Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature
Brittany Froese Hamfeldt
Communications on Pure & Applied Analysis 2018, 17(2): 671-707 doi: 10.3934/cpaa.2018036

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.

keywords: Gaussian curvature elliptic partial differential equations Monge-Ampère equations Gaussian curvature viscosity solutions finite difference methods

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