On Neumann problems for elliptic and parabolic equations on bounded manifolds

Sheng Guo

Published 2020-01-03Version 1

In this paper, we study fully nonlinear second-order elliptic and parabolic equations with Neumann boundary conditions on compact Riemannian manifolds with smooth boundary. We derive oscillation bounds for admissible solutions with Neumann boundary condition $u_\nu = \phi(x)$ assuming the existence of suitable $\mathcal{C}$-subsolutions. We use a parabolic approach to derive a solution of a $k$-Hessian equation with Neumann boundary condition $u_\nu = \phi(x)$ under suitable assumptions.