Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space

TL;DR

This paper develops a large-scale boundary regularity theory for solutions to random elliptic equations on the half-space, establishing decay estimates and Liouville theorems using homogenization correctors adapted to the boundary.

Contribution

It introduces a novel boundary regularity framework for random elliptic operators, including the construction of homogenization correctors specifically adapted to the half-space setting.

Findings

Established $C^{1,eta}$-type regularity at the boundary for random elliptic equations.

Derived decay estimates for the homogenization-adapted tilt-excess near the boundary.

Proved a Liouville theorem characterizing solutions with polynomial growth.

Abstract

We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated $C^{1,\alpha}$-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are based on the existence of homogenization correctors adapted to the half-space setting, which we construct - by an entirely deterministic argument - as a modification of the homogenization corrector on the whole space. This adaption procedure is carried out inductively on larger scales, crucially relying on the regularity theory already established on smaller scales.