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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 under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.2010 Mathematics Subject Classification. 35J60, 35J70, 49L25 .

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We prove Hölder regularity of the gradient, up to the boundary for solutions of some fully-nonlinear, degenerate elliptic equations, with degeneracy coming from the gradient.