On resolvent approximations of elliptic differential operators with periodic coefficients.

We consider resolvents (A ϵ + 1) − 1 of elliptic second-order differential operators A ϵ = − div a (x / ϵ) ∇ in R d with ε-periodic measurable matrix a (x / ϵ) and study the asymptotic behaviour of (A ϵ + 1) − 1 , as the period ε goes to zero. We provide a construction for the leading terms of the 'operator asymptotics' of (A ϵ + 1) − 1 in the sense of L 2 -operator-norm convergence and prove order ϵ 2 remainder estimates. We apply the modified method of the first approximation with the usage of Steklov's smoothing. The class of operators covered by our analysis includes uniformly elliptic families with bounded coefficients and also with unbounded coefficients from the John–Nirenberg space BMO (bounded mean oscillation). [ABSTRACT FROM AUTHOR]