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

Abstract

We consider an asymptotic spectral problem for a second order differential operator, with piecewise constants coefficients, in a two-dimensional domain Ω ɛ . Here Ω ɛ is Ω ɛ = Ω ∪ ω ɛ ∪ Γ , where Ω is a fixed open bounded domain with boundary Γ, ω ɛ 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 Ω ɛ , 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.