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

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