The authors consider the following uniformly elliptic partial differential equations (PDEs) of the form
\[ \begin{aligned} F(D^{2} u ) = f & \quad\text{in }U,\\ u = g & \quad \text{on }\partial U,\\ \end{aligned} \]
where \(U\) is an open subset of \(R^{n}\) with regular boundary, \(F\) is uniformly elliptic with ellipticity constants \(\lambda\) and \(\Lambda\) such that \(\lambda \leq \Lambda\), \(f\in C^{0,1}(\overline{U})\), and \(g\in C^{1,\eta'}(\partial U)\) for some \(\eta' \in (0,1]\).
The authors obtain the algebraic rate of convergence for monotone and consistent finite difference approximations to Lipschitz-continuous viscosity solutions of the stated fully nonlinear boundary value problem. The key point of the analysis is their regularity result for Lipschitz-continuous viscosity solutions. The analysis asserts that outside sets of an arbitrary small measure, Lipschitz-continuous solutions have pointwise second-order expansions with an error that is controlled by the size of the exceptional set and the quadratic expansion or that, at a given scale, solutions have uniform polynomial approximations.
The authors’ result can be used also in other contexts. For example, it is used by L. Caffarelli and P. E. Souganidis [Rate of convergence for stochastic homogenization of uniformly elliptic partial differential equations, preprint] to establish a rate of convergence for the homogenization of fully nonlinear, uniformly elliptic, second-order PDEs in random environments.