Note on quantitative homogenization results for parabolic systems in $\mathbb{R}^d$

TL;DR

This paper presents a new, concise proof for homogenization results of parabolic systems with rapidly oscillating periodic coefficients, utilizing contour integral representation and existing resolvent approximations.

Contribution

It offers an alternative proof method for homogenization of parabolic systems, simplifying previous spectral and shift method approaches.

Findings

Provides a new proof based on contour integral representation.

Uses existing resolvent approximation results with two-parametric error estimates.

Simplifies the derivation of homogenization results for parabolic systems.

Abstract

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a semigroup $e^{-tA_\varepsilon}$, $t\geqslant 0$, generated by a matrix elliptic second order differential operator $A_\varepsilon \geqslant 0$. Coefficients of $A_\varepsilon$ are periodic, depend on $\mathbf{x}/\varepsilon$ and oscillate rapidly as $\varepsilon \rightarrow 0$. Approximations for $e^{-tA_\varepsilon}$ were obtained by T. A. Suslina (2004, 2010) via the spectral method and by V. V. Zhikov and S. E. Pastukhova (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 T. A. Suslina (2015).