Boundary Behavior of Nonparametric Minimal Surfaces—Some Theorems and Conjectures

Abstract

Suppose D is a domain in the plane which is locally convex at every point of its boundary except possibly one, say (0,0), and φ is continuous on ∂D except possibly at (0,0), where it might have a jump discontinuity. Then for all directions from (0,0) into D, the radial limits of f exist, where f is the solution of the minimal surface equation in D or of an equation of prescribed (bounded) mean curvature in D with $f\,\epsilon\,C^0\,(\bar D\,\backslash\{(0,0)\})$ and $f=\phi\,\text{on}\,\partial D\backslash\{(0,0)\})$. Some conjectures which would generalize this result are mentioned.