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, which then implies the one dimensional symmetry of minimizers in ${\mathbb{R}}^n$ for $n\le 7$. This generalizes Allard's $\epsilon$-regularity theorem for stationary varifolds to the setting of Allen-Cahn equations.

63 pages, author's final version. To appear in J. Eur. Math. Soc

References in corpus (1)

• Harmonic approximation and improvement of flatness in a singular perturbation problem

Cited by in corpus (3)

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• One-dimensional symmetry for integral systems in two dimensions