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De Giorgi's conjecture in dimension eight

Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C

^{2}

solution

u:\mathbb R^8\to(-1,1)

\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.

Manuscripts in this family · 1

Sep 26, 2026·article.pdf

We resolve De Giorgi's conjecture positively in dimension eight: every entire C

^{2}

solution

u:\mathbb R^8\to(-1,1)

\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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