Higher-order estimates for fully nonlinear difference equations. I

Abstract

The purpose of this work is to establish a priori C 2 ' a 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 Th, 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 C 2 ' a semi-norm in terms of the C° norm.

Schauder estimates for fully nonlinear elliptic difference operators

“…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.…”

“…As in the continuous case (see [10,11,15]), the interior k, ® semi-norms in (3.11) where d = diam « . By choosing°su¯ciently small and using the interpolation inequalities [3,13] (as in the continuous case [10,11]), we nally arrive at the interior [u] ¤ k;« + [u] ¤ 2;® ;« 6 C(juj 0;« + (diam « ) 2+ ® ); (3.14) where C depends on n, ¤ = ¶ , • = ¶ , • h=h, a 0 =» h,ã= ¶ 0 ,`0, ® and K. Accordingly, we have the following theorem.…”

Higher-Order Estimates for Fully Nonlinear Difference Equations. Ii

“…(See [5] for an exposition of Kunkle's construction and details of the said modification, as well as proofs of the facts contained in the above lemma. )…”

“…In this paper the setting and notation are as in [6], where we derived a discrete a priori C 2,α estimate for solutions of difference equations involving operators of the form…”

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.…”