Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems

Abstract

We consider different stiff spectral problems with a small parameter for the Laplace operator in two different domains of the plane Ω and Ω ε , respectively. Here Ω ε = Ω ∪ ω ε ∪ Γ , where Ω is a fixed open bounded domain with boundary Γ, ω ε is a curvilinear strip of variable width O ( ε ) , and Γ = Ω ¯ ∩ ω ¯ ε . ε and δ ε are small parameters that converge towards zero. The first problem is a Wentzell spectral problem in the fixed domain Ω, with the parameter δ ε appearing on the boundary condition, multiplying the normal derivative on Γ. For the second problem, posed in Ω ε with a Neumann condition on the boundary of Ω ε , the density and stiffness constants are of order O ( ε − t ) in the strip ω ε , with t > 1 , while they are of order O ( 1 ) in the fixed domain Ω. We provide asymptotic expansions for the eigenvalues and eigenfunctions of both problems and obtain bounds for convergence rates of these eigenelements as ε → 0 . In addition, we seek out the connection between both problems, which have a common limiting eigenvalue problem (cf. (2.15)–(2.16)), and notice an asymptotic dissociation in two spectral problems on Ω and Γ. We also show that the Wentzell spectral problem can be considered as an alternative approach for the stiff problem in the perturbed domain Ω ε when δ ε = ε t − 1 , as ε → 0 .