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

Abstract

We consider the homogenization of a singularly perturbed selfadjoint 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 jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order order effective operators with constant coefficients, depending on the large parameters.

Spectral Asymptotics for the Schrödinger Operator on the Line with Spreading and Oscillating Potentials

“…This transpires for instance in [2,3,1] where, in order to deal with large potentials, the authors introduce a factorization principle which is similar to our normal form transformation (see below), although without the change of variable. We would also like to mention the recent works [25,7,26], where similar multiscale problem as ours are studied.…”

“…, where ϕ eff,0 n,ε,α is the n th L 2 -normalized eigenfunction of the effective operator L eff,0 ε,α defined in (7). Moreover, we have the approximation…”

Spectral asymptotics for the Schrödinger operator on the line with spreading and oscillating potentials

“…This transpires for instance in [2,3,1] where, in order to deal with large potentials, the authors introduce a factorization principle which is similar to our normal form transformation (see below), although without the change of variable. We would also like to mention the recent works [22,6,23], where similar multiscale problem as ours are studied. 1.4.2.…”

Subcritical perturbation of a locally periodic elliptic operator

“…In particular, the main contribution to the energy of the eigenfunctions is given by the second term in the asymptotic series approximation. The analysis of this case has not been included in the previous studies of the subcritical case [18,11], and our purpose here is to fill this gap between (i) and (iii).…”

“…Examples of scalar second order operators have been given in [3,4,8,7,20,12,13,15,14,19,18]. A transport equation has been studied in [2], and scalar fourth order operators have been considered in [5,11]. In [5] a system of equations have been studied with purely periodic singular perturbation and locally periodic continuous perturbation.…”