A new proof of Savin's theorem on Allen–Cahn equations

Abstract

In this paper we establish an improvement of tilt-excess decay estimate for the Allen–Cahn equation, and use this to give a new proof of Savin's theorem on the uniform $C^{1,\alpha }$ regularity of flat level sets. This generalizes Allard’s $\epsilon$-regularity theorem for stationary varifolds to the setting of Allen–Cahn equations. A new proof of Savin’s theorem on the one-dimensional symmetry of minimizers in ${\mathbb{R}}^n$ for $n\le 7$ is also given.