Two-parametric error estimates in homogenization of second order elliptic systems in $\mathbb{R}^d$ including lower order terms

TL;DR

This paper develops two-parametric error estimates for the homogenization of second-order elliptic systems with lower order terms, providing approximations for the generalized resolvent in various operator norms.

Contribution

It introduces novel two-parametric error bounds for the homogenization process of elliptic systems including lower order terms, extending existing results to more general operators.

Findings

Derived two-parametric error estimates for the generalized resolvent.

Provided approximations in both $L_2$ and $H^1$ operator norms.

Extended homogenization techniques to systems with unbounded lower order terms.

Abstract

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we consider a selfadjoint operator ${\mathcal B}_\varepsilon$, $0< \varepsilon \leqslant 1$, given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D}) + \sum_{j=1}^d (a_j({\mathbf x}/\varepsilon) D_j +D_j a_j({\mathbf x}/\varepsilon)^*) + Q({\mathbf x}/\varepsilon)$, where $b({\mathbf D}) = \sum_{l=1}^d b_l D_l$ is the first order differential operator, and $g, a_j, Q$ are matrix-valued functions in ${\mathbb R}^d$ periodic with respect to some lattice $\Gamma$. It is assumed that $g$ is bounded and positive definite, while $a_j$ and $Q$ are, in general, unbounded. We study the generalized resolvent $({\mathcal B}_\varepsilon - \zeta Q_0({\mathbf x}/\varepsilon))^{-1}$, where $Q_0$ is a $\Gamma$-periodic, bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. Approximations for the generalized resolvent in the $(L_2 \to L_2)$- and $(L_2 \to H^1)$-norms with two-parametric error estimates (with respect to the parameters $\varepsilon$ and $\zeta$) are obtained.