Result 375, Partial differential equations

De Giorgi's conjecture in dimension eight

Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution u:R8→(−1,1)u:\mathbb R^8\to(-1,1) of Δu=u3−u\Delta u=u^3-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:R7→[−1,1]v:\mathbb R^7\to[-1,1] as constant wells or planar transitions, without an energy-growth assumption.

Proof

The bigger picture

Why it matters

A transition between two states might bend through space, but this manuscript claims that certain transitions must be flat. The result identifies conditions under which an eight-dimensional equation permits only one-dimensional transition profiles.

What changes?

The unreviewed manuscript reports that every twice continuously differentiable solution, defined throughout eight-dimensional space, of 'Laplacian of u equals u cubed minus u' is a planar transition if its values lie strictly between -1 and 1 and its derivative in one fixed direction is everywhere positive. Planar means depending on just one linear coordinate. It also classifies all stable entire solutions in seven dimensions, valued in [-1,1], as constant wells or planar transitions, without any energy-growth assumption.

What does that help mathematicians do?

The seven-dimensional claim rules out stable curved or more complicated transition patterns under the stated boundedness condition. Here stability means nonnegative second-order energy change under localized perturbations, and constant wells are the uniform states -1 and 1. Removing an energy-growth assumption matters because researchers would not need a separate large-scale energy estimate to apply this classification.

Are there practical applications?

Its immediate value is foundational: it connects a solution's stability or directional increase to the geometry of its transitions. Researchers studying this equation can use the claimed classification to constrain possible solutions before trying to construct them. The supplied material does not establish a computational method or a direct practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A positive resolution of De Giorgi's conjecture in dimension eight

September 26, 2026 97 pages

We resolve De Giorgi's conjecture positively in dimension eight: every entire C2 solution u:R8→(−1,1)u:\mathbb R^8\to(-1,1) of Δu=u3−u\Delta u=u^3-u with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution v:R7→[−1,1]v:\mathbb R^7\to[-1,1] of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.

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Cite (BibTeX) 
@misc{OAI:De-Giorgis-conjecture-in-dimension-eight-September-26-2026,
  author = {{OpenAI}},
  title = {{A positive resolution of De Giorgi's conjecture in dimension eight}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf}{OAI:De-Giorgis-conjecture-in-dimension-eight-September-26-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.