Schauder estimates for fully nonlinear elliptic difference operators

Abstract

In this paper, we are concerned with discrete Schauder estimates for solutions of fully nonlinear elliptic difference equations. Our estimates are discrete versions of second derivative Hölder estimates of Evans, Krylov and Safonov for fully nonlinear elliptic partial differential equations. They extend previous results of Holtby for the special case of functions of pure second-order differences on cubic meshes. As with Holtby's work, the fundamental ingredients are the pointwise estimates of Kuo-Trudinger for linear difference schemes on general meshes.

Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs

“…The previous assumptions are essentially the same ones in [8], except that in (C 6 ) we assume the regularity of the solutions in ( 23) and (24) does not depend on λ and the constant in the estimate of W hλ − W ε hλ ∞ is of the form C/λ. Note that with respect to the notation of [8], we make explicit in (23) the dependence on the zero-order term λW (y).…”

“…where l θ (x, y) = l θ (x, y) + tr(a θ (x, y)X) + f θ (x, y) • p. The approximate equation ( 23) and its perturbation (24) are λW (y) + H(y, 2…”

“…The point is to check that W hλ , W ε hλ ∈ C 0, δ with δ independent of λ, while in [8] δ depends on λ and goes to 0 for λ going to 0. To show the requested regularity of W hλ , W ε hλ we use a result in [23,24] but we need to rewrite the scheme in a slightly different way.…”

“…An important ingredient of the proof is an estimate of the rate of convergence of λW h λ towards λw λ as h → 0, where W h λ is the numerical solution and h is the approximation parameter. In order to obtain the previous estimate, we adapt the technique introduced in [8] and we use the regularity theory for finite difference schemes in [23,24].…”