Spectral Stiff Problems in Domains with a Strongly Oscillating Boundary
Delfina Gómez, С. А. Назаров, Eugenia Pérez · Birkhäuser Boston eBooks · 2011
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.