Nonlinear elliptic equations

该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，即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，该项目也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，这是描述我们的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Isaacs方程的解导致某些随机过程的最优策略，例如，在工程和金融中;海森方程也与力学中的非线性弹性理论有关，它研究的机制，使材料被拉伸返回到其原始大小和形状。特殊的拉格朗日方程的目标是推导Schauder和Calderon-Zygmund估计，回答了高维齐次二阶解是否平凡的问题，并研究了次临界方程连续粘性解的低正则性.平均曲率流自相似解的目的是对拉格朗日平移解进行分类，并研究三维欧氏空间中嵌入球收缩器的唯一性。本文的主要目的是研究四维及高维二次Hessian方程和标量曲率方程的Hessian估计，得到三维二次Hessian方程的Schauder和Calderon-Zygmund估计，以及k-对称Hessian方程的Liouville问题.对于完全非线性椭圆型方程如三维Isaacs方程的研究，主要是研究一般的三维完全非线性椭圆型方程，特别是k-对称Hessian方程和k-分段线性Isaacs方程的线性组合形式的方程的正则性。复杂的Monge-Ampere方程的计划是显示复杂的Monge-Ampere方程的任何全局解的平凡性，包括具有某些必要限制的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.