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

Grégoire Allaire (1, 2) , Y. Capdeboscq (3) , Marjolaine Puel (4)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 DeFI - Shape reconstruction and identification
3 MI - Mathematical Institute [Oxford]
4 IMT - Institut de Mathématiques de Toulouse UMR5219

We study the asymptotic behavior of the first eigenvalue and eigenfunctionof a one-dimensional periodic elliptic operator with Neumann boundaryconditions. The second order elliptic equation is not self-adjointand is singularly perturbed since, denoting by $\epsilon$ the period,each derivative is scaled by an $\epsilon$ factor.The main difficulty is that the domain size is not an integer multipleof the period. More precisely, for a domain of size $1$ and a givenfractional part $0\leq\delta<1$, we consider a sequence of periods$\epsilon_n=1/(n+\delta)$ with $n\in \mathbb{N}$. In other words, the domaincontains $n$ entire periodic cells and a fraction $\delta$ of a cellcut by the domain boundary. According to the value of the fractionalpart $\delta$, different asymptotic behaviors are possible: in somecases an homogenized limit is obtained, while in other cases thefirst eigenfunction is exponentially localized at one of theextreme points of the domain.

Domaines: Analyse numérique [cs.NA]

Grégoire Allaire, Y. Capdeboscq, Marjolaine Puel. Homogenization of a One-Dimensional Spectral Problem for a Singularly Perturbed Elliptic Operator with Neumann Boundary Conditions. Discrete and Continuous Dynamical Systems - Series B, 2012, 17, pp.1-31. ⟨10.3934/dcdsb.2012.17.1⟩. ⟨hal-00784042⟩