Stationarity of Lipschitz weak solutions of the minimal-surface system
Let $n \ge 3$, $m \ge 2$, let $B_1 \subset \mathbb{R}^n$ be the unit ball and let $u \in W^{1,\infty}(B_1; \mathbb{R}^m)$. Put $g_{ij} = \delta_{ij} + \sum_{\alpha=1}^m \partial_i u^\alpha \partial_j u^\alpha$, the metric induced on the graph of $u$, and $(g^{ij}) = g^{-1}$. The minimal surface system $$\sum_{i,j=1}^n \partial_i\bigl(\sqrt{\det g}\, g^{ij} \partial_j u^\alpha\bigr) = 0 \qquad (1 \le \alpha \le m)$$ says that the area of the graph is critical under variations of the values of $u$, and the system $$\sum_{i=1}^n \partial_i\bigl(\sqrt{\det g}\, g^{ij}\bigr) = 0 \qquad (1 \le j \le n)$$ that it is critical under variations of the domain. For $u \in C^2$ the first system implies the second. For $m = 1$, Lipschitz weak solutions of the first system are smooth by the De Giorgi–Nash–Moser theorem, and Hirsch, Mooney and Tione proved that they are smooth when $n = 2$, for every $m$. Tsai and Wang proved that a Lipschitz weak solution satisfies the second system, in every dimension and codimension, under a bound on the products of pairs of singular values of $Du$. Lawson and Osserman constructed a non-flat minimal cone in $\mathbb{R}^7$ that is the graph of a Lipschitz map from $\mathbb{R}^4$ to $\mathbb{R}^3$.
Problem. If $u$ satisfies the first system in the sense of distributions, must it also satisfy the second system in the sense of distributions? No small-slope or area-decreasing condition is imposed.
Source: Lawson and Osserman (1977), Conjecture 2.1, as cited by Hirsch, Mooney and Tione (2026) and by Mooney (2024), Remark 3.2. Stated by AIM — Open Applied Problems for n ≥ 3 and m ≥ 2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, minimal graphs, weak solutions
References
@article{lawson1977,
author = {Lawson, H. B. and Osserman, R.},
title = {Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system},
journal = {Acta Math.},
volume = {139},
year = {1977},
pages = {1--17},
doi = {10.1007/BF02392232},
}
@article{hirsch2026,
author = {Hirsch, Jonas and Mooney, Connor and Tione, Riccardo},
title = {On the {L}awson--{O}sserman conjecture},
journal = {Invent. Math.},
year = {2026},
doi = {10.1007/s00222-026-01442-4},
eprint = {2308.04997},
archivePrefix = {arXiv},
}
@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},
}
@misc{tsai2026,
author = {Tsai, Chung-Jun and Wang, Mu-Tao},
title = {Calibrating forms for minimal graphs in arbitrary codimension},
year = {2026},
eprint = {2604.04336},
archivePrefix = {arXiv},
}
@misc{aim-lawson-osserman-stationarity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Stationarity of Lipschitz weak solutions of the minimal-surface system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/406-lawson-osserman-stationarity.md},
note = {Entry 406 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-00875, "Stationarity of Lipschitz weak solutions of the minimal-surface system", clopen.solutions.