本計畫以研究黎曼流形上的 Dirichlet 固有值問題，特別是關於高階固有值的估計問題，以及相關的應用為主要研究對象。我們藉由探討 P.Li 教授與丘成桐教授對於歐式空間中有界區域的 Dirichlet 問題與高階固有值的估計所採用的方法，研究其推廣至一般黎曼流形的可能性。同時，我們也討論有關黎曼流行上的隨機測地方程式的相關問題，利用 S.Helgason 教授對於對稱空間上之傅立葉轉換的相關探討，我們給出這類流形上之隨機測地方程式的初步結果。 This project focuses on the investigations of the Dirichlet eigenvalue problem in Riemannian manifolds, especially for the estimate of higher eigenvalues of Laplace operator and its applivations. By the methods in P.Li and Yau？？ s work about studying the Dirichlet problem and the estimate of higher eigenvalues in a bounded domain of Euclidean space, we investigate the possibility of extending their work to general Riemannian manifolds. Meanwhile, we also discuss the related problems about the stochastic geodesic on Riemannian manifolds. By S.Helgason？？ s results about Fourier transforms on symmetric spaces, we give partial results about the stochastic geodesic equations on these manifolds.
