A corrector result for thewave equation with high oscillating periodic coefficients

Abstract

In the homogenization of a wave problem with oscillating coefficients in the diffusion term it is well known that the corresponding limit equation has the same structure with a diffusion term which agrees with the elliptic homogenized limit. Thus one can think that the oscillations of the solution of the wave equation are similar to the ones of the corresponding elliptic problem and then that the corrector for the elliptic problem is still a corrector for the wave problem. However in a paper by Brahim-Otsmane, Francfort and Murat, 1992, it was proved that this only holds if the initial data are “well posed”. In general, it is necessary to add to the elliptic corrector another term depending on the initial data. In this paper we obtain this term in the case of a wave problem posed in ℝN with periodic coefficients. This term is obtained using the two-scale convergence theory. It oscillates periodically in the space variable but almost periodically in the time one.