Publication 15-CNA-020
Slow motion for the nonlocal Allen-Cahn equation in $n$-dimensions
Ryan Murray
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA, USA
rwmurray@andrew.cmu.edu
Matteo Rinaldi
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
matteor@andrew.cmu.edu
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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