Fully Nonlinear Elliptic and Parabolic Equations

这个项目的研究活动将加深我们对两个密切相关的数学领域的理解：偏微分方程和微分几何，它们只是超级微积分。同时，该项目还将对将要研究的方程所在的领域产生影响：特殊的拉格朗日方程和复杂的蒙日-安培方程为现代物理学弦理论中的镜像对称提供了数学基础，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理解；平均曲率流是材料科学中的一个有效模型；黑森方程还与力学中的非线性弹性理论有关，该理论研究材料在拉伸后恢复其原始尺寸和形状的机制。特殊拉格朗日方程的研究目标是推导临界相和超临界相方程的Schauder估计和Calderon-Zygmund估计，回答五维或更高维的齐次二阶解是否平凡，研究亚临界相方程连续粘度解的低正则性。求解带约束的外部Liouville问题以及特殊拉格朗日方程复版本的（整个）Liouville问题。对称sigma-k方程的研究目的是研究四维及四维以上的sigma-2方程和sigma-2主曲率方程的Hessian估计，获得三维sigma-2方程的Schauder估计和Calderon-Zygmund估计，研究sigma-k方程的Liouville问题。复实蒙日-安培方程的计划是证明具有一定必要限制条件的Kahler - Ricci流的复蒙日-安培方程（包括自缩方程）的任何全局解的平凡性，并推导出实蒙日-安培方程在必要非坍缩条件下解的正则性。最大曲面方程的尝试是研究Bernstein问题的外解和非坍缩条件下解的正则性。研究完全非线性抛物型方程的目的是在一定必要条件下证明抛物型蒙日-安佩方程黏度解的唯一性和存在性，并在一定凸性条件下导出拉格朗日平均曲率流的Hessian估计。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.