Asymptotic Analysis of the Steklov Spectral Problem in Thin Perforated Domains with Rapidly Varying Thickness and Different Limit Dimensions

2015 ◽  
pp. 495-507
Author(s):  
A. Popov
Keyword(s):  
Asymptotic Analysis ◽  
Spectral Problem ◽  
Perforated Domains ◽  
Varying Thickness

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

2013 ◽  
Vol 143 (6) ◽  
pp. 1255-1289 ◽  
Author(s):  
Andrii Khrabustovskyi
Keyword(s):  
Riemannian Manifold ◽  
Small Parameter ◽  
Asymptotic Behaviour ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Beltrami Operator ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Accumulation Points ◽  
Image Position

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limitsare positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

ASYMPTOTIC ANALYSIS OF PERIODICALLY-PERFORATED NONLINEAR MEDIA AT THE CRITICAL EXPONENT

2009 ◽  
Vol 11 (06) ◽  
pp. 1009-1033 ◽  
Author(s):  
LAURA SIGALOTTI
Keyword(s):  
Asymptotic Analysis ◽  
Critical Exponent ◽  
Space Dimension ◽  
Convergence Result ◽  
Nonlinear Media ◽  
Perforated Domains ◽  
Extra Term ◽  
Γ Convergence ◽  
Vector Valued ◽  
Critical Scale

We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with n-growth where n is the space dimension, showing that there exists a critical scale for the perforations such that the Γ-limit is non-trivial. We prove that the limit extra-term is given by a formula of homogenization type, which simplifies in the case of n-homogeneous energy densities.

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