On Homogenization for Piecewise Locally Periodic Operators

We discuss homogenization of a strongly elliptic operator A(epsilon) = - div Lambda(x, x/epsilon#)del on a bounded C-1,C-1 domain in R-d with either Dirichlet or Neumann boundary condition. The function A is piecewise Lipschitz in the first variable and periodic in the second one, and the function epsilon(#) is identically equal to epsilon(i)(epsilon) on each piece Omega(i), with epsilon(i)(epsilon) -> 0 as epsilon -> 0. For mu in a resolvent set, we show that the resolvent (A(epsilon) - mu)(-1) converges, as epsilon -> 0, in the operator norm on L-2(Omega)(n) to the resolvent (A(0) - mu)(-1) of the effective operator at the rate epsilon(nu), where epsilon(nu) stands for the largest of epsilon(i)(epsilon). We also obtain an approximation for the resolvent in the operator norm from L-2(Omega)(n) to H-1 (Omega)(n) with error of order epsilon(1/2)(nu).