Regularity of the singular set in the fully nonlinear obstacle problem

Abstract

For the obstacle problem involving a convex fully nonlinear elliptic operator, we show that the singular set in the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha }$ manifold, and the lower strata are covered by $C^{1,{\log }^{\epsilon }}$ manifolds. This recovers some of the recent regularity results due to Colombo–Spolaor–Velichkov (2018) and Figalli–Serra (2019) when the operator is the Laplacian.