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

Abstract

The paper deals with the asymptotic behavior as $\eps\to 0$ of the spectrum of Laplace-Beltrami operator $\Delta\e$ on the Riemannian manifold $M\e$ ($\mathrm{\dim} M\e=N\geq 2$) depending on a small parameter $\eps>0$. $M\e$ consists of two perforated domains which are connected by array of tubes of the length $q\e$. Each perforated domain is obtained by removing from the fix domain $\Omega\subset \mathbb{R}^N$ the system of $\eps$-periodically distributed balls of the radius $d\e=\bar{o}(\eps)$. We obtain a variety of homogenized spectral problems in $\Omega$, their type depends on some relations between $\eps$, $d\e$ and $q\e$. In particular if the limits $\liml_{\eps\to 0}q\e$ and $\liml_{\eps\to 0}\ds{(d\e)^{N-1}q\e \eps^{-N}}$ are positive then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.