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

Ryan Murray, Matteo Rinaldi

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


The goal of this paper is to study the slow motion of solutions of the nonlocal Allen–Cahn equation in a bounded domain Ω ⊂ Rn, 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 ε- 1 is deduced, where ε is the interaction length parameter. The key tool is a second-order Γ -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.

Original languageEnglish (US)
Article number147
JournalCalculus of Variations and Partial Differential Equations
Issue number6
StatePublished - Dec 1 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


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