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

This article addresses the construction and analysis of the Green's function for the Neumann boundary value problem associated with the operator $-\Delta + a$ on a smooth bounded domain $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) with $a\in L^\infty(\Omega)$. Under the assumption that $-\Delta + a$ 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 $\Omega$, $G$ is compared to the Green's function of the laplacian, with pointwise estimates. Other properties, like symmetry and positivity among other things, are established.