First-order homogenization

TL;DR

This paper establishes a first-order homogenization result for quadratic functionals, explicitly characterizing the limiting behavior using correctors and a novel duality approach with PDEs.

Contribution

It introduces a new method combining duality and a refined Riemann-Lebesgue Lemma to analyze quadratic functionals in homogenization.

Findings

Explicit form of the limiting functional derived

Scaling of energy identified

New duality-based approach developed

Abstract

We provide a first-order homogenization result for quadratic functionals. In particular, we identify the scaling of the energy and the explicit form of the limiting functional in terms of the first-order correctors. The main novelty of the paper is the use of the dual correspondence between quadratic functionals and PDEs, combined with a refinement of the classical Riemann-Lebesgue Lemma.