### Abstract

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 |

### Fingerprint

### Cite this

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

**Singular perturbation problems involving curvature.** / Moser, Roger.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Differential Geometry and Continuum Mechanics.*Springer Proceedings in Mathematics & Statistics, vol. 137, Springer, pp. 49-75.

}

TY - CHAP

T1 - Singular perturbation problems involving curvature

AU - Moser, Roger

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

UR - http://www.springer.com/gb/book/9783319185729

M3 - Chapter

SN - 978-3-319-18572-9

T3 - Springer Proceedings in Mathematics & Statistics

SP - 49

EP - 75

BT - Differential Geometry and Continuum Mechanics

A2 - Chen, Gui-Qiang

A2 - Grinfeld, Michael

A2 - Knops, R. J.

PB - Springer

ER -