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

Blanchard, Dominique; Gaudiello, Antonio

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 449-460.

We investigate the asymptotic behaviour, as $\epsilon \to 0$, of a class of monotone nonlinear Neumann problems, with growth $p-1$ ($p\in ]1,+\infty [$), on a bounded multidomain ${\Omega }_{\epsilon }\subset {ℝ}^N$ $(N\ge 2)$. 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.

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