Homogenization of a One-Dimensional Spectral Problem for a Singularly Perturbed Elliptic Operator with Neumann Boundary Conditions

We study the asymptotic behavior of the rst eigenvalue and eigenfunction of a one-dimensional periodic elliptic operator with Neumann boundary conditions. The second order elliptic equation is not self-adjoint and is singularly perturbed since, denoting by ε the period, each derivative is scaled by an ε factor. The main di culty is that the domain size is not an integer multiple of the period. More precisely, for a domain of size 1 and a given fractional part 0 ≤ δ < 1, we consider a sequence of periods n = 1/(n + δ) with n ∈ N. In other words, the domain contains n entire periodic cells and a fraction δ of a cell cut by the domain boundary. According to the value of the fractional part δ, di erent asymptotic behaviors are possible: in some cases an homogenized limit is obtained, while in other cases the rst eigenfunction is exponentially localized at one of the extreme points of the domain.