The Dirichlet problem for prescribed principal curvature equations

In this paper the Dirichlet problem for the equation which prescribes the ith principal curvature of the graph of a function u is considered. A Comparison principle is obtained within the class of semiconvex subsolutions by a local perturbation procedure combined with a fine Lipschitz estimate on the elliptic operator. Existence of solutions is stated for the Dirichlet problem with boundary conditions in the viscosity sense; further assumptions guarantee that no loss of boundary data occurs. Some conditions relating the geometry of the domain and the prescribing data which are sufficient for existence and uniqueness of solutions are presented.