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

Theorem 1.1: every $u\in C^2(\mathbb R^8,(-1,1))$ with $\Delta u=u^3-u$ and $\partial_8u>0$ equals $\tanh((e\cdot x-c)/\sqrt2)$ with $e_8>0$ . Theorem 1.2: every stable solution $v:\mathbb R^7\to[-1,1]$ is $\pm1$ or planar, without an energy-growth assumption; this is sharp by Pacard-Wei's nonplanar stable solutions in $\mathbb R^8$ . 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.

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Occurred: Sep 26, 2026

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Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: De Giorgi's conjecture on monotone solutions of the Allen-Cahn equation · De Giorgi's conjecture

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

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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 $n=8$ with values in $(-1,1)$ and strict positivity of one partial derivative, with no limit or energy assumption. The paper states only the dimension-eight case; the lower dimensions $n\le7$ 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 $\mathbb R^7$ 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.

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