Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem
Dominique Blanchard; Antonio Gaudiello
ESAIM: Control, Optimisation and Calculus of Variations (2010)
- Volume: 9, page 449-460
- ISSN: 1292-8119
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topBlanchard, Dominique, and Gaudiello, Antonio. "Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 449-460. <http://eudml.org/doc/90705>.
@article{Blanchard2010,
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
$\Omega_\varepsilon\subset
\mathbb\{R\}^N$ (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:
$\{\lim_\{\varepsilon\rightarrow 0\}
\{\varepsilon^p\over
h_\varepsilon\}=0\}$.
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.
},
author = {Blanchard, Dominique, Gaudiello, Antonio},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Homogenization; oscillating boundaries; multidomain; monotone problem.; monotone nonlinear Neumann problems; Dirichlet transmission condition},
language = {eng},
month = {3},
pages = {449-460},
publisher = {EDP Sciences},
title = {Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem},
url = {http://eudml.org/doc/90705},
volume = {9},
year = {2010},
}
TY - JOUR
AU - Blanchard, Dominique
AU - Gaudiello, Antonio
TI - Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 449
EP - 460
AB -
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
$\Omega_\varepsilon\subset
\mathbb{R}^N$ (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:
${\lim_{\varepsilon\rightarrow 0}
{\varepsilon^p\over
h_\varepsilon}=0}$.
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.
LA - eng
KW - Homogenization; oscillating boundaries; multidomain; monotone problem.; monotone nonlinear Neumann problems; Dirichlet transmission condition
UR - http://eudml.org/doc/90705
ER -
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