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Trapped modes in an elastic plate with a hole


Authors: C. Förster and T. Weidl
Translated by: V. Sloushch
Original publication: Algebra i Analiz, tom 23 (2011), nomer 1.
Journal: St. Petersburg Math. J. 23 (2012), 179-202
MSC (2010): Primary 74B05
DOI: https://doi.org/10.1090/S1061-0022-2011-01192-6
Published electronically: November 10, 2011
MathSciNet review: 2760154
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Abstract: For an infinite linear elastic plate with stress-free boundary, the trapped modes arising around holes in the plate are investigated. These are $ L^2$-eigenvalues of the elastostatic operator in the punched plate subject to Neumann type stress-free boundary conditions at the surface of the hole. It is proved that the perturbation gives rise to infinitely many eigenvalues embedded into the essential spectrum. The eigenvalues accumulate to a positive threshold. An estimate of the accumulation rate is given.


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Additional Information

C. Förster
Affiliation: Institute for Analysis, Dynamics, and Modeling, Department of Mathematics and Physics, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: foerster@mathematik.uni-stuttgart.de

T. Weidl
Affiliation: Institute for Analysis, Dynamics, and Modeling, Department of Mathematics and Physics, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: weidl@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/S1061-0022-2011-01192-6
Keywords: Elasticity operator, trapped modes
Received by editor(s): October 1, 2010
Published electronically: November 10, 2011
Dedicated: In the memory of our dear teacher M. Sh. Birman
Article copyright: © Copyright 2011 American Mathematical Society