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

Abstract We study interior $$C^{2,\alpha }$$ C 2 , α regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form $$F(D^2u)=0$$ 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 $$C^2$$ C 2 solutions of the previous equation are $$C^{2,\bar{\alpha }(\lambda /\Lambda )}$$ C 2 , α ¯ ( λ / Λ ) in the interior of the domain, where $$0<\lambda \le \Lambda $$ 0 < λ ≤ Λ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain $$C^{2,\tilde{\alpha }}$$ C 2 , α ~ regularity for an explicit exponent $$\tilde{\alpha }=\tilde{\alpha }(\lambda /\Lambda )>\lambda /\Lambda $$ α ~ = α ~ ( λ / Λ ) > λ / Λ .