Transactions of the Moscow Mathematical Society

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Asymptotic expansion of eigenelements of the Laplace operator in a domain with a large number of `light' concentrated masses sparsely situated on the boundary. Two-dimensional case


Author: G. A. Chechkin
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 70 (2009).
Journal: Trans. Moscow Math. Soc. 2009, 71-134
MSC (2000): Primary 35J25; Secondary 35B25, 35B27, 35B40
Published electronically: December 3, 2009
MathSciNet review: 2573638
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Abstract: This paper looks at eigenoscillations of a membrane containing a large number of concentrated masses on the boundary. The asymptotic behaviour of the frequencies of eigenoscillations is studied when a small parameter characterizing the diameter and density of the concentrated masses tends to zero. Asymptotic expansions of eigenelements of the corresponding problems are constructed and the expansions are accurately substantiated. The case where the diameter of the masses is much smaller than the distance between them is investigated under the assumption that the limit boundary condition is still a Dirichlet condition.


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Additional Information

G. A. Chechkin
Affiliation: Moscow State University, Moscow, Russia
Email: chechkin@mech.math.msu.su

DOI: http://dx.doi.org/10.1090/S0077-1554-09-00177-0
Keywords: Laplace operator, eigenoscillations, asymptotic expansions, singular perturbations
Published electronically: December 3, 2009
Additional Notes: This research was partially supported by the Russian Foundation for Basic Research (grant # 09–01–00530a) and by the Programme for Support of Leading Scientific Schools (grant # NSh–1698.2008.1).
Article copyright: © Copyright 2009 American Mathematical Society