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Spectral Stiff Problems in Domains with a Strongly Oscillating Boundary

  • D. Gómez
  • S. A. Nazarov
  • E. Pérez

Abstract

We consider an asymptotic spectral problem for a second order differential operator, with piecewise constant coefficients, in a two-dimensional domain Ω ε which depends on a small parameter ε. Here, Ω ε is Ω ε =Ω∪ω ε ∪Γ, where Ω is a fixed open bounded domain, ω ε is a curvilinear strip of variable width O(ε), and \(\Gamma={\overline{\Omega}}\cap {\overline{\omega}_{\varepsilon}}\) is the boundary of Ω. The density and stiffness constants are of order O(ε −1) and O(ε t ), respectively, in this strip, while they are of order O(1) in the fixed domain Ω; t is a parameter such that 0≤t<1. Imposing the Neumann condition on the boundary of Ω ε , we study the asymptotic behavior, as ε→0, of the eigenvalues and eigenfunctions. Denoting by (ν,τ) the orthogonal curvilinear coordinates in a neighborhood of Γ, we consider two different types of bands ω ε . First, the case where ω ε ={x : 0<ν<εh(τ)} with h a positive function of the τ variable. Second, the case where ω ε ={x : 0<ν<εh ε (τ)} with h ε (τ)=h(τ/ε) and h a positive and periodic function; namely, a domain with strongly oscillating boundary.

Keywords

Scalar Product Spectral Problem Neumann Problem Order Differential Operator Thin Band 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Universidad de CantabriaSantanderSpain
  2. 2.Institute for Problems in Mechanical EngineeringSt. PetersburgRussia

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