Stable De Giorgi conjecture of the Allen--Cahn equation in ${\mathbb{R}}^3$

Yong Liu, Tianci Luo, Kelei Wang +3

We prove that every bounded stable entire solution $v$ of the Allen--Cahn equation in ${\mathbb{R}}^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in ${\mathbb{R}}^4$ i…