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The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions
Author(s) -
Wang MuTao
Publication year - 2004
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10117
Subject(s) - mathematics , submanifold , dirichlet problem , bounded function , lipschitz continuity , domain (mathematical analysis) , codimension , dirichlet distribution , graph , pure mathematics , combinatorics , regular polygon , mathematical analysis , boundary value problem , geometry
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.