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

TL;DR

This paper analyzes the energy asymptotics of a Dirichlet to Neumann operator related to water waves, revealing a transition from local to nonlocal behavior and establishing the Gamma-convergence of associated energies.

Contribution

It provides a detailed Fourier analysis of the operator and characterizes the Gamma-limit of the energy, introducing a new nonlocal perimeter operator for positive nonlocal parameters.

Findings

For negative and zero parameters, the energy Gamma-converges to the classical perimeter.

For positive parameters, the Gamma-limit is a new nonlocal operator.

In one dimension, the limit interpolates between classical and nonlocal perimeters.

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'(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.