Stability of Bernstein type theorem for the minimal surface equation

TL;DR

This paper investigates the stability of solutions to the minimal surface equation in unbounded convex domains, proving uniqueness in non-half spaces and a foliation property in half spaces, extending Bernstein-type results.

Contribution

It establishes a stability theorem for Bernstein-type solutions in unbounded convex domains, including uniqueness and foliation properties, and introduces a comparison principle.

Findings

Solutions are unique in non-half space domains.

Graphs of solutions form a foliation in half space domains.

A comparison principle for the minimal surface equation is established.

Abstract

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 $\Omega$.