A priori estimates for solutions of fully nonlinear equations with convex level set

Abstract

We derive an a priori C2,α estimate for solutions of the fully non-linear elliptic equation F(D2u) = 0, provided the level set ∑ = {M | F(M) = 0} satisfies: (a) ∑ ∩ {M | TrM = t} is strictly convex for all constants t; (b) the angle between the identity matrix I and the normal Fij to ∑ is strictly positive on the non-convex part of ∑. Moreover, we do not need any convexity assumption on F in the course of the proof for the two dimensional case, as the classical result indicates.