Slow motion for the nonlocal Allen-Cahn equation in n-dimensions

The goal of this paper is to study the slow motion of solutions of the nonlocal Allen-Cahn equation in a bounded domain $\Omega \subset \mathbb{R}^n$, for $n > 1$. The initial data is assumed to be close to a configuration whose interface separating the states minimizes the surface area (or perimeter); both local and global perimeter minimizers are taken into account. The evolution of interfaces on a time scale $\varepsilon^{-1}$ is deduced, where $\varepsilon$ is the interaction length parameter. The key tool is a second-order $\Gamma$-convergence analysis of the energy functional, which provides sharp energy estimates. New regularity results are derived for the isoperimetric function of a domain. Slow motion of solutions for the Cahn-Hilliard equation starting close to global perimeter minimizers is proved as well.