*Accepted Paper*

**Inserted:** 5 dec 2015

**Last Updated:** 17 oct 2016

**Journal:** Calc. Var. Partial Differential Equations

**Year:** 2016

**Abstract:**

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 $\epsilon^{-1}$ is deduced, where $\epsilon$ 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.

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