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 $-\kappa$ for $\kappa \ge 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} 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 {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.

35 pages, 1 figure

References in corpus (2)

• Harnack inequality for nondivergent parabolic operators on Riemannian manifolds
• Maximum principle for viscosity solutions on Riemannian manifolds

Cited by in corpus (2)

• Harnack inequality for degenerate and singular operators of $p$-Laplacian type on Riemannian manifolds
• Second derivative estimates for uniformly elliptic operators on Riemannian manifolds