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

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: 10.1080/00036810802213249

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