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arXiv · 1811.04914

A proof of the Krylov-Safonov theorem without localization

Connor Mooney
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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 $W^{2,\,ε}$ estimate where $ε$ decays with the ellipticity ratio of the coefficients at a rate that improves previous results, and is optimal in two dimensions.

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Connor Mooney. 2019-01-23. A proof of the Krylov-Safonov theorem without localization. https://arxiv.org/abs/1811.04914

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