  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.
