Stochastic homogenization and geometric singularities : a study on corners

TL;DR

This paper investigates the effects of geometric singularities like corners on the homogenization of elliptic equations, introducing new correctors and estimates to better understand and approximate solutions in polygonal domains.

Contribution

It constructs and analyzes corner correctors for homogenization in polygonal domains, providing growth estimates and a novel 2-scale expansion adapted to corners.

Findings

Constructed extended homogenization correctors for corners.

Proved growth estimates with logarithmic loss.

Developed a quasi-optimal error estimate for the new expansion.

Abstract

In this contribution we are interested in the quantitative homogenization properties of linear elliptic equations with homogeneous Dirichlet boundary data in polygonal domains with corners. To begin our study of this situation, we consider the setting of an angular sector in 2 dimensions : Unlike in the whole-space, on such a sector there exist non-smooth harmonic functions (these depend on the angle of the sector). Here, we construct extended homogenization correctors corresponding to these harmonic functions and prove growth estimates for these which are quasi-optimal, namely optimal up to a logarithmic loss. Our construction of the corner correctors relies on a large-scale regularity theory for a-harmonic functions in the sector, which we also prove and which, as a by-product, yields a Liouville principle. We also propose a nonstandard 2-scale expansion, which is adapted to the sectoral domain and incorporates the corner correctors. Our final result is a quasi-optimal error estimate for this adapted 2-scale expansion.