键词: Fully; Nonlinear; Elliptic; Equations; Related

This proposal concerns fully nonlinear elliptic equations that arise from or have strong connections with problems in geometry. It focuses on three topics: estimates for fully nonlinear elliptic equations on real or complex manifolds; asymptotic Plateau problems in hyperbolic space; and questions on regularity and weak solutions of the complex Monge-Ampere equation. These topics are all closely related, with the strong common feature that most of the major questions are centred at or reduce to establishing certain a priori estimates for solutions of some fully nonlinear partial differential equations. Yet each of these questions presents unique technical challenges.Fully nonlinear elliptic equations play key roles in understanding difficult problems in mathematics, and have important applications such as in image processing, optical reflector designs, optimal mass transport, and mathematical physics. A central issue in order to solve a fully nonlinear second order equation is to establish a priori estimates for perspective solutions. Our goal is to search for techniques to derive such estimates under conditions which are close to optimal for general elliptic (but not assumed uniformly elliptic) equations on real or complex manifolds. Breakthroughs in this direction would have broad impacts to the whole field of fully nonlinear PDEs and their applications.

该提案涉及由几何问题产生或与几何问题有密切联系的完全非线性椭圆方程。它重点关注三个主题：实数或复流形上完全非线性椭圆方程的估计；双曲空间中的渐近高原问题；以及关于复杂 Monge-Ampere 方程的正则性和弱解的问题。这些主题都密切相关，具有很强的共同特征，即大多数主要问题都集中于或简化为某些完全非线性偏微分方程的解建立某些先验估计。然而，这些问题中的每一个都提出了独特的技术挑战。完全非线性椭圆方程在理解数学难题中发挥着关键作用，并且在图像处理、光学反射器设计、最优质量传输和数学物理等领域具有重要的应用。求解完全非线性二阶方程的一个中心问题是建立透视解的先验估计。我们的目标是寻找在接近实数或复流形上的一般椭圆（但不假设均匀椭圆）方程的最佳条件下导出此类估计的技术。这个方向的突破将对全非线性偏微分方程及其应用的整个领域产生广泛的影响。