The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. In particular, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the $W^{s,1}$-seminorm. In particular, we show that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for the fractional seminorm; we also provide an existence result for minimisers of the fractional seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang result describing the full and void patterns of nonlocal minimal surfaces.

中文翻译：

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分数半范数和非局部极小曲面的极小值

最近的文献深入研究了两类非局部变分问题，即与作用于合适 Sobolev-Gagliardo 空间中函数的能量泛函的最小化相关的问题，以及与作用于可测集的分数周长的最小化相关的问题。欧几里得空间。在本文中，我们将这两种变分问题联系起来。特别是，我们研究了非局部极小曲面与 $W^{s,1}$-seminorm 的极小值之间的联系。特别地，我们证明了一个函数是分数半范数的最小值当且仅当它的水平集是分数周长的最小值，并且非局部极小曲面的特征函数是分数半范数的最小值；