Error estimates taking account of correctors in homogenization of elliptic operators

S. E. Pastukhova

Sbornik. Mathematics, Tome 215 (2024) no. 7, pp. 932-952

Résumé

For divergence-form second-order elliptic operators with measurable $\varepsilon$-periodic coefficients in $\mathbb{R}^d$ resolvent approximations with error term of order $\varepsilon^2$ as $\varepsilon\to 0$ in the operator norm $\|\cdot\|_{H^1{\to}H^1}$ are constructed. The method of two-scale expansions in powers of $\varepsilon$ up to order two inclusive is used. The lack of smoothness in the data of the problem is overcome by use of Steklov smoothing or its iterates. First scalar differential operators with real matrix of coefficients which act on functions $u\colon \mathbb{R}^d\to \mathbb{R}$, and then matrix differential operators with complex-valued tensor of order four which act on functions $u\colon \mathbb{R}^d\to \mathbb{C}^n$ are considered. Bibliography: 20 titles.

Détail

Keywords: periodic differential operators, homogenization, correctors, resolvent approximations, operator error estimates.