The Dirichlet problem for the minimal surface system in arbitrary codimension

Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10062

Communications on Pure and Applied Mathematics , 57, (2), 67-281, 2004
Let $\Omega $ be a bounded $C^2$ domain in $R^n$ and $\phi : \partial \Omega \to \mathbb{R}^m$ be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map $f : \Omega \to \mathbb{R}^m$ with $ f |_{\partial \Omega}$ and with the graph of $f$ a minimal submanifold in $R^{n+m}$. For $m = 1$, the Dirichlet problem was solved more than thirty years ago by Jenkins-Serrin [13] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension $m$. We prove if $\psi : \bar{\Omega} \to \mathbb{R}^m$ satisfies $8n\delta sup_{\Omega}||D^2 \psi| + \sqrt{2} sup_{\partial \Omega} |D\psi| <1$, then the Dirichlet problem for $ \psi |_{\partial \Omega}$ is solvable in smooth maps. Here $\delta$ is the diameter of $\Omega$ . Such a condition is necessary in view of an example of Lawson-Osserman [15]. In order to prove this result, we study the associated parabolic system and solve the Cauchy-Dirichlet problem with $\psi$ as initial data.
Dirichlet problem, minimal surface, codimension
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@inproceedings{mu-tao2004the,
  title={The Dirichlet problem for the minimal surface system in arbitrary codimension},
  author={Mu-Tao Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822150908676213377},
  booktitle={Communications on Pure and Applied Mathematics },
  volume={57},
  number={2},
  pages={67-281},
  year={2004},
}
Mu-Tao Wang. The Dirichlet problem for the minimal surface system in arbitrary codimension. 2004. Vol. 57. In Communications on Pure and Applied Mathematics . pp.67-281. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822150908676213377.
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