Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator

Article type: Research Article

Authors: Chechkina, Alexandra^a | Pankratova, Iryna^{b; *} | Pettersson, Klas^b

Affiliations: [a] Lomonosov Moscow State University, Moscow, Russia | [b] Narvik University College, Narvik, Norway

Correspondence: [*] Corresponding author. E-mail: Irina.Pankratova@hin.no.

Abstract: We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.

Keywords: homogenization, spectral problem, higher order equations, localization of eigenfunctions

DOI: 10.3233/ASY-151291

Journal: Asymptotic Analysis, vol. 93, no. 1-2, pp. 141-160, 2015