Nonlinear elliptic equations

非线性椭圆方程

• 批准号: 1362168
• 负责人: Yu Yuan
• 金额: $ 27.64万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362168&HistoricalAwards=false
• 关键词: Nonlinear; elliptic; equations

The research activity into this proposed project will deepen our understanding of two intimately connected mathematical fields, partial differential equations and differential geometry, which may be viewed as extensions of advanced calculus. Simultaneously, the project will also have impact on the areas on which the equations studied in the project rest: some equations provide the mathematical foundation for mirror symmetry in the string theory of modern physics, which is a unified way to describe our physical universe; another equation is an effective model in material science; solutions to the so called Isaacs equations lead to the optimal strategy for certain random processes, for example, in engineering and finance; Also Hessian equations are related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape.The objectives for special Lagrangian equations are to derive Schauder and Calderon-Zygmund estimates for the equations with critical and supercritical phases, to answer the question whether any homogeneous order two solution in dimension five or higher is trivial or not, and to study low regularity of continuous viscosity solutions to the equations with subcritical phases. The purposes for self similar solutions to mean curvature flows are to classify Lagrangian translating solutions and study uniqueness of embedded sphere shrinker in 3-d Euclidean space. The aim for symmetric Hessian equations is to investigate Hessian estimates for quadratic Hessian equations in dimension four and higher and also scalar curvature equations, to obtain Schauder and Calderon-Zygmund estimates for 3-d quadratic Hessian equations, and to study the Liouville problem for k-symmetric Hessian equations. The attempt for fully nonlinear elliptic equations such as Isaacs equations in 3-d is to study the regularity for general fully nonlinear elliptic equations in 3-d, in particular for equations in the form of linear combinations of k-symmetric Hessians and finitely piecewise linear Isaacs equations. The plan for complex Monge-Ampere equations is to show the triviality of any global solution to complex Monge-Ampere equations including self-shrinking equations for the Kahler Ricci flow with certain necessary restrictions.

该提出的项目的研究活动将加深我们对两个密切相关的数学领域，部分微分方程和差异几何形状的理解，这可能被视为高级演算的扩展。同时，该项目还将影响项目休息中研究的方程式的领域：某些方程式为现代物理弦理论中的镜像对称性提供了数学基础，这是描述我们物理宇宙的统一方法；另一个方程是材料科学中的有效模型。所谓的ISAACS方程的解决方案为某些随机过程（例如工程和金融方面）带来了最佳策略； Also Hessian equations are related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape.The objectives for special Lagrangian equations are to derive Schauder and Calderon-Zygmund estimates for the equations with critical and supercritical phases, to answer the question whether any homogeneous order two solution in dimension five or higher is trivial or not, and to study low regularity具有亚临界相的方程式的连续粘度解。自我类似解决方案的曲率流的目的是对拉格朗日翻译解决方案进行分类，并研究3-D欧几里得空间中嵌入式球体收缩器的独特性。对称的Hessian方程的目的是研究四个及更高和标量曲率方程的二次HESSIAN方程的Hessian估计值，以获取Schauder和Calderon-Zygmund估计3-D Quadratic Hessian方程的估计，并研究K-Memmetric Hessian方程的Liouville问题。 3-D中的全非线性椭圆方程（例如ISAACS方程）的尝试是为了研究3-D中一般非线性椭圆方程的规律性，尤其是以k-对称Hessians的线性组合形式的方程式和有限分段的线性线性ISAACS方程。复杂的Monge-Ampere方程的计划是表明对复杂的Monge-Ampere方程的任何全球解决方案的微不足道，包括Kahler Ricci流的自我缩减方程，并具有某些必要的限制。