Green’s function normal-derivative asymptotics via thickness

Establish the asymptotic relation between the boundary normal derivative of the Green’s function G_Ω and the thickness function τ_Ω at boundary points with a well-defined normal, namely that ∂G_Ω/∂ν(x,y) ∼ 1/τ_Ω(x) as y approaches x along the inward normal.

Background

The authors introduce the thickness function τΩ as a quantitative descriptor of how far ∂Ω lies from the reference convex set C along inward normals. They then propose a potential-theoretic connection linking τΩ to the singular behavior of the Green’s function’s normal derivative near the boundary.

Establishing this relation would quantitatively tie geometric thickness to analytic boundary singularities in potential theory.

References

Conjecture Let $G_{\Omega}(x,y)$ be the Green's function of $\Omega$. Then for $x \in \partial \Omega$ where the normal exists, we have:

\frac{\partial G_{\Omega}{\partial \nu}(x,y) \sim \frac{1}{\tau_{\Omega}(x)} \quad \text{as } y \to x \text{ along the normal}.

This relation quantifies the singularity of the normal derivative near the boundary.

— Geometric Properties of Level Sets for Domains under Geometric Normal Property  (2603.30026 - Barkatou, 31 Mar 2026) in Applications and Perspectives (Potential Theory subsection)