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Existence and nonexistence of radial limits of minimal surfaces


Author: Kirk E. Lancaster
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
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Abstract: A bounded solution of the minimal surface equation is constructed which has no radial limits at a boundary point.


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  • [1] R. Courant, Dirichlet principle, conformal mapping, and minimal surfaces, Interscience, New York, 1950. MR 0036317 (12:90a)
  • [2] P. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [3] A. Elcrat and K. Lancaster, On the behavior of a nonparametric minimal surface in a nonconvex quadrilateral, Arch. Rational Mech. Anal. 94 (1986), 209-226. MR 846061 (87i:53010)
  • [4] -, Boundary behavior of nonparametric surfaces of prescribed mean curvature near a reentrant corner, Trans. Amer. Math. Soc. 297 (1986), 645-650. MR 854090 (87h:35098)
  • [5] R. Finn, Remarks relevant to minimal surfaces and to surfaces of prescribed mean curvature, J. Analyse Math. 14 (1965), 139-160. MR 0188909 (32:6337)
  • [6] C. Gerhardt, Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z. 139 (1974), 173-198. MR 0437925 (55:10846)
  • [7] E. Giusti, Boundary behavior of nonparametric minimal surfaces, Indiana Univ. Math. J. 22 (1972), 435-444. MR 0305253 (46:4383)
  • [8] S. Hildebrandt, Boundary behavior of minimal surfaces, Arch. Rational Mech. Anal. 35 (1969), 47-82. MR 0248650 (40:1901)
  • [9] H. Jenkins and J. Serrin, Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal. 32 (1966), 321-342. MR 0190811 (32:8221)
  • [10] K. Lancaster, Boundary behavior of a nonparametric minimal surface in $ {R^3}$ at a nonconvex point, Analysis 5 (1985), 61-69. MR 791492 (86m:49053)
  • [11] -, 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.
  • [12] -, Nonparametric minimal surfaces in $ {R^3}$ whose boundaries have a jump discontinuity, Internat. J. Math. Sci. 11 (1988), 651-656. MR 959444 (89m:35072)
  • [13] J. C. C. Nitsche, On new results in the theory of minimal surfaces, Bull. Amer. Math. Soc. 71 (1965), 195-270. MR 0173993 (30:4200)
  • [14] -, Über ein verallgemeinertes Dirichletsches Problem für die Minimal flächengleichung und hebbare Unstetigkeiten ihrer Lösungen, Math. Ann. 158 (1965), 203-214. MR 0175047 (30:5233)
  • [15] -, The boundary behavior of minimal surfaces. Kellogg's theorem and branch points on the boundary, Invent. Math. 8 (1969), 313-333. MR 0259766 (41:4399a)
  • [16] T. Radó, The problem of least area and the problem of Plateau, Math. Z. 32 (1930), 763-796. MR 1545197
  • [17] W. Ramey and D. Ullrich, The pointwise Fatou theorem and its converse for positive pluriharmonic functions, Duke Math. J. 49 (1982), 655-675. MR 672501 (84a:32003)
  • [18] D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana Univ. Math. J. 26 (1977), 817-838. MR 0463968 (57:3906)
  • [19] L. Simon, Boundary regularity for solutions of the nonparametric least area problem, Ann. of Math. 103 (1976), 429-455. MR 0638358 (58:30681)
  • [20] -, Boundary behavior of solutions of the nonparametric least area problem, Bull. Austral. Math. Soc. 26 (1982), 17-27. MR 679917 (84f:53006)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0969523-6
Keywords: Nonparametric minimal surface, radial limit
Article copyright: © Copyright 1989 American Mathematical Society

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