# Science at the triple point between mathematics, mechanics and materials science

## Publication 123

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

##### Authors:

Ryan Murray
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA, USA

Matteo Rinaldi
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA, USA

##### 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.
##### Get the paper in its entirety
15-CNA-020.pdf

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