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

TL;DR

This paper studies the spectral behavior of a singularly perturbed, locally periodic, fourth order elliptic operator, revealing how eigenfunctions localize and can be approximated by effective operators with constant coefficients.

Contribution

It introduces a homogenization approach for a complex fourth order elliptic operator with large parameters, providing eigenfunction approximations in terms of effective operators.

Findings

Eigenfunctions exhibit localization due to large parameters.

Eigenfunctions can be approximated by functions from effective operators.

The spectral asymptotics depend on the interplay of local periodicity and large parameters.

Abstract

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation 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 give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective operators with constant coefficients, depending on the large parameters.