scholarly journals For the minimal surface equation, the set of solvable boundary values need not be convex

1996 ◽  
Vol 53 (3) ◽  
pp. 369-372
Author(s):  
Frank Morgan
Keyword(s):  
Minimal Surface ◽  
Smooth Domain ◽  
Boundary Values ◽  
Surface Equation ◽  
Minimal Surface Equation

One might think that if the minimal surface equation had a solution on a smooth domain D ⊂ Rn with boundary values φ, it would have a solution with boundary values tφ for all 0 ≤ t ≤ 1. We give a counterexample in R2.

scholarly journals Nonparametric minimal surfaces inR3whose boundaries have a jump discontinuity

1988 ◽  
Vol 11 (4) ◽  
pp. 651-656 ◽  
Author(s):  
Kirk E. Lancaster
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Variational Solution ◽  
Jump Discontinuity ◽  
Boundary Values ◽  
Surface Equation ◽  
Radial Limits ◽  
Minimal Surface Equation ◽  
Upper Limits ◽  
Locally Convex

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.

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