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

We study interior C2,α regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form F(D2u)=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 solutions of the previous equation are C2,α¯(λ/Λ) in the interior of the domain, where 0<λ≤Λ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain C2,α~ regularity for an explicit exponent α~=α~(λ/Λ)>λ/Λ.