Reiterated homogenization for elliptic operators

Meunier, N Van Schaftingen, Jean [UCL]

In this Note, using the periodic unfolding method (see D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. 1335 (1) (2002) 99-104), we study reiterated homogenization for equations of the form -div(a(epsilon)(x, Du(epsilon))) = f, where a(epsilon) is Caratheodory and satisfies some monotone and growth conditions. We show that if we assume that T-delta((epsilon))' (T-epsilon (a(epsilon))) (x, y, z,xi) converges, for almost all (x, y, z) epsilon Omega x Y x Z, to a Caratheodory operator, then the sequences u(epsilon) and Du,5 converge in a certain sense to the solution (u(0), (u) over cap, (u) over tilde) of a limit variational problem, as epsilon --> 0. In particular this contains the case a(epsilon)(x,xi) = a(x, x/epsilon, {x/epsilon}y/delta(epsilon) xi), where a is periodic in the second and third arguments, and continuous in each argument. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.