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ABSTRACT. We derive an a prioriC 2, α estimate for solutions of the fully non-linear elliptic equationF (D 2 u) = 0, provided the level setΣ=M|F (M) =0 satisfies: (a) Σ∩M|TrM= t is strictly convex for all constants t; (b) the angle between the identity matrixI and the normalFij toΣis strictly positive on the non-convex part of Σ. Moreover, we do not need any convexity assumption onF in the course of the proof for the two dimensional case, as the classical result indicates. 1.
Luis Caffarelli (Sat,) studied this question.