On homogenization for a periodic elliptic operator in a strip

Abstract:

In a strip $\Pi = \mathbb {R}\times (0,a)$, the operator \begin{equation*} A_\varepsilon = D_1 g_1(x_1/\varepsilon ,x_2) D_1 + D_2 g_2(x_1/\varepsilon ,x_2) D_2 \end{equation*} 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 $x_2$.