Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type∗

We study fully nonlinear partial differential equations of Monge-Ampère type involving the derivates with respect to a family X of vector fields. The main result is a comparison principle among viscosity subsolutions, convex with respect to X , and viscosity supersolutions (in a weaker sense than usual), which implies the uniqueness of solution to the Dirichlet problem. Its assumptions include the equation of prescribed horizontal Gauss curvature in Carnot groups. By Perron method we also prove the existence of a solution either under a growth condition of the nonlinearity with respect to the gradient of the solution, or assuming the existence of a subsolution attaining continuously the boundary data, therefore generalizing some classical result for Euclidean Monge-Ampère equations. MSC 2010: 35J96, 35H20, 35R03, 49L25.