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

Aritro Pathak

TL;DR

The paper addresses whether uniform pointwise upper bounds for the Dirichlet Green's function of elliptic operators with boundary-singular drift can hold in the unit ball. It reduces the problem to a radial ODE for the Green's function along the axis and uses an integrating factor argument to show that, for a drift strength $C=1$, the Green's function $\mathcal{G}_m(x,0)$ becomes unbounded at a fixed point such as $(\tfrac{1}{2},0,\dots,0)$ as the truncation parameter $m\to\infty$. It also proves a uniform-in-$m$ lower bound in balls away from the pole and notes that the result fails when the drift constant satisfies $C<1$, where uniform bounds can hold. These findings contradict a prior claim by Hofmann and Lewis and clarify the threshold behavior between well-posedness with bounded drift and blow-up with boundary-singular drift, with implications for the $L^p$ Dirichlet problem and elliptic measure.

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.

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

TL;DR

The paper addresses whether uniform pointwise upper bounds for the Dirichlet Green's function of elliptic operators with boundary-singular drift can hold in the unit ball. It reduces the problem to a radial ODE for the Green's function along the axis and uses an integrating factor argument to show that, for a drift strength , the Green's function becomes unbounded at a fixed point such as as the truncation parameter . It also proves a uniform-in- lower bound in balls away from the pole and notes that the result fails when the drift constant satisfies , where uniform bounds can hold. These findings contradict a prior claim by Hofmann and Lewis and clarify the threshold behavior between well-posedness with bounded drift and blow-up with boundary-singular drift, with implications for the Dirichlet problem and elliptic measure.

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 Dirichlet problem for both the parabolic and elliptic operators. Our construction provides a counterexample to an earlier claim of Hofmann and Lewis.
Paper Structure (4 sections, 5 theorems, 28 equations)

This paper contains 4 sections, 5 theorems, 28 equations.

Table of Contents

  1. Introduction.
  2. Lower bounds on the Green's function.
  3. Counterexample for uniform upper bounds.
  4. Acknowledgment.

Key Result

Theorem 1

For any bounded domain $\Omega\subset \mathbb{R}^{n}$, there exists a unique non-negative function $\mathcal{G}_m:\Omega\times \Omega\to \mathbb{R}\cup \{\infty\}$, called the Green's function associated with $L_m$, such that the following holds:

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof : Proof of \ref{['thm1']}
  • Corollary 5
  • proof : Proof of Theorem 2
  • Remark 1
  • Remark 2
  • ...and 1 more