Asymptotic behaviour of spectrum of Laplace-Beltrami operator on Riemannian manifolds with complex microstructure

Author: Khrabustovskyi, Andrii

Source: Applicable Analysis, Volume 87, Number 6, June 2008 , pp. 735-750(16)

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Abstract:

The article deals with a convergence of the spectrum of the Laplace-Beltrami operator Δε on a Riemannian manifold depending on a small parameter ε > 0. This manifold consists of a domain Ω ⊂ n with a large number of small 'holes' whose boundaries are glued to the boundaries of the n-dimensional spheres with small truncated segment. The number of the 'holes' increases, as ε → 0, while their radii tend to zero. We prove that the spectrum converges to the spectrum of the homogenized operator having (in contrast to Δε) a non-empty essential spectrum.

Keywords: homogenization; Laplace-Beltrami operator; spectrum; Riemannian manifold

Document Type: Research article

DOI: http://dx.doi.org/10.1080/00036810802213249

Affiliations: 1: Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Kharkiv, Ukraine

Publication date: 2008-06-01

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