| Shrnutí: | We study the asymptotic behavior of the first eigenvalue and eigen- function 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 (Epsilon) the period, each derivative is scaled by an (Epsilon) factor. The main dificulty 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 (<=) (Delta) < 1, we consider a sequence of periods (Epsilon) = 1 =(n + (Delta)) with n (is a member of) N. In other words, the domain contains n entire periodic cells and a fraction (Delta) of a cell cut by the domain boundary. According to the value of the fractional part (Delta), different asymptotic behaviors are possible: in some cases an homogenized limit is obtained, while in other cases the first eigenfunction is exponentially localized at one of the extreme points of the domain.
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