Numerical Solution of the Simple Monge-Amp\`ere Equation with Non-convex Dirichlet Data on Non-convex Domains

The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Amp\`ere equation is known independently of the convexity of the domain or Dirichlet boundary data -- when the Monge-Amp\`ere equation is posed as Bellman problem. However, the convergence to the viscosity solution has only been proved on strictly convex domains. In this paper we provide numerical evidence that convergence of numerical solutions is observed more generally without convexity assumptions. We illustrate how in the limit multi-valued functions may be approximated to satisfy the Dirichlet conditions on the boundary as well as local convexity in the interior of the domain.