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A note on the Neumann eigenvalues of the biharmonic operator
Author(s) -
Provenzano Luigi
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4063
Subject(s) - biharmonic equation , mathematics , eigenvalues and eigenvectors , neumann boundary condition , lipschitz continuity , mathematical analysis , boundary value problem , lipschitz domain , operator (biology) , pure mathematics , biochemistry , chemistry , physics , repressor , quantum mechanics , transcription factor , gene
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.