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Higher-order estimates for fully nonlinear difference equations. I

Published online by Cambridge University Press:  20 January 2009

Derek W. Holtby
Derek W. Holtby
Affiliation:
CSIRO Telecommunications and Industrial Physics, Epping, NSW 1710, Australia
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Abstract

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The purpose of this work is to establish a priori C2, α estimates for mesh function solutions of nonlinear positive difference equations in fully nonlinear form on a uniform mesh, where the fully nonlinear finite-difference operator ℱh 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. We do this 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 part II. We also derive the discrete interpolation inequalities needed to obtain estimates for the interior C2, α semi-norm in terms of the C0 norm.

Keywords

fully nonlinear diference equationsdiscrete a priori Hölder estimatesdiscrete seminormsdiscrete interpolation inequalities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 3 , October 2000 , pp. 485 - 510
Copyright
Copyright © Edinburgh Mathematical Society 2000

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