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 .

International audienceGiven a coercive Hamiltonian which is quasi-convex with respect to the gradient variable and periodic with respect to time and space at least ''far away from the origin'', we consider the solution of the Cauchy problem of the corresponding Hamilton-Jacobi equation posed on the real line. Compact perturbations of coercive periodic quasi-convex Hamiltonians enter into this framework for example. We prove that the rescaled solution converges towards the solution of the expected effective Hamilton-Jacobi equation, but whose ''flux'' at the origin is ''limited'' in a sense made precise by the authors in \cite{im}. In other words, the homogenization of such a Hamilton-Jacobi equation yields to supplement the expected homogenized Hamilton-Jacobi equation with a junction condition at the single discontinuous point of the effective Hamiltonian. We also illustrate possible applications of such a result by deriving, for a traffic flow problem, the effective flux limiter generated by the presence of a finite number of traffic lights on an ideal road. We also provide meaningful qualitative properties of the effective limiter

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators Pk ±, defined respectively as the sum of the largest and the smallest k eigenvalues of the Hessian matrix. For the operator Pk + we obtain results analogous to those which hold for the Laplace operator in space dimension k. Whereas, owing to the stronger degeneracy of the operator Pk −, we get totally different results

In this paper we give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in R N , and equipped with homogeneous Dirichlet boundary conditions. 2010 Mathematics Subject Classification. 35J60; 35B33; 34B53.

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator \mathcal{P}^+_{1} mapping a function u to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian.