Nonparametric minimal surfaces inR3whose boundaries have a jump discontinuity

Abstract

LetΩbe a domain inR2which is locally convex at each point of its boundary except possibly one, say(0,0),ϕbe continuous on∂Ω/{(0,0)}with a jump discontinuity at(0,0)andfbe the unique variational solution of the minimal surface equation with boundary valuesϕ. Then the radial limits offat(0,0)from all directions inΩexist. If the radial limits all lie between the lower and upper limits ofϕat(0,0), then the radial limits offare 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's is proven.