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Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes

Published online by Cambridge University Press:  03 December 2013

Andrii Khrabustovskyi*
Affiliation:
B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Avenue, Kharkiv 61103, Ukraine
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Abstract

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The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limits

are positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2013