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

The operator A(epsilon) = D(1)g(1)(x(1)/epsilon,x(2))D(1) + D(2)g(2)(x(1)/epsilon,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(epsilon)(x(1),x(2))=Q(x(1)/epsilon,x(2)). The behavior of the operator (A epsilon+Q(epsilon))(-1) as epsilon -> 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)(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 Schrodinger operator with a singular potential periodic in one direction. Copyright (C) 2011 John Wiley & Sons, Ltd.