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

1. Problem Statement

The paper investigates the pointwise upper bounds of the Dirichlet Green's function for a class of elliptic operators defined on convex bounded domains ($K \subset \mathbb{R}^n, n \geq 3$).

The operator $L$ is given by:
$$Lu = -\Delta u + B \cdot \nabla u = 0$$
where the drift term $B(X)$ exhibits a singular behavior near the boundary $\partial K$. Specifically, the drift satisfies:
$$\delta(X)^{1-\beta} |B(X)| \leq M$$
where $\delta(X) = \text{dist}(X, \partial K)$, $0 < \beta < 1$, and $M < \infty$.

Crucially, the drift satisfies a pointwise divergence condition:
$$-\frac{M}{\delta(x)^{2-\beta}} \leq \nabla \cdot B \leq 0$$
This implies that $\nabla \cdot B$ is negative in the distributional sense. The drift is strong enough that standard perturbative techniques (treating the drift as a lower-order perturbation of the Laplacian) fail, as the drift does not satisfy the condition $|B|^2 \in K_n$ (Kato class) required by previous literature (e.g., Cranston-Zhao, Ifra-Riahi).

The Goal: Establish an interior pointwise upper bound for the Green's function $G(x, y)$ of the form:
$$G(x, 0) \leq \frac{K'}{|x|^{n-2}}$$
for $x$ near the origin (the pole), where $K'$ depends on $M$ and the ellipticity constants.

2. Methodology

The paper extends and streamlines a previous result by the same author ([Pat25]), which was restricted to the unit ball. The primary challenge in moving from the unit ball to general convex domains is the geometry: in a non-spherical domain, moving infinitesimally inward from the boundary does not guarantee a uniform distance from the boundary, potentially causing the drift term to accumulate uncontrollably.

The author employs the following methodological innovations:

A. Minkowski Interpolation

To handle the geometry of general convex domains, the author introduces a family of convex bodies $K_y$ defined by the Minkowski sum:
$$K_y = (1-y)K + yB_1$$
where $K$ is the original domain and $B_1$ is an interior ball. This creates a continuous deformation from the boundary of $K$ to the interior ball. This allows the author to track maximum and minimum points of the Green's function on the boundaries $\partial K_y$ as $y$ varies, effectively creating a "spherical-like" coordinate system adapted to the convex geometry.

B. Near-Field and Far-Field Analysis

The proof is divided into two regimes based on the distance from the pole (origin):

1. Far-Field Effect ($K \setminus B_1$): The author analyzes the decay of the gradient of the Green's function as one moves from the boundary of $K$ toward the interior ball. By using the support functions of the convex bodies $K_y$, the author derives differential inequalities for the gradient along the normal direction.
2. Near-Field Effect ($B_1$): The analysis focuses on the behavior near the pole, utilizing Harnack inequalities and specific sequences of minimum points.

C. Differential Inequalities and Iteration

The core of the proof involves establishing differential inequalities for the radial derivative of the Green's function ($G'$) along the normal vectors of the surfaces $\partial K_y$.

• Maximum Principle: At a maximum point on a convex surface, the tangential second derivatives are non-positive.
• Drift Control: The author uses the condition on $\nabla \cdot B$ to bound the term $B \cdot \nabla G$.
• Iteration: By iterating these inequalities over infinitesimal steps $dy$ (or $dy$ in the support function space), the author shows that the gradient cannot grow arbitrarily large. The integral of the drift term along these paths is shown to be bounded due to the convexity and the specific power-law decay of the drift.

D. Contradiction Argument

The proof proceeds by contradiction. It assumes that the Green's function can grow faster than the fundamental solution ($1/|x|^{n-2}$). This assumption implies the existence of a sequence of drifts and points where the measure of the level sets of the Green's function behaves in a specific way (Case 1 or Case 2). By combining the upper bounds derived from the far-field analysis with lower bounds derived from the near-field analysis (using Lemma 4 and Lemma 5), the author demonstrates that the assumed growth leads to a mathematical contradiction.

3. Key Contributions

1. Extension to Convex Domains: The paper successfully generalizes the upper bound results for singular drifts from the unit ball to arbitrary convex bounded domains. This is significant because the geometry of convex domains introduces complexities in tracking the distance to the boundary that are absent in the ball.
2. Streamlined Proof: The author provides a more streamlined proof compared to the unit ball case by utilizing the Minkowski interpolation technique. This avoids the need for complex tracking of accumulating level sets in the far field, replacing it with a geometric argument based on the boundaries of the interpolated domains $K_y$.
3. Handling Non-Perturbative Drifts: The work addresses drifts that are too singular for standard $L^p$ or Kato class perturbation theories. It establishes that even with drifts diverging as $\delta(x)^{-(1-\beta)}$, the Green's function retains the same singularity order as the fundamental solution of the Laplacian ($|x|^{2-n}$).
4. New Lemmas: The paper introduces Lemma 5, which provides an alternative, streamlined proof for the control of gradients in the far-field case, ensuring the existence of minimum points with controlled gradient decay.

4. Main Results

Theorem 1 (Main Result):
For the operator $L = -\Delta + B \cdot \nabla$ in a convex bounded domain $K \subset \mathbb{R}^n$ ($n \geq 3$), with a drift $B$ satisfying the singular condition $\delta(X)^{1-\beta}|B(X)| \leq M$ and the divergence condition $\nabla \cdot B \leq 0$, the Dirichlet Green's function $G(x, 0)$ satisfies:
$$G(x, 0) \leq \frac{K'(M, \lambda)}{|x|^{n-2}}$$
for $x$ in the interior (specifically $|x| \leq 1/2$), where $K'$ is a constant depending only on $M$ and the ellipticity constants.

Theorem 2 & 3:
The paper recalls and confirms the lower bound (Theorem 2) and the bound with a moving pole (Theorem 3) from the previous work [Pat25], noting that these hold in the convex domain setting as well, provided the drift satisfies the necessary regularity in compact subdomains.

5. Significance

• Theoretical Advancement: This work bridges a gap in the theory of elliptic PDEs with singular coefficients. It demonstrates that the "worst-case" singularity of the Green's function is determined by the principal part (Laplacian) even in the presence of strong, singular drifts, provided the drift has a specific negative divergence structure.
• Geometric Robustness: By proving these estimates for convex domains, the results are applicable to a much wider class of physical and geometric problems than just spherical domains. The use of Minkowski interpolation offers a powerful tool for analyzing PDEs in non-smooth or non-symmetric convex geometries.
• Foundation for Future Work: The author notes that these methods could be extended to non-convex domains via bi-Lipschitz mappings and are relevant for problems involving landscape functions and singular potentials in other areas of PDEs. The results also clarify the limits of perturbative methods, showing that non-perturbative techniques are necessary for this class of drifts.

In summary, the paper establishes that the fundamental singularity structure of the Green's function is robust against singular drifts in convex domains, providing rigorous upper bounds that match the Laplacian's fundamental solution.