A note on the Neumann eigenvalues of the biharmonic operator

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.