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

The convergence, as ε ↓ 0, of the functional F_ε (Ψ ) = ∫ RN u_ε (x)Ψ (x,x / ε ) associated with a given L² function u_ε with support in a fixed compact set is studied. The test functions Ψ (x,y) are continuous on RN × RN and periodic in y. A convergence theorem is proved under the weaker assumption that u_ε 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.