Mathematics – Spectral Theory
Scientific paper
2010-11-17
Mathematics
Spectral Theory
The second version involves an additional theorem (Theorem 4.7) and some other improvements. The manuscript submitted for publ
Scientific paper
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.
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