Stochastic Homogenization and Geometric Singularities: A Study on Corners

.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 two dimensions: Unlike in the whole-space, on such a sector there exist nonsmooth 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.