### Abstract

We study bounded, monotone solutions of Δu = W′(u) in the whole of ℝn, where W is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, u is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of Γ-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and wishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.

Original language | English |
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Pages (from-to) | 665-682 |

Number of pages | 18 |

Journal | Communications in Partial Differential Equations |

Volume | 41 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 |

Externally published | Yes |