[Valeurs asymptotiques de graphes minimaux dans un disque]
Nous considérons les solutions de l´équation de la courbure moyenne prescrite sur le disque unité ouvert de l’espace euclidien. Nous prouvons qu’une telle solution a une limite radiale presque partout qui, éventuellement, peut-être infinie. Nous donnons l´exemple d´une solution de l´équation des surfaces minimales en dimension deux, qui admet des limites radiales finies sur un ensemble de mesure nulle. Ce travail répond à une question de Nitsche.
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.
Keywords: Minimal graphs, radial limits, Fatou theorem
Mot clés : graphe minimal, limite radiale, théorème de Fatou
@article{AIF_2010__60_7_2357_0,
author = {Collin, Pascal and Rosenberg, Harold},
title = {Asymptotic values of minimal graphs in~a~disc},
journal = {Annales de l'Institut Fourier},
pages = {2357--2372},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {60},
number = {7},
year = {2010},
doi = {10.5802/aif.2610},
zbl = {1239.53004},
mrnumber = {2849267},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2610/}
}
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 SP - 2357 EP - 2372 VL - 60 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2610/ DO - 10.5802/aif.2610 LA - en ID - AIF_2010__60_7_2357_0 ER -
%0 Journal Article %A Collin, Pascal %A Rosenberg, Harold %T Asymptotic values of minimal graphs in a disc %J Annales de l'Institut Fourier %D 2010 %P 2357-2372 %V 60 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2610/ %R 10.5802/aif.2610 %G en %F AIF_2010__60_7_2357_0
Collin, Pascal; Rosenberg, Harold. Asymptotic values of minimal graphs in a disc. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2357-2372. doi : 10.5802/aif.2610. http://www.numdam.org/articles/10.5802/aif.2610/
[1] Construction of harmonic diffeomorphisms and minimal graphs (To appear in Annals of Math)
[2] Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal., Volume 21 (1966), pp. 321-342 | DOI | MR | Zbl
[3] Two theorems on boundary properties of minimal surfaces in nonparametric form, Math. Notes, Volume 21 (1977), pp. 307-310 | DOI | MR | Zbl
[4] On new results in the theory of minimal surfaces, B. Amer. Math. Soc., Volume 71 (1965), pp. 195-270 | DOI | MR | Zbl
Cité par Sources :
