共形几何中的非局部偏微分方程

The conformal mapping is one of the most important topics in complex analysis of one variable, and has been used widely in the areas of hydromechanics, aerodynamics, elastomechanics, electromagnetic field and thermal field theory and etc. The classical Riemann mapping theorem asserts that every simply connected domain in the complex plane, whose boundary is more than one point, is conformal to the disk. This project considers a generalization of Riemann mapping theorem: The constant Q-curvature problem at the asymptotical boundaries of Poincaré-Einstein manifolds. This problem is equivalent to solving a nonlocal partial differential equation (PDE). If Q-curvature is coincident to mean curvature, Escobar [Ann Math 92] and a series of work of Professor F. Marques at Princeton gave a satisfactory anwer. Based on the applicant’s previous work on the Nirenberg problem, this project takes the classical theory of second order elliptic PDEs and parabolic PDEs into account, develops analysis tools for nonlocal PDEs, proposes a variational method to solve the constant Q-curvature problem, establishes a priori estimates by searching a geometry quantity, and proves the long time existence and convergence of the nonlocal heat flow.

共形映射是单复变函数论的最重要研究课题之一, 被广泛地应用于流体力学、空气动力学、弹性力学、电磁场与热场理论等领域. 著名的黎曼映射定理说, 复平面上任何一个边界不止一点的单连通的区域都共形于单位圆盘. 本项目考虑黎曼映射定理推广形式: Poincaré-Einstein 流形的渐近边界上的常 Q-曲率问题. 这个问题等价于求解非局部偏微分方程. 当Q-曲率为平均曲率时, Escobar [Ann Math 92]和美国Princeton 大学教授F. Maruqes一系列工作给出满意回答. 基于申请人在 Nirenberg 问题的研究工作, 本项目从二阶椭圆型或抛物型偏微分方程的经典理论出发, 对非局部偏微分方程发展分析工具, 拟采用变分法解决存在性, 寻找整体几何量用以建立先验估计, 证明非局部热流的长时间存在性和收敛性.

共形映射是单复变函数论的最重要研究课题之一, 被广泛地应用流体力学、空气动力学、弹性力学、电磁场与热场理论等领域. 著名的黎曼映射定理说, 复平面上任何一个边界不止一点的单连通的区域都共形于单位圆盘. 本项目考虑黎曼映射定理推广形式: Poincaré-Einstein 流形的渐近边界上的常 Q-曲率问题. 这个问题等价于求解非局部偏微分方程. 当Q-曲率为平均曲率时, Escobar [Ann Math 92]和美国Princeton 大学教授F. Maruqes一系列工作给出满意回答. 基于申请人在 Nirenberg 问题的研究工作, 本项目从二阶椭圆型或抛物型偏微分方程的经典理论出发, 对非局部偏微分方程发展分析工具, 采用变分法解决存在性, 寻找整体几何量用以建立先验估计, 证明非局部热流的长时间存在性和收敛性.