On homogenization for a periodic elliptic operator in a strip

In a strip $\Pi = \mathbb{R}\times (0,a)$, the operator

is considered, where $g_1$, $g_2$ are periodic with respect to the first variable. Periodic boundary conditions are put on the boundary of the strip. The behavior of the operator $A_\varepsilon$in the limit $\varepsilon\to 0$ is studied. It is proved that, with respect to the operator norm in $L_2(\Pi)$, the resolvent $(A_\varepsilon+I)^{-1}$ tends to the resolvent of the effective operator $A^0$. A sharp order estimate for the norm of the difference of the resolvents is obtained. The operator $A^0$ is of the same type, but its coefficients depend only on $x_2$.