# 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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