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For scalar fully nonlinear partial differential equations depending on theHessian andspatial coordinates, we present a general theory for obtainingcomparison principles and well posedness for the associated Dirichlet problemwith continuous boundary data. In particular, we treat admissible viscositysolutions of elliptic branches of the equation, where the nonlinearity need notbe monotone on all of the space of symmetric N by N matrices. An ellipticbranch (in the sense of Krylov, 1995) of the equation is encoded by a setvalued map from the coordinate domain into the elliptic subsets of thesymmetric matrices (an elliptic map). The nonlinearity will be monotone alongthis map and the degenerate elliptic PDE is replaced by the a differentialinclusion. Weak solutions to such differential inclusions are defined by usingthe notion given by Harvey-Lawson (2009) in a pointwise manner. If the ellipticmap is uniformly upper semicontinuous, we show that the comparison principleholds for these weak solutions and that Perron's method yields a uniquecontinuous solution to the associated abstract Dirichlet problem provided thatthe boundary is suitably convex with respect to the elliptic mapand its dual inthe sense of Harvey and Lawson. When the map encodes an elliptic branch of agiven PDE, these soluitions are shown to be admissible viscosity solutions ofthe PDE problem. Various applications are described in terms of structuralconditions which ensure the existence of the needed elliptic map. Examplesinclude non-totally degenerate equations and equations involving theeigenvalues of the Hessian and their perturbations. In certain situations, themethods employed here will be shown to operate freely, while classicalviscosity approaches may not.

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