A positive resolution of De Giorgi's conjecture in dimension eight
We resolve De Giorgi's conjecture positively in dimension eight: every entire C 2 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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