A General Convergence Result for a Functional Related to the Theory of Homogenization

The convergence, as 0, of the functional F_ () = ₑ₍ u_ (x) (x, x /) associated with a given L² function u_ with support in a fixed compact set is studied. The test functions (x, y) are continuous on RN RN and periodic in y. A convergence theorem is proved under the weaker assumption that u_ remains in a bounded subset of L². Finally, the use of multiple-scale expansions in homogenization is justified, and a new approach is proposed for the mathematical analysis of homogenization problems.