Note on quantitative homogenization results for parabolic systems in R d.

In L 2 ( R d ; C n ) , we consider a semigroup e - t A ε , t ⩾ 0 , generated by a matrix elliptic second-order differential operator A ε ⩾ 0 . Coefficients of A ε are periodic, depend on x / ε , and oscillate rapidly as ε → 0 . Approximations for e - t A ε were obtained by Suslina (Funktsional Analiz i ego Prilozhen 38(4):86-90, 2004) and Suslina (Math Model Nat Phenom 5(4):390-447, 2010) via the spectral method and by Zhikov and Pastukhova (Russ J Math Phys 13(2):224-237, 2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by Suslina (2015).