The vanishing discount problem and viscosity Mather measures. Part 2:   boundary value problems
  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. 
Hitoshi Ishii, Hiroyoshi Mitake, Hung V. Tran
math.AP