De Giorgi's conjecture in dimension eight

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.

Manuscript

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

September 26, 2026 · source · cite as \cite{OAI:De-Giorgis-conjecture-in-dimension-eight-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.

Result 375 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.