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 $u:{\mathbb{R}}^n\to \mathbb{R}$ 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 $|\nabla u(x)|=o(|x{|}^{\epsilon })$ for $\epsilon \le {\epsilon }_0(n,\Phi )$ with the constant ${\epsilon }_0(n,\Phi )>0$.

arXiv commentFinal version, to appear in Mathematische Annalen