Higher-Order Estimates for Fully Nonlinear Difference Equations. Ii

Abstract

The purpose of this work is to establish a priori C 2,α estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator F h is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author's knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh n-plane. In a subsequent paper a truly interior estimate will be established in a mesh n-box.

Schauder estimates for fully nonlinear elliptic difference operators

“…and assume initially that x = 0 and m i > 0; i = 1; : : : ; n. Expanding, as in the one-dimensional case [2,13], we have u(y) = (u + m 1 j± 1 j¯1u)(0; y 2 ; : : : ; y n ) + m1¡2 X k= 0…”

“…We pass from the frozen to the general case though a method employed by Safonov [10,11,15] for fully nonlinear partial di¬erential equations, which was already extended to di¬erence equations by Holtby in [2,3]. As remarked earlier, our equations are more general than those considered by Holtby, and overall our approach in simpler.…”

“…To illustrate our conditions, we describe the particular example treated by Holtby in [2]. Here, the mesh E is the cubic mesh of width h in R n , that is, E = fh(m 1 ; : : : ; m n ) 2 R n j m i 2 Z; i = 1; : : : ; ng;…”

Higher-order estimates for fully nonlinear difference equations. I

“…Ifd@u is not bounded (for instance if Q is not compact), then for every compact subset K of Q there is a compact K' C Q such that (This is a straightforward modification of Kunkle's result, which has an important consequence for the proof of the general estimate for Th depending explicitly upon x, to feature in a subsequent paper. See [7] for an exposition of Kunkle's construction and details of the modification. Kunkle's work generalizes that of Favard [4] and DeBoor [3] from mesh functions of one variable to mesh functions of several variables.)…”

Favard interpolation from subsets of a rectangular lattice

“…In two papers [2,3], Holtby used a multivariate Favard-like theorem from [4] to arrive at bounds on solutions to multivariate difference equations. The results presented here have potential for similar applications.…”