Energy asymptotics of a Dirichlet to Neumann problem related to water waves

Pietro Miraglio, Enrico Valdinoci

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Abstract

We consider a Dirichlet to Neumann operator La 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 La(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 languageEnglish
Pages (from-to)5997-6025
Number of pages29
JournalNonlinearity
Volume33
Issue number11
DOIs
Publication statusPublished - Nov 2020

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