Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues

Abstract

AbstractWe consider an asymptotic spectral problem for a second order differential operator, with piecewise constants coefficients, in a two-dimensional domain Ωɛ. Here Ωɛ is Ωɛ=Ω∪ωɛ∪Γ, where Ω is a fixed open bounded domain with boundary Γ, ωɛ is a curvilinear strip of variable width O(ɛ), and Γ=Ω¯∩ω¯ɛ. The density and stiffness constants are of order O(ɛ−m−t) and O(ɛ−t) respectively in this strip, while they are of order O(1) in the fixed domain Ω; t and t+m are positive parameters and ɛ∈(0,1). Imposing the Neumann condition on the boundary of Ωɛ, for t⩾0 and m⩾−t we provide asymptotics for the eigenvalues and eigenfunctions as ɛ→0. We obtain sharp estimates of convergence rates for the eigenpairs in the case where t=1 and m=0, which can, in fact, be extended to other cases