TY - JOUR

T1 - Higher-order estimates for fully nonlinear difference equations. II

AU - Holtby, Derek W.

PY - 2001

Y1 - 2001

N2 - The purpose of this work is to establish a priori C2'Q 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 J-/, 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 Holder 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.

AB - The purpose of this work is to establish a priori C2'Q 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 J-/, 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 Holder 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.

KW - Discrete a priori holder estimates

KW - Discrete seminorms

KW - Fully nonlinear difference equations

KW - Non-local discrete hessian

UR - http://www.scopus.com/inward/record.url?scp=23044534711&partnerID=8YFLogxK

U2 - 10.1017/s0013091598000200

DO - 10.1017/s0013091598000200

M3 - Article

SN - 0013-0915

VL - 44

SP - 87

EP - 102

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

IS - 1

ER -