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.

中文翻译：

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与水波相关的 Dirichlet 到 Neumann 问题的能量渐近

我们考虑在水波模型中出现的 Dirichlet 到 Neumann 算子 $\mathcal{L}_a$，具有非局部参数 $a\in(-1,1)$。我们根据傅立叶变换推导出算子的表达式，突出了小频率的局部行为和大频率的非局部行为。我们进一步研究了与方程 $ \mathcal{L}_a(u)=W'(u) $ 相关的能量的 $ \Gamma $-收敛，其中 $W$ 是双阱势。当 $a\in(-1,0]$ 能量 $\Gamma$-收敛到经典周长时，而对于 $a\in(0,1)$，$\Gamma$-limit 是一个新的非局部算子，在维度 $n=1$ 内插经典和非局部周长。