### Abstract

Consider an anisotropic area functional, giving rise to a variational principle for the shape of crystal surfaces. Sometimes such a functional is regularised with an additional curvature term to avoid difficulties coming from a lack of convexity. We study the asymptotic behaviour of the resulting functional as the strength of the regularisation tends to 0. We consider two cases. The first corresponds to a cubic crystal structure. The expected shapes of the crystal surfaces are polyhedra with faces parallel to the coordinate planes, and for the regularised functionals, we discover a limiting energy depending on the lengths of the edges. In the second case, we have a uniaxial anisotropy. We calculate the limiting energy for surfaces of revolution and give a lower bound for topological spheres.

Original language | English |
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Title of host publication | Differential Geometry and Continuum Mechanics |

Editors | Gui-Qiang Chen, Michael Grinfeld, R. J. Knops |

Publisher | Springer |

Pages | 49-75 |

ISBN (Print) | 978-3-319-18572-9 |

Publication status | Published - 2015 |

### Publication series

Name | Springer Proceedings in Mathematics & Statistics |
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Publisher | Springer |

Volume | 137 |

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## Cite this

Moser, R. (2015). Singular perturbation problems involving curvature. In G-Q. Chen, M. Grinfeld, & R. J. Knops (Eds.),

*Differential Geometry and Continuum Mechanics*(pp. 49-75). (Springer Proceedings in Mathematics & Statistics; Vol. 137). Springer.