Construction and properties for the Green's function with Neumann boundary condition

TL;DR

The paper constructs and analyzes the Green's function $G$ for the Neumann problem associated with the coercive operator $- abla^2 + a$ on a smooth bounded domain $\Omega$, with $a\in L^{\infty}(\Omega)$ and $N\ge 3$. It develops an explicit iterative construction using the Laplacian fundamental solution $\Gamma$ and auxiliary kernels $\Gamma_i$, then corrects to satisfy Neumann boundary conditions, and extends the function to the boundary. The authors prove existence, uniqueness, symmetry, positivity, a representation formula for solutions, and sharp interior and boundary pointwise estimates, including precise comparisons to $\Gamma$ and to $\Gamma_\nu$ near the boundary. These results provide quantitative tools for explicit solution representations and will support further work on nonlinear problems under Neumann boundary conditions.

Abstract

This article addresses the construction and analysis of the Green's function for the Neumann boundary value problem associated with the operator $-Δ+ a$ on a smooth bounded domain $Ω\subset \mathbb{R}^N$ ($N \geq 3$) with $a\in L^\infty(Ω)$. Under the assumption that $-Δ+ a$ is coercive, we obtain the existence, uniqueness, and qualitative properties of the Green's function $G(x,y)$. The Green's function $G(x,y)$ is constructed explicitly, satisfying pointwise estimates and derivative estimates near the singularity. Also, near the boundary of $Ω$, $G$ is compared to the Green's function of the laplacian, with pointwise estimates. Other properties, like symmetry and positivity among other things, are established.

Theorems & Definitions (37)

• Definition 1: Green's function
• Theorem 1.1
• Definition 2
• Proposition 2.1
• proof
• Definition 3
• Definition 4
• Definition 5
• Lemma 2.2
• proof