Compound Asymptotic Expansions for Spectral Problems

Abstract

In this chapter the elliptic spectral problems in singularly perturbed domains are analyzed. The asymptotic expansion of simple and multiple eigenvalues and of the associated eigenfunctions with respect to the small parameter which governs the size of singular perturbations are derived. The compound asymptotic expansions method [148, 170] is applied to this end. The singular and nonsmooth perturbations far and close to the boundary of a smooth reference domain are considered [131, 160, 177, 178, 179]. We point out that the specific asymptotic expansions cannot be derived by the classic shape sensitivity technique of [62, 196, 210]. However, the obtained results in the case of the boundary perturbations may be compared with the related results obtained by the speed method of the shape sensitivity analysis (cf. example for the topological derivative on the boundary associated with the energy functional in Section 1.2). We refer also to [8, 65, 66, 67, 68, 69] for the related results on the asymptotic analysis of spectral problems by the matched asymptotic expansions method of [100].