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

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 $\varepsilon_{0}(n, \Phi)>0$.