Fully nonlinear geometric partial differential equations

本计画的主要目标是为微分几何问题中的全非线性方程式开发基本的分析工具。第一个项目将建立在PI与Dinew和Zhang的工作基础上，该工作证明了复蒙日-安培方程弱解二阶导数的内Holder连续性。这种估计在研究卡勒度规时是必要的，因为它的势是弱正则的。PI提出在更优条件下建立该正则性估计。沿着这个方向，PI还将研究复杂蒙日-安培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关于一般黎曼流形的Alexandrov-Bakelman-Pucci （ABP）估计的工作。PI将研究这种ABP方法在几何分析中尚未广泛利用的更有趣的应用。这一技术有可能应用于研究第一个提出的复蒙日-安培方程项目。第三个项目涉及具有边界的拉格朗日子流形的平均曲率流，这与一个完全非线性方程的边值问题有关。如果解析部分得到很好的理解，它将为构造有边界的特殊拉格朗日子流形和研究有界域间保面积最小映射的存在性提供有力的工具。本课题旨在研究由完全非线性椭圆型和抛物型方程引起的微分几何问题。目的是更好地理解几何量和这些重要的几何全非线性方程的性质之间的关系。在这个建议中发展的创新技术将导致解决几何中的重要问题，并丰富现有的全非线性偏微分方程理论。在这个过程中，这项工作对物理学和应用科学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.