Nonlinear elliptic 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，它们可以被视为高等微积分的扩展。同时，该项目还将对项目中所研究的方程所依赖的领域产生影响：一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，这是描述我们物理宇宙的统一方式；另一个方程是材料科学中的有效模型；所谓的艾萨克斯方程的解可以为某些随机过程（例如工程和金融）提供最优策略；此外，海森方程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，回答五维或更高维齐次二阶解是否平凡的问题，以及研究亚临界相方程连续粘度解的低正则性。平均曲率流的自相似解的目的是对拉格朗日平移解进行分类，研究嵌入球收缩器在三维欧几里德空间中的唯一性。对称Hessian方程的目的是研究四维及四维以上的二次Hessian方程和标量曲率方程的Hessian估计，获得三维二次Hessian方程的Schauder和Calderon-Zygmund估计，研究k对称Hessian方程的Liouville问题。研究三维全非线性椭圆方程（如三维艾萨克方程）的正则性是研究一般三维全非线性椭圆方程，特别是k对称Hessians方程与有限分段线性艾萨克方程线性组合形式的方程的正则性。复杂蒙日-安培方程的计划是显示任何复杂蒙日-安培方程的整体解的琐碎性，包括具有一定必要限制的Kahler Ricci流的自收缩方程。

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.