Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature

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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.

Original languageEnglish (US)
Pages (from-to)671-707
Number of pages37
JournalCommunications on Pure and Applied Analysis
Volume17
Issue number2
DOIs
StatePublished - Mar 2018

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Keywords

  • Elliptic partial differential equations
  • Finite difference methods.
  • Gaussian curvature
  • Monge-Ampére equations
  • Viscosity solutions

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