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

TL;DR

This paper constructs a counterexample demonstrating that uniform upper bounds for Green's functions fail when elliptic operators have singular boundary drifts diverging as inverse distance to the boundary.

Contribution

It provides the first explicit counterexample to previous claims of uniform Green's function bounds under singular boundary drifts.

Findings

Counterexample invalidates previous uniform upper bound claims

Drifts diverging as inverse boundary distance cause failure of bounds

Highlights limitations in existing elliptic PDE estimates

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.