A proof of the Krylov–Safonov theorem without localization
Connor Mooney · Communications in Partial Differential Equations · 2019
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 W2, ε estimate where ε decays with the ellipticity ratio of the coefficients at a rate that improves previous results, and is optimal in two dimensions.