New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

Abstract

We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the Hörmander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton–Jacobi–Bellman form, such as those involving the Pucci’s extremal operators over Hörmander vector fields.