Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem

¹ Université de Rouen, UMR 6085, 76821 Mont-Saint-Aignan Cedex, France, and Laboratoire d'Analyse Numérique, Université P. et M. Curie, Case Courrier 187, 75252 Paris Cedex 05, France; blanchar@ann.jussieu.fr.
² Università degli Studi di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale, via G. di Biasio 43, 03043 Cassino (FR), Italy; gaudiell@unina.it.

Received: 22 November 2002
Revised: 7 February 2003

Abstract

We investigate the asymptotic behaviour, as ε → 0, of a class of monotone nonlinear Neumann problems, with growth p-1 (p ∈]1, +∞[), on a bounded multidomain (N ≥ 2). The multidomain Ω_E is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness h_E in the x_N direction, as ε → 0. The second one is a “forest" of cylinders distributed with ε-periodicity in the first N - 1 directions on the upper side of the plate. Each cylinder has a small cross section of size ε and fixed height (for the case N=3, see the figure). We identify the limit problem, under the assumption: . After rescaling the equation, with respect to h_E, on the plate, we prove that, in the limit domain corresponding to the “forest" of cylinders, the limit problem identifies with a diffusion operator with respect to x_N, coupled with an algebraic system. Moreover, the limit solution is independent of x_N in the rescaled plate and meets a Dirichlet transmission condition between the limit domain of the “forest" of cylinders and the upper boundary of the plate.