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

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 \(\varepsilon ^{-1}\) 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.