The Dirichlet Problem for the Minimal Surface System in Arbitrary Codimension
نویسنده
چکیده
Let Ω be a bounded C2 domain in R and φ : ∂Ω → R be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → R with f |∂Ω = φ and with the graph of f a minimal submanifold in Rn+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 codimensionm. We prove if ψ : Ω → R satisfies 8nδ supΩ |D2ψ| + √ 2 sup∂Ω |Dψ| < 1 , then the Dirichlet problem for ψ|∂Ω is solvable in smooth maps. Here δ is the diameter of Ω. 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 ψ as initial data.
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