Dirichlet Green's functions with singular drifts at the boundary of convex domains

TL;DR

This paper derives upper bounds for Dirichlet Green's functions of elliptic operators with singular boundary drifts in convex domains, extending previous results and simplifying proofs.

Contribution

It extends prior work on Green's functions with boundary singular drifts from the unit ball to general convex domains, providing streamlined proofs.

Findings

Established interior pointwise upper bounds for Green's functions

Extended results from the unit ball to convex domains

Simplified the proof methodology

Abstract

In convex bounded domains in R^n with n >= 3, we establish interior pointwise upper bounds for the Dirichlet Green's function of elliptic operators in the unit ball B(0,1) in R^n, n >= 3, whose principal part is the Laplacian and which include a drift term that diverges near the boundary like a negative power of the distance with exponent strictly less than 1. This work extends an earlier result for operators with such drifts in the unit ball, and streamlines the proof in particular to adopt it to the question in convex domains.