HOMOGENIZATION OF THE NEUMANN PROBLEM FOR ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS

Let O subset of R-d be a bounded domain with the boundary of class C-1,C-1. In L-2(O; C-n), a matrix elliptic second order differential operator A(N),(epsilon) with the Neumann boundary condition is considered. Here epsilon > 0 is a small parameter; the coefficients of A(N),(epsilon) are periodic and depend on x/epsilon. There are no regularity assumptions on the coefficients. It is shown that the resolvent (A(N,epsilon)-lambda I)(-1), lambda is an element of C\R+, converges in the L-2-operator norm to the resolvent of the effective operator A(N)(0) with constant coefficients, as epsilon -> 0. A sharp order estimate parallel to( A(N),(epsilon) -lambda I)(-1) - (A(N)(0) - lambda I)(-1) parallel to (L2 -> L2) <= C epsilon is obtained. Approximation for the resolvent (A(N),(epsilon) -lambda I)(-1) in the norm of operators acting from L-2(O;C-n) to the Sobolev space H-1(O; C-n) with an error O(root epsilon) is found. Approximation is given by the sum of the operator (A(N)(0) -lambda I)(-1) and the first order corrector. In a strictly interior subdomain O' a similar approximation with an error O(epsilon) is obtained. The case where lambda = 0 is also studied.