Asymptotic values of minimal graphs in a disc
Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2357-2372.

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.

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.

Received:
Accepted:
DOI: 10.5802/aif.2610
Classification: 53A10,  53C43
Keywords: Minimal graphs, radial limits, Fatou theorem
Collin, Pascal 1; Rosenberg, Harold 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)
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Collin, Pascal; Rosenberg, Harold. Asymptotic values of minimal graphs in a disc. Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2357-2372. doi : 10.5802/aif.2610. https://aif.centre-mersenne.org/articles/10.5802/aif.2610/

[1] Collin, P.; Rosenberg, H. Construction of harmonic diffeomorphisms and minimal graphs (To appear in Annals of Math) | Zbl

[2] Jenkins, H.; Serrin, J. 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] Mikyukov, V. Two theorems on boundary properties of minimal surfaces in nonparametric form, Math. Notes, Volume 21 (1977), pp. 307-310 | DOI | MR | Zbl

[4] Nitsche, J. On new results in the theory of minimal surfaces, B. Amer. Math. Soc., Volume 71 (1965), pp. 195-270 | DOI | MR | Zbl

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