Higher order interior estimates for fully nonlinear difference equations I

Research Report MRR96-053

Higher order interior estimates for fully nonlinear difference equations I

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 $\F$ 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 deal here with the special case that the operator does not depend explicitly upon the independent variables by discretizing the approach of Evans for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger. The result in this special case forms the basis for a more general result in a subsequent paper. We also derive the discrete interpolation inequalities needed to obtain estimates for the interior $C^{2,\,\alpha}$ semi-norm in terms of the

norm.

Select this link for a text-only version of this abstract.

This service is maintained by the Mathematical Sciences Institute (MSI)
Comments to webmaster@maths.anu.edu.au URL: http://wwwmaths.anu.edu.au/