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

The operator A" = D1g1(x1/", x2)D1 + D2g2(x1/", x2)D2 is considered in L2(R 2 ), where gj(x1, x2), j = 1,2, are periodic in x1 with period 1, bounded and positive definite. Let function Q(x1, x2) be bounded, positive definite and periodic in x1 with period 1. Let Q " (x1, x2) = Q(x1/", x2). The behavior of the operator (A" + Q " ) 1 as " ! 0 is studied. It is proved that the operator (A" + Q " ) 1 tends to (A 0 + Q 0 ) 1 in the operator norm in L2(R 2 ). Here A 0 is the effective operator whose coefficients depend only onx2, Q 0 is the mean value of Q in x1. A sharp order estimate for the norm of the difference (A" + Q " ) 1 (A 0 + Q 0 ) 1 is obtained. The �

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