PreprintarXiv licencearXiv
A C2-Perturbative Bernstein Theorem for Anisotropic Entire Minimal Graphs
Abstract
We prove a Bernstein theorem for Φ-anisotropic minimal hypersurfaces in dimensions 1≤ n≤ 7 that the only entire smooth solutions u of Φ-anisotropic minimal hypersurfaces equation are affine functions provided the anisotropic area functional integrand Φ is sufficiently C2--close to the Euclidean area integrand. This settles the C2 entire--graph version of anisotropic Bernstein problem posed by Mooney and Yang \cite{MooneyYang2024}, and the proof uses a compactness--rigidity argument combined with Figalli's regularity theorem established in \cite{Figalli2017}.
Related
- Bernstein theorem for $3$-dimensional anisotropic minimal graphs with free boundary
- Lehto-type growth conditions and a Picard theorem for the gradient of entire solutions to the Euler -- Lagrange equations of variational integrals with linear growth in the plane including a small extension of Moser's Bernstein result
- Calabi-Bernstein type theorems for entire solutions of certain nonlinear elliptic equations in the plane
- Sharp Bernstein--Nikolskii inequalities for polynomials and entire functions of exponential type
- Existence of positive solutions to anisotropic elliptic equations in the entire Euclidean space
BibTeX
@misc{chen2026c2perturbative,
title = {{A $C^2$-Perturbative Bernstein Theorem for Anisotropic Entire Minimal Graphs}},
author = {Lu Chen and Jiali Lan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.29117},
eprint = {2608.29117},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29117},
}
RIS
TY - UNPB
TI - A C^2-Perturbative Bernstein Theorem for Anisotropic Entire Minimal Graphs
AU - Lu Chen
AU - Jiali Lan
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2608.29117
UR - https://arxiv.org/abs/2608.29117
AB - We prove a Bernstein theorem for Φ-anisotropic minimal hypersurfaces in dimensions 1≤ n≤ 7 that the only entire smooth solutions u of Φ-anisotropic minimal hypersurfaces equation are affine functions provided the anisotropic area functional integrand Φ is sufficiently C^2--close to the Euclidean area integrand. This settles the C^2 entire--graph version of anisotropic Bernstein problem posed by Mooney and Yang \cite{MooneyYang2024}, and the proof uses a compactness--rigidity argument combined with Figalli's regularity theorem established in \cite{Figalli2017}.
ER -