A Bernstein-type theorem for minimal graphs over convex domains

Abstract

Given any n 2, we show that if ¨Rn is an open convex domain (e.g. a half-space), and uW ! R is a solution to the minimal surface equation which agrees with a linear function on @ , then u must itself be linear.

Stability of Bernstein type theorem for the minimal surface equation

“…By the Bernstein type theorem in [9], we know v(x) = l(x) + cx n for some constant c. Hence, Proof. Note for any x ∈ Ω,…”

“…Suppose X := {x ∈ Ω : u(x) − l(x) > 0} = ∅ and Y ⊂ X be one of its connected component. Then u| Y : Y → R is a solution of the minimal surface equation with Dirichlet boundary value l. Since Y is contained in a convex cone or a slab, then by Edelen and Wang [9] (actually their arguments still work for domains contained in slabs), u| Y = l, which is a contradiction. Hence, X = ∅.…”

“…Based on the work in [9], we can reduce above theorem to the following special one. We say that a domain Ω satisfies the exterior cone property if there is an infinity cone outside Ω.…”

“…If Ω is not a half space, we prove that the solution is unique. If Ω is a half space, we prove that graphs of all solutions form a foliation of Ω × R. This can be viewed as a stability type theorem for Edelen-Wang's Bernstein type theorem in [9]. We also establish a comparison principle for the minimal surface equation in Ω.…”

“…For n ≥ 8, there were non-planar entire minimal graphs constructed by Bombieri, De Giorgi and Giusti [5] and so the Bernstein theorem fails in higher dimensions. Just recently, Edelen and the second named author [9] established a Bernstein type theorem for minimal graphs over unbounded convex domains. In any dimension they can show that if the boundary of a minimal graph over an unbounded convex domain Ω R n is contained in a hyperplane, then so is the minimal graph itself.…”