Fully nonlinear elliptic and parabolic equations

该项目专注于特殊拉格朗日方程、对称 Hessian 方程、Isaacs 方程、复杂 Monge-Ampere 方程及其抛物线版本（例如拉格朗日平均曲率流）的研究。全非线性一致椭圆方程和抛物方程（在一般维度上有凸性条件，在二维上没有凸性假设）的正则性和可解性理论已经得到很好的发展。刚刚列出的具体方程要么不满足凸性条件，要么不表现出均匀的椭圆性或抛物线性。仅在一般鞍座情况下进行了初步尝试。对称Hessian方程和复Monge-Ampere方程已经取得了实质性进展，但仍然没有针对这些方程的Schauder或Calderon-Zygmund理论；令人惊讶的是，一般维度的二次对称 Hessian 方程的正则性问题仍然悬而未决。该项目旨在解决这些基本问题。对上述方程的研究将进一步加深我们对两个密切相关的数学领域：偏微分方程和微分几何的了解。此外，该项目还将对出现这些方程的领域产生影响。特殊的拉格朗日方程和复杂的蒙日-安培方程为现代物理学弦论中的镜像对称性提供了数学基础，这是描述我们物理宇宙的统一方式。艾萨克斯方程的解可以得出某些随机过程的最优策略，例如在工程和金融领域。 Hessian 方程还与力学中的非线性弹性理论相关，该理论研究被拉伸的材料恢复到其原始尺寸和形状的机制。部分研究还涉及研究生的参与。

This project concentrates on the study of special Lagrangian equations, symmetric Hessian equations, Isaacs equations, complex Monge-Ampere equations, and their parabolic versions (e.g., Lagrangian mean curvature flows). The theory of regularity and solvability for fully nonlinear uniformly elliptic and parabolic equations (with the convexity condition in general dimensions and without the convexity hypothesis in dimension two) is well developed. The concrete equations just listed either do not satisfy the convexity condition or do not exhibit uniform ellipticity or parabolicity. Only preliminary attempts have been made in the general saddle cases. Substantial advances have been achieved for the symmetric Hessian equations and the complex Monge-Ampere equations, yet there is still no Schauder or Calderon-Zygmund theory for these equations; and surprisingly the regularity problem for the quadratic symmetric Hessian equations in general dimension still remains open. This project seeks to address these fundamental issues.Investigations into the aforementioned equations will further our knowledge of two closely related mathematical fields, partial differential equations and differential geometry. Moreover, the project will also have impact on the areas where these equations arise. Special Lagrangian equations and complex Monge-Ampere 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. Solutions to Isaacs equations lead to the optimal strategy for certain random processes, for example, in engineering and finance. 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. Part of the research also involves participation of graduate students.