Existence and nonexistence of radial limits of minimal surfaces
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- by Kirk E. Lancaster PDF
- Proc. Amer. Math. Soc. 106 (1989), 757-762 Request permission
Abstract:
A bounded solution of the minimal surface equation is constructed which has no radial limits at a boundary point.References
- R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR 0036317
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Alan R. Elcrat and Kirk E. Lancaster, On the behavior of a nonparametric minimal surface in a nonconvex quadrilateral, Arch. Rational Mech. Anal. 94 (1986), no. 3, 209–226. MR 846061, DOI 10.1007/BF00279863
- Alan R. Elcrat and Kirk E. Lancaster, Boundary behavior of a nonparametric surface of prescribed mean curvature near a reentrant corner, Trans. Amer. Math. Soc. 297 (1986), no. 2, 645–650. MR 854090, DOI 10.1090/S0002-9947-1986-0854090-X
- Robert Finn, Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature, J. Analyse Math. 14 (1965), 139–160. MR 188909, DOI 10.1007/BF02806384
- Claus Gerhardt, Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z. 139 (1974), 173–198. MR 437925, DOI 10.1007/BF01418314
- Enrico Giusti, Boundary behavior of non-parametric minimal surfaces, Indiana Univ. Math. J. 22 (1972/73), 435–444. MR 305253, DOI 10.1512/iumj.1972.22.22039
- Stefan Hildebrandt, Boundary behavior of minimal surfaces, Arch. Rational Mech. Anal. 35 (1969), 47–82. MR 248650, DOI 10.1007/BF00248494
- Howard Jenkins and James Serrin, Variational problems of minimal surface type. II. Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal. 21 (1966), 321–342. MR 190811, DOI 10.1007/BF00282252
- Kirk E. Lancaster, Boundary behavior of a nonparametric minimal surface in $\textbf {R}^3$ at a nonconvex point, Analysis 5 (1985), no. 1-2, 61–69. MR 791492, DOI 10.1524/anly.1985.5.12.61 —, Boundary behavior of nonparametric minimal surfaces—some theorems and conjectures, 37-41, in Variational Methods for Free Surface Interfaces (P. Concus and R. Finn, editors), Springer-Verlag, New York, 1987.
- Kirk E. Lancaster, Nonparametric minimal surfaces in $\textbf {R}^3$ whose boundaries have a jump discontinuity, Internat. J. Math. Math. Sci. 11 (1988), no. 4, 651–656. MR 959444, DOI 10.1155/S0161171288000791
- Johannes C. C. Nitsche, On new results in the theory of minimal surfaces, Bull. Amer. Math. Soc. 71 (1965), 195–270. MR 173993, DOI 10.1090/S0002-9904-1965-11276-9
- Johannes C. C. Nitsche, Über ein verallgemeinertes Dirichletsches Problem für die Minimalflächengleichung und hebbare Unstetigkeiten ihrer Lösungen, Math. Ann. 158 (1965), 203–214 (German). MR 175047, DOI 10.1007/BF01360040
- Johannes C. C. Nitsche, The boundary behavior of minimal surfaces. Kellogg’s theorem and Branch points on the boundary, Invent. Math. 8 (1969), 313–333. MR 259766, DOI 10.1007/BF01404636
- Tibor Radó, The problem of the least area and the problem of Plateau, Math. Z. 32 (1930), no. 1, 763–796. MR 1545197, DOI 10.1007/BF01194665
- Wade Ramey and David Ullrich, The pointwise Fatou theorem and its converse for positive pluriharmonic functions, Duke Math. J. 49 (1982), no. 3, 655–675. MR 672501
- Donald Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana Univ. Math. J. 26 (1977), no. 5, 817–838. MR 463968, DOI 10.1512/iumj.1977.26.26066
- Leon Simon, Boundary regularity for solutions of the non-parametric least area problem, Ann. of Math. (2) 103 (1976), no. 3, 429–455. MR 638358, DOI 10.2307/1970947
- Leon Simon, Boundary behaviour of solutions of the nonparametric least area problem, Bull. Austral. Math. Soc. 26 (1982), no. 1, 17–27. MR 679917, DOI 10.1017/S0004972700005566
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 106 (1989), 757-762
- MSC: Primary 35J60; Secondary 53A10
- DOI: https://doi.org/10.1090/S0002-9939-1989-0969523-6
- MathSciNet review: 969523