On the smoothness of solutions of fully nonlinear second order equations in the plane

We study interior C2,α C 2 , α regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form F(D²u)=0 F ( D 2 u ) = 0 in two independent variables and without any geometric condition on F . By means of the theory of divergence form equations we prove that C² C 2 solutions of the previous equation are C2,α ̄(λ /Λ ) C 2 , α ¯ ( λ / Λ ) in the interior of the domain, where 0<λ ≤ Λ 0 < λ ≤ Λ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain C2,α ̃ C 2 , α ~ regularity for an explicit exponent α ̃=α ̃(λ /Λ )>λ /Λ α ~ = α ~ ( λ / Λ ) > λ / Λ .