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 | References | Similar Articles | Additional Information

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:
https://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