Homogenization in general periodically perforated domains by a spectral approach. (English) Zbl 1028.35018

Briane, Marc

Calc. Var. Partial Differ. Equ. 15, No. 1, 1-23 (2002).

The author considers the classical Neumann problem in an \(\varepsilon\)-periodically perforated bounded set, showing that the homogenization process depends on the asymptotic behaviour of the spectrum of the local problem. He studies the case when there exists an eigenvalue \(\Lambda_n(\varepsilon)\) of the above mentioned spectrum such that \(\Lambda_n(\varepsilon)\gg \varepsilon^2\), proving that if \(n\) is the smallest integer with this property, then the homogenized system is a linear system of \(n\) second-order equations.

Reviewer: Dan Polisevski (Bucureşti)

MSC:

35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
35J25	Boundary value problems for second-order elliptic equations
74Q15	Effective constitutive equations in solid mechanics
76M50	Homogenization applied to problems in fluid mechanics

Keywords:

\(n\)th-eigenvalue; Neumann problem