A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equations

Discrete Green’s functions

“…The results of this theorem are a significant improvement over the estimates in [5] where the order of the singularity in the zzzth order difference quotient was Ppq~' with e > 0.…”

“…Our estimates, in this section, on the orders of growth of difference quotients of the solution to (1) will be an improvement and an extension of the results in [5, p. 31]. Our proof will rest heavily on the method of proof in [5,Theorem 3]. We will also use a result of Bramble and Thomée [1, Theorem, p. 585] on the rate of growth of GiP; Q); in particular, their result says that {G(P; Q)\p is summable for any power p 2: 0.…”

“…Let U(Z) = GRiZ; Q) and V(Z) = GR(P; Z). Since we may make, by reflection, U(Z) = 0 on dRh, our result follows by an application of the discrete Green's identity; see [5].…”

“…If a(P) and c(P) are a-Holder continuous over R with common Holder constant La and if the condition in (2) is satisfied, then there exist constants Sm and Tm, which depend upon L, X, La, diam R and a but are independent of A, such that (3) |£>(m)(P, 0)1 g SjPmPQ; \D(m\P;Q)\ g Tm mm{diP), diQ)}/P%\ Proof. We reflect G(P; Q) into a region Û'h D Ö with ti'h described in [5]. About ß G tih and each of its reflected images, we construct squares Mh(Q) of sidelength NQh where 7VQ is independent of A and Q.…”