本项目主要研究树上生灭过程的收敛速度和p-Laplacian主特征值的定量估计问题。. 首先,从Poisson方程出发，利用变分公式、对偶方法等研究树上生灭过程的收敛速度估计问题, 也即 p=2时p-Laplacian特征值估计；其次,借鉴树上生灭过程收敛速度的方法，以泛函不等式为工具，研究p-Laplacian特征值(1<p<\infty)定量估计。这些内容可类比一维空间中生灭过程的相关结论，为克服由一维空间到树上导致的拓扑结构改变带来的问题，我们主要采取首先考虑特殊情形、类比方法，对树结构进行分解等方法展开研究。.基于p-Lapalcian特征值与泛函不等式的关系，本项目的研究内容一定程度上弥补了定量估计泛函不等式最优常数的理论空隙，另外，最近发现收敛速度与用于设计网络算法的cutoff现象等密切相关，因此，本项目的研究内容从理论和应用上均有重要价值。

The program is proposed to investigate the estimates of coverage speed for birth-death processes on trees and p-Laplacian principal eigenvalues.. Firstly, we study the coverage speed for birth-death processes on trees by taking advantage of variational formulas and dual methods. That is the p-Laplacian principal eigenvalues with p=2. Secondly, we will estimate the p-Laplacian(1<p<\infty) principal eigenvalues on trees by functional inequalities. These contents studied can be fully considered by comparing them with corresponding results obtained for birth-death processes in dimension one. In order to overcome the difficulties encountered duo to the changes of the Topo structure, we can consider the simple cases first, and then the general cases, splitting technic and so on..We know that the p-Laplacian eigenvalues are closely related to the best constant of functional inequalities, to some extent, the contents studies in the program complement the gap of estimating the best constant of functional inequalities. Besides, a cutoff phenomenon which is useful for algorithm designed in web and so on are closely related to the coverage speed. The research contents of this project from the theory to applications have important value.