Reiterated homogenization for elliptic operators
Nicolas L.J. Meunier, Jean Van Schaftingen · Comptes Rendus Mathématique · 2005
In this Note, using the periodic unfolding method (see D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 335 (1) (2002) 99–104), we study reiterated homogenization for equations of the form − div ( a ε ( x , D u ε ) ) = f , where a ε is Carathéodory and satisfies some monotone and growth conditions. We show that if we assume that T δ ( ε ) ′ ( T ε ( a ε ) ) ( x , y , z , ξ ) converges, for almost all ( x , y , z ) ∈ Ω × Y × Z , to a Carathéodory operator, then the sequences u ε and D u ε converge in a certain sense to the solution ( u 0 , u ˆ , u ˜ ) of a limit variational problem, as ε → 0 . In particular this contains the case a ε ( x , ξ ) = a ( x , x ε , { x / ε } Y δ ( ε ) , ξ ) , where a is periodic in the second and third arguments, and continuous in each argument.