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

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 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 ε → 0 ε p h ε = 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.


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 (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

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