Inserted: 31 jul 2020
Last Updated: 31 jul 2020
We investigate strong maximum (and minimum) principles for fully nonlinear second order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci's extremal operators, some singular operators like those modeled on the $p$- and $\infty$-Laplacian, and mean curvature type problems. As a byproduct, we establish new strong comparison principles for some second order uniformly elliptic problems when the manifold has nonnegative sectional curvature.
Keywords: degenerate elliptic equation, Riemannian manifold, Fully nonlinear equation, Hopf boundary lemma, strong maximum principle, strong comparison principle