Fully nonlinear geometric partial differential equations

这个项目的主要目标是为微分几何问题中的完全非线性方程开发基本的分析工具。第一个项目将建立在PI的工作与Dinew和张证明了内部保持器连续性的二阶导数的弱解复杂的蒙赫-安培方程。这个估计对于研究势函数弱正则的Kahler度量是必不可少的。PI建议在更优的条件下建立这种规律性估计。沿着这个方向，PI也将研究复Monge-Ampere型方程的一些内部估计。在第二个项目中，PI决定扩展他最近与Wang关于一般黎曼流形上的Alexandrov-Bakelman-Pucci（ABP）估计的工作。PI将研究这种ABP方法的更有趣的应用，这种方法在几何分析中尚未得到广泛利用。这是可能的，应用这种技术来研究第一个提出的项目上的复杂的蒙赫-安培方程。第三个项目是关于一个完全非线性方程边值问题的有边拉格朗日子流形的平均曲率流。如果对解析部分有很好的理解，它将为构造特殊的有边界Lagrange子流形和研究有界区域之间的保面积极小映射的存在性提供有力的工具。本项目旨在通过完全非线性椭圆和抛物方程研究微分几何中的问题。我们的目标是更好地了解这些重要的几何完全非线性方程的几何量和属性之间的关系。在这个建议中开发的创新技术将导致几何中的重要问题的解决方案，并丰富现有的完全非线性偏微分方程理论。在这个过程中，这项工作在物理学和应用科学方面都产生了重要的影响。

The primary goal of this project is to develop fundamental analytic tools for fully nonlinear equations airing from problems in differential geometry. The first project will build on the PI's work with Dinew and Zhang which proved the interior Holder continuity of the second order derivative for the weak solution of complex Monge-Ampere equations. This estimate is essential in studying the Kahler metric whose potential is of weak regularity. The PI proposes to establish this regularity estimate under more optimal condition. Along this direction, the PI will also study some interior estimates for the complex Monge-Ampere type equations. In the second project, the PI is determined to extend his recent work with Wang about the Alexandrov-Bakelman-Pucci (ABP) estimate on general Riemannian manifolds. The PI will investigate more interesting applications of this ABP method that is not broadly exploited in geometric analysis yet. It is possible to apply this technique to study the first proposed project on the complex Monge-Ampere equations. The third project concerns the mean curvature flows for Lagrangian submanifolds with boundaries which is related to a boundary value problem of a fully nonlinear equation. If the analytic parts were well understood, it would provide a powerful tool to construct special Lagrangian submanifolds with boundaries and to study the existence of the area-preserving minimal maps between bounded domains.This project aims at studying problems arising from differential geometry via fully nonlinear elliptic and parabolic equations. The goal is to better understand relations between geometric quantities and properties of these important geometric fully nonlinear equations. The innovative techniques developed in this proposal will lead to the solutions of important problems in geometry and enriching the existing theory of fully nonlinear partial differential equations in general. In the process, this work is having important consequences in both physics and applied sciences.