Nonlinear elliptic equations

对这一拟议项目的研究活动将加深我们对两个紧密相连的数学领域--偏微分方程组和微分几何--的理解，这两个领域可以被视为高等微积分的扩展。同时，该项目还将对项目中研究的方程所在的领域产生影响：一些方程为现代物理的弦理论中的镜像对称提供了数学基础，弦理论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Isaacs方程的解导致了某些随机过程的最优策略，例如在工程和金融领域；Hessian方程也与力学中的非线性弹性理论有关，该理论研究材料被拉伸后恢复其原始尺寸和形状的机制。特殊的拉格朗日方程的目标是推导具有临界和超临界相的方程的Schauder和Calderon-Zygmund估计，回答五维或更高维的任何齐次二阶解是否平凡的问题，以及研究具有亚临界相的方程连续粘性解的低正则性。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，并研究三维欧氏空间中嵌入球面收缩算子的唯一性。对称Hessian方程的目的是研究四维及以上二次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，并研究k-对称Hessian方程的Liouville问题。对于完全非线性椭圆型方程，例如三维中的Isaacs方程，研究一般的三维完全非线性椭圆型方程的正则性，特别是k-对称Hessians方程和有限分段线性Isaacs方程的线性组合形式的方程的正则性。复Monge-Ampere方程的计划是显示复Monge-Ampere方程的任何整体解的平凡，包括具有某些必要限制的Kahler Ricci流的自缩方程。

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.