The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus

TL;DR

This paper introduces viscosity Mather measures to analyze the vanishing discount problem for fully nonlinear degenerate elliptic PDEs on a torus, proving convergence of solutions as the discount factor approaches zero.

Contribution

It extends Mather measures to viscosity solutions and establishes convergence results for the discount problem in a new variational framework.

Findings

Viscosity Mather measures are introduced for nonlinear PDEs.

Solutions of the discount problem converge to an ergodic solution as discount vanishes.

The approach applies to fully nonlinear degenerate elliptic equations on a torus.

Abstract

We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions $v^\lambda$ of the discount problem with the factor $\lambda>0$ converges to a solution of the ergodic problem as $\lambda \to 0$.