High‐contrast homogenization of linear systems of partial differential equations

We give some integrability conditions for the coefficients of a sequence of elliptic systems with varying coefficients in order to obtain the stability for homogenization. In the case of equations, it is well known that equi‐integrability and bound in L¹ are enough for this purpose; however, this is based on the maximum principle, and then, it does not work for systems. Here, we use an extension of the Murat–Tartar div‐curl lemma because of M. Briane, J. Casado‐Díaz, and F. Murat in order to obtain the stability by homogenization for systems uniformly elliptic, with bounded coefficients in , with N the dimension of the space. We also show that a weaker ellipticity condition can be assumed, but then, we need a stronger integrability for the coefficients. Copyright © 2016 John Wiley & Sons, Ltd.