Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient

The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a suitable potential theory. The approach combines the notions of "proper elliptic branches" inspired by Krylov [Trans Amer Math Soc, 1995] with the "monotonicity-duality method" initiated by Harvey and Lawson [Comm Pure Appl Math, 2009]. In the variable coefficient nonlinear potential theory, a special role is played by the Hausdorff continuity of the proper elliptic map Theta which defines the potential theory. In the applications to nonlinear equations defined by an operator F, structural conditions on F will be determined for which there is a correspondence principle between Theta-subharmonics/superharmonics and admissible viscosity sub and supersolutions of the nonlinear equation and for which comparison for the equation follows from the associated compatible potential theory. General results and explicit models of interest in differential geometry will be examined. Examples of improvements with respect to existing results on comparison principles will be given.