A high-order corrector estimate for a semi-linear elliptic system in perforated domains

TL;DR

This paper develops a high-order corrector estimate for the homogenization of semi-linear elliptic systems in perforated domains, addressing nonlinear reactions and validating the convergence rate of asymptotic expansions.

Contribution

It introduces a novel high-order corrector estimate for semi-linear elliptic systems in perforated domains, extending previous homogenization results.

Findings

Provides a rigorous error estimate between micro and macro concentrations.

Validates the convergence rate of formal asymptotic expansions.

Addresses nonlinear volume and surface reactions in homogenization.

Abstract

We derive in this note a high-order corrector estimate for the homogenization of a microscopic semi-linear elliptic system posed in perforated domains. The major challenges are the presence of nonlinear volume and surface reaction rates. This type of correctors justifies mathematically the convergence rate of formal asymptotic expansions for the two-scale homogenization settings. As main tool, we follow the standard approach by the energy-like method to investigate the error estimate between the micro and macro concentrations and micro and macro concentration gradients. This work aims at generalizing the results reported in [2, 7].