Polynomial anisotropic minimal graphs in dimensions four and five
Let $\Phi$ be a positive even function on $S^n$, identified with its positively one-homogeneous extension to $\mathbb{R}^{n+1} \setminus \{0\}$, with uniformly convex level sets: $D^2\Phi(\nu)$ restricted to $\nu^\perp$ is positive definite, with a positive lower bound uniform over $\nu \in S^n$. A function $u$ on $\mathbb{R}^n$ whose graph is critical for the anisotropic area $\int_\Sigma \Phi(\nu)\, dA$ solves $$\operatorname{div}\bigl(DF(\nabla u)\bigr) = 0 \quad \text{on } \mathbb{R}^n, \qquad F(p) = \Phi((-p, 1)).$$ Jenkins ($n = 2$) and Simon ($n = 3$) proved that every entire solution is affine. Mooney constructed a quadratic polynomial on $\mathbb{R}^6$ that solves this equation for a uniformly convex integrand $\Phi \in C^{2,1}(S^6)$. Mooney and Yang constructed non-affine smooth entire solutions on $\mathbb{R}^4$, for smooth integrands, with $\sup_{B_r} u \sim r^{1 + \mu}$ for large $r$, for each $\mu \in (0, 1/2)$.
Problem. Does there exist $n \in \{4, 5\}$, a non-affine real polynomial $u$ on $\mathbb{R}^n$, and a positive even integrand $\Phi \in C^{2,1}(S^n)$ whose positively one-homogeneous extension has uniformly convex level sets, such that $\operatorname{div}\bigl(DF(\nabla u)\bigr) = 0$ on $\mathbb{R}^n$, where $F(p) = \Phi((-p, 1))$?
Source: Mooney (2024), §7.2, after Mooney (2022), §1; stated for integrands in C^{2,1} by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, anisotropic minimal hypersurfaces, Bernstein theorems
References
@article{mooney2024,
author = {Mooney, Connor},
title = {Bernstein theorems for nonlinear geometric {PDE}s},
journal = {Commun. Pure Appl. Anal.},
volume = {23},
number = {12},
year = {2024},
pages = {1958--1989},
doi = {10.3934/cpaa.2024072},
eprint = {2407.11903},
archivePrefix = {arXiv},
}
@article{mooney2022,
author = {Mooney, Connor},
title = {Entire solutions to equations of minimal surface type in six dimensions},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {24},
number = {12},
year = {2022},
pages = {4353--4361},
doi = {10.4171/JEMS/1202},
eprint = {1910.11297},
archivePrefix = {arXiv},
}
@article{mooney2024anisotropic,
author = {Mooney, Connor and Yang, Yang},
title = {The anisotropic {B}ernstein problem},
journal = {Invent. Math.},
volume = {235},
number = {1},
year = {2024},
pages = {211--232},
doi = {10.1007/s00222-023-01222-4},
eprint = {2209.15551},
archivePrefix = {arXiv},
}
@article{jenkins1961,
author = {Jenkins, H. B.},
title = {On two-dimensional variational problems in parametric form},
journal = {Arch. Rational Mech. Anal.},
volume = {8},
number = {1},
year = {1961},
pages = {181--206},
doi = {10.1007/BF00277437},
}
@article{simon1977,
author = {Simon, Leon},
title = {On some extensions of {B}ernstein's theorem},
journal = {Math. Z.},
volume = {154},
number = {3},
year = {1977},
pages = {265--273},
doi = {10.1007/BF01214329},
}
@misc{aim-polynomial-anisotropic-minimal-graphs,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Polynomial anisotropic minimal graphs in dimensions four and five}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/408-polynomial-anisotropic-minimal-graphs.md},
note = {Entry 408 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00889, "Polynomial anisotropic minimal graphs in dimensions four and five", clopen.solutions.