The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems

TL;DR

This paper extends a variational approach to boundary value problems for fully nonlinear elliptic PDEs, providing new representation formulas and convergence results for the vanishing discount problem.

Contribution

It develops the variational framework to handle boundary conditions and establishes new formulas and convergence results for solutions.

Findings

New representation formulas for solutions and critical values

Convergence results for the vanishing discount problem in boundary value settings

Extension of the variational approach to boundary problems

Abstract

In arXiv:1603.01051 (Part 1 of this series), we have introduced a variational approach to studying the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations in a torus. We develop this approach further here to handle boundary value problems. In particular, we establish new representation formulas for solutions of discount problems, critical values, and use them to prove convergence results for the vanishing discount problems.