Compactness and stable regularity in multiscale homogenization

In this paper, we develop some new techniques to study the multiscale elliptic equations in the form of -div(A(epsilon) del u(epsilon)) = 0, where A(epsilon) (x) = A( x, x/epsilon(1), ..., x/epsilon(n)) is an n-scale oscillating periodic coefficient matrix, and (epsilon(i))(1 <= i <= n) are scale parameters. We show that the C-a-Holder continuity with any alpha is an element of (0, 1) for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary (epsilon(1), epsilon(2), ..., epsilon(n)) is an element of (0, 1](n) and particularly is independent of the ratios between epsilon(i)'s. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by H-convergence. The Lipschitz estimate for arbitrary (epsilon(i))(1 <= i <= n) still remains open. However, for special laminate structures, i.e., A(e) (x) = A( x, x(1)/epsilon(1), ..., x(d)/epsilon(d)), we show that the Lipschitz estimate is stable for arbitrary (epsilon(1), epsilon(2),..., epsilon(d)) is an element of (0, 1](d). This is proved by a technique of reperiodization.