Abstract
We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampere 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.
Highlights
The Gaussian curvature of a hypersurface is given by the product of the principle curvatures of the surface
We developed a proof that surfaces of prescribed Gaussian curvature can be constructed through the use of monotone approximations of a Monge-Ampere type equation
Typical convergence proofs for the approximation of weak solutions of nonlinear degenerate elliptic equations require on a comparison principle that ensures that sub-solutions lie below super-solutions
Summary
The Gaussian curvature of a hypersurface is given by the product of the principle curvatures of the surface. One of the goals of this work is to show that these weak solutions are equivalent, so that numerical convergence results for viscosity solutions will apply to generalised surfaces of prescribed Gaussian curvature. This equivalence is established in Theorems 5 and 8. X→y and if v is any other generalised solution of the Monge-Ampere PDE that satisfies (5) v ≤ u on Ω This weaker notion of Dirichlet boundary conditions leads to an existence result for the problem of prescribed Gaussian curvature. If Hypothesis 2.4 holds, the Monge-Ampere equation (2) has a unique generalised solution that satisfies the Dirichlet boundary conditions in the weak sense.
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