关于分数次Laplacian的一些问题的研究

We will study several problems related to fractional Laplacian. The main contents of this project are as follow: (1) consider the regularity lifting estimate of Dirichlet problem of fractional Laplacian,(2)consider the regularity lifting estimate of boundary problem of fractional second order elliptic equation,(3) study the sharp fractional hardy type inequality on mean convex domain, (4) Some applications of fractional Hardy inequality in PDE: We will focus on the regularity, Liouville type property, eigenvalue problem of fractional elliptic equation with singular lower order term and the analysis on symmetric group of several kinds of nonlinear equations. .The development of the present project is useful to understand some relationship between the subjects such as Harmonic analysis , PDE and related subjects, moreover it is useful for the future research in related fields.

本项目研究与分数次Laplacian相关的几个问题。主要研究内容为:（1）分数次Laplacian算子的Dirichlet边值问题解的正则提升估计；（2）分数次二阶椭圆型算子的边值问题解的正则提升估计；（3）在平均凸区域上研究带余项的最佳分数次Hardy不等式；（4）分数次Hardy型不等式在分数次Laplacian方程边值问题中的应用研究，如带奇异低阶项的分数次椭圆型方程的正则性，Liouville型结果,特征值问题以及几类非线性方程的对称群分析。本项目的研究将有助于探索调和分析,PDE和相关学科之间的某些联系，并为今后的相关研究奠定基础。

项目组主要研究了与分数次Laplacian相关的几个问题。主要研究成果为:（1）与分数次Laplacian相关的几类PDE定解问题适定性研究工作；（2）与分数次Laplacian相关的积分算子的有界性研究工作；（3）在平均凸区域上研究带余项的最佳分数次Hardy不等式及其应用研究。项目组共发表与本项目相关的学术论文13篇，主办了多次调和分析学术会议，培养了21名研究生。本项目的研究工作加强了调和分析,PDE和相关学科之间的联系，并为今后的相关研究奠定基础。