A Bernstein Type Theorem for Minimal Graphs over Convex Domains

TL;DR

This paper proves that minimal graphs over convex domains that match a linear boundary condition must be linear themselves, extending Bernstein's theorem to higher dimensions and convex settings.

Contribution

It establishes a Bernstein type theorem for minimal graphs over convex domains, showing linearity under boundary conditions, which was previously known mainly for entire solutions.

Findings

Minimal solutions over convex domains with linear boundary data are linear.

The result generalizes Bernstein's theorem to convex domains in higher dimensions.

The proof applies to domains like half-spaces, confirming linearity in these cases.

Abstract

Given any $n \geq 2$, we show that if $\Omega \subsetneq \mathbb{R}^n$ is an open convex domain (e.g. a half-space), and $u : \Omega \to \mathbb{R}$ is a solution to the minimal surface equation which agrees with a linear function on $\partial \Omega$, then $u$ must itself be linear.