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Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots

✍ Scribed by Youcef Amirat; Gregory A. Chechkin; Rustem R. Gadyl'shin

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
383 KB
Volume
33
Category
Article
ISSN
0170-4214

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✦ Synopsis

Abstract

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a two‐dimensional bounded domain with thin shoots, depending on a small parameter Ξ΅. Under the assumption that the width of the shoots goes to zero, as Ξ΅ tends to zero, we construct the limit (homogenized) problem and prove the convergence of the eigenvalues and eigenfunctions to the eigenvalues and eigenfunctions of the limit problem, respectively. Under the additional assumption that the shoots, in a fixed vicinity of the basis, are straight and periodic, and their width and the distance between the neighboring shoots are of order Ξ΅, we construct the two‐term asymptotics of the eigenvalues of the problem, as Ξ΅β†’0. Copyright Β© 2009 John Wiley & Sons, Ltd.

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## Abstract For certain unbounded domains the Laplace operator with Dirichlet condition is shown to have an unbounded sequence of eigenvalues which are embedded into the essential spectrum. A typical example of such a domain is a locally perturbed cylinder with circular cross‐section whose diameter