Stability of Bernstein type theorem for the minimal surface equation

Let $\mathrm{\Omega }⊊{\mathbf{R}}^n(n\ge 2)$ be an unbounded convex domain. We study the minimal surface equation in $\mathrm{\Omega }$ with boundary value given by the sum of a linear function and a bounded uniformly continuous function in ${\mathbf{R}}^n$. If $\mathrm{\Omega }$ is not a half space, we prove that the solution is unique. If $\mathrm{\Omega }$ is a half space, we prove that graphs of all solutions form a foliation of $\mathrm{\Omega }\times \mathbf{R}$. This can be viewed as a stability type theorem for Edelen-Wang’s Bernstein type theorem in \cite{EW2021}. We also establish a comparison principle for the minimal surface equation in $\mathrm{\Omega }$.