Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature

Document Type

Article

Publication Date

3-1-2018

Abstract

We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampére 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 conver-gence theorems break down. By relying on a geometric interpretation of weak solutions, we prove a relaxed comparison principle that applies only in the in-terior 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 pres-cribed Gaussian curvature equation and present several challenging examples to validate these results.

Identifier

85055801231 (Scopus)

Publication Title

Communications on Pure and Applied Analysis

External Full Text Location

https://doi.org/10.3934/cpaa.2018036

e-ISSN

15535258

ISSN

15340392

First Page

671

Last Page

707

Issue

2

Volume

17

Grant

1619807

Fund Ref

National Science Foundation

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