Homogenization and Two-Scale Convergence

TLDR

Two‑scale convergence, introduced by Nguetseng, provides a refined description of oscillating function sequences and is particularly valuable for homogenizing PDEs with periodically oscillating coefficients. The paper aims to introduce two‑scale convergence as a tool for better describing oscillating sequences and to propose a new method for proving homogenization convergence, offering an alternative to Tartar’s energy method. The authors propose a new two‑scale convergence method that replaces Tartar’s energy method for proving homogenization convergence. They prove that bounded \(L^{2}\) sequences are relatively compact under two‑scale convergence, establish a corrector theorem enabling replacement by the two‑scale limit, and demonstrate the method’s power and simplicity on linear and nonlinear second‑order elliptic equations.

Abstract

Following an idea of G. Nguetseng, 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 Tartar. 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.