Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions

TL;DR

This paper explores the stability, comparison, and existence of solutions for convex Hamilton-Jacobi equations with Neumann boundary conditions, introducing the Aubry-Mather set and analyzing its properties.

Contribution

It develops new stability and existence results for convex Hamilton-Jacobi equations with Neumann boundary conditions, and defines the Aubry-Mather set in this context.

Findings

Stability under infimum and convex combination of subsolutions

Existence of solutions for convex, coercive Hamilton-Jacobi equations with Neumann boundary conditions

Properties and existence of Aubry-Mather sets and calibrated extremals

Abstract

We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry-Mather set associated with the Neumann type boundary problem and establish some properties of the Aubry-Mather set including the existence results for the ``calibrated'' extremals for the corresponding action functional (or variational problem).