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 ε0, of a class of monotone nonlinear Neumann problems, with growth p-1 (p]1,+[), on a bounded multidomain Ω ε N (N2). 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 ε0. 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 ε0 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.

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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