Fully Nonlinear Elliptic Equations

该项目的研究活动将继续深化和拓宽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。该项目将对特殊拉格朗日方程、复杂的蒙赫-安培方程和哈密顿定常方程的研究产生影响，这些方程为现代物理学弦理论中的镜像对称提供了数学基础，并对广义相对论中的最大表面系统产生了影响。这些海森方程也与力学中的非线性弹性理论有关，该理论研究拉伸材料恢复其原始尺寸和形状的机制。该项目为研究生提供研究培训机会。对特殊拉格朗日方程的研究目的是得到临界相和超临界相方程的Schauder和Calderón-Zygmund估计，回答在5维或更高维中的齐次二阶解是否平凡，研究亚临界相方程的连续粘性解的低正则性，研究具有连续变量相位的特殊Lagrange方程Dirichlet问题解的存在唯一性，以及求解带约束的周期Liouville问题和特殊Lagrange方程的复形式的（整体）Liouville问题。本文的目的是研究4维及更高维sigma-k方程的Hessian估计和正则性，得到三维sigma-k方程的Schauder和Calderón-Zygmund估计，以及sigma-k方程的Liouville问题。复杂和真实的Monge-Ampère方程的计划是证明复杂Monge-Ampère方程的任何整体解的平凡性，包括具有某些必要限制的Kähler-Ricci流的自收缩方程，并在非崩溃条件下导出真实的Monge-Ampère方程的解的正则性。对于极大曲面系统，我们的目标是研究非塌陷条件下的伯恩斯坦问题的外解和解的正则性。该项目还将研究哈密顿定常方程，旨在确定第二边值问题的解的存在性和哈密顿定常方程的刚性。该奖项反映了NSF的法定使命，并通过使用基金会的智力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 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.