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

2025/11/16 by Aritro Pathak, Pathak, Aritro
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2511.12741

openalex publication_date 2025/11/16 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

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) ⊂ ℝⁿ, n ≥ 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,α_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.

Citations

• A counterexample for pointwise upper bounds on Green's function with a singular drift at boundary
• Global Hölder solvability of second order elliptic equations with locally integrable lower-order coefficients
• The Fundamental Solution of an Elliptic Equation with Singular Drift
• Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields
• Uniform rectifiability and elliptic operators satisfying a Carleson measure condition
• Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains
• Harnack inequality for degenerate and singular elliptic equations with\n unbounded drift
• The Harnack inequality and related properties for solutions to elliptic\n and parabolic equations with divergence-free lower-order coefficients
• On Divergence-free Drifts

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