Stability of Bernstein type theorem for the minimal surface equation

Let $\Omega \subsetneq \mathbf{R}^n\,(n\geq 2)$ be an unbounded convex domain. We study the minimal surface equation in $\Omega$ with boundary value given by the sum of a linear function and a bounded uniformly continuous function in $\mathbf{R}^n$. If $\Omega$ is not a half space, we prove that the solution is unique. If $\Omega$ is a half space, we prove that graphs of all solutions form a foliation of $\Omega\times\mathbf{R}$. 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 $\Omega$.