关于几何偏微分方程的若干研究

发布时间：2018-07-06 15:06

本文选题：先验估计 + 四元数Monge-Ampere方程　；参考：《中国科学技术大学》2017年博士论文

【摘要】：本文的主要内容有两个部分,分三章.第一部分包括第二章和第三章,分别研究了二阶椭圆方程中做先验估计的两种方法——下解方法和P-函数方法.第二章以四元数Monge-Ampere方程的Dirichlet问题为例,利用下解方法我们推导出解在一般区域上的先验估计,并以此得到一个存在性定理.从而,我们可以回答S.Alesker在文献[4]中提出的第四个问题.第三章主要考虑三维Minkowski空间中常平均曲率方程的Dirichlet问题.基于解的临界点的唯一性,我们利用P-函数方法改进文献[14]中的梯度估计,从而,获得解的高度估计.最后,在本文的第二部分(即第四章),我们集中讨论双曲空间中的浸入超曲面和球的区域上共形度量之间的关系,并将文献[17]中的整体对应关系推广至二维。作为应用,我们得到一个Liouville型定理和一个新的更强的Bernstein型定理.
[Abstract]:The main content of this paper has two parts, divided into three chapters. The first part includes the second chapter and the third chapter. We study two methods for the priori estimation of the second order elliptic equation, the lower solution method and the P- function method, respectively. In chapter 2, taking the Dirichlet problem of the quaternion Monge-Ampere equation as an example, we derive a priori estimate of the solution in a general domain by using the lower solution method, and obtain a existence theorem. Thus, we can answer the fourth question raised by S. Alesker in [4]. In chapter 3, the Dirichlet problem for the equation of constant mean curvature in three dimensional Minkowski space is considered. Based on the uniqueness of the critical point of the solution, we use the P- function method to improve the gradient estimation in reference [14] and obtain the height estimate of the solution. Finally, in the second part of this paper (chapter 4), we focus on the relationship between the conformal metric in hyperbolic space and the conformal metric of the sphere, and generalize the global correspondence relation in [17] to two dimensions. As applications, we obtain a Liouville type theorem and a new stronger Bernstein type theorem.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.2;O186.1

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