PDE and mathematical physics · Oct 6, 2026

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)

\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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