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.

$ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations

“…When the oscillation of the linearized operator F ij depends continuously on the oscillation of D 2 u, there will be a δ 0 such that if the oscillation of the Hessian is smaller than δ 0 the oscillation of F ij will be less than ε 0 , thus C 2,α estimates apply. In particular, any modulus of continuity on the Hessian can be used to derive Hölder continuity: Essentially, the results in [6] can be "quantized". (Keep in mind that we may alway use a transformation like the one following (2.41), locally, so that the equation satisfies a K ε -condition nearby).…”

$C^{2,α}$ estimates for solutions to almost linear elliptic equations

“…When the oscillation of the linearized operator F ij depends continuously on the oscillation of D 2 u, there will be a δ 0 such that if the oscillation of the Hessian is smaller than δ 0 the oscillation of F ij will be less than ε 0 , thus C 2,α estimates apply. In particular, any modulus of continuity on the Hessian can be used to derive Hölder continuity: Essentially, the results in [CLW11] can be "quantized". (Keep in mind that we may alway use a transformation like the one following (2.43), locally, so that the equation satisfies a K ′ ε -condition nearby).…”