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**Homogenization and two-scale convergence.**
*(English)*
Zbl 0770.35005

Summary: Following an idea of G. Nguetseng [SIAM J. Math. Anal. 20, No. 3, 608-623 (1989; Zbl 0688.35007)], the author defines a notion of “two- scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in \(L^ 2(\Omega)\) are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in \(L^ 2(\Omega))\) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of L. Tartar [Cours Peccot, Collège de France (1977)]. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

### MSC:

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

35B40 | Asymptotic behavior of solutions to PDEs |