Fully Nonlinear Elliptic Equations

完全非线性椭圆方程

• 批准号: 2054973
• 负责人: Yu Yuan
• 金额: $ 29.07万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054973&HistoricalAwards=false
• 关键词: Fully; Nonlinear; Elliptic; Equations; Fully; Nonlinear; Elliptic; Equations

The research activities of this project will continue to deepen and broaden our understanding of two intimately connected mathematical fields: partial differential equations and differential geometry. The project will have an impact in the study of special Lagrangian equations, complex Monge-Ampère equations, and Hamiltonian stationary equations, which provide the mathematical foundation for mirror symmetry in the string theory of modern physics, and of maximal surface systems, which have the roots in general relativity. These 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 project provides research training opportunities for graduate students. The objectives for special Lagrangian equations are to derive Schauder and Calderón-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, to investigate the existence and uniqueness of solutions to the Dirichlet problem for the special Lagrangian equation with continuous variable phase, and to resolve periodic Liouville problems with constraints as well as (entire) Liouville problem for the complex version of the special Lagrangian equation. The aim for symmetric sigma-k equations is to investigate Hessian estimates and regularity for sigma-2 equations in dimension four and higher, to obtain Schauder and Calderón-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-Ampère equations is to demonstrate the triviality of any global solution to complex Monge-Ampère equations including self-shrinking equations for the Kähler-Ricci flow with certain necessary restrictions and to derive regularity of solutions to the real Monge-Ampère equations under a noncollapsing condition. For the case of maximal surface systems the goal is to study the Bernstein problems for exterior solutions and regularity for solutions under a noncollapsing condition. The project will also take on Hamiltonian stationary equations, where it aims to establish existence of the solutions to the second boundary value problem and rigidity for the Hamiltonian stationary equations.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.

该项目的研究活动将继续加深和扩大我们对两个紧密联系的数学领域的理解：部分微分方程和差异几何形状。该项目将对特殊拉格朗日方程，复杂的蒙格 - 安培方程和汉密尔顿固定方程式的研究产生影响，这些方程为现代物理学的弦理论和最大表面系统的数学基础提供了数学基础，这些基础具有一般相对论的根。这些HESSIAN方程也与机械中的非线性弹性理论有关，该理论研究了一种机制，该机制将拉伸的材料返回其原始大小和形状。该项目为研究生提供了研究培训机会。 The objectives for special Lagrangian equations are to derive Schauder and Calderón-Zygmund estimates for equations with critical and supercritical phases, to answer whether any homogeneous order two solutions in dimension five or higher is trivial, to study low regularity of continuous viscosity solutions to the equations with subcritical phases, to investigate the existence and uniqueness of solutions to the Dirichlet problem for the special Lagrangian equation with连续可变阶段，并解决特殊Lagrangian方程的复杂版本的限制以及（整个）Liouville问题的定期问题。对称Sigma-k方程的目的是研究四个及更高尺寸的Sigma-2方程的Hessian估计和规律性，以获取Schauder和Calderón-Zygmund估计3-D Sigma-2方程，并研究Sigma-K方程的Liouville问题。复杂和真实的Monge-Ampère方程的计划是证明对复杂的Monge-Ampère方程的任何全球解决方案的微不足道，包括Kähler-Icci流动的自我缩减方程，并在非收获条件下对真实的Monge-Ampère方程进行某些必要的限制，并为实现某些必要的限制。对于最大表面系统的情况，目标是研究外部解决方案的伯恩斯坦问题，并在非策略条件下研究解决方案的规律性。该项目还将采用汉密尔顿固定方程式，旨在建立解决第二个边界价值问题的解决方案和汉密尔顿固定方程式的僵化。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来通过评估来诚实地支持。