Discrete Green’s functions

Abstract:

Let $G(P;Q)$ be the discrete Green’s function over a discrete h-convex region $\Omega$ of the plane; i.e., $a(P){G_{x\bar x}}(P;Q) + c(P){G_{y\bar y}}(P;Q) = - \delta (P;Q)/{h^2}$ for $P \in {\Omega _h},G(P;Q) = 0$ for $P \in \partial {\Omega _h}$. Assume that $a(P)$ and $c(P)$ are Hölder continuous over $\Omega$ and positive. We show that $|{D^{(m)}}G(P;Q)| \leqq {A_m}/\rho _{P\;Q}^m$ and $|{\tilde D^{(m)}}G(P;Q)| \leqq {B_m}d(Q)/\rho _{P\;Q}^{m + 1}$, where ${D^{(m)}}$ is an mth order difference quotient with respect to the components of P or Q, and ${\tilde D^{(m)}}$ denotes an mth order difference quotient only with respect to the components of P.