几何中完全非线性椭圆偏微分方程的斜边值问题

In this project, we mainly study three oblique boundary value problems for nonlinear elliptic equations in geometry: One is the k-Yamabe problem with boundary in Euclidean space; Two is semilinear oblique boundary value problem of Monge-Ampere type equation; Three is the prescribed contact angle problem of Hessian equation. k-Yamabe problem with boundary is derived from the classical Yamabe problem in conformal geometry, which is in fact the Neumann boundary value problem; Semilinear oblique boundary value problem is the generalization of the Neumann boundary value problem; The prescribed contact angle problem arises from capillary problems in physics. The study of the existence of solutions for the boundary value problems in PDEs is one of the basic problems. By choosing different suitable auxiliary functions, we establish the a prior estimates up to the second order derivatives of solutions separately. And we obtain the existence results following the standard elliptic theory. The study of this project will fulfill the development of the basic theory of elliptic equations and promote the development of conformal geometry. Besides that, our methods or techniques are relatively new. So we strive to make progress on these issues.

本项目主要研究几何中完全非线性椭圆方程的以下三个斜边值问题的解的存在性：一是欧氏空间中带边 k-Yamabe 问题；二是球上 Monge-Ampere 方程半线性斜边值问题；三是 Hessian 方程的预定夹角问题。带边 k-Yamabe 问题来自于共形几何中的经典 Yamabe 问题，本质上是 Neumann 问题；半线性斜边值问题是 Neumann 边值问题的推广；预定夹角问题来自于物理中的毛细问题。解的存在性是偏微分方程的边值问题研究中的基本问题之一。通过选取不同的适当的辅助函数，分别建立解的直到二阶导数的先验估计。运用椭圆方程的标准理论，分别得到这三个边值问题的解的存在性。此课题的研究将完善椭圆方程基本理论的发展，并且对共形几何的发展起到很好的推动作用。同时研究的方法或技巧有较多创新，力求对这些问题有所进展。

本项目主要研究几何中完全非线性椭圆方程的斜边值问题的解的存在唯一性，主要包括平均曲率方程及平均曲率流的的预定夹角边值问题、.Neumann边值问题及一般的斜导数边值问题。半线性斜边值问题是 Neumann 边值问题的推广，预定夹角问题来自于物理中的毛细问题。.解的存在唯一性是偏微分方程边值问题研究中的基本并且非常重要的问题之一。通过选取不同的适当的辅助函数，运用极大值原理的方法.分别建立解的最大模估计、梯度估计、二阶导数估计等的先验估计。然后再由椭圆偏微分方程的标准理论，将分别得到这三个边值问题的解的存在性。.并将这些结果推广到黎曼流形上及相应的抛物方程。此课题的研究将完善椭圆偏微分方程的基本理论，并且对其他完全非线性椭圆方程比如Monge-Ampere 方程、K-Hessian 方程的边值问题的解决起到很好的推动作用。