We consider the linearized elasticity system in a multidomain of . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and tend to zero simultaneously, with , we identify the limit problem. This limit problem involves six junction conditions.

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.

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 (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 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: . 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 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.