Estimates in homogenization of higher-order elliptic operators

A divergent-type elliptic operator A(epsilon) of arbitrary even order 2m is studied. Coefficients of the operator are epsilon-periodic, epsilon > 0 is a small parameter. The resolvent equation A(epsilon)u(epsilon) + lambda u(epsilon) = f is solvable in the Sobolev space H-m(R-d) of order in m for any f is an element of L-2(R-d), provided the parameter lambda is sufficiently large, lambda > Lambda, where the bound A depends only on constants from ellipticity condition. The limit equation is of the same type but with constant coefficients, that is, (A) over capu + lambda u = f. The limit operator (A) over cap can be considered here, for instance, in the sense of G-convergence. We prove that the resolvent ((A) over cap + lambda)(-1) approximates (A(epsilon) + lambda)(-1) in operator (L-2 -> L-2)-norm with the estimate parallel to(A(epsilon) + lambda)(-1)-((A) over cap +lambda)(-1)parallel to(L2(Rd)-> L2(Rd)) = O(epsilon), as epsilon -> 0. We find also the approximation of the resolvent (A(epsilon) + lambda)(-1) operator (L-2 -> H-m)-norm. This is the sum ((A) over cap + lambda)(-1) + K-epsilon, where K-epsilon is a correcting Qperat r whose structure is given'We Pr ve the estimate parallel to(A(epsilon) + lambda)(-1)-((A) over cap +lambda)(-1)-K-epsilon parallel to(L2(Rd))-> Hm(Rd) = O(epsilon), as epsilon -> 0.