Estimate For Eigenvalue Problems

Eigenvalues problem has been widely used in mathematical physics. In order to study them, we need some methods in Geometric Analysis. Therefore how to estimate the eigenvalues becomes an important issue in Geometric Analysis. During the past decades, many authors had paid attention to this problem and made a comprehensive study on it, including Jeff cheeger, Shing-Tung Yau[3,15,16], Shiu-Yuen Cheng, etc. In this paper, we do research on the eigenvalues problem based on the previous theories.Main work in this paper are summarized as follows:In chapter 1, At first, we present a delicate introduction to the problem of the Dirichlet eigenvalues. At the same time the Laplace-Beltrami operator is introduced. Then we also study some basic properties of the Dirichlet eigen-values. Finally we discuss about some theories of partial differential equations which will be used in this paper.In chapter 2 we mainly make research on the problem of estimation of the Dirichlet eigenvalues. At the beginning, we enumerate some theories which has been proved many years ago. For example, Q.-H. Yu and J.-Q. Zhang had given a lower-bound estimation of the gap between the first and second eigenvalues of the Schrodinger Operator, that isλ2-λ1≥π2/d2. Based on it, I do some research on the problem of the Dirichlet eigenvalues. That is, the estimation can be improved toλ2-λ1≥3π2/d2 on the condition of sup u2/u1≥11 inf u2/u1 or sup u2/u1≤1/11 inf u2/u1.