黎曼流形上特征值问题的研究

The project mainly studies the eigenvalue problems of elliptic operators in Riemannian manifolds, the main contents and objectives are as follows...1.For the clamped plate problem of Dirichlet biharmonic operator on a bounded domain in the 2-dimensional Euclidean space, the project studies the estimates of lower order eigenvalues and their application. The difficulties of the problems are coming over the difficultiy of changing sign of the first eigenvalue, choosing suitable trial functions in admissible function class and reducing the error taken by using the Cauchy-Schwarz inequality...Firstly, this project studies the estimates of the upper bounds of the ratio of the second and third eigenvalues and the first one. Secondly, this project studies the estimates of the higher order eigenvlues. The estimate of the ratio of the second eigenvalue and the first one is one of the important contents of Payne-Pólya–Weinberger type Conjecture...2. For the Buckling problem of biharmonic operator on a bounded domain in the n-dimensional Riemannian manifolds, the project aims to come over the difficulty of Dirichlet inner product and get better eigenvalue inequality, using comparison theorems in Riemannian manifolds and integration on orthogonal groups. On this base, we can obtain the estimates of lower and upper bounds of higher order eigenvalue which are best possible in the sense of the order.

本项目旨在研究黎曼流形上椭圆算子的特征值问题，我们的主要内容和目标是：..1. 对于二维欧氏空间有界域上重调和算子的clamped plate问题，考虑其低阶特征值的上界估计及其应用。这个问题的难点在于如何克服第一特征函数变号的困难，在容许函数类中选出合适的测试函数，以及如何尽量减少在使用柯西-施瓦兹不等式时带来的误差。..本项目将首先考虑的是第二、三特征值与第一特征值比率的上界估计，然后利用此估计得到高阶特征值的更好上界估计，其中第二特征值与第一特征值比率的上界估计是Payne-Pólya–Weinberger 型猜想的主要内容之一。..2. 对于一般黎曼流形上的重调和算子Buckling特征值问题，我们希望克服狄里克莱内积带来的困难，利用黎曼几何的比较定理和正交群上的积分等方法，得到更好的特征值不等式，在此基础上得到高阶特征值在阶的意义下最优的上、下界估计。

特征值理论从20世纪69年代以来就是一个非常活跃的研究课题，它在微分几何、几何分析、数学物理以及复几何中有着广泛的应用。申请人在该项目的支持下，围绕着特征值及其相关问题展开研究，在Kodai Mathematical Journal, Pacific Journal of Mathematics, Journal of Mathematical Analysis and Applications, Manuscripta Mathematica, Journal of Functional Analysis, Canadian Journal of Mathematics,Journal of Geometry and Physics, Differential Geometry and its Applications,The Journal of Geometric Analysis等学术期刊上发表论文9篇，还有一篇论文被学术期刊Inthernational Mathematical Research Notice (IMRN)接受发表。