Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

Abstract

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by −κ for κ≥0. In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such M, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.