Compactness and stable regularity in multiscale homogenization

Abstract

In this paper we develop some new techniques to study the multiscale elliptic equations in the form of −div Aε∇uε = 0, where Aε(x) = A(x, x/ε1, • • • , x/εn) is an n-scale oscillating periodic coefficient matrix, and (εi) 1≤i≤n are scale parameters. We show that the C α -Hölder continuity with any α ∈ (0, 1) for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary (ε1, ε2, • • • , εn) ∈ (0, 1] n and particularly is independent of the ratios between εi's. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by H-convergence. The Lipschitz estimate for arbitrary (εi) 1≤i≤n still remains open. However, for special laminate structures, i.e., Aε(x) = A(x, x1/ε1, • • • , x d /εn), we show that the Lipschitz estimate is stable for arbitrary (ε1, ε2, • • • , εn) ∈ (0, 1] n . This is proved by a technique of reperiodization.