实和复的非线性椭圆偏微分方程的诺伊曼边值问题及几何应用

The nonlinear elliptic partial differential equation is a basic study project, including the existence、 the regularity and the uniqueness of solutions，etc., which is closed related to some geometric problems. By the late 1970s, the theory of nonlinear elliptic equations is applied to the study of Geometry, and it plays an important role. A seires of important geometric problems related to nonlinear elliptic equations were solved, including the Yamabe problem, the positive mass conjecture and the Calabi conjecture. A priori estimates and the regularity theory of elliptic equaitons play a crucial role in the solutions of these problems. Till now, the Dirichlet problems of fully nonlinear elliptic PDEs have been widely studied and there are many important applications. But for the corresponding Neumann problems, there are still many open problems. This project studies the a priori estimates and the regularity theory of some real and complex fully nonlinear elliptic partial differential equations, including the interior and the global a priori estimates of Neumann problems, and establish the existence theory and the regularity theory. Also, we apply these theories to study some relevant geometric problems such as the Yamabe problems with boundary and the geometric inequalities.

非线性椭圆偏微分方程是一个基本研究对象，其中解存在性、正则性及唯一性等都是重要的研究课题，并且与一些几何问题有密切联系。从上世纪七十年代末期开始，非线性椭圆型方程理论应用到了几何学的研究中，并且起了重要作用。一系列与非线性椭圆型方程相关的重要几何问题得到了解决，这包括 Yamabe 问题，正质量猜想以及 Calabi 猜想。椭圆型方程解的先验估计和正则性在这些问题的解决中起了关键作用。目前，完全非线性椭圆偏微分方程的 Dirichlet 问题得到比较充分的研究和应用，相应的诺伊曼边值问题仍有很多问题未知。本项目将研究实和复的完全非线性偏微分方程的先验估计和正则性，包括内部先验估计和诺伊曼边值问题解的整体先验估计等，建立解的存在性和正则性的理论，以及应用这些理论研究带边的高阶 Yamabe 问题和几何不等式等相关的几何问题。

非线性椭圆偏微分方程是一个基本研究对象，其中解存在性、正则性及唯一性等都是重要的研究课题。椭圆型方程解的先验估计在这些问题的解决中起了关键作用。目前，完全非线性椭圆偏微分方程的 Dirichlet 问题得到比较充分的研究和应用，相应的诺伊曼边值问题仍有很多问题未知。本项目研究实和复的完全非线性偏微分方程诺伊曼边值问题的先验估计，进而得到解的存在性和正则性理论。并研究一些相关的几何问题。..项目执行期间，项目负责人及其团队认真深入地研究了相关问题。项目负责人发表SCI 论文5 篇，均在国际权威杂志 Math. Annalen, Bulletin of Mathematical Sciences J. Differential Equations, International Mathematics Research Notices等。