Heisenberg群与Minkowski空间中的非线性椭圆方程

项目名称： Heisenberg群与Minkowski空间中的非线性椭圆方程

项目编号： No.11471188

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 王培合

作者单位： 曲阜师范大学

项目金额： 60万元

中文摘要： 本项目拟做三个方面的工作。 1.Heisenberg群上的Hessian测度最近由Trudinger和张伟给出了定义，我们希望在此基础上研究相关的积分估计，进而得到Hessian测度的弱连续性和Wolfff位势估计。同时也考虑有关完全非线性方程的临界指标和次临界指标的Liouville定理和奇性刻画。 2.Minkowski空间中预定曲率的类空超曲面是长久以来人们关心的问题，预定平均曲率方程的Dirichlet问题已经有充要条件，我们拟对预定平均曲率方程的Neumann问题以及预定高阶（大于2阶）曲率的Dirichlet边值问题解的存在性进行研究。 3.在Minkowski空间中，极大类空超曲面有着重要几何、物理背景和应用。我们希望得到该曲面水平集的定性和定量的凸性刻画。我们也将考虑Minkowski空间中常平均曲率类空超曲面的水平集的凸性问题，该问题在欧氏空间中有反例说明水平集没有凸性。

中文关键词： Heisenberg群；类空超曲面；外在曲率；水平集

英文摘要： We prepare to discuss the following three problems. 1. Recently, Prof. Trudinger and Zhangwei gave the welldefined Hessian measure on Heisenberg group. We want to follow this to study the integral estimate and to deduce its weak continuity and the Wolff potential estimate. Also, we want to discuss Liouville theorems and the properties of the sigularities with respect to the critical and subcritical index problems of fully nonlinear equations on the Heisenberg group. 2.In the Minkowski space,the equations of spacelike hypersurface with prescribed curvature is an interesting problem for a long history. Now, we have already known the sufficient and the necessary conditions for the Dirichlet problem of mean extrinsic curvature. Here, we sduty the corresponding Neumann problem and we also want to discuss the Dirichlet problem of prescribed high oder (bigger than 2) curvature equations. 3.In the Minkowski space,the maximal spacelike hypersurface is one of the important subjects and it has important geometric and physical background. We expect to draw out the properties of the level sets of this object from qualitative and quantitative aspects. Also, we will consider the convexity of the level sets of the constant mean extrinsic curvature spacelike surfaces, the similar problem in Euclidean space turned to have no convex level sets by a famous counterexample.

英文关键词： Heisenberg Group;spacelike hypersurface;extrinsic curvature;level sets