Stability of Edelen-Wang?s Bernstein type theorem for the minimal surface equation

Let 12 C Rn (n >= 2) be an unbounded convex domain. We study the minimal surface equation in 12 with boundary value given by the sum of a linear function and a bounded uniformly continuous function in Rn. If 12 is not a half space, we prove that the solution is unique. If 12 is a half space, we prove that graphs of all solutions form a foliation of 12 x R. This can be viewed as a stability type theorem for Edelen-Wang's Bernstein type theorem in [10]. We also establish a comparison principle for the minimal surface equation in 12. (c) 2022 Elsevier Inc. All rights reserved.