Regularity of elliptic systems in divergence form with directional homogenization

In this paper, we study regularity of solutions of elliptic systems in divergence form with directional homogenization. Here directional homogenization means that the coefficients of equations are rapidly oscillating only in some directions. We will investigate the different regularity of solutions on directions with homogenization and without homogenization. Actually, we obtain uniform interior $W^{1, p}$ estimates in all directions and uniform interior $C^{1, γ}$ estimates in the directions without homogenization.