On an extension of the method of two-scale convergence and its applications

The concept of two-scale convergence associated with a fixed periodic Borel measure $\mu$ is introduced. In the case when $d\mu=dx$ is Lebesgue measure on the torus convergence in the sense of Nguetseng–Allaire is obtained. The main properties of two-scale convergence are revealed by the simultaneous consideration of a sequence of functions and a sequence of their gradients. An application of two-scale convergence to the homogenization of some problems in the theory of porous media (the double-porosity model) is presented. A mathematical notion of “softly or weakly coupled parallel flows” is worked out. A homogenized operator is constructed and the convergence result itself is interpreted as a “strong two-scale resolvent convergence”. Problems concerning the behaviour of the spectrum under homogenization are touched upon in this connection.