Fully nonlinear elliptic equations

主要研究人员将继续研究特殊的拉格朗日方程、艾萨克斯方程、对称的海森方程和复蒙格-安培方程。完全非线性一致椭圆型方程(在任意维上满足凸性条件，在二维上不满足凸性假设)的先验估计和可解性理论得到了很好的发展。刚刚列出的具体方程要么不满足凸性条件，要么不表现出一致的椭圆性。只有初步尝试来处理这类问题，主要是在马鞍案中。对称Hessian方程和复Monge-Ampere方程已经取得了实质性的进展，但仍没有Schauder或Calderon-Zygmund理论。这个项目试图纠正这种状况。对上述具体方程的研究将加深我们对两个相关数学领域--偏微分方程组和微分几何--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密切相关，该理论研究材料被拉伸后恢复其原始尺寸和形状的机制。

The principle investigator will continue his study of special Lagrangian equations, Isaacs equations, symmetric Hessian equations, and complex Monge-Ampere equations. The theory of a priori estimates and solvability for fully nonlinear uniformly elliptic equations (with the convexity condition in arbitrary 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. Only preliminary attempts have been made to deal with such issues, mainly in the saddle case. 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. This project seeks to remedy that state of affairs. Investigations into the aforementioned concrete equations will further our knowledge of two 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 closely related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape.