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
Department of Mathematics and Statistics, Wichita State University, Wichita 67208, KS, USA
Received 21 January 1987; Revised 18 February 1987
Copyright © 1988 Kirk E. Lancaster. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.