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

Allaire, Grégoire

SIAM J. Math. Anal. 23, No. 6, 1482-1518 (1992).

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

Keywords:

media with periodic structure; sequences of oscillating functions; corrector-type theorem; periodically oscillating coefficients; linear and nonlinear second-order elliptic equations

Citations:

Zbl 0688.35007