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Issue ESAIM: COCV
Volume 9, 2003
Page(s) 449 - 460
DOI 10.1051/cocv:2003022

ESAIM: COCV, August 2003, Vol. 9, pp. 449-460
DOI: 10.1051/cocv:2003022

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

Dominique Blanchard1 and Antonio Gaudiello2

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 November 22, 2002. Revised February 7, 2003.)

Abstract
We investigate the asymptotic behaviour, as $\varepsilon\rightarrow 0$ , of a class of monotone nonlinear Neumann problems, with growth p-1 ( $p\in ]1,+\infty[$), on a bounded multidomain $\Omega_\varepsilon\subset \mathbb{R} ^N$ $(N\geq2)$. The multidomain $\Omega_\varepsilon$ is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness $h_\varepsilon$ in the xN direction, as $\varepsilon\rightarrow 0$ . The second one is a "forest" of cylinders distributed with $\varepsilon$-periodicity in the first N-1 directions on the upper side of the plate. Each cylinder has a small cross section of size $\varepsilon$ and fixed height (for the case N=3, see the figure). We identify the limit problem, under the assumption: ${\lim_{\varepsilon\rightarrow 0} {\varepsilon^p\over h_\varepsilon}=0}$ . After rescaling the equation, with respect to $h_\varepsilon$, 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 xN, coupled with an algebraic system. Moreover, the limit solution is independent of xN 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.


Mathematics Subject Classification. 35B27, 35J60.

Key words: Homogenization, oscillating boundaries, multidomain, monotone problem.


© EDP Sciences, SMAI 2003


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