On saddle solutions of the Allen–Cahn equation I: Pointwise estimates

Abstract

Saddle solutions of the Allen–Cahn equation in ${\mathbb{R}}^{2m}$ are characterized by the property that they vanish precisely on the Simons cones, a family of classical minimal surfaces with one singularity at the origin. Their existence and uniqueness are known, by results of Cabré–Terra (2009–2012) and Dang (1992). Schatzman (1995) proved that the saddle solution is unstable for $m=1$. Cabré–Terra (2009–2012) showed the instability for $m=2,3$ and stability for $m\ge 7$. This left open the case of $m=4,5,6$, which is conjectured to be stable (even energy minimizing). Towards this conjecture, here we establish some pointwise estimates for the saddle solutions.