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 $-\text{div} \big(A_\varepsilon \nabla u_{\varepsilon} \big) = 0$, where $A_\varepsilon(x) = A(x, x/\varepsilon_1,\cdots, x/\varepsilon_n)$ is an $n$-scale oscillating periodic coefficient matrix, and $(\varepsilon_i)_{1\le i\le n}$ are scale parameters. We show that the $C^\alpha$-H\"{o}lder continuity with any $\alpha\in (0,1)$ for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary $(\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n) \in (0,1]^n$ and particularly is independent of the ratios between $\varepsilon_i$'s. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by $H$-convergence. The Lipschitz estimate for arbitrary $(\varepsilon_i)_{1\le i\le n}$ still remains open. However, for special laminate structures, i.e., $A_\varepsilon(x) = A(x,x_1/\varepsilon_1, \cdots, x_d/\varepsilon_n)$, we show that the Lipschitz estimate is stable for arbitrary $(\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n) \in (0,1]^n$. This is proved by a technique of reperiodization.