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arXiv · 1512.01706

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

Ryan Murray·Matteo Rinaldi
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Abstract

The goal of this paper is to study the slow motion of solutions of the nonlocal Allen-Cahn equation in a bounded domain $Ω\subset \mathbb{R}^n$, 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 $Γ$-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.

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Ryan Murray, Matteo Rinaldi. 2015-12-05. Slow motion for the nonlocal Allen-Cahn equation in n-dimensions. https://arxiv.org/abs/1512.01706

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