The first dimension with a singular one-phase minimizing cone
For $d \ge 2$ and a ball $B \subset \mathbb{R}^d$, the one-phase Bernoulli (Alt–Caffarelli) energy is $$J_B(u) = \int_B \bigl(|\nabla u|^2 + \mathbf{1}_{\{u > 0\}}\bigr) \, dx.$$ Let $d_*$ be the least $d$ for which there is a nonnegative, nonzero $u \in H^1_{\mathrm{loc}}(\mathbb{R}^d) \cap C(\mathbb{R}^d)$, homogeneous of degree one, with $0 \in \partial\{u > 0\}$, that minimizes $J_B$ in every ball $B$ among nonnegative competitors $v$ with $v - u \in H^1_0(B)$, and that is not of the form $(x \cdot e)_+$ for a unit vector $e$. De Silva and Jerison constructed such a minimizer in $\mathbb{R}^7$. Jerison and Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions $d \le 4$, the case $d = 3$ being due to Caffarelli, Jerison and Kenig. Together these give $5 \le d_* \le 7$.
Problem. Determine $d_*$.
Source: Jerison and Savin (2015), recalling the conjecture of Caffarelli, Jerison and Kenig that stable cones are flat up to dimension six; as the critical dimension d_* with 5 ≤ d_* ≤ 7, Firester, Tsiamis and Wang (2026). Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Alt–Caffarelli functional, minimizing cones, one-phase free boundary problem
References
@incollection{caffarelli2004,
author = {Caffarelli, Luis A. and Jerison, David and Kenig, Carlos E.},
title = {Global energy minimizers for free boundary problems and full regularity in three dimensions},
booktitle = {Noncompact Problems at the Intersection of Geometry, Analysis, and Topology},
series = {Contemp. Math.},
volume = {350},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2004},
pages = {83--97},
doi = {10.1090/conm/350/06339},
}
@article{desilva2009,
author = {De Silva, Daniela and Jerison, David},
title = {A singular energy minimizing free boundary},
journal = {J. Reine Angew. Math.},
volume = {635},
year = {2009},
pages = {1--21},
doi = {10.1515/CRELLE.2009.074},
}
@article{jerison2015,
author = {Jerison, David and Savin, Ovidiu},
title = {Some remarks on stability of cones for the one-phase free boundary problem},
journal = {Geom. Funct. Anal.},
volume = {25},
number = {4},
year = {2015},
pages = {1240--1257},
doi = {10.1007/s00039-015-0335-6},
eprint = {1410.7463},
archivePrefix = {arXiv},
}
@misc{firester2026,
author = {Firester, Benjy and Tsiamis, Raphael and Wang, Yipeng},
title = {Stability inequalities for one-phase cones},
year = {2026},
eprint = {2601.16966},
archivePrefix = {arXiv},
}
@misc{aim-one-phase-critical-dimension,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The first dimension with a singular one-phase minimizing cone}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/157-one-phase-critical-dimension.md},
note = {Entry 157 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension $n\ge7$, its singular set has …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00703, "The first dimension with a singular one-phase minimizing cone", clopen.solutions.