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.

A note on the Neumann eigenvalues of the biharmonic operator / Provenzano, Luigi. - In: MATHEMATICAL METHODS IN THE APPLIED SCIENCES. - ISSN 0170-4214. - 41:3(2018), pp. 1005-1012. [10.1002/mma.4063]

A note on the Neumann eigenvalues of the biharmonic operator

Luigi Provenzano
2018

Abstract

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.
2018
biharmonic operator; eigenvalues; Neumann boundary conditions; Poisson's ratio; Mathematics (all); Engineering (all)
01 Pubblicazione su rivista::01a Articolo in rivista
A note on the Neumann eigenvalues of the biharmonic operator / Provenzano, Luigi. - In: MATHEMATICAL METHODS IN THE APPLIED SCIENCES. - ISSN 0170-4214. - 41:3(2018), pp. 1005-1012. [10.1002/mma.4063]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1446688
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