The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions

Source abstract

Abstract Let Ω be a bounded C 2 domain in ℝ n and ϕ ∂Ω → ℝ m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ℝ m with f | ∂Ω = ϕ and with the graph of f a minimal submanifold in ℝ 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 → ℝ m satisfies 8 n δ sup Ω | D 2 ψ | + √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.