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

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