One-dimensionality of monotone Allen–Cahn fronts in dimension five
Let $u \in C^2(\mathbb{R}^5; (-1, 1))$ solve $$-\Delta u = u - u^3 \quad \text{in } \mathbb{R}^5, \qquad \partial_{x_5} u > 0 \quad \text{everywhere}.$$ De Giorgi conjectured in 1978 that the level sets of every solution of $\Delta u = u^3 - u$ on $\mathbb{R}^n$ with $|u| \le 1$ and $\partial_{x_n} u > 0$ are hyperplanes, at least for $n \le 8$. Ghoussoub and Gui proved this for $n = 2$ and Ambrosio and Cabré for $n = 3$. Savin proved it for $n \le 8$ under the additional assumption $\lim_{x_n \to \pm\infty} u(x', x_n) = \pm 1$, and del Pino, Kowalczyk and Wei constructed counterexamples for every $n \ge 9$. Florit-Simon and Serra proved that stable solutions $u : \mathbb{R}^4 \to (-1, 1)$ with bounded energy density are one-dimensional. In 2026 Liu, Luo, Wang, Wei, Wei and Wu, and also Chan, Fernández-Real, Figalli, Florit-Simon and Serra, proved that bounded stable solutions on $\mathbb{R}^3$ are one-dimensional, which gives the conjecture for $n = 4$.
Problem. Must there exist $a \in \mathbb{R}^5$ with $|a| = 1$ and $a_5 > 0$, and $b \in \mathbb{R}$, such that $$u(x) = \tanh\Bigl(\frac{a \cdot x - b}{\sqrt2}\Bigr) \qquad \text{for every } x \in \mathbb{R}^5?$$ No limits as $x_5 \to \pm\infty$, energy minimization or energy-growth bound is assumed.
Source: De Giorgi (1979), for every dimension n ≤ 8; stated for n = 5 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Allen–Cahn equation, De Giorgi conjecture, phase transitions
References
@incollection{degiorgi1979,
author = {De Giorgi, Ennio},
title = {Convergence problems for functionals and operators},
booktitle = {Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978)},
editor = {De Giorgi, Ennio and Magenes, Enrico and Mosco, Umberto},
publisher = {Pitagora},
address = {Bologna},
year = {1979},
pages = {131--188},
}
@article{ghoussoub1998,
author = {Ghoussoub, Nassif and Gui, Changfeng},
title = {On a conjecture of {D}e {G}iorgi and some related problems},
journal = {Math. Ann.},
volume = {311},
number = {3},
year = {1998},
pages = {481--491},
doi = {10.1007/s002080050196},
}
@article{ambrosio2000,
author = {Ambrosio, Luigi and Cabr{\'e}, Xavier},
title = {Entire solutions of semilinear elliptic equations in {$\mathbb{R}^3$} and a conjecture of {D}e {G}iorgi},
journal = {J. Amer. Math. Soc.},
volume = {13},
number = {4},
year = {2000},
pages = {725--739},
doi = {10.1090/S0894-0347-00-00345-3},
}
@article{savin2009,
author = {Savin, Ovidiu},
title = {Regularity of flat level sets in phase transitions},
journal = {Ann. of Math. (2)},
volume = {169},
number = {1},
year = {2009},
pages = {41--78},
doi = {10.4007/annals.2009.169.41},
}
@article{delpino2011,
author = {del Pino, Manuel and Kowalczyk, Micha{\l} and Wei, Juncheng},
title = {On {D}e {G}iorgi's conjecture in dimension {$N \ge 9$}},
journal = {Ann. of Math. (2)},
volume = {174},
number = {3},
year = {2011},
pages = {1485--1569},
doi = {10.4007/annals.2011.174.3.3},
}
@misc{floritsimon2025,
author = {Florit-Simon, Enric and Serra, Joaquim},
title = {On stable solutions to the {A}llen--{C}ahn equation with bounded energy density in {$\mathbb{R}^4$}},
year = {2025},
eprint = {2509.02739},
archivePrefix = {arXiv},
}
@misc{liu2026,
author = {Liu, Yong and Luo, Tianci and Wang, Kelei and Wei, Juncheng and Wei, Yong and Wu, Ke},
title = {Stable {D}e {G}iorgi conjecture of the {A}llen--{C}ahn equation in {$\mathbb{R}^3$}},
year = {2026},
eprint = {2609.21680},
archivePrefix = {arXiv},
}
@misc{chan2026,
author = {Chan, Hardy and Fern{\'a}ndez-Real, Xavier and Figalli, Alessio and Florit-Simon, Enric and Serra, Joaquim},
title = {Stable solutions of the {A}llen--{C}ahn equation in dimension three are one-dimensional},
year = {2026},
eprint = {2609.30194},
archivePrefix = {arXiv},
}
@misc{aim-de-giorgi-dimension-five,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{One-dimensionality of monotone Allen–Cahn fronts in dimension five}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/089-de-giorgi-dimension-five.md},
note = {Entry 089 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We resolve De Giorgi's conjecture positively in dimension eight: every entire $C^2$ solution $u:\R^8\to(-1,1)$ of $\Delta u=u^3-u$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $v:\R^7\to[-1,1]$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00674, "One-dimensionality of monotone Allen–Cahn fronts in dimension five", clopen.solutions.