  In this paper we investigate the validity and the consequences of the maximum
principle for degenerate elliptic operators whose higher order term is the sum
of "k" eigenvalues of the Hessian. In particular we shed some light on some
very unusual phenomena due to the degeneracy of the operator. We prove moreover
Lipschitz regularity results and boundary estimates for problems in convex
domains which are intersections of ball of same radius. We prove also that
smooth bounded strictly convex domains are part of this class. As a consequence
we obtain the existence of solutions of the Dirichlet problem and of principal
eigenfunctions.
