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
2 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR7641
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 $\eps$ the period,each derivative is scaled by an $\eps$ 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 $n\in \NN$. 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.
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Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2012, 17, pp.1-31. 〈10.3934/dcdsb.2012.17.1〉
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Soumis le : dimanche 3 février 2013 - 12:27:02
Dernière modification le : mercredi 23 mai 2018 - 17:58:04

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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〉

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