[Paper Review] Convergent approximation of surfaces of prescribed Gaussian curvature with weak Dirichlet conditions

Brittany D. Froese|arXiv (Cornell University)|Jan 23, 2016

Geometric Analysis and Curvature Flows21 references3 citations

TL;DR

This paper develops a monotone numerical scheme for approximating surfaces with prescribed Gaussian curvature by solving a fully nonlinear Monge-Ampère equation. It establishes convergence to the viscosity solution in the interior of the domain despite weak Dirichlet conditions that prevent standard comparison principles, using a relaxed interior comparison principle and an adapted Barles-Souganidis framework.

ABSTRACT

We consider the construction of surfaces of prescribed Gaussian curvature via the numerical solution of a fully nonlinear partial differential equation of Monge-Amp\`ere type. In this setting, solutions may not be able to satisfy even a simple Dirichlet boundary condition, and the resulting weak solutions need not be continuous up to the boundary. As a consequence, sub-solutions do not always lie below super-solutions, and existing convergence proofs do not apply. 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 then modify the Barles-Souganidis convergence framework to show that monotone approximations converge to the viscosity solution in the interior of the domain. We describe a monotone approximation of the equation and present several challenging computational examples to validate these theoretical results.

Motivation & Objective

• To address the challenge of constructing surfaces with prescribed Gaussian curvature when standard Dirichlet boundary conditions fail.
• To overcome the breakdown of classical comparison principles due to discontinuous weak solutions at the boundary.
• To develop a convergence framework for monotone numerical approximations in the interior of the domain despite lack of boundary regularity.
• To validate the theoretical framework with computational examples involving complex curvature behaviors.

Proposed method

• Adapting the Barles-Souganidis convergence framework to handle weak Dirichlet conditions by restricting the comparison principle to the interior of the domain.
• Introducing a geometric interpretation of weak solutions to define a relaxed comparison principle valid only in the interior.
• Constructing a monotone finite difference approximation of the fully nonlinear Monge-Ampère equation for prescribed Gaussian curvature.
• Using viscosity solution theory to justify convergence of the monotone scheme to the unique viscosity solution in the interior.
• Implementing computational experiments to test the scheme on surfaces with challenging curvature distributions and boundary behaviors.

Experimental results

Research questions

• RQ1 How can convergence be established for numerical approximations of the Monge-Ampère equation when standard comparison principles fail due to weak boundary conditions?
• RQ2 What geometric properties of weak solutions allow for a relaxed comparison principle restricted to the interior of the domain?
• RQ3 Can a monotone finite difference scheme converge to the viscosity solution under weak Dirichlet conditions, and how can this be proven?
• RQ4 What computational evidence supports the theoretical convergence and robustness of the scheme in non-smooth settings?

Key findings

• A relaxed comparison principle is established that applies only in the interior of the domain, enabling convergence analysis despite boundary discontinuities.
• The modified Barles-Souganidis framework ensures that monotone numerical approximations converge to the viscosity solution in the interior of the domain.
• The proposed monotone scheme successfully computes solutions for surfaces with prescribed Gaussian curvature even when boundary data are weak or discontinuous.
• Computational examples demonstrate the scheme's robustness on complex curvature configurations, validating the theoretical convergence results.