A proof of the Krylov-Safonov theorem without localization

Abstract

The Krylov-Safonov theorem says that solutions to non-divergence uniformly elliptic equations with rough coefficients are Hölder continuous. The proof combines a basic measure estimate with delicate localization and covering arguments. Here we give a "global" proof based on convex analysis that avoids the localization and covering arguments. As an application of the technique we prove a $W^{2,\epsilon }$ estimate where $\epsilon$ decays with the ellipticity ratio of the coefficients at a rate that improves previous results, and is optimal in two dimensions.

arXiv commentTo appear in Comm. Partial Differential Equations

References in corpus (1)

• Global $W^{2,\delta }$ estimates for a type of singular fully nonlinear elliptic equations