A priori estimates for classical solutions of fully nonlinear elliptic equations

Abstract

For the fully nonlinear uniformly elliptic equation F(D ² u) = 0, it is well known that the viscosity solutions are C ^2,α if the nonlinear operator F is convex (or concave). In this paper, we study the classical solutions for the fully nonlinear elliptic equation where the nonlinear operator F is locally C ^1,β a.e. for any 0 < β < 1. We will prove that the classical solutions u are C ^2,α. Moreover, the C ^2,α norm of u depends on n, F and the continuous modulus of D ² u.