First-order homogenization
Riccardo Cristoferi, Lorenza D'Elia
TL;DR
This work delivers a first-order $\Gamma$-convergence result for quadratic energies in a periodic setting by isolating bulk behavior and overcoming boundary effects via a dual PDE perspective and a refined $\text{Riemann-Lebesgue}$ lemma. Central to the analysis is the identification of the bulk energy scale $\varepsilon_n$ and an explicit first-order limiting functional $F^1_{ ext{hom}}$ expressed through the gradient of the homogenized minimizer $u^{\min}_0$ and the first-order correctors $\psi_i$. The authors also show that for general $L^p$-type functionals, the first-order expansion collapses, yielding no additional information beyond the zeroth-order homogenization, under suitable convexity and periodicity assumptions. The results rely on a careful two-scale expansion, detailed corrector structures, and sharp energy estimates enabled by a quantitative RL lemma, with potential implications for thin-structure models and phase-field-type systems in periodic media.
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.