A comparison theorem for eigenvalues of the Newton potential.

Abstract

Dedicated to Prof Mukhtarbay Otelbaev on the occasion of his seventieth birthday There is a wealth of interesting results comparing between Dirichlet and Neumann eigenvalues. In this paper we compare the eigenvalues of the Newton potential with the Dirichlet eigenvalues and the Neumann eigenvalues in a bounded domain in Rd. First we show that the spectral problem of the Newton potential is equivalent to a spectral problem of a non-local boundary value problem of the Laplace operator then it is proved that the nth eigenvalue of the Laplacian with the non-local boundary condition is between the nth eigenvalue of Neumann Laplacian and the nth eigenvalue of Dirichlet Laplacian in a bounded domain of any dimensional euclidian space.