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

Abstract

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 Ωε ⊂ R N (N ≥ 2). The multidomain Ωε is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness hε in the xN 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: limε→0 ε p hε = 0. After rescaling the equation, with respect to hε, 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 xN , coupled with an algebraic system. Moreover, the limit solution is independent of xN 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.

Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1

“…Since the classes of admissible controls K (1) ε and K (1) ε are defined on variable spaces depending on ε, we should introduce the special convergence of controls. Definition 3.1.…”

“…From definitions of the sets K (1) ε and K (2) ε (see (0.1), (0.2)) it follows that these controls are restrictions of functions from the following sets…”

“…Here the control sets K (1) 0 and K (2) 0 are defined in (2.1) and (2.2) respectively, the cost functional J 0 is defined by (3.8).…”

“…Therefore boundary-value problems in thick junctions of different types are very extensively investigated at present (see [1][2][3]6,8,10], [19,20,24,25] and references therein). The aim of these researches is to develop rigorous methods to study the asymptotic behavior of solutions when the number of the attached thin domains of a thick junction infinitely increases and their thickness vanishes.…”

“…The thin cylinders G ε are divided into two levels G (1) ε and G (2) ε . Cylinders G (m) ε (k) are ε-periodically alternated along the Ox 1 -direction and Ox 2 -direction and they are joined with Ω 0 over the ε-homothetic images…”