Flatness of anisotropic minimal graphs in $\mathbb{R}^{n+1}$

TL;DR

This paper establishes a Bernstein-type theorem for anisotropic minimal hypersurfaces in all dimensions, showing that under certain conditions, entire solutions are necessarily linear functions.

Contribution

It proves a Bernstein theorem for $ ext{Phi}$-anisotropic minimal graphs in all dimensions, extending classical results to anisotropic settings with specific closeness and growth conditions.

Findings

Entire solutions are linear under the given conditions.

The result applies to all dimensions in Euclidean space.

Conditions involve closeness of $ ext{Phi}$ to classical integrand and growth constraints.

Abstract

We prove a Bernstein theorem for $\Phi$-anisotropic minimal hypersurfaces in all dimensional Euclidean spaces that the only entire smooth solutions $u: \mathbb{R}^{n}\rightarrow \mathbb{R}$ of $\Phi$-anisotropic minimal hypersurfaces equation are linear functions provided the anisotropic area functional integrand $\Phi$ is sufficiently $C^{3}$-close to classical area functional integrand and $|\nabla u(x)|=o(|x|^{\varepsilon})$ for $\varepsilon\leq \varepsilon_{0}(n, \Phi)$ with the constant $\varepsilon_{0}(n, \Phi)>0$.