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Erschienen in:
1987 | OriginalPaper | Buchkapitel
Boundary Behavior of Nonparametric Minimal Surfaces—Some Theorems and Conjectures
verfasst von : Kirk E. Lancaster
Erschienen in: Variational Methods for Free Surface Interfaces
Verlag: Springer New York
Enthalten in: Professional Book Archive
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Suppose D is a domain in the plane which is locally convex at every point of its boundary except possibly one, say (0,0), and φ is continuous on ∂D except possibly at (0,0), where it might have a jump discontinuity. Then for all directions from (0,0) into D, the radial limits of f exist, where f is the solution of the minimal surface equation in D or of an equation of prescribed (bounded) mean curvature in D with $$f\,\epsilon\,C^0\,(\bar D\,\backslash\{(0,0)\})$$ and $$f=\phi\,\text{on}\,\partial D\backslash\{(0,0)\})$$. Some conjectures which would generalize this result are mentioned.