Higher order interior estimates for fully nonlinear difference equations in a domain

Research Report MRR98-007

Higher order interior estimates for fully nonlinear difference equations in a domain

Derek W. Holtby

Abstract: The purpose of this work is to establish a priori $C^{2,\alpha}$ 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 $\Cal F_h$ 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 explicity upon the independent variables (the so-called frozen case) established in a previous paper to approach the general case of explicit dependence upon the independent variables. We make our approach 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. In the paper we establish the desired interior Hölder estimate in a mesh n-cube the corners of which are lattice points.

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Download paper:	PDF file (242K)
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Download cover sheet:	PDF file (58K)
	DVI file (3K)