Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators. We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.