We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Ωε in the plane R2. Here Ωε is Ω∪ωε∪Γ, where Ω is a fixed bounded domain with boundary Γ, ωε is a curvilinear band of width O(ε), and Γ=Ω‾∩ω‾ε. The density and stiffness constants are of order ε−m−t and ε−t respectively in this band, while they are of order 1 in Ω; t≥1, m>2, and ε is a small positive parameter. We address the asymptotic behavior, as ε→0, for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Γ.
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- Прикладная математика