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

Abstract We study interior C^2, 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 C^2, (/) C 2, α ¯ (λ / Λ) in the interior of the domain, where 0 0 λ ≤ Λ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain C^2, C 2, α ~ regularity for an explicit exponent = (/) > / α ~ = α ~ (λ / Λ) > λ / Λ.