A note on the Neumann eigenvalues of the biharmonic operator

TL;DR

This paper investigates how Neumann eigenvalues of the biharmonic operator depend on Poisson's ratio, showing Lipschitz continuity and the convergence behavior as the ratio approaches 1, revealing connections to Dirichlet eigenvalues.

Contribution

It establishes the Lipschitz continuity of Neumann eigenvalues with respect to Poisson's ratio and describes their asymptotic behavior as the ratio approaches 1.

Findings

Neumann eigenvalues are Lipschitz continuous in Poisson's ratio.

All Neumann eigenvalues tend to zero as Poisson's ratio approaches 1.

Eigenvalues at Poisson's ratio 1 include positive eigenvalues not limiting to those at lower ratios.

Abstract

We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to $\sigma\in [0,1[$ and that all the Neumann eigenvalues tend to zero as $\sigma\rightarrow 1^-$. 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 $\sigma\in[0,1[$ as $\sigma\rightarrow 1^-$, and which coincide with the Dirichlet eigenvalues of the biharmonic operator.