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

2 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : 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.
Document type :
Journal articles

Cited literature [16 references]

https://hal.inria.fr/hal-00784042
Contributor : Houssem Haddar Connect in order to contact the contributor
Submitted on : Wednesday, February 5, 2020 - 12:18:53 AM
Last modification on : Friday, October 22, 2021 - 4:41:40 AM
Long-term archiving on: : Wednesday, May 6, 2020 - 3:39:12 PM

### File

ACP-rev2.pdf
Files produced by the author(s)

`

### Citation

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, American Institute of Mathematical Sciences, 2012, 17, pp.1-31. ⟨10.3934/dcdsb.2012.17.1⟩. ⟨hal-00784042⟩

Record views