# Precised approximations in elliptic homogenization beyond the periodic setting

## TL;DR

This paper advances elliptic homogenization theory by establishing explicit convergence rates for two-scale expansions in non-periodic settings with local perturbations, using adapted correctors and generalized assumptions.

## Contribution

It introduces a novel approach to homogenization beyond periodic structures, providing explicit convergence rates and a generalized framework for local perturbations.

## Key findings

- Proved $W^{1,p}$ and Lipschitz convergence with explicit rates
- Developed an adapted corrector for non-periodic perturbations
- Generalized assumptions for broader applicability

## Abstract

We consider homogenization problems for linear elliptic equations in divergence form. The coecients are assumed to be a local perturbation of some periodic background. We prove $W^{1,p}$ and Lipschitz convergence of the two-scale expansion, with explicit rates. For this purpose, we use a corrector adapted to this particular setting, and dened in [10, 11], and apply the same strategy of proof as Avellaneda and Lin in [1]. We also propose an abstract setting generalizing our particular assumptions for which the same estimates hold.