Abstrakt:
I will discuss certain homogenization and large-scale regularity results in degenerate and/or singular random media. First I describe a quenched invariance principle and local limit theorem for the random conductance model with unbounded conductances satisfying certain moment bounds. For this, deterministic regularity results for non-uniformly elliptic equations play an important role.
Finally I discuss homogenization and regularity results for uniformly elliptic equations on two-dimensional sector with a non-convex corner. On these domains exist non-smooth harmonic functions satisfying homogeneous Dirichlet boundary conditions. For those functions, we construct ‚corner-correctors‘ and give quasi-optimal estimates on the growth of those ‚corner-correctors‘. These estimates are then used to develope a corresponding large-scale regularity theory and to establish quasi-optimal error estimates for a non-standard two-scale expansion adapted to sectorial domains.