Viscosity solutions to second order parabolic PDEs on Riemannian manifolds

TL;DR

This paper develops a theory for viscosity solutions to second order parabolic PDEs on compact Riemannian manifolds, establishing key properties like comparison, uniqueness, and existence, with results depending on curvature conditions.

Contribution

It extends viscosity solution theory to Riemannian manifolds, providing new existence and uniqueness results under curvature and regularity assumptions.

Findings

Comparison principle established

Uniqueness and existence proven

Results depend on manifold curvature conditions

Abstract

In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold $M$ has nonnegative sectional curvature, we get the finest results. If one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, the assumptions on curvature can be thrown away.