幂零流形上的热核分析与应用

项目名称： 幂零流形上的热核分析与应用

项目编号： No.11426109

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 冯声涯

作者单位： 华东理工大学

项目金额： 3万元

中文摘要： 幂零流形是现代数学研究的重要领域, 并与其它数学分支建立了广泛的联系. 该项目的研究将围绕幂零流形上次Laplace算子的热核展开. 研究内容包括三部分: (1)次Laplace热核的Gauss估计和梯度估计; (2) Poincaré等式和Log–Sobolev型不等式; (3)热核测度的Talagrand型不等式. 该研究将使我们看到黎曼流形上经典的分析结果在幂零流形上的推广和刻画, 更深入地了解幂零流形上算子和方程的结构. 本项目的独到之处在于: (1) 次Laplace热核的精确解以及相关估计将完整地得到研究; (2)Hamilton-Jacobi理论统一了各项研究内容; (3)研究方法和技术手段与多个学科分支交叉渗透.

中文关键词： 幂零流形；Hamilton系统；热核；周期解；泛函不等式

英文摘要： In modern mathematics, nilmanifolds are important research areas, and they are extensively linked with other branches of mathematics. The objectives of this research project are to study the heat kernels of sub-Laplacians on nilmanifolds. This project consists of three parts. The first part is dedicated to Gaussian estimates and gradient estimates for the heat kernels of sublaplacians based on their explicit formulae. Inequlities of Poincaréype and Log–Sobolev type will be considered in the second part. In the last part, we discuss Talagrand type inequalities associated to the heat kernel measures on nilmanifolds. This research generalises the classical results on Riemannian manifolds to the sub-Riemannian setting, and leads our deep understanding to the structure of operators and equations on sub-Riemannian manifolds. The innovation of this research consists in (1) that the exact formulae of the heat kernels and related estimates are completely studied, (2) that Hamilton-Jacobi theory unifies each part of this research and (3) that the research methods and technical points cross many other branches of mathematics.

英文关键词： nilmanifold；Hamilton system；heat kernel；periodic solution；functional inequality