Borderline regularity for fully nonlinear equations in Dini domains

In this paper, we prove borderline gradient continuity of viscosity solutions to Fully nonlinear elliptic equations at the boundary of a $C^{1,\dini}$-domain. Our main result Theorem 3.1 is a sharpening of the boundary gradient estimate proved by Ma-Wang following the borderline interior gradient regularity estimates established Daskalopoulos-Kuusi-Mingione. We however mention that, differently from the approach in the interior case which depends on $W^{1,q}$ estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Caffarelli.