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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary


Authors: V. A. Kozlov and S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
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
Published electronically: August 19, 2011
MathSciNet review: 2760089
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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.


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Additional 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
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/S1061-0022-2011-01178-1
Keywords: Biharmonic operator, Dirichlet problem, asymptotic expansions of eigenvalues, eigenoscillations of the Kirchhoff plate, rapid oscillation and nonregular perturbation of the boundary.
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)
Dedicated: Dedicated to Vasiliĭ Mikhaĭlovich Babich
Article copyright: © Copyright 2011 American Mathematical Society