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

Chechkin, G.

doi:10.1090/S0077-1554-09-00177-0

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
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by G. A. Chechkin
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2009, 71-134

DOI: https://doi.org/10.1090/S0077-1554-09-00177-0

Published electronically: December 3, 2009

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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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