Fully nonlinear elliptic equations

主要调查员将继续他的研究特殊拉格朗日方程，艾萨克方程，对称海森方程和复杂的蒙格-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已经得到了很好的发展。上面所列的具体方程要么不满足凸性条件，要么不具有一致椭圆性。只对处理这些问题作了初步尝试，主要是在saddle案中。对称Hessian方程和复Monge-Ampere方程的研究已经取得了很大的进展，但仍然没有Schauder或Calderon-Zygmund理论。本项目旨在纠正这种状况。对上述具体方程的研究将加深我们对偏微分方程和微分几何这两个相关数学领域的认识。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格-安培方程为现代物理学弦理论中的镜像对称提供了数学基础，这是描述我们物理宇宙的统一方式。Isaacs方程的解导致某些随机过程的最优策略，例如，在工程和金融领域。海森方程也与力学中的非线性弹性理论密切相关，该理论研究拉伸材料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.