On correctors for linear elliptic homogenization in the presence of local defects

TL;DR

This paper investigates the existence and uniqueness of correctors in linear elliptic homogenization with local defects, introducing a new approach based on a priori estimates and extending to various related equations.

Contribution

It provides a more general and versatile method for establishing corrector existence in perturbed periodic media, including non-divergence form and advection-diffusion equations.

Findings

Established existence and uniqueness of correctors with gradient in L^r for locally perturbed coefficients.

Developed an alternative approach based on a priori estimates for well-posedness.

Extended results to non-divergence form equations and various generalizations.

Abstract

We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form --$\partial$i(aij$\partial$ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in L r if the local perturbation is itself L r , r < +$\infty$. The present work follows up on our works [7, 8, 9], providing an alternative, more general and versatile approach , based on an a priori estimate, for this well-posedness result. Equations in non-divergence form such as --aij$\partial$iju = f are also considered, along with various extensions. The case of general advection-diffusion equations --aij$\partial$iju + bj$\partial$ju = f is postponed until our future work [10]. An appendix contains a corrigendum to our earlier publication [9].