# American Institute of Mathematical Sciences

January  2012, 17(1): 1-31. doi: 10.3934/dcdsb.2012.17.1

## 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, France 2 Mathematical Institute, 24-29 St Giles’, OXFORD OX1 3LB, United Kingdom 3 Institut de Mathématiques, Université de Toulouse and CNRS, Université Paul Sabatier, 31062 Toulouse Cedex 9, France

Received  April 2011 Revised  July 2011 Published  October 2011

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.
Citation: Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
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