LIOUVILLE PRINCIPLES AND A LARGE-SCALE REGULARITY THEORY FOR RANDOM ELLIPTIC OPERATORS ON THE HALF-SPACE

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,C-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.