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 $sigma$. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to $sigmain[0,1[$ and that all the Neumann eigenvalues tend to zero as $sigma ightarrow 1^-$. Moreover, we show that the Neumann problem defined by setting $sigma=1$ admits a sequence of positive eigenvalues of finite multiplicity which are not limiting points for the Neumann eigenvalues with $sigmain[0,1[$ as $sigma ightarrow 1^-$, and which coincide with the Dirichlet eigenvalues of the biharmonic operator.