A note on the Neumann eigenvalues of the biharmonic operator

Abstract

We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio σ. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to σ∈[0,1[and that all the Neumann eigenvalues tend to zero as σ→1−. Moreover, we show that the Neumann problem defined by setting σ = 1 admits a sequence of positive eigenvalues of finite multiplicity that are not limiting points for the Neumann eigenvalues with σ∈[0,1[as σ→1− and that coincide with the Dirichlet eigenvalues of the biharmonic operator. Copyright © 2016 John Wiley & Sons, Ltd.