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.

Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator

“…In particular, the boundary conditions do not satisfy the complementing conditions, see [1,27]. We refer to [42] for considerations on the spectrum of problem (12).…”

Spectral Analysis of the Biharmonic Operator Subject to Neumann Boundary Conditions on Dumbbell Domains

“…[28,Example 2.15] where problem (1.3) with τ = 0 is referred to as the Neumann problem for the biharmonic operator. Moreover, we point out that a number of recent papers devoted to the analysis of (1.2) have con rmed that problem (1.2) can be considered as the natural Neumann problem for the biharmonic operator, see [7], [8], [10], [11], [12], [13], [14], [29]. We also refer to [22] for an extensive discussion on boundary value problems for higher order elliptic operators.…”

The Bilaplacian with Robin boundary conditions

“…While in the Dirichlet case this coefficient has no effect whatsoever on the problem, in the other cases it is an important parameter. We refer to [45] for further discussion of the Poisson coefficient (see also [72]).…”