Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature
It provides a rigorous convergence guarantee for a class of fully nonlinear PDEs where standard comparison principles fail, benefiting researchers in geometric PDEs and numerical analysis.
The paper develops a convergent numerical scheme for surfaces of prescribed Gaussian curvature, even when solutions are discontinuous at the boundary, by proving a relaxed comparison principle and modifying the Barles-Souganidis convergence framework. The method is validated on challenging examples.
We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampère 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.