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De Giorgi's conjecture on monotone solutions of the Allen-Cahn equation

De Giorgi conjectured in 1978 that if u∈C2(Rn)u\in C^2(\mathbb R^n) solves the Allen-Cahn equation Δu=u3−u\Delta u=u^3-u with ∣u∣≤1|u|\le1 and ∂xnu>0\partial_{x_n}u>0 everywhere, then for n≤8n\le8 the level sets of uu are hyperplanes, i.e. u(x)=tanh⁡((e⋅x−c)/2)u(x)=\tanh((e\cdot x-c)/\sqrt2) for a unit vector ee and c∈Rc\in\mathbb R. It is the phase-transition analogue of the Bernstein problem for minimal graphs. Ghoussoub-Gui proved it for n=2n=2, Ambrosio-Cabre for n=3n=3, Savin for 4≤n≤84\le n\le8 under the extra assumption that u(x′,t)→±1u(x',t)\to\pm1 as t→±∞t\to\pm\infty, and del Pino-Kowalczyk-Wei gave counterexamples for n≥9n\ge9. Is every such monotone solution one-dimensional in every dimension n≤8n\le8, without the limit assumption?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Elliptic PDE; Allen-Cahn equation and phase transitions
Posed by
Ennio De Giorgi
Year posed
1978
Years open
48y
Solved
2026-09-26
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every u∈C2(R8,(−1,1))u\in C^2(\mathbb R^8,(-1,1)) with Δu=u3−u\Delta u=u^3-u and ∂8u>0\partial_8u>0 equals tanh⁡((e⋅x−c)/2)\tanh((e\cdot x-c)/\sqrt2) with e8>0e_8>0. Theorem 1.2: every stable solution v:R7→[−1,1]v:\mathbb R^7\to[-1,1] is ±1\pm1 or planar, without an energy-growth assumption; this is sharp by Pacard-Wei's nonplanar stable solutions in R8\mathbb R^8. A final section explains why the argument fails in dimension nine, consistent with del Pino-Kowalczyk-Wei. The paper does not treat other potentials, the stable De Giorgi problem in dimension eight, or the Gibbons and Savin variants beyond what the monotone theorem implies.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 26, 2026) has no companion in the release.

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the manuscript (TeX source) were read against De Giorgi's conjecture. Theorem 1.1 treats exactly n=8n=8 with values in (−1,1)(-1,1) and strict positivity of one partial derivative, with no limit or energy assumption. The paper states only the dimension-eight case; the lower dimensions n≤7n\le7 follow by extending a solution constantly in the missing variables, an observation made here, not in the paper. The proof reduces the monotone case to Theorem 1.2 (every stable solution on R7\mathbb R^7 is constant or planar, with no energy-growth hypothesis) using classical inputs (Alberti-Ambrosio-Cabre, Jerison-Monneau, Federer, Wang's near-unit-density theorem); the stable classification is the new part. Not refereed. No Lean formalization in the release.

Source

Changelog1 change

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