Quantitative stratification for the fractional Allen-Cahn equation and stationary nonlocal minimal surface

Abstract

We study properties of solutions to the fractional Allen-Cahn equation when $s\in (0, 1/2)$ and dimension $n\geq 2$. By applying the quantitative stratification principle developed by Naber and Valtorta, we obtain an optimal quantitative estimate on the transition set. As an application of this estimate, we improve the potential energy estimates of Cabre, Cinti, and Serra (2021), providing sharp versions for the fractional Allen-Cahn equation. Similarly, we obtain optimal perimeter estimates for stationary nonlocal minimal surfaces, extending previous results of Cinti, Serra, and Valdinoci (2019) from the stable case.