HOMOGENIZATION OF A ONE-DIMENSIONAL SPECTRAL PROBLEM FOR A SINGULARLY PERTURBED ELLIPTIC OPERATOR WITH NEUMANN BOUNDARY CONDITIONS

作者：Allaire Gregoire^*; Capdeboscq Yves; Puel Marjolaine

来源：Discrete and Continuous Dynamical Systems-Series B, 2012, 17(1): 1-31.

DOI：10.3934/dcdsb.2012.17.1

• Homogenization
• spectral problem
• localization

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 E 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 %26lt;= delta %26lt; 1, we consider a sequence of periods epsilon(n) = 1/(n + delta) with n is an element of 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.

• 出版日期2012-1