### Abstract

We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of F-convergence. The approach is demonstrated through the model problem [formula ommittes] It is shown that in certain nonconvex domains The approach is demonstrated through the model problem Ω ⊂R^{n} and for ε small, there exist nonconstant localminimisers u^{ε} satisfying u^{ε≃±}1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit u^{ε→}u^{()} the hypersurface separating the states u_{0}= 1 and u_{0}=-1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and anisotropic perturbations replacing’∇_{u}’^{2}.

Original language | English (US) |
---|---|

Pages (from-to) | 69-84 |

Number of pages | 16 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 111 |

Issue number | 1-2 |

DOIs | |

State | Published - 1989 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*,

*111*(1-2), 69-84. https://doi.org/10.1017/S0308210500025026

**Local minimisers and singular perturbations.** / Kohn, Robert; Sternberg, Peter.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*, vol. 111, no. 1-2, pp. 69-84. https://doi.org/10.1017/S0308210500025026

}

TY - JOUR

T1 - Local minimisers and singular perturbations

AU - Kohn, Robert

AU - Sternberg, Peter

PY - 1989

Y1 - 1989

N2 - We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of F-convergence. The approach is demonstrated through the model problem [formula ommittes] It is shown that in certain nonconvex domains The approach is demonstrated through the model problem Ω ⊂Rn and for ε small, there exist nonconstant localminimisers uε satisfying uε≃±1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit uε→u() the hypersurface separating the states u0= 1 and u0=-1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and anisotropic perturbations replacing’∇u’2.

AB - We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of F-convergence. The approach is demonstrated through the model problem [formula ommittes] It is shown that in certain nonconvex domains The approach is demonstrated through the model problem Ω ⊂Rn and for ε small, there exist nonconstant localminimisers uε satisfying uε≃±1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit uε→u() the hypersurface separating the states u0= 1 and u0=-1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and anisotropic perturbations replacing’∇u’2.

UR - http://www.scopus.com/inward/record.url?scp=84975988204&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975988204&partnerID=8YFLogxK

U2 - 10.1017/S0308210500025026

DO - 10.1017/S0308210500025026

M3 - Article

AN - SCOPUS:84975988204

VL - 111

SP - 69

EP - 84

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 1-2

ER -