Stability of Bernstein type theorem for the minimal surface equation

Let $Ω\subsetneq \mathbf{R}^n\,(n\geq 2)$ 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 $\mathbf{R}^n$. 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 $Ω\times\mathbf{R}$. 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 $Ω$.

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