Nonlocal limits in the study of linear elliptic systems arising in periodic homogenization

In the present paper, we obtain the two-scale limit system of a sequence of linear elliptic periodic problems with varying coefficients. We show that this system has not the same structure than the classical one, obtained when the coefficients are fixed. This is due to the apparition of nonlocal effects. Our results give an example showing that the homogenization of elliptic problems with varying coefficients, depending on one parameter, gives in general a nonlocal limit problem.