HOMOGENIZATION OF HIGH ORDER ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS

A selfadjoint strongly elliptic operator A(epsilon) of order 2p given by the expression b(D)* g(x/epsilon) b(D), epsilon > 0, is studied in L-2(R-d; C-n). Here g(x) is a bounded and positive definite (m x m)-matrix-valued function on R-d; it is assumed that g(x) is periodic with respect to some lattice. Next, b(D) = Sigma(vertical bar alpha vertical bar=p) b(alpha)D(alpha) is a differential operator of order p with constant coefficients; the b(alpha) are constant (m x n)-matrices. It is assumed that m >= n and that the symbol b(xi) has maximal rank. For the resolvent (A(epsilon) - zeta I)(-1) with zeta is an element of C \ [ 0,infinity), approximations are obtained in the norm of operators in L-2(R-d; C-n) and in the norm of operators acting from L-2(R-d; C-n) to the Sobolev space H-p(R-d; C-n), with error estimates depending on epsilon and zeta.