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In this paper we investigate the validity and the consequences of the maximumprinciple for degenerate elliptic operators whose higher order term is the sumof "k" eigenvalues of the Hessian. In particular we shed some light on somevery unusual phenomena due to the degeneracy of the operator. We prove moreoverLipschitz regularity results and boundary estimates for problems in convexdomains which are intersections of ball of same radius. We prove also thatsmooth bounded strictly convex domains are part of this class. As a consequencewe obtain the existence of solutions of the Dirichlet problem and of principaleigenfunctions.

A family of degenerate elliptic operators: maximum principle and its consequences