On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations

For scalar fully nonlinear partial differential equations F(x, D^2u(x)) = 0 with x in Omega a bounded open domain in N dimensional Euclidian space, we present a general theory for obtaining N by N matrices. We treat admissible viscosity solutions u of elliptic branches of the equation in the sense of Krylov [1995] and extend the program initiated by Harvey and Lawson [2009] in the homogeneous case when F does not depend on x. In particular, for the set valued map Theta defining the elliptic branch by way of the differential inclusion D^2u(x) belongs to the boundary of Theta(x), we identify a uniform continuity property which ensures the validity of the comparison principle and the applicability of Perron's method for the differential inclusion on suitably convex domains, where the needed boundary convexity is characterized by Theta. Structural conditions on F are then derived which ensure the existence of an elliptic map Theta with the needed regularity. Concrete applications are given in which standard structural conditions on F may fail and without the request of convexity conditions in the equation. Examples include perturbed Monge-Ampere equations and equations prescribing eigenvalues of the Hessian.