Minimal Graphs in ℍ n × ℝ and ℝ n + 1

Ricardo Sà Earp[1]; Eric Toubiana[2]

  • [1] Pontifícia Universidade Católica do Rio de Janeiro Departamento de Matemática Rio de Janeiro, 22453-900 RJ (Brazil)
  • [2] Université Paris VII, Denis Diderot Institut de Mathématiques de Jussieu Case 7012, 2 place Jussieu 75251 Paris Cedex 05 (France)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 7, page 2373-2402
  • ISSN: 0373-0956

Abstract

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We construct geometric barriers for minimal graphs in ℍ n × ℝ . We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in ℍ n extending continuously to the interior of each face, taking infinite boundary data on one face and zero boundary value data on the other faces.In ℍ n × ℝ , we solve the Dirichlet problem for the vertical minimal equation in a C 0 convex domain Ω ⊂ ℍ n taking arbitrarily continuous finite boundary and asymptotic boundary data.We prove the existence of another Scherk type hypersurface, given by the solution of the vertical minimal equation in the interior of certain admissible polyhedron taking alternatively infinite values + ∞ and - ∞ on adjacent faces of this polyhedron.We establish analogous results for minimal graphs when the ambient is the Euclidean space ℝ n + 1 .

How to cite

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Sà Earp, Ricardo, and Toubiana, Eric. "Minimal Graphs in $\mathbb{H}^n\times \mathbb{R}$ and $\mathbb{R}^{n+1}$." Annales de l’institut Fourier 60.7 (2010): 2373-2402. <http://eudml.org/doc/116338>.

@article{SàEarp2010,
abstract = {We construct geometric barriers for minimal graphs in $\mathbb\{H\}^n\times \mathbb\{R\}.$We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in $\mathbb\{H\}^n$ extending continuously to the interior of each face, taking infinite boundary data on one face and zero boundary value data on the other faces.In $\mathbb\{H\}^n\times \mathbb\{R\}$, we solve the Dirichlet problem for the vertical minimal equation in a $C^0$ convex domain $\Omega \subset \mathbb\{H\}^n$ taking arbitrarily continuous finite boundary and asymptotic boundary data.We prove the existence of another Scherk type hypersurface, given by the solution of the vertical minimal equation in the interior of certain admissible polyhedron taking alternatively infinite values $+\infty $ and $-\infty $ on adjacent faces of this polyhedron.We establish analogous results for minimal graphs when the ambient is the Euclidean space $\mathbb\{R\}^\{n+1\}$.},
affiliation = {Pontifícia Universidade Católica do Rio de Janeiro Departamento de Matemática Rio de Janeiro, 22453-900 RJ (Brazil); Université Paris VII, Denis Diderot Institut de Mathématiques de Jussieu Case 7012, 2 place Jussieu 75251 Paris Cedex 05 (France)},
author = {Sà Earp, Ricardo, Toubiana, Eric},
journal = {Annales de l’institut Fourier},
keywords = {Dirichlet problem; minimal equation; vertical graph; Perron process; barrier; convex domain; asymptotic boundary; translation hypersurface; Scherk hypersurface},
language = {eng},
number = {7},
pages = {2373-2402},
publisher = {Association des Annales de l’institut Fourier},
title = {Minimal Graphs in $\mathbb\{H\}^n\times \mathbb\{R\}$ and $\mathbb\{R\}^\{n+1\}$},
url = {http://eudml.org/doc/116338},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Sà Earp, Ricardo
AU - Toubiana, Eric
TI - Minimal Graphs in $\mathbb{H}^n\times \mathbb{R}$ and $\mathbb{R}^{n+1}$
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 7
SP - 2373
EP - 2402
AB - We construct geometric barriers for minimal graphs in $\mathbb{H}^n\times \mathbb{R}.$We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in $\mathbb{H}^n$ extending continuously to the interior of each face, taking infinite boundary data on one face and zero boundary value data on the other faces.In $\mathbb{H}^n\times \mathbb{R}$, we solve the Dirichlet problem for the vertical minimal equation in a $C^0$ convex domain $\Omega \subset \mathbb{H}^n$ taking arbitrarily continuous finite boundary and asymptotic boundary data.We prove the existence of another Scherk type hypersurface, given by the solution of the vertical minimal equation in the interior of certain admissible polyhedron taking alternatively infinite values $+\infty $ and $-\infty $ on adjacent faces of this polyhedron.We establish analogous results for minimal graphs when the ambient is the Euclidean space $\mathbb{R}^{n+1}$.
LA - eng
KW - Dirichlet problem; minimal equation; vertical graph; Perron process; barrier; convex domain; asymptotic boundary; translation hypersurface; Scherk hypersurface
UR - http://eudml.org/doc/116338
ER -

References

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  11. Laurent Hauswirth, Harold Rosenberg, Joel Spruck, Infinite boundary value problems for constant mean curvature graphs in ℍ 2 × ℝ and 𝕊 2 × ℝ , Amer. J. Math. 131 (2009), 195-226 Zbl1178.53062MR2488489
  12. Howard Jenkins, James Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math. 229 (1968), 170-187 Zbl0159.40204MR222467
  13. L. Mazet, M. M. Rodriguez, H. Rosenberg, The Dirichlet problem for the minimal surface equation – with possible infinite boundary data – over domains in a Riemannian manifold Zbl1235.53007
  14. Barbara Nelli, Harold Rosenberg, Minimal surfaces in ℍ 2 × ℝ , Bull. Braz. Math. Soc. (N.S.) 33 (2002), 263-292 Zbl1038.53011MR1940353
  15. Ye-Lin Ou, p -harmonic functions and the minimal graph equation in a Riemannian manifold, Illinois J. Math. 49 (2005), 911-927 (electronic) Zbl1089.58010MR2210268
  16. J. Spruck, Interior gradient estimates and existence theorems for constant mean curvature graphs in M n × ℝ , Pure Appl. Math. Q. 3 (2007), 785-800 Zbl1145.53048MR2351645

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