# Homogenization of spectral problem on Riemannian manifold consisting of two domains connected by many tubes - Mathematics > Spectral Theory

Homogenization of spectral problem on Riemannian manifold consisting of two domains connected by many tubes - Mathematics > Spectral Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: The paper deals with the asymptotic behavior as $\eps\to 0$ of the spectrumof 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 thelength $q\e$. Each perforated domain is obtained by removing from the fixdomain $\Omega\subset \mathbb{R}^N$ the system of $\eps$-periodicallydistributed balls of the radius $d\e=\bar{o}\eps$. We obtain a variety ofhomogenized spectral problems in $\Omega$, their type depends on some relationsbetween $\eps$, $d\e$ and $q\e$. In particular if the limits $\liml {\eps\to0}q\e$ and $\liml {\eps\to 0}\ds{d\e^{N-1}q\e \eps^{-N}}$ are positive thenthe homogenized spectral problem contains the spectral parameter in a nonlinearmanner, and its spectrum has a sequence of accumulation points.

Author: Andrii Khrabustovskyi

Source: https://arxiv.org/