N維球上笛里西與諾伊曼特徵值之比較

Abstract

本文旨在討論球面上凸區域之笛里西與諾伊曼特徵值之比較.我們導出以 下結果:如果球面上凸區域邊界的均區率非正,則每組相對應的特徵值,諾 伊曼將小於等於笛里西.而且,如果有一組特徵值諾伊曼等於笛里西,則此 邊界為球面上的一個極小曲面. Let M be a compact domain in the n-sphere with smooth boundary. Assume that the mean curvature h of the boundary of M is nonpositive. We prove that the k-th Neumann eigenvalue is less than or equal to the k-th Dirichlet eigenvalue of M. Moreover, these inequalites are strict unless the boundary of M is minimal.