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The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions. (English) Zbl 1071.35050
Author’s abstract: Let $$\Omega$$ be a bounded $$C^2$$ domain in $$\mathbb R^n$$ and $$\Phi: \partial\Omega\rightarrow \mathbb R^m$$ be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map $$f:\Omega \rightarrow \mathbb R^m$$ with $$f| _{\partial\Omega} = \Phi$$ and with the graph of $$f$$ a minimal submanifold in $$\mathbb R^{n+m}.$$ For $$m=1,$$ the Dirichlet problem was solved more than 30 years ago by H. Jenkins and J. Serrin [J. Reine Angew. Math. 229, 170–187 (1968; Zbl 0159.40204)] for any mean convex domains and the solutions are all smooth.
This paper considers the Dirichlet problem for convex domains in arbitrary codimention $$m.$$ We prove that if $$\psi:\overline\Omega\rightarrow \mathbb R^m$$ 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 H. B. Lawson and R. Osserman [Acta Math. 139, 1–17 (1977; Zbl 0376.49016)]. In order to prove this result, we study the associated parabolic system and solve the Cauchy-Dirichlet problem with $$\psi$$ as initial data.

MSC:
 35J60 Nonlinear elliptic equations 49Q05 Minimal surfaces and optimization 35K55 Nonlinear parabolic equations
Keywords:
Dirichlet problem; minimal surface
Citations:
Zbl 0159.40204; Zbl 0376.49016
Full Text:
References:
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