A counterexample for pointwise upper bounds on Green's function with a singular drift at boundary

Abstract

We show an example of a sequence of elliptic operators in the unit ball with drifts that diverge as the inverse distance to the boundary, for which we do not get uniform upper estimates for the Green's function with the pole at the origin. Such drifts have been considered in the literature in the study of the $L^{p}$ Dirichlet problem for both the parabolic and elliptic operators. Our construction provides a counterexample to an earlier claim of Hofmann and Lewis.