Numerical solutions of boundary value problems for Κ-surfaces inR3

Abstract

A K-surface is a surface whose Gauss curvature K is a positive constant. In this article, we will consider K-surfaces that are defined by a nonlinear boundary value problem. In this setting, existence follows from some recent results on nonlinear second-order elliptic partial differential equations. The analytical techniques used to establish these results motivate effective numerical methods for computing K-surfaces. In theory, the solvability of the boundary value problem reduces to the existence of a subsolution. In an analogous way, we find that if an approximate numerical subsolution can be determined, then the corresponding K-surface can be computed. We will consider two boundary value problems. In the first problem, the K-surface is a graph over a plane. In the second, the K-surface is a radial graph over a sphere. From certain geometrical considerations, it follows that there is a maximum Gauss curvature Kmax for these problems. Using a continuation method, we estimate Kmax and determine numerically the unique one-parameter family of K-surfaces that exist for K E (0,Kmax). This is the first time that this numerical method has been applied to the nonlinear partial differential equations for a K -surface. Sharp estimates for Kmax are not available analytically, except in special situations such as a surface of revolution, where the parametrization can be obtained explicitly in terms of elliptic functions. We find that our numerical estimates for Kmax are in close agreement with the expected values in these cases. © 1996 John Wiley & Sons, Inc.