Stability of Bernstein type theorem for the minimal surface equation

Abstract

Let Ω R n (n ≥ 2) be an unbounded convex domain. We study the minimal surface equation in Ω with boundary value given by the sum of a linear function and a bounded uniformly continuous function in R n . 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.