非线性偏微分方程解的微观凸性及其几何应用

项目名称： 非线性偏微分方程解的微观凸性及其几何应用

项目编号： No.11301497

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 陈传强

作者单位： 浙江工业大学

项目金额： 22万元

中文摘要： 解的几何性质是非线性偏微分方程理论中的一个基本问题，它与方程解的正则性、存在性及唯一性等有密切联系。对方程解的凸性研究，既是分析研究的重要内容，也是研究方程本身的需要。到目前为止，偏微分方程的凸性的研究主要有两个方法：一是从宏观角度利用弱极值原理的宏观凸性方法；二是从微观角度出发利用强极值原理的微观凸性方法（即常秩定理）。对椭圆方程解的凸性已经有一系列的研究，对抛物方程的研究更复杂，结果较少。本项目将讨论一些抛物方程的解的时空凸性，包括解的时空凸性、空间水平集和时空水平集的凸性，并研究其几何应用；同时对热球和几何流的时空第二基本形式进行研究。进而将加深对抛物偏微分方程解的形状的了解，对方程和几何的研究都有意义。

中文关键词： 时空凸性；水平集；常秩定理；正则性；曲率

英文摘要： The geometric property of solutions to partial differential equations is a fundamental issue, which is in close connect with the regularity, the existence and the uniqueness of solutions. The study of the convexity of solutions is both the important content of the analysis research and the study of partial differential equations. Until now, there are two important methods to approach the convexity of the solutions of partial differential equations, which are the macroscopic method with the weak maximum principle and the microscopic method with the strong maximum principle (that is contant rank theorem method). There are series of literature devoted to the covexity of solutions to elliptic equations, but fewer to the parabolic equations. This project aims to study the spacetime convexity of solutions to parabolic partial differential equaions, inculding the spacetime convexity of solutions, the convexity of spatial level sets and spacetime level sets, and the geometric applications. Also the heat ball and the spacetime second fundamental form of geometric flows will be concerned. This research will promote the understanding of the geometric shape of the solutions to parabolic partial differential equations, which has great meanings to the study of partial differential equations and geometry.

英文关键词： spacetime convexity；level set；constant rank theorem；regularity；curvature