The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary
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V. A. Kozlov and S. A. Nazarov
Translated by: A. Plotkin - St. Petersburg Math. J. 22 (2011), 941-983
- DOI: https://doi.org/10.1090/S1061-0022-2011-01178-1
- Published electronically: August 19, 2011
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Abstract:
Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharmonic operator in a domain with highly indented and rapidly oscillating boundary (the Kirchhoff model of a thin plate). The asymptotic constructions depend heavily on the quantity $\gamma$ that describes the depth $O(\varepsilon ^\gamma )$ of irregularity ($\varepsilon$ is the oscillation period). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible, in particular, to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula.References
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Bibliographic Information
- V. A. Kozlov
- Affiliation: Department of Mathematics, Linkoping University, 581 83 Linkoping, Sweden
- Email: vlkoz@mai.liu.se
- S. A. Nazarov
- Affiliation: Institute of Mechanical Engineering Problems, Russian Academy of Sciences, Bol′shoĭ Pr. V.O. 61, St. Petersburg 199178, Russia
- MR Author ID: 196508
- Email: srgnazarov@yahoo.co.uk
- Received by editor(s): June 15, 2010
- Published electronically: August 19, 2011
- Additional Notes: The paper was written during S. A. Nazarov’s visit to the University of Linköping, whose financial support is acknowledged gratefully. Also, V. A. Kozlov thanks the Swedish Research Council (VR), and S. A. Nazarov thanks RFBR (project no. 09-01-00759)
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 941-983
- MSC (2010): Primary 35J40; Secondary 31A25, 31A30, 35P20
- DOI: https://doi.org/10.1090/S1061-0022-2011-01178-1
- MathSciNet review: 2760089
Dedicated: Dedicated to Vasiliĭ Mikhaĭlovich Babich