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

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 Ω_E is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness h_E 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: ${\lim_{\varepsilon\rightarrow 0} {\varepsilon^p\over h_\varepsilon}=0}$. After rescaling the equation, with respect to h_E, 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.