Viscosity solutions to second order partial differential equations on Riemannian manifolds

Abstract

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F ( x , u , d u , d 2 u ) = 0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable x) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d 2 u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.