# 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

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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 (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 -

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