# 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

## Access Full Article

top## Abstract

top## How to cite

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 -

## References

top- G. Allaire, Homogenization and Two-Scale Convergence. SIAM J. Math Anal.23 (1992) 1482-1518.
- G. Allaire and M. Amar, Boundary Layer Tails in Periodic Homogenization. ESAIM: COCV4 (1999) 209-243.
- Y. Amirat and O. Bodart, Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary. J. Anal. Appl.20 (2001) 929-940.
- N. Ansini and A. Braides, Homogenization of Oscillating Boundaries and Applications to Thin Films. J. Anal. Math.83 (2001) 151-183.
- D. Blanchard, L. Carbone and A. Gaudiello, Homogenization of a Monotone Problem in a Domain with Oscillating Boundary. ESAIM: M2AN33 (1999) 1057-1070.
- R. Brizzi and J.P. Chalot, Boundary Homogenization and Neumann Boundary Value Problem. Ricerche Mat.46 (1997) 341-387.
- G. Buttazzo and R.V. Kohn, Reinforcement by a Thin Layer with Oscillating Thickness. Appl. Math. Optim.16 (1987) 247-261.
- G.A. Chechkin, A. Friedman and A.L. Piatniski, The Boundary Value Problem in a Domain with Very Rapidly Oscillating Boundary. J. Math. Anal. Appl.231 (1999) 213-234.
- P.G. Ciarlet and P. Destuynder, A Justification of the Two-Dimensional Linear Plate Model. J. Mécanique18 (1979) 315-344.
- D. Cioranescu and J. Saint Jean Paulin, Homogenization in Open Sets with Holes. J. Math. Anal. Appl.71 (1979) 590-607.
- A. Corbo Esposito, P. Donato, A. Gaudiello and C. Picard, Homogenization of the p-Laplacian in a Domain with Oscillating Boundary. Comm. Appl. Nonlinear Anal.4 (1997) 1-23.
- A. Gaudiello, Asymptotic Behaviour of non-Homogeneous Neumann Problems in Domains with Oscillating Boundary. Ricerche Mat.43 (1994) 239-292.
- A. Gaudiello, Homogenization of an Elliptic Transmission Problem. Adv. Math Sci. Appl.5 (1995) 639-657.
- A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino, Asymptotic Analysis for Monotone Quasilinear Problems in Thin Multidomains. Differential Integral Equations15 (2002) 623-640.
- A. Gaudiello, R. Hadiji and C. Picard, Homogenization of the Ginzburg-Landau Equation in a Domain with Oscillating Boundary. Commun. Appl. Anal. (to appear).
- A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili, On the Junction of Elastic Plates and Beams. C. R. Acad. Sci. Paris Sér. I335 (2002) 717-722.
- H. Le Dret, Problèmes variationnels dans les multi-domaines : modélisation des jonctions et applications. Masson, Paris (1991).
- J.L. Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires. Dunod, Paris (1969).
- T.A. Mel'nyk, Homogenization of the Poisson Equations in a Thick Periodic Junction. ZAA J. Anal. Appl. 18 (1999) 953-975.
- T.A. Mel'nyk and S.A. Nazarov, Asymptotics of the Neumann Spectral Problem Solution in a Domain of ``Thick Comb" Type. J. Math. Sci.85 (1997) 2326-2346.
- G. Nguetseng, A General Convergence Result for a Functional Related to the Theory of Homogenization. SIAM J. Math Anal.20 (1989) 608-623.
- L. Tartar, Cours Peccot, Collège de France (March 1977). Partially written in F. Murat, H-Convergence, Séminaire d'analyse fonctionnelle et numérique de l'Université d'Alger (1977-78). English translation in Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R.V. Kohn, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser-Verlag (1997) 21-44.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.