On Neumann problems for elliptic and parabolic equations on bounded manifolds

TL;DR

This paper investigates fully nonlinear elliptic and parabolic equations with Neumann boundary conditions on compact Riemannian manifolds, establishing bounds and solutions using a parabolic approach under certain assumptions.

Contribution

It introduces new oscillation bounds and solution existence results for Neumann problems on manifolds, employing a parabolic method and subsolution assumptions.

Findings

Derived oscillation bounds for solutions with Neumann boundary conditions.

Established existence of solutions for k-Hessian equations with Neumann boundary conditions.

Applied a parabolic approach to solve boundary value problems on manifolds.

Abstract

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.