Fully Nonlinear Elliptic and Parabolic Equations

完全非线性椭圆和抛物线方程

• 批准号: 1800495
• 负责人: Yu Yuan
• 金额: $ 24万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800495&HistoricalAwards=false
• 关键词: Fully; Nonlinear; Elliptic; Parabolic; Equations

The research activity in this project will deepen our understanding of two intimately connected mathematical fields: partial differential equations and differential geometry, which are just super calculus. Simultaneously, the project will also have impact on the areas where the equations to be investigated rest: special Lagrangian equations and complex Monge-Ampere 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; maximal surface equations are directly from the fascinating general relativity, which fundamentally changed our understanding of the world; mean curvature flow is an effective model in material science; Hessian equations are also 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 of research on special Lagrangian equations are to derive Schauder and Calderon-Zygmund estimates for equations with critical and supercritical phases, to answer whether any homogeneous order two solution in dimension five or higher is trivial, to study low regularity of continuous viscosity solutions to the equations with subcritical phases, and to resolve exterior Liouville problems with constraints as well as (entire) Liouville problem for the complex version of the special Lagrangian equation. The aim of research on symmetric sigma-k equations is to investigate Hessian estimates for sigma-2 equations in dimension four and higher and also sigma-2 principle curvature equations, to obtain Schauder and Calderon-Zygmund estimates for 3-d sigma-2 equations, and to study the Liouville problem for sigma-k equations. The plan for complex and real Monge-Ampere equations is to demonstrate the triviality of any global solution to complex Monge-Ampere equations including self-shrinking equations for the Kahler Ricci flow with certain necessary restrictions and to derive regularity of solutions to the real Monge-Ampere equations under a necessary noncollapsing condition. The attempt for maximal surface equations is to study the Bernstein problems for exterior solutions and regularity for solutions under a noncollapsing condition. The purposes for the study on fully nonlinear parabolic equations are to show uniqueness and existence for viscosity solutions to parabolic Monge-Ampere equations under certain necessary conditions and to derive Hessian estimates for Lagrangian mean curvature flow under certain convexity condition.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目的研究活动将加深我们对两个紧密相连的数学领域的理解：偏微分方程组和微分几何，这两个领域只是超级微积分。同时，该项目还将对方程研究的其他领域产生影响：特殊的拉格朗日方程和复的Monge-Ampere方程为现代物理的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式；极大曲面方程直接来自迷人的广义相对论，它从根本上改变了我们对世界的认识；平均曲率流是材料科学中的一个有效模型；Hessian方程也与力学中的非线性弹性理论有关，该理论研究被拉伸的材料恢复到其原始尺寸和形状的机制。研究特殊拉格朗日方程的目的是推导具有临界和超临界相的方程的Schauder和Calderon-Zygmund估计，回答任何五维或更高维的齐次二阶解是否平凡，研究具有亚临界相的方程连续粘性解的低正则性，以及解决带约束的外部Liouville问题和特殊拉格朗日方程的复形式的(完整)Liouville问题。研究对称sigma-k方程的目的是研究四维及更高维sigma-2方程和sigma-2主曲率方程的Hessian估计，得到三维sigma-2方程的Schauder估计和Calderon-Zygmund估计，并研究sigma-k方程的Liouville问题。复数和实数Monge-Ampere方程的计划是证明复数Monge-Ampere方程(包括Kahler Ricci流的自缩方程)的任何整体解的平凡性，并在一个必要的非折叠条件下导出实数Monge-Ampere方程解的正则性。极大曲面方程的尝试是在非折叠条件下研究外解的Bernstein问题和解的正则性。研究完全非线性抛物方程的目的是证明在某些必要条件下抛物型Monge-Ampere方程粘性解的唯一性和存在性，并在某些凸性条件下推导拉格朗日平均曲率流的Hessian估计。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Maximal Hypersurfaces over Exterior Domains

• DOI: 10.1002/cpa.21929
• 发表时间: 2019-03
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Guanghao Hong;Yu Yuan

Regularity for Almost Convex Viscosity Solutions of the Sigma-2 Equation

Sigma-2 方程的近似凸粘度解的正则性

• DOI: 10.4208/jms.v54n2.21.03
• 发表时间: 2020
• 期刊: Journal of Mathematical Study
• 影响因子: 0.8
• 作者: Yuan, Ravi Shankar

Hessian estimates for convex solutions to quadratic Hessian equation

二次 Hessian 方程凸解的 Hessian 估计

• DOI: 10.1016/j.anihpc.2018.07.001
• 发表时间: 2019
• 期刊: Analyse non linéaire
• 作者: McGonagle, Matt;Song, Chong;Yuan, Yu
• 通讯作者: Yuan, Yu

A Bernstein problem for special Lagrangian equations in exterior domains

外域特殊拉格朗日方程的伯恩斯坦问题

• DOI: 10.1016/j.aim.2019.106927
• 发表时间: 2020
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Dongsheng Li;Zhisu Li;Yu Yuan
• 通讯作者: Yu Yuan

Hessian estimate for semiconvex solutions to the sigma-2 equation

sigma-2 方程半凸解的 Hessian 估计

• 发表时间: 2020
• 期刊: Calculus of variations and partial diffferential equations
• 作者: Shankar, Ravi;Yuan, Yu
• 通讯作者: Yuan, Yu