Homogenization of periodic parabolic systems in the $L_2({\mathbb{R}}^d)$-norm with the corrector taken into account

Abstract

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, consider a self-adjoint matrix second order elliptic differential operator ${\mathcal{B}}_{\epsilon }$, $0<\epsilon ⩽1$. The principal part of the operator is given in a factorised form, the operator contains first and zero order terms. The operator ${\mathcal{B}}_{\epsilon }$ is positive definite, its coefficients are periodic and depend on $\mathbf{x}/\epsilon$. We study the behaviour in the small period limit of the operator exponential $e^{-{\mathcal{B}}_{\epsilon }t}$, $t⩾0$. The approximation in the $(L_2\to L_2)$-operator norm with error estimate of order $O({\epsilon }^2)$ is obtained. The corrector is taken into account in this approximation. The results are applied to homogenization of the solutions for the Cauchy problem for parabolic systems.

41 pages. This is an English translation of the paper originally published in Russian. Translated version: St. Petersburg Math. J. 31 (2020), to appear. Minor revision of the first version. Some typos removed