Issue |
ESAIM: COCV
Volume 9, February 2003
|
|
---|---|---|
Page(s) | 449 - 460 | |
DOI | https://doi.org/10.1051/cocv:2003022 | |
Published online | 15 September 2003 |
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
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
ΩE is
composed of two domains. The first one
is a plate which becomes
asymptotically flat, with thickness
hE in the
xN 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 hE, 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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