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

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.