Pointwise bounds on Dirichlet Green's functions for a singular drift term

TL;DR

This paper develops pointwise bounds for Green's functions of elliptic operators with singular boundary-diverging drift terms, extending previous results to non-coercive and singular cases.

Contribution

Introduces a novel technique to bound Green's functions with singular, non-coercive drifts near the boundary of the domain.

Findings

Bounds are uniform in subdomains of the unit ball for drifts with controlled singularity.

Explicit dependence of bounds on the radius of the subdomain is provided.

First estimates for non-coercive drifts, applicable even with singular potentials.

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.