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

Abstract

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 ${\mathcal{L}}_a(u)=W^{\prime }(u)$, 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.