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Partial differential equations · Result 375

De Giorgi's conjecture in dimension eight

1 paper

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

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Papers 1

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

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

OpenAI · September 26, 2026

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

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@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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