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

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 ⊂ ℝ 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.