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

Abstract

The operatorvbox is considered in L2(ℝ2), 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 Qε(x1, x2) = Q(x1/ε, x2). The behavior of the operator (Aε + Qε)−1 as ε→0 is studied. It is proved that the operator (Aε + Qε)−1 tends to (A0 + Q0)−1 in the operator norm in L2(ℝ2). Here, A0 is the effective operator whose coefficients depend only on x2, Q0 is the mean value of Q in x1. A sharp order estimate for the norm of the difference (Aε + Qε)−1−(A0 + Q0)−1 is obtained. The result is applied to homogenization of the Schrödinger operator with a singular potential periodic in one direction. Copyright © 2011 John Wiley & Sons, Ltd.