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

TL;DR

This paper investigates the asymptotic spectral behavior of the Laplace-Beltrami operator on a Riemannian manifold composed of two perforated domains connected by tubes, revealing complex homogenized problems depending on geometric parameters.

Contribution

It introduces new homogenized spectral problems for manifolds with perforations connected by tubes, especially when certain geometric limits are positive, leading to nonlinear spectral parameters.

Findings

Homogenized spectral problems depend on relations between parameters.

Presence of accumulation points in the spectrum under certain limits.

Nonlinear spectral parameter in the homogenized problem.

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.