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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.

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Acknowledgements

This paper forms a portion of the Ph.D. theses of both authors, at Carnegie Mellon University. The authors would like to thank Irene Fonseca and Giovanni Leoni for careful readings of the manuscript and the Center for Nonlinear Analysis at Carnegie Mellon University. They would also like to thank Nicola Fusco and Massimiliano Morini for helpful discussions on the subject of this paper. The first author’s research was partially supported by the awards NSF PIRE Grant Nos. OISE-0967140, DMS 1211161 and DMS 0905723, while the second author was partially supported by the awards DMS 0905778 and DMS 1412095.

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Correspondence to Matteo Rinaldi.

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Communicated by L. Ambrosio.

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Murray, R., Rinaldi, M. Slow motion for the nonlocal Allen–Cahn equation in n dimensions. Calc. Var. 55, 147 (2016). https://doi.org/10.1007/s00526-016-1086-4

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Mathematics Subject Classification

  • 35K91
  • 35A15
  • 35B25