Existence and Uniqueness of Lipschitz Continuous Graphs with Prescribed Levi Curvature

In this paper we prove comparison principles between viscosity semicontinuous sub- and supersolutions of the generalized Dirichlet problem (in the sense of viscosity solutions) for the {\it Levi Monge-Amp\`{e}re} equation. As a consequence of this result and of the Perron's method we get the existence of a continuous solution of the Dirichlet problem related to the prescribed Levi curvature equation under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by building Lipschitz continuous barriers and by applying a weak Bernstein method.