International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 651-656
doi:10.1155/S0161171288000791

Nonparametric minimal surfaces in R3 whose boundaries have a jump discontinuity

Abstract

Let Ω be a domain in R2 which is locally convex at each point of its boundary except possibly one, say (0,0), ϕ be continuous on ∂Ω/{(0,0)} with a jump discontinuity at (0,0) and f be the unique variational solution of the minimal surface equation with boundary values ϕ. Then the radial limits of f at (0,0) from all directions in Ω exist. If the radial limits all lie between the lower and upper limits of ϕ 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's is proven.