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

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 $L_{2} (R^{2})$ , 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 $Q^{ε} (x_{1}, x_{2}) = Q (x_{1} / ε, x_{2})$ . The behavior of the operator $(A_{ε} + Q^{ε}$ as $ε \to 0$ is studied. It is proved that the operator $(A_{\varepsilon}+ Q^{\varepsilon})^{-1} $t e n d s t o$ (A^{0} + Q^{0})^{-1} $i n t h e o p e r a t o r n o r m i n$ L_{2}( $R$ ^{2}) $. H e r e$ 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 $(A_{ε} + Q^{ε})^{- 1} - (A^{0} + Q^{0})^{- 1}$ is obtained. The result is applied to homogenization of the Schr"odinger operator with a singular potential periodic in one direction.

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 $L_{2} (R^{2})$ , 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 $Q^{ε} (x_{1}, x_{2}) = Q (x_{1} / ε, x_{2})$ . The behavior of the operator $(A_{ε} + Q^{ε}$ as $ε \to 0$ is studied. It is proved that the operator $(A_{\varepsilon}+ Q^{\varepsilon})^{-1} $t e n d s t o$ (A^{0} + Q^{0})^{-1} $i n t h e o p e r a t o r n o r m i n$ L_{2}( $R$ ^{2}) $. H e r e$ 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 $(A_{ε} + Q^{ε})^{- 1} - (A^{0} + Q^{0})^{- 1}$ is obtained. The result is applied to homogenization of the Schr"odinger operator with a singular potential periodic in one direction.

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