De Giorgi's conjecture on monotone solutions of the Allen-Cahn equation

Theorem 1.1: every $u\in C^2(\mathbb R^8,(-1,1))$ with $\Delta u=u^3-u$ and $\partial_8u>0$ equals $\tanh((e\cdot x-c)/\sqrt2)$ with $e_8>0$. Theorem 1.2: every stable solution $v:\mathbb R^7\to[-1,1]$ is $\pm1$ or planar, without an energy-growth assumption; this is sharp by Pacard-Wei's nonplanar stable solutions in $\mathbb R^8$. A final section explains why the argument fails in dimension nine, consistent with del Pino-Kowalczyk-Wei. The paper does not treat other potentials, the stable De Giorgi problem in dimension eight, or the Gibbons and Savin variants beyond what the monotone theorem implies.

Theorem 1.1: every $u\in C^2(\mathbb R^8,(-1,1))$ with $\Delta u=u^3-u$ and $\partial_8u>0$ equals $\tanh((e\cdot x-c)/\sqrt2)$ with $e_8>0$. Theorem 1.2: every stable solution $v:\mathbb R^7\to[-1,1]$ is $\pm1$ or planar, without an energy-growth assumption; this is sharp by Pacard-Wei's nonplanar stable solutions in $\mathbb R^8$. A final section explains why the argument fails in dimension nine, consistent with del Pino-Kowalczyk-Wei. The paper does not treat other potentials, the stable De Giorgi problem in dimension eight, or the Gibbons and Savin variants beyond what the monotone theorem implies.