Note on quantitative homogenization results for\xa0parabolic systems in {mathbb {R}}^d

Yulia Meshkova

doi:10.1007/s00028-020-00600-2

Abstract

In L_2({mathbb {R}}^d;{mathbb {C}}^n), we consider a semigroup e^{-tA_varepsilon }, tgeqslant 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 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).

Highlights

The subject of this note is quantitative estimates in periodic homogenization, i.e. approximations for the corresponding resolving operator in the uniform operator topology
The homogenization problem is to describe the behaviour of the solution uε in the small period limit ε → 0
Since we derive the parabolic estimates from the elliptic ones, the achievement of the present paper can be interpreted as a quantitative Trotter–Kato like result in homogenization context

Summary

Introduction

The subject of this note is quantitative estimates in periodic homogenization, i.e. approximations for the corresponding resolving operator in the uniform operator topology. The classical result is that uε → u0 in the L2-norm, where the limit function u0 is the solution of the equation of the same type A0u0 + u0 = F, where A0 = −div g0∇ is the so-called effective operator with the constant matrix g0. The limit behaviour of its resolvent or the semigroup e−t Aε is given by the corresponding function of the so-called effective operator A0 = b(D)∗g0b(D) with the constant matrix g0. For the scalar elliptic operator Aε = −div gε(x)∇, where g(x) is a symmetric matrix with real entries, one has ∈ L∞ and it is possible to replace the smoothing operator Sε in the corrector by the identity operator In this case, estimate (10) was obtained in [12, Theorem 1.3]. For the matrix elliptic operator, (L2 → H 1)-approximation for e−t Aε was proved in [9, Theorem 11.1] (but with another smoothing operator in the corrector)

New proof

Discussion