Asymptotic values of minimal graphs in a disc

Pascal Collin[1]; Harold Rosenberg[2]

  • [1] Institut de Mathématiques de Toulouse Université Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex 9 (France)
  • [2] IMPA, 22460-320 Estrada Dona Castorina 110 Rio de Janeiro (Brasil)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 7, page 2357-2372
  • ISSN: 0373-0956

Abstract

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We consider solutions of the prescribed mean curvature equation in the open unit disc of euclidean n-dimensional space. We prove that such a solution has radial limits almost everywhere; which may be infinite. We give an example of a solution to the minimal surface equation that has finite radial limits on a set of measure zero, in dimension two. This answers a question of Nitsche.

How to cite

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Collin, Pascal, and Rosenberg, Harold. "Asymptotic values of minimal graphs in a disc." Annales de l’institut Fourier 60.7 (2010): 2357-2372. <http://eudml.org/doc/116337>.

@article{Collin2010,
abstract = {We consider solutions of the prescribed mean curvature equation in the open unit disc of euclidean n-dimensional space. We prove that such a solution has radial limits almost everywhere; which may be infinite. We give an example of a solution to the minimal surface equation that has finite radial limits on a set of measure zero, in dimension two. This answers a question of Nitsche.},
affiliation = {Institut de Mathématiques de Toulouse Université Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex 9 (France); IMPA, 22460-320 Estrada Dona Castorina 110 Rio de Janeiro (Brasil)},
author = {Collin, Pascal, Rosenberg, Harold},
journal = {Annales de l’institut Fourier},
keywords = {Minimal graphs; radial limits; Fatou theorem; minimal graphs},
language = {eng},
number = {7},
pages = {2357-2372},
publisher = {Association des Annales de l’institut Fourier},
title = {Asymptotic values of minimal graphs in a disc},
url = {http://eudml.org/doc/116337},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Collin, Pascal
AU - Rosenberg, Harold
TI - Asymptotic values of minimal graphs in a disc
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 7
SP - 2357
EP - 2372
AB - We consider solutions of the prescribed mean curvature equation in the open unit disc of euclidean n-dimensional space. We prove that such a solution has radial limits almost everywhere; which may be infinite. We give an example of a solution to the minimal surface equation that has finite radial limits on a set of measure zero, in dimension two. This answers a question of Nitsche.
LA - eng
KW - Minimal graphs; radial limits; Fatou theorem; minimal graphs
UR - http://eudml.org/doc/116337
ER -

References

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  1. P. Collin, H. Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs Zbl1209.53010
  2. H. Jenkins, J. Serrin, Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal. 21 (1966), 321-342 Zbl0171.08301MR190811
  3. V. Mikyukov, Two theorems on boundary properties of minimal surfaces in nonparametric form, Math. Notes 21 (1977), 307-310 Zbl0402.53003MR474055
  4. J. Nitsche, On new results in the theory of minimal surfaces, B. Amer. Math. Soc. 71 (1965), 195-270 Zbl0135.21701MR173993

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