Spectral approach to homogenization of an elliptic operator periodic in some directions

Abstract

The operator \[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \] is considered in $L_2({\mathbb{R}}^2)$, where $g_j(x_1,x_2)$, $j=1,2,$ are periodic in $x_1$ with period 1, bounded and positive definite. Let function $Q(x_1,x_2)$ be bounded, positive definite and periodic in $x_1$ with period 1. Let $Q^{\epsilon }(x_1,x_2)=Q(x_1/\epsilon ,x_2)$. The behavior of the operator $(A_{\epsilon }+Q^{\epsilon }$ as $\epsilon \to 0$ is studied. It is proved that the operator $(A_{\epsilon }+Q^{\epsilon }{)}^{-1}$ tends to $(A^0+Q^0{)}^{-1}$ in the operator norm in $L_2({\mathbb{R}}^2)$. Here $A^0$ is the effective operator whose coefficients depend only on $x_2$, $Q^0$ is the mean value of $Q$ in $x_1$. A sharp order estimate for the norm of the difference $(A_{\epsilon }+Q^{\epsilon }{)}^{-1}-(A^0+Q^0{)}^{-1}$ is obtained. The result is applied to homogenization of the Schrödinger operator with a singular potential periodic in one direction.

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