Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space

In L2(ℝd; ℂn), we consider a wide class of matrix elliptic second order differential operators $\mathcal{A}$ε with rapidly oscillating coefficients (depending on x/ε). For a fixed τ > 0 and small ε > 0, we find approximation of the operator exponential exp(− $\mathcal{A}$ετ) in the (L2(ℝd; ℂn) → H1(ℝd; ℂn))-operator norm with an error term of order ε. In this approximation, the corrector is taken into account. The results are applied to homogenization of a periodic parabolic Cauchy problem