Homogenization of parabolic and elliptic periodic operators in L2(?d) with the first and second correctors taken into account

摘要

In the space L2(?d;?n), a wide class of matrix elliptic second order differential operators (DO's) Aε(is studied; the A(are assumed to admit a factorization of the form A(= X* ε Xε where Xε is a homogeneous first order DO. The coefficients of these operators are periodic and depend on x/ε,ε > 0. The behavior of the operator exponential e-A ετ > 0, and of the resolvent (Aε + I)-1 for small ε is investigated. An approximation for the exponential e-A ετ in the operator norm in L2(Rd;Cn) with an error term of order τ-3/2ε3 is obtained. For the resolvent (Aε + I)-1, approximation in the norm of operators acting from H1(?d;?n) to L2(?d;?n) is found with an error term of order ε3. In these approximations, the first and second order correctors are taken into account.