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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
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