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In this work we establish local $C^{2,\alpha}$ regularity estimates for flatsolutions to non-convex fully nonlinear elliptic equations provided thecoefficients and the source function are of class $C^{0,\alpha}$. For problemswith merely continuous data, we prove that flat solutions are locally$C^{1,\text{Log-Lip}}$.

Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems