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## Abstract

We consider a Dirichlet to Neumann operator L_{a} arising in a model for water waves, with a nonlocal parameter a ∈ (−1, 1). We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behaviour for small frequencies and a nonlocal behaviour for large frequencies. We further investigate the Γ-convergence of the energy associated to the equation L_{a}(u) = W(u), where W is a double-well potential. When a ∈ (−1, 0] the energy Γconverges to the classical perimeter, while for a ∈ (0, 1) the Γ-limit is a new nonlocal operator, that in dimension n = 1 interpolates the classical and the nonlocal perimeter.

Original language | English |
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Pages (from-to) | 5997-6025 |

Number of pages | 29 |

Journal | Nonlinearity |

Volume | 33 |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 2020 |

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