Pointwise bounds on Dirichlet Green's functions for a singular drift term
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,\alpha}_{\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.
Cite
@article{arxiv.2511.12741,
title = {Pointwise bounds on Dirichlet Green's functions for a singular drift term},
author = {Aritro Pathak},
journal= {arXiv preprint arXiv:2511.12741},
year = {2026}
}
Comments
Added Eq 93, explaining the subsequent argument in greater detail. Further, a minor correction at the end of Lemma 8. arXiv admin note: text overlap with arXiv:2405.13313