HOMOGENIZATION OF ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS: OPERATOR ERROR ESTIMATES IN L2(Rd) WITH CORRECTOR TAKEN INTO ACCOUNT

A matrix elliptic selfadjoint second order differential operator (DO) B-epsilon with rapidly oscillating coefficients is considered in L-2(R-d; C-n). The principal part b(D)* g(epsilon(-1)x) b(D) of this operator is given in a factorized form, where g is a periodic, bounded, and positive definite matrix-valued function and b(D) is a matrix first order DO whose symbol is a matrix of maximal rank. The operator B-epsilon also includes first and zero order terms with unbounded coefficients. The problem of homogenization in the small period limit is studied. For the generalized resolvent of B-epsilon, approximation in the L-2(R-d; C-n)-operator norm with an error O(epsilon(2)) is obtained. The principal term of this approximation is given by the generalized resolvent of the effective operator B-0 with constant coefficients. The first order corrector is taken into account. The error estimate obtained is order sharp; the constants in estimates are controlled in terms of the problem data. General results are applied to homogenization problems for the Schrodinger operator and the two-dimensional Pauli operator with singular rapidly oscillating potentials.