The Neumann Green function of the annulus

TL;DR

This work constructs an explicit Neumann Green function for the Laplacian in an annulus in $\mathbb{R}^N$ ($N\ge3$) by expanding in zonal harmonics and Gegenbauer polynomials. The key result expresses $G_a(x,y)$ as the sum of the singular kernel $\frac{1}{\omega_{N-1}(N-2)|x-y|^{N-2}}$ and a carefully computed regular part $H_a(x,y)$, with explicit radial coefficients $A_m(\rho), B_m(\rho), C_0$ tied to the inner and outer boundary conditions. This representation accounts for the zero-average normalization required by Neumann problems and enables precise analysis of Green and Robin functions, with implications for Lyapunov-Schmidt reductions and multi-bubble phenomena in annular domains. The approach parallels Dirichlet results (Grossi–Vujadinovi\'c) but reveals essential Neumann-specific features, and aligns with prior work on critical elliptic problems in related geometries (Salazar, PRV).

Abstract

Using Gegenbauer polynomials and the zonal harmonic functions we build an explicit representation formula for the Green function with Neumann boundary conditions in the annulus.

Theorems & Definitions (10)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['principale']}.
• Remark 3.3
• Remark 3.4