Flatness of three-dimensional fractional minimizing cones for every interaction order
For $s \in (0,1)$, a measurable set $E \subset \mathbb{R}^3$ and a bounded open set $U \subset \mathbb{R}^3$, let $$\operatorname{Per}_s(E;U) = \int_{E \cap U}\!\int_{E^c} \frac{dy\,dx}{|x-y|^{3+s}} + \int_{E \setminus U}\!\int_{E^c \cap U} \frac{dy\,dx}{|x-y|^{3+s}}$$ be the fractional perimeter of Caffarelli, Roquejoffre and Savin. Call $E$ a minimizing cone if $rE = E$ up to null sets for every $r > 0$, $\operatorname{Per}_s(E;U) < \infty$ for every bounded smooth $U$, and $\operatorname{Per}_s(E;U) \le \operatorname{Per}_s(F;U)$ for every measurable $F$ agreeing with $E$ outside $U$; it is nontrivial if both $E$ and $E^c$ have positive volume in the unit ball. Minimizing cones arise as blow-up limits of minimizers of the fractional perimeter. Savin and Valdinoci proved that minimizing cones in $\mathbb{R}^2$ are half-planes for every $s \in (0,1)$. In $\mathbb{R}^3$, Caffarelli and Valdinoci proved that minimizing cones are half-spaces for $s$ sufficiently close to $1$; Cabré, Cinti and Serra proved the same for stable cones with boundary smooth away from the origin, for $s$ sufficiently close to $1$; and Tsiamis proved that minimizing cones are half-spaces for $s$ sufficiently close to $0$.
Problem. Fix any $s \in (0,1)$ and let $E \subset \mathbb{R}^3$ be a nontrivial minimizing cone. Must $E$ agree almost everywhere with a half-space whose boundary passes through the origin? The question is for every $s \in (0,1)$, including orders not close to either endpoint.
Source: Stated by AIM — Open Applied Problems, for the minimizing cones of the fractional perimeter of Caffarelli, Roquejoffre and Savin (2010), after the flatness results of Caffarelli and Valdinoci (2013) and Cabré, Cinti and Serra (2020) for s close to 1 and of Tsiamis (2026) for s close to 0.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG, OC
- Keywords
- AIM Open Applied Problems, fractional perimeters, minimal cones, nonlocal minimal surfaces
References
@article{caffarelli2010,
author = {Caffarelli, L. and Roquejoffre, J.-M. and Savin, O.},
title = {Nonlocal minimal surfaces},
journal = {Comm. Pure Appl. Math.},
volume = {63},
number = {9},
year = {2010},
pages = {1111--1144},
doi = {10.1002/cpa.20331},
}
@article{cabre2020,
author = {Cabr{\'e}, Xavier and Cinti, Eleonora and Serra, Joaquim},
title = {Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s \sim 1$},
journal = {J. Reine Angew. Math.},
volume = {764},
year = {2020},
pages = {157--180},
doi = {10.1515/crelle-2019-0005},
eprint = {1710.08722},
archivePrefix = {arXiv},
}
@misc{tsiamis2026,
author = {Tsiamis, Raphael},
title = {Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s$ close to zero},
year = {2026},
eprint = {2608.21340},
archivePrefix = {arXiv},
}
@article{savin2013,
author = {Savin, Ovidiu and Valdinoci, Enrico},
title = {Regularity of nonlocal minimal cones in dimension 2},
journal = {Calc. Var. Partial Differential Equations},
volume = {48},
number = {1-2},
year = {2013},
pages = {33--39},
doi = {10.1007/s00526-012-0539-7},
eprint = {1202.0973},
archivePrefix = {arXiv},
}
@article{caffarelli2013,
author = {Caffarelli, Luis and Valdinoci, Enrico},
title = {Regularity properties of nonlocal minimal surfaces via limiting arguments},
journal = {Adv. Math.},
volume = {248},
year = {2013},
pages = {843--871},
doi = {10.1016/j.aim.2013.08.007},
eprint = {1105.1158},
archivePrefix = {arXiv},
}
@misc{aim-fractional-minimizing-cones-three-dimensional,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Flatness of three-dimensional fractional minimizing cones for every interaction order}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/471-fractional-minimizing-cones-three-dimensional.md},
note = {Entry 471 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-00593, "Flatness of three-dimensional fractional minimizing cones for every interaction order", clopen.solutions.