TY - JOUR
T1 - Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature
AU - Hamfeldt, Brittany Froese
N1 - Funding Information:
65N22; Secondary: 35J66, 35J67, 35J70, 35J96, 53A99. Key words and phrases. Gaussian curvature, elliptic partial differential equations, Monge-Ampère equations, viscosity solutions, finite difference methods. This work was partially supported by NSF DMS-1619807.
Funding Information:
This work was partially supported by NSF DMS-1619807.
Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.
PY - 2018/3
Y1 - 2018/3
N2 - 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.
AB - 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.
KW - Elliptic partial differential equations
KW - Finite difference methods.
KW - Gaussian curvature
KW - Monge-Ampére equations
KW - Viscosity solutions
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U2 - 10.3934/cpaa.2018036
DO - 10.3934/cpaa.2018036
M3 - Article
AN - SCOPUS:85055801231
SN - 1534-0392
VL - 17
SP - 671
EP - 707
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
IS - 2
ER -