Pointwise bounds on Dirichlet Green's functions for a singular drift term
Aritro Pathak
TL;DR
This work establishes sharp, pointwise bounds for the Dirichlet Green's function of elliptic operators with a Laplacian principal part and a singular drift that blows up near the boundary in the unit ball, under a negative-divergence condition and $0<\beta<1$. The authors develop a lower bound using a Grönwall-type energy method and introduce a novel upper-bound strategy based on a modified Lorentz-norm and level-set analysis, producing bounds that are uniform on $B(0,r)$ with explicit $r$-dependence. Remarkably, these bounds hold even for noncoercive drifts that are majorized by radial functions, marking a first step in pointwise Green's-function estimates for such drifts and suggesting extensions to singular potentials and Schrödinger-type problems. The approach combines maximum principles, weak formulations, and delicate control of Green's-function level sets, and lays groundwork for generalizations to other domains and parabolic analogues.
Abstract
We introduce a technique to obtain pointwise upper and lower bounds for the Green's function of elliptic operators whose principal part is the Laplacian and that include a drift term diverging near the boundary like a power of the inverse distance with exponent less than 1, in the unit ball B(0,1) \subset \mathbb{R}^n, n \ge 3. The constants in the upper estimates are uniform in B(0,r) for each r < 1, with explicit dependence on r. The drift here belongs to C^{1,α}_{\mathrm{loc}} and may, more generally, be majorized by a function radially integrable up to the boundary. These appear to be the first such estimates for non-coercive drifts and remain new even for smooth drifts, suggesting extensions to singular potentials and other settings where energy methods fail.