### Abstract

We examine the singularly perturbed variational problem E_{ε}(ψ) = ∫ ε^{-1}(1 - |∇ψ|^{2})^{2} + ε|∇∇ψ|^{2} in the plane. As ε → 0, this functional favours |∇ψ| = 1 and penalizes singularities where |∇∇ψ| concentrates. Our main result is a compactness theorem: if {E_{ε}(ψ_{ε})}_{ε↓0} is uniformly bounded, then {∇ψ_{ε}}_{ε↓0} is compact in L^{2}. Thus, in the limit ε → 0, ψ solves the eikonal equation |∇ψ| = 1 almost everywhere. Our analysis uses 'entropy relations' and the 'div-curl lemma,' adopting Tartar's approach to the interaction of linear differential equations and nonlinear algebraic relations.

Original language | English (US) |
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Pages (from-to) | 833-844 |

Number of pages | 12 |

Journal | Royal Society of Edinburgh - Proceedings A |

Volume | 131 |

Issue number | 6 |

DOIs | |

State | Published - 2001 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

DeSimone, A., Müller, S., Kohn, R. V., & Otto, F. (2001). A compactness result in the gradient theory of phase transitions.

*Royal Society of Edinburgh - Proceedings A*,*131*(6), 833-844. https://doi.org/10.1017/s030821050000113x