Inequalities between Dirichlet and Neumann eigenvalues for domains in spheres

Abstract

Let M be a domain in the unit n-sphere with smooth boundary. The purpose of this paper is to describe some inequalities between Dirichlet and Neumann eigenvalues for M under certain convex restrictions on the boundary. We prove that if the mean curvature of the boundary is nonpositive, then the kth nonzero Neumann eigenvalue is less than or equal to the kth Dirichlet eigenvalue for k = 1, 2, Furthermore, if the second fundamental form of the boundary is nonpositive, then the (k + [n-1/2])th nonzero Neumann eigenvalue is less than or equal to the kth Dirichlet eigenvalue for k = 1, 2, (. . .).