Nonparametric minimal surfaces in \(R^ 3\) whose boundaries have a jump discontinuity (Q910561)

Summary: Let \(\Omega\) be a domain in \({\mathbb{R}}^ 2\) which is locally convex at each point of its boundary except possibly one, say (0,0), \(\phi\) be continuous on \(\partial \Omega /\{(0,0)\}\) with a jump discontinuity at (0,0) and f be the unique variational solution of the minimal surface equation with boundary values \(\phi\). Then the radial limits of f at (0,0) from all directions in \(\Omega\) exist. If the radial limits all lie between the lower and upper limits of \(\phi\) at (0,0), then the radial limits of f are weakly monotonic; if not, they are weakly increasing and then decreasing (or the reverse). Additionally, their behavior near the extreme directions is examined and a conjecture of the author is proven.