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. Bibliography: 25 titles.