  We consider the numerical approximation of surfaces of prescribed Gaussian
curvature via the solution of a fully nonlinear partial differential equation
of Monge-Amp\`ere type. These surfaces need not be continuous up to the
boundary of the domain and the Dirichlet boundary condition must be interpreted
in a weak sense. As a consequence, sub-solutions do not always lie below
super-solutions, standard comparison principles fail, and existing convergence
theorems break down. By relying on a geometric interpretation of weak
solutions, we prove a relaxed comparison principle that applies only in the
interior of the domain. We provide a general framework for proving existence
and stability results for consistent, monotone finite difference approximations
and modify the Barles-Souganidis convergence framework to show convergence in
the interior of the domain. We describe a convergent scheme for the prescribed
Gaussian curvature equation and present several challenging examples to
validate these results.
