Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 9  (2003), p. 449-460

We investigate the asymptotic behaviour, as $\epsilon \to 0$, of a class of monotone nonlinear Neumann problems, with growth $p-1$ ($p\in \right]1,+\infty \left[$), on a bounded multidomain ${\Omega }_{\epsilon }\subset {ℝ}^{N}$ $\left(N\ge 2\right)$. The multidomain ${\Omega }_{\epsilon }$ is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness ${h}_{\epsilon }$ in the ${x}_{N}$ direction, as $\epsilon \to 0$. The second one is a “forest” of cylinders distributed with $\epsilon$-periodicity in the first $N-1$ directions on the upper side of the plate. Each cylinder has a small cross section of size $\epsilon$ and fixed height (for the case $N=3$, see the figure). We identify the limit problem, under the assumption: ${lim}_{\epsilon \to 0}\frac{{\epsilon }^{p}}{{h}_{\epsilon }}=0$. After rescaling the equation, with respect to ${h}_{\epsilon }$, 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.

DOI : https://doi.org/10.1051/cocv:2003022
Classification:  35B27,  35J60
Keywords: homogenization, oscillating boundaries, multidomain, monotone problem
@article{COCV_2003__9__449_0,
author = {Blanchard, Dominique and Gaudiello, Antonio},
title = {Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {9},
year = {2003},
pages = {449-460},
doi = {10.1051/cocv:2003022},
zbl = {1071.35012},
mrnumber = {1998710},
language = {en},
url = {http://www.numdam.org/item/COCV_2003__9__449_0}
}

Blanchard, Dominique; Gaudiello, Antonio. Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003) , pp. 449-460. doi : 10.1051/cocv:2003022. http://www.numdam.org/item/COCV_2003__9__449_0/

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