Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions

Grégoire Allaire; Yves Capdeboscq; Marjolaine Puel

doi:10.3934/dcdsb.2012.17.1

Article Contents

2012, Volume 17, Issue 1: 1-31. Doi: 10.3934/dcdsb.2012.17.1

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Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions

1.
CMAP, CNRS UMR 7641, École Polytechnique, Route de Saclay, Palaiseau F91128
2.
Mathematical Institute, 24-29 St Giles’, OXFORD OX1 3LB
3.
Institut de Mathématiques, Université de Toulouse and CNRS, Université Paul Sabatier, 31062 Toulouse Cedex 9

Received: April 2011

Revised: July 2011

Published: January 2012

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Abstract

We study the asymptotic behavior of the first 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 $\epsilon$ the period, each derivative is scaled by an $\epsilon$ factor. The main difficulty 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 $0\leq\delta<1$, we consider a sequence of periods $\epsilon_n=1/(n+\delta)$ with $n\in \mathbb{N}$. 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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Mathematics Subject Classification: Primary: 35B27; Secondary: 74Q05.

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