Weak KAM theory for discount Hamilton-Jacobi equations and its application

TL;DR

This paper develops weak KAM theory for discount Hamilton-Jacobi equations, introduces $oldsymbol{ extit{ extalpha}}$-limit points of minimizing curves, and applies these concepts to derive error estimates for viscosity solutions in the vanishing discount process.

Contribution

It extends weak KAM theory to discount equations and introduces $ extit{ extalpha}$-limit points, providing new tools for analyzing viscosity solutions and their error estimates.

Findings

Introduced $ extit{ extalpha}$-limit points of minimizing curves.

Established properties of the associated dynamical systems.

Applied theory to obtain error estimates in the vanishing discount limit.

Abstract

Weak KAM theory for discount Hamilton-Jacobi equations and corresponding discount Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of $\alpha$-limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of $\alpha$-limit points is effectively exploited with properties of the corresponding dynamical systems.