Localization effects for Dirichlet problems in domains surrounded by thin stiff and heavy bands

We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Ω_ε in the plane R². Here Ω_ε is Ω∪ω_ε∪Γ, where Ω is a fixed bounded domain with boundary Γ, ω_ε 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.