Fully nonlinear elliptic and parabolic equations

本项目致力于研究特殊的拉格朗日方程、对称海森方程、艾萨克斯方程、复Monge-Ampere方程及其抛物形式(例如，拉格朗日平均曲率流)。完全非线性一致椭圆型和抛物型方程的正则性和可解性理论(在一般维上满足凸性条件，在二维上不满足凸性假设)得到了很好的发展。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性。在一般的马鞍案中只进行了初步的尝试。对称Hessian方程和复Monge-Ampere方程已经取得了很大的进展，但这些方程仍然没有Schauder或Calderon-Zygmund理论；令人惊讶的是，一般维上的二次对称Hessian方程的正则性问题仍然是开放的。这个项目试图解决这些基本问题。对上述方程的研究将加深我们对两个密切相关的数学领域--偏微分方程式和微分几何--的了解。此外，该项目还将对出现这些方程的地区产生影响。特殊的拉格朗日方程和复的Monge-Ampere方程为现代物理学弦理论中的镜像对称性提供了数学基础，弦理论是描述我们物理宇宙的统一方式。对于某些随机过程，例如在工程和金融领域，求解Isaacs方程可以得到最优策略。赫斯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。这项研究的一部分也涉及研究生的参与。

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.