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

Dominique Blanchard; Antonio Gaudiello[1]

  • [1] Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, via G. Di Biasio 43, 03043 Cassino (FR), Italy;

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 449-460
  • ISSN: 1292-8119

Abstract

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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 Ω ε ⊂ ℝ N ( N ≥ 2 ) . The multidomain Ω ε is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness h ε in the x N 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 ε → 0 ε p h ε = 0 . After rescaling the equation, with respect to h ε , 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 x N , coupled with an algebraic system. Moreover, the limit solution is independent of x N 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.

How to cite

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Blanchard, 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 (2003): 449-460. <http://eudml.org/doc/246069>.

@article{Blanchard2003,
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\ge 2)$. 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 $x_N$ 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 $x_N$, coupled with an algebraic system. Moreover, the limit solution is independent of $x_N$ 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.},
affiliation = {Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, via G. Di Biasio 43, 03043 Cassino (FR), Italy;},
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},
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/246069},
volume = {9},
year = {2003},
}

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
PY - 2003
PB - EDP-Sciences
VL - 9
SP - 449
EP - 460
AB - 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\ge 2)$. 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 $x_N$ 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 $x_N$, coupled with an algebraic system. Moreover, the limit solution is independent of $x_N$ 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/246069
ER -

References

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