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

We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We analyse the spectral behaviour of the operator, characterizing the limit of the eigenvalues and of the eig...

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Bibliographic Details
Published inIntegral equations and operator theory Vol. 89; no. 3; pp. 377 - 408
Main Authors Arrieta, José M., Ferraresso, Francesco, Lamberti, Pier Domenico
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.11.2017
Springer Nature B.V
Subjects
Analysis
Boundary conditions
Eigenvalues
Elastic deformation
Elasticity
Mathematics
Mathematics and Statistics
Secondary 35P15
35B25
Dumbbell domains
35J35
Spectral analysis
Biharmonic operator
Primary 35J40
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Summary:We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We analyse the spectral behaviour of the operator, characterizing the limit of the eigenvalues and of the eigenprojections as the thickness of the channel goes to zero. In applications to linear elasticity, the fourth order operator under consideration is related to the deformation of a free elastic plate, a part of which shrinks to a segment. In contrast to what happens with the classical second order case, it turns out that the limiting equation is here distorted by a strange factor depending on a parameter which plays the role of the Poisson coefficient of the represented plate.
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ISSN:0378-620X
1420-8989
DOI:10.1007/s00020-017-2391-9