Cell averaging two-scale convergence: Applications to periodic homogenization

TL;DR

This paper introduces a new form of two-scale convergence that simplifies homogenization proofs for PDEs with periodic coefficients, demonstrating its effectiveness on classical problems and discussing limitations on nonlinear cases.

Contribution

It proposes an alternative two-scale convergence method that streamlines homogenization analysis and proves compactness results more naturally.

Findings

Simplifies the proof of classical homogenization results.

Establishes first-order boundary corrector results.

Identifies limitations in nonlinear problems.

Abstract

The aim of the paper is to introduce an alternative notion of two-scale convergence which gives a more natural modeling approach to the homogenization of partial differential equations with periodically oscillating coefficients: while removing the bother of the admissibility of test functions, it nevertheless simplifies the proof of all the standard compactness results which made classical two-scale convergence very worthy of interest: bounded sequences in $L^2_{\sharp}[Y,L^2(\Omega)]$ and $L^2_{\sharp}[Y,H^1(\Omega)]$ are proven to be relatively compact with respect to this new type of convergence. The strengths of the notion are highlighted on the classical homogenization problem of linear second-order elliptic equations for which first order boundary corrector-type results are also established. Eventually, possible weaknesses of the method are pointed out on a nonlinear problem: the weak two-scale compactness result for $\mathbb{S}^2$-valued stationary harmonic maps.