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Boundary layer homogenization for periodic oscillating boundaries

Habbal A.

Control and Cybernetics

2001

Vol. 30, no 3

279-301

The paper is devoted to the study of the boundary layer behaviour of solutions to partial differential equations occurring in domains with periodic oscillating boundaries, the frequency and the amplitude of the oscillations being the same. First, the transport method, a classical one from the optimal design theory, is used in order to state the problem in a fixed domain ; then, an adapted two-scale boundary layer convergence is developed. Apart from this new hybrid approach, the main difference with related works is consideration of a bounded unit-cell, yielding a simple functional framework. Convergence, as well as a homogenized equation for the first order boundary layer term are given, and a first order corrector result is proved. This a priori boundingg is very well suited to problems of control, and to numerical implementation considerations. The difficulty in obtaining higher order correctors due to the bounding of the unit-cell is finally discussed.