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

Technical Summary: A Counterexample for Pointwise Upper Bounds on Green's Function with Singular Drift at the Boundary

Problem Statement
This paper addresses the behavior of the Green's function associated with a sequence of elliptic operators $L_m = -\nabla \cdot (\nabla u) + B_m \cdot \nabla u$ defined on the unit ball $\Omega = B(0, 1) \subset \mathbb{R}^n$ ($n \geq 3$). The focus is on operators where the drift term $B_m$ diverges as the inverse distance to the boundary, specifically behaving like $C/(1-|x|)$ near $\partial \Omega$.

While pointwise upper and lower bounds for Green's functions are well-established for operators with bounded drifts or drifts bounded by $\epsilon/\delta(x)$ (where $\epsilon$ is sufficiently small), the behavior for drifts with a larger constant $C$ remains a point of contention. The paper specifically targets a claim made by Hofmann and Lewis regarding the existence of uniform upper estimates for such singular drifts in the elliptic setting.

Methodology
The author constructs a specific sequence of counterexamples using a radial drift field $B_m$ defined as:
$$B_m = \begin{cases} -\frac{C}{1-|r|} \hat{r} & \text{if } |r| \leq 1 - \frac{1}{m} \\ -C_m \hat{r} & \text{if } 1 - \frac{1}{m} < |r| < 1 \end{cases}$$
where $C \geq 1$ is an arbitrary constant and $C_m$ is chosen to ensure continuity or specific behavior in the boundary layer. As $m \to \infty$, this sequence approximates a drift $B(x)$ satisfying $|B(x)| \leq C/\delta(x)$.

The methodology proceeds in three main stages:

1. Symmetry Reduction: The author proves that for the specific pole at the origin, the Green's function $G_m(x, 0)$ depends only on the radial variable $r = |x|$. This reduces the partial differential equation to an ordinary differential equation (ODE) in the radial coordinate.
2. Establishment of Lower Bounds: Using arguments similar to those in existing literature (specifically referencing [2]), the paper establishes a uniform lower bound for the Green's function in the interior region. It is shown that $G_m(y, z) \geq c_0 |z-y|^{-(n-2)}$ for points sufficiently close to each other and away from the boundary, where the constant $c_0$ is independent of $m$. This relies on the Harnack inequality and careful estimation of the drift term's effect within annuli bounded away from the boundary.
3. Explicit Integration and Divergence Analysis: By solving the radial ODE explicitly using an integrating factor, the author derives an expression for the derivative of the Green's function. The solution is integrated from a point near the boundary to the center (specifically evaluating at $x = (1/2, 0, \dots, 0)$). The analysis distinguishes between the region where the drift follows the inverse-distance law and the boundary layer where it is truncated.

Key Results
The central result, Theorem 2, demonstrates that for the operator defined with $C=1$, the value of the Green's function at the point $(1/2, 0, \dots, 0)$ with a pole at the origin diverges to infinity as $m \to \infty$:
$$\lim_{m \to \infty} G_m\left(\frac{1}{2}, 0, \dots, 0; 0\right) = \infty$$
The proof relies on the fact that while the lower bound remains uniform, the upper bound fails. The explicit integration shows that the term involving the exponential of the drift integral grows unbounded as $m$ increases, specifically due to the contribution from the boundary layer where the drift is truncated at $C_m$.

Consequently, the paper establishes that uniform pointwise upper estimates for the Green's function do not exist for this class of singular drifts when the constant $C$ is not sufficiently small.

Significance and Claims
The paper claims to provide a definitive counterexample to an earlier assertion by Hofmann and Lewis (specifically Theorem 4.3(a) in Chapter III of their work [4]). The author argues that the purported proof in the cited work fails because it relies on an elliptic version of a lemma that cannot be proved without assuming the very upper bound on the Green's function that is being sought. The cited proof attempts to generalize a result from the parabolic case to the elliptic case, a step the author demonstrates is invalid for drifts of this magnitude.

Furthermore, the paper notes that if the constant $C$ in the drift is chosen to be sufficiently small (specifically $C < 1$ in the context of the constructed example, or generally $\epsilon(\Omega)$ as per [9]), the expected pointwise upper bounds are recovered. This highlights a sharp threshold in the behavior of solutions based on the magnitude of the singular drift.

Finally, the author observes that in the limit as $m \to \infty$, the solution to the Poisson-Dirichlet problem with bounded data (e.g., $f = \mathbf{1}_{B(1/2, \epsilon)}$) ceases to be well-defined in $W^{1,2}(B(0, 1))$, as the solution values diverge to infinity everywhere in a neighborhood of the origin.

Conclusion
The paper concludes that for elliptic operators with drifts diverging as the inverse distance to the boundary, uniform upper bounds on the Green's function cannot be guaranteed if the drift coefficient exceeds a certain threshold. This result corrects the literature regarding the validity of extending parabolic estimates to the elliptic setting for such singular coefficients.