Discrete Green’s functions

Abstract

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