Abstract: | Let Ω be a bounded C2 domain in ?n and ? ?Ω → ?m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ?m with f|?Ω = ? and with the graph of f a minimal submanifold in ?n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin 12] 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 that if ψ : ¯Ω → ?m 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 and Osserman 13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc. |