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 Δ^ε 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 Δ^ε) a non-empty essential spectrum.

Keywords:

• homogenization
• Laplace–Beltrami operator
• spectrum
• Riemannian manifold

AMS Subject Classifications: :

• 35B27
• 35P20
• 58G25
• 58G30

Acknowledgements

The author is grateful to Prof. E.Ya. Khruslov for the statement of this problem and the attention he paid to this work. The work is partially supported by the Grant of NASU for the young scientists No. 20207.