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

Abstract

Let be a bounded C 2 domain in R n and φ : ∂ → R m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : → R m with f | ∂ = 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 :→ R m satisfies 8nδ sup |D 2 ψ| + √ 2 sup ∂ |Dψ| < 1, then the Dirichlet problem for ψ| ∂ is solvable in smooth maps. Here δ is the diameter of . 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 CauchyDirichlet problem with ψ as initial data.

A stability criterion for nonparametric minimal submanifolds

“…In [8], the second author constructs solutions to the Dirichlet problem of minimal surface systems in higher dimension and codimension and the solutions satisfies * Ω > 1 2 for Ω the volume form of an n-dimensional subspace. When Σ is the graph of f : D ⊂ R n → R m and Ω is the volume form of the domain R n extending to the whole R n+m , we have the relation * Ω =…”

Minimal cones and self-expanding solutions for mean curvature flows

“…Note that the cone C is of C 3,α class. Analog to the step 1 in the proof of Theorem 1.1 in [38] (with Schauder fixed-point theorem and Schauder estimates for linear parabolic equations), we get √ tT converging to the regular cone C in C 3 -norm. Now let us show the claim (5.15).…”

Remarks on a class of solutions to the minimal surface system

“…A corollary of the Bernstein-type result is the following regularity theorem proved in [30]. The proof of the Bernstein type theorem is based on calculating the Laplacian of the following quantity: * Ω = 1…”